Subgroups — C₂×C₄₁×#3294 #23837313

Order 10496.

2858 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄²×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3294	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#480	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×#466	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#447	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#102×C₄₁	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₄₁	—	2	1	no	—
14	C₂³×D₄×C₄₁	—	2	1	no	—
15	C₂³×D₄×C₄₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₄₁	C₂²	1	2	yes	complemented
18	C₂³×C₈×C₄₁	C₂²	1	2	yes	complemented
19	C₄₁×#547	—	2	4	no	—
20	C₄₁×#547	C₂²	1	8	yes	complemented
21	D₄²×C₄₁	—	2	4	no	—
22	D₄²×C₄₁	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
24	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#123×C₄₁	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₄₁	—	2	1	no	—
27	C₂×#121×C₄₁	—	2	1	no	—
28	C₂×#121×C₄₁	C₂²	1	2	yes	—
29	C₂×#121×C₄₁	C₂²	1	2	yes	complemented
30	C₂×#120×C₄₁	—	2	2	no	—
31	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
35	C₄₁×#486	C₂²	1	8	yes	complemented
36	C₄₁×#480	C₂²	1	16	yes	complemented
37	C₄₁×#474	C₂²	1	8	yes	complemented
38	C₄₁×#466	C₂²	1	16	yes	complemented
39	C₄₁×#459	C₂²	1	8	yes	complemented
40	C₄₁×#447	C₂²	1	16	yes	complemented
41	C₄×D₈×C₄₁	C₂²	1	8	yes	complemented
42	C₈×D₄×C₄₁	C₂²	1	8	yes	complemented
43	C₂×#107×C₄₁	C₂²	1	2	yes	complemented
44	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
46	C₂×#102×C₄₁	C₂²	1	2	yes	—
47	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
48	C₂×#102×C₄₁	C₂²	1	16	yes	complemented
49	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic, complemented
50	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
51	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
52	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
53	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	complemented
54	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
55	C₂⁵×C₄₁	—	2	2	no	—
56	C₂×#41×C₄₁	—	2	8	no	—
57	C₂×#41×C₄₁	—	4	8	no	—
58	C₂²×Q₈×C₄₁	—	2	1	no	—
59	C₂²×D₄×C₄₁	—	2	8	no	—
60	C₂²×D₄×C₄₁	—	2	2	no	—
61	C₂²×D₄×C₄₁	—	2	2	no	—
62	C₂²×D₄×C₄₁	—	2	8	no	—
63	C₂²×D₄×C₄₁	—	2	2	no	—
64	C₂²×D₄×C₄₁	—	2	1	no	—
65	C₂²×D₄×C₄₁	—	2	4	no	—
66	C₂²×D₄×C₄₁	—	2	1	no	—
67	C₂²×D₄×C₄₁	—	4	4	no	—
68	C₂²×D₄×C₄₁	—	4	8	no	—
69	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
70	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₄₁	C₂³	1	8	yes	—
73	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₄₁	—	2	2	no	—
75	C₂³×C₄×C₄₁	—	2	1	no	—
76	C₂³×C₄×C₄₁	—	2	1	no	—
77	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
78	C₂×D₈×C₄₁	—	2	4	no	—
79	C₂×D₈×C₄₁	—	4	4	no	—
80	C₂×D₈×C₄₁	C₂³	1	4	yes	—
81	C₂×D₈×C₄₁	C₂³	1	8	yes	complemented
82	C₂²×C₈×C₄₁	—	4	4	no	—
83	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
85	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₄₁	C₂³	1	8	yes	complemented
87	C₄⋊Q₈×C₄₁	—	2	2	no	—
88	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	complemented
89	#127×C₄₁	—	2	2	no	—
90	#127×C₄₁	C₂³	1	4	yes	—
91	#123×C₄₁	—	2	16	no	—
92	C₂²⋊Q₈×C₄₁	—	2	8	no	—
93	#121×C₄₁	—	2	4	no	—
94	#121×C₄₁	—	2	4	no	—
95	#121×C₄₁	—	2	8	no	—
96	#121×C₄₁	C₂³	1	8	yes	—
97	#121×C₄₁	C₂³	1	8	yes	complemented
98	#120×C₄₁	—	2	16	no	—
99	C₄×D₄×C₄₁	—	2	2	no	—
100	C₄×D₄×C₄₁	—	2	2	no	—
101	C₄×D₄×C₄₁	D₄	1	4	yes	—
102	C₄×D₄×C₄₁	C₂³	1	4	yes	—
103	C₄×D₄×C₄₁	C₂³	1	4	yes	—
104	C₄×D₄×C₄₁	C₂³	1	4	yes	complemented
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
107	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	complemented
113	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
118	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
119	#107×C₄₁	C₂³	1	4	yes	complemented
120	#107×C₄₁	C₂³	1	8	yes	complemented
121	C₄⋊C₈×C₄₁	C₂³	1	4	yes	complemented
122	#102×C₄₁	C₂³	1	4	yes	—
123	#102×C₄₁	C₂³	1	8	yes	—
124	#102×C₄₁	C₂³	1	8	yes	complemented
125	#102×C₄₁	C₂³	1	8	yes	complemented
126	#102×C₄₁	C₂³	1	4	yes	complemented
127	#102×C₄₁	C₂³	1	8	yes	complemented
128	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	complemented
129	C₄×C₈×C₄₁	C₂³	1	4	yes	complemented
130	C₂⁴×C₄₁	—	2	1	no	—
131	C₂⁴×C₄₁	—	2	1	no	—
132	C₂⁴×C₄₁	—	2	2	no	—
133	C₂⁴×C₄₁	—	2	8	no	—
134	C₂⁴×C₄₁	—	4	8	no	—
135	C₂⁴×C₄₁	—	4	8	no	—
136	C₂⁴×C₄₁	—	8	4	no	—
137	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
138	#41×C₄₁	—	4	32	no	—
139	C₂×Q₈×C₄₁	—	2	4	no	—
140	C₂×Q₈×C₄₁	—	4	4	no	—
141	C₂×D₄×C₄₁	—	2	16	no	—
142	C₂×D₄×C₄₁	—	2	8	no	—
143	C₂×D₄×C₄₁	—	2	8	no	—
144	C₂×D₄×C₄₁	—	2	1	no	—
145	C₂×D₄×C₄₁	—	2	1	no	—
146	C₂×D₄×C₄₁	—	2	8	no	—
147	C₂×D₄×C₄₁	—	2	2	no	—
148	C₂×D₄×C₄₁	—	2	4	no	—
149	C₂×D₄×C₄₁	—	2	4	no	—
150	C₂×D₄×C₄₁	—	2	2	no	—
151	C₂×D₄×C₄₁	—	2	4	no	—
152	C₂×D₄×C₄₁	—	2	2	no	—
153	C₂×D₄×C₄₁	—	2	2	no	—
154	C₂×D₄×C₄₁	—	2	4	no	—
155	C₂×D₄×C₄₁	—	4	8	no	—
156	C₂×D₄×C₄₁	—	4	16	no	—
157	C₂×D₄×C₄₁	—	4	8	no	—
158	C₂×D₄×C₄₁	—	4	8	no	—
159	C₂×D₄×C₄₁	—	4	8	no	—
160	C₂×D₄×C₄₁	—	4	32	no	—
161	C₂×D₄×C₄₁	—	4	4	no	—
162	C₂×D₄×C₄₁	—	8	2	no	—
163	C₂×D₄×C₄₁	D₈	1	8	yes	complemented
164	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₄₁	#41	1	4	yes	complemented
166	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
167	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
168	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₄₁	—	2	4	no	—
170	C₂²×C₄×C₄₁	—	2	4	no	—
171	C₂²×C₄×C₄₁	—	2	8	no	—
172	C₂²×C₄×C₄₁	—	2	1	no	—
173	C₂²×C₄×C₄₁	—	2	1	no	—
174	C₂²×C₄×C₄₁	—	2	2	no	—
175	C₂²×C₄×C₄₁	—	2	4	no	—
176	C₂²×C₄×C₄₁	—	2	4	no	—
177	C₂²×C₄×C₄₁	—	4	8	no	—
178	C₂²×C₄×C₄₁	—	4	4	no	—
179	C₂²×C₄×C₄₁	—	4	4	no	—
180	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
186	D₈×C₄₁	—	2	8	no	—
187	D₈×C₄₁	—	4	8	no	—
188	C₂×C₈×C₄₁	—	2	2	no	—
189	C₂×C₈×C₄₁	—	2	2	no	—
190	C₂×C₈×C₄₁	—	2	4	no	—
191	C₂×C₈×C₄₁	—	4	8	no	—
192	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
193	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
194	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	2	no	—
197	C₄⋊C₄×C₄₁	—	2	8	no	—
198	C₄⋊C₄×C₄₁	—	2	4	no	—
199	C₄⋊C₄×C₄₁	—	2	1	no	—
200	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
201	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
203	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
204	C₂²⋊C₄×C₄₁	—	2	4	no	—
205	C₂²⋊C₄×C₄₁	—	2	4	no	—
206	C₂²⋊C₄×C₄₁	—	2	8	no	—
207	C₂²⋊C₄×C₄₁	—	2	8	no	—
208	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
209	C₄²×C₄₁	C₂×D₄	1	2	yes	—
210	C₄²×C₄₁	C₂⁴	1	2	yes	—
211	C₂³×C₄₁	—	2	4	no	—
212	C₂³×C₄₁	—	2	4	no	—
213	C₂³×C₄₁	—	2	4	no	—
214	C₂³×C₄₁	—	2	2	no	—
215	C₂³×C₄₁	—	2	2	no	—
216	C₂³×C₄₁	—	2	4	no	—
217	C₂³×C₄₁	—	4	16	no	—
218	C₂³×C₄₁	—	4	2	no	—
219	C₂³×C₄₁	—	4	2	no	—
220	C₂³×C₄₁	—	4	4	no	—
221	C₂³×C₄₁	—	4	2	no	—
222	C₂³×C₄₁	—	4	2	no	—
223	C₂³×C₄₁	—	4	16	no	—
224	C₂³×C₄₁	—	4	4	no	—
225	C₂³×C₄₁	—	4	4	no	—
226	C₂³×C₄₁	—	4	2	no	—
227	C₂³×C₄₁	—	8	8	no	—
228	C₂³×C₄₁	—	8	32	no	—
229	C₂³×C₄₁	C₂×D₈	1	8	yes	—
230	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
231	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₄₁	—	4	8	no	—
233	D₄×C₄₁	—	2	4	no	—
234	D₄×C₄₁	—	2	2	no	—
235	D₄×C₄₁	—	2	4	no	—
236	D₄×C₄₁	—	2	2	no	—
237	D₄×C₄₁	—	2	2	no	—
238	D₄×C₄₁	—	2	4	no	—
239	D₄×C₄₁	—	4	16	no	—
240	D₄×C₄₁	—	4	16	no	—
241	D₄×C₄₁	—	4	16	no	—
242	D₄×C₄₁	—	4	2	no	—
243	D₄×C₄₁	—	4	4	no	—
244	D₄×C₄₁	—	4	8	no	—
245	D₄×C₄₁	—	8	4	no	—
246	D₄×C₄₁	C₂×D₈	1	16	yes	complemented
247	D₄×C₄₁	C₂×#41	1	4	yes	complemented
248	C₂×C₄×C₄₁	—	2	4	no	—
249	C₂×C₄×C₄₁	—	2	2	no	—
250	C₂×C₄×C₄₁	—	2	8	no	—
251	C₂×C₄×C₄₁	—	2	4	no	—
252	C₂×C₄×C₄₁	—	2	4	no	—
253	C₂×C₄×C₄₁	—	2	2	no	—
254	C₂×C₄×C₄₁	—	2	2	no	—
255	C₂×C₄×C₄₁	—	2	4	no	—
256	C₂×C₄×C₄₁	—	4	2	no	—
257	C₂×C₄×C₄₁	—	4	2	no	—
258	C₂×C₄×C₄₁	—	4	4	no	—
259	C₂×C₄×C₄₁	—	4	16	no	—
260	C₂×C₄×C₄₁	—	4	8	no	—
261	C₂×C₄×C₄₁	—	4	8	no	—
262	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
263	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
265	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
268	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
269	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
270	C₈×C₄₁	—	2	2	no	—
271	C₈×C₄₁	—	2	2	no	—
272	C₈×C₄₁	—	2	4	no	—
273	C₂×#3294	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₄₁	—	2	4	no	—
275	C₂²×C₄₁	—	2	16	no	—
276	C₂²×C₄₁	—	2	4	no	—
277	C₂²×C₄₁	—	2	4	no	—
278	C₂²×C₄₁	—	4	8	no	—
279	C₂²×C₄₁	—	4	2	no	—
280	C₂²×C₄₁	—	4	2	no	—
281	C₂²×C₄₁	—	4	4	no	—
282	C₂²×C₄₁	—	4	2	no	—
283	C₂²×C₄₁	—	4	2	no	—
284	C₂²×C₄₁	—	4	4	no	—
285	C₂²×C₄₁	—	4	4	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	8	32	no	—
288	C₂²×C₄₁	—	8	4	no	—
289	C₂²×C₄₁	—	8	8	no	—
290	C₂²×C₄₁	#546	1	2	yes	—
291	C₂²×C₄₁	C₂²×D₈	1	8	yes	—
292	C₂²×C₄₁	C₂²×D₈	1	2	yes	—
293	C₂²×C₄₁	#577	1	2	yes	—
294	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₄₁	—	2	4	no	—
296	C₄×C₄₁	—	2	2	no	—
297	C₄×C₄₁	—	4	2	no	—
298	C₄×C₄₁	—	4	2	no	—
299	C₄×C₄₁	—	4	4	no	—
300	C₄×C₄₁	#546	1	2	yes	—
301	C₄×C₄₁	C₂²×D₈	1	4	yes	—
302	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
303	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×#547	C₂×C₄₁	1	1	yes	characteristic, complemented
305	C₂×D₄²	C₂×C₄₁	1	1	yes	characteristic, complemented
306	#3294	C₂×C₄₁	1	16	yes	direct product factor
307	C₂×#486	C₂×C₄₁	1	1	yes	characteristic, complemented
308	C₂×#480	C₂×C₄₁	1	2	yes	complemented
309	C₂×#474	C₂×C₄₁	1	1	yes	characteristic, complemented
310	C₂×#466	C₂×C₄₁	1	2	yes	complemented
311	C₂×#459	C₂×C₄₁	1	1	yes	characteristic, complemented
312	C₂×#447	C₂×C₄₁	1	2	yes	complemented
313	C₂×C₄×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
314	C₂×C₈×D₄	C₂×C₄₁	1	1	yes	characteristic, complemented
315	C₂²×#102	C₂×C₄₁	1	2	yes	complemented
316	C₂×C₄₁	—	2	8	no	—
317	C₂×C₄₁	—	4	4	no	—
318	C₂×C₄₁	—	8	2	no	—
319	C₂×C₄₁	—	8	4	no	—
320	C₂×C₄₁	#3294	1	4	yes	direct product factor
321	C₂×C₄₁	C₂×#546	1	1	yes	characteristic
322	C₂×C₄₁	C₂³×D₈	1	1	yes	characteristic
323	C₂×C₄₁	C₂×#577	1	1	yes	characteristic
324	C₂²×#41	—	2	1	no	—
325	C₂³×D₄	—	2	1	no	—
326	C₂³×D₄	C₂²×C₄₁	1	2	yes	—
327	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
328	C₂²×D₈	C₂²×C₄₁	1	2	yes	complemented
329	C₂³×C₈	C₂²×C₄₁	1	2	yes	complemented
330	#547	—	2	4	no	—
331	#547	C₂²×C₄₁	1	8	yes	complemented
332	D₄²	—	2	4	no	—
333	D₄²	C₂²×C₄₁	1	8	yes	complemented
334	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
335	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×#123	—	2	2	no	—
337	C₂×C₂²⋊Q₈	—	2	1	no	—
338	C₂×#121	—	2	1	no	—
339	C₂×#121	C₂²×C₄₁	1	2	yes	—
340	C₂×#121	C₂²×C₄₁	1	2	yes	complemented
341	C₂×#120	—	2	2	no	—
342	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
345	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
346	#486	C₂²×C₄₁	1	8	yes	complemented
347	#480	C₂²×C₄₁	1	16	yes	complemented
348	#474	C₂²×C₄₁	1	8	yes	complemented
349	#466	C₂²×C₄₁	1	16	yes	complemented
350	#459	C₂²×C₄₁	1	8	yes	complemented
351	#447	C₂²×C₄₁	1	16	yes	complemented
352	C₄×D₈	C₂²×C₄₁	1	8	yes	complemented
353	C₈×D₄	C₂²×C₄₁	1	8	yes	complemented
354	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic, complemented
355	C₂×#107	C₂²×C₄₁	1	2	yes	complemented
356	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
357	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
358	C₂×#102	C₂²×C₄₁	1	2	yes	—
359	C₂×#102	C₂²×C₄₁	1	16	yes	complemented
360	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
361	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
362	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
363	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic, complemented
364	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	complemented
365	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
366	C₄₁	C₂×#3294	1	1	yes	characteristic, direct product factor, Sylow
367	C₂⁵	—	2	2	no	—
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	4	8	no	—
370	C₂²×Q₈	—	2	1	no	—
371	C₂²×D₄	—	2	8	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	2	2	no	—
374	C₂²×D₄	—	2	8	no	—
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	1	no	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	—	4	4	no	—
380	C₂²×D₄	—	4	8	no	—
381	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
382	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
383	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
384	C₂²×D₄	C₂³×C₄₁	1	8	yes	—
385	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
386	C₂³×C₄	—	2	2	no	—
387	C₂³×C₄	—	2	1	no	—
388	C₂³×C₄	—	2	1	no	—
389	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
390	C₂×D₈	—	2	4	no	—
391	C₂×D₈	—	4	4	no	—
392	C₂×D₈	C₂³×C₄₁	1	4	yes	—
393	C₂×D₈	C₂³×C₄₁	1	8	yes	complemented
394	C₂²×C₈	—	4	4	no	—
395	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
397	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₄₁	1	8	yes	complemented
399	C₄⋊Q₈	—	2	2	no	—
400	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	complemented
401	#127	—	2	2	no	—
402	#127	C₂³×C₄₁	1	4	yes	—
403	#123	—	2	16	no	—
404	C₂²⋊Q₈	—	2	8	no	—
405	#121	—	2	4	no	—
406	#121	—	2	4	no	—
407	#121	—	2	8	no	—
408	#121	C₂³×C₄₁	1	8	yes	—
409	#121	C₂³×C₄₁	1	8	yes	complemented
410	#120	—	2	16	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	D₄×C₄₁	1	4	yes	—
414	C₄×D₄	C₂³×C₄₁	1	4	yes	—
415	C₄×D₄	C₂³×C₄₁	1	4	yes	—
416	C₄×D₄	C₂³×C₄₁	1	4	yes	complemented
417	C₂×C₄⋊C₄	—	2	2	no	—
418	C₂×C₄⋊C₄	—	2	4	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	complemented
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	2	no	—
428	C₂×C₂²⋊C₄	—	2	2	no	—
429	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
430	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
431	#107	C₂³×C₄₁	1	4	yes	complemented
432	#107	C₂³×C₄₁	1	8	yes	complemented
433	C₄⋊C₈	C₂³×C₄₁	1	4	yes	complemented
434	#102	C₂³×C₄₁	1	4	yes	—
435	#102	C₂³×C₄₁	1	8	yes	—
436	#102	C₂³×C₄₁	1	8	yes	complemented
437	#102	C₂³×C₄₁	1	8	yes	complemented
438	#102	C₂³×C₄₁	1	4	yes	complemented
439	#102	C₂³×C₄₁	1	8	yes	complemented
440	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	complemented
441	C₄×C₈	C₂³×C₄₁	1	4	yes	complemented
442	C₂⁴	—	2	1	no	—
443	C₂⁴	—	2	1	no	—
444	C₂⁴	—	2	2	no	—
445	C₂⁴	—	2	8	no	—
446	C₂⁴	—	4	8	no	—
447	C₂⁴	—	4	8	no	—
448	C₂⁴	—	8	4	no	—
449	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
450	#41	—	4	32	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	4	4	no	—
453	C₂×D₄	—	2	16	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	4	8	no	—
468	C₂×D₄	—	4	16	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	4	8	no	—
471	C₂×D₄	—	4	8	no	—
472	C₂×D₄	—	4	32	no	—
473	C₂×D₄	—	4	4	no	—
474	C₂×D₄	—	8	2	no	—
475	C₂×D₄	D₈×C₄₁	1	8	yes	complemented
476	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
477	C₂×D₄	#41×C₄₁	1	4	yes	complemented
478	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
479	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
480	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	4	8	no	—
490	C₂²×C₄	—	4	4	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
498	D₈	—	2	8	no	—
499	D₈	—	4	8	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	4	no	—
503	C₂×C₈	—	4	8	no	—
504	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
505	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
506	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
513	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
514	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
515	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
521	C₄²	C₂×D₄×C₄₁	1	2	yes	—
522	C₄²	C₂⁴×C₄₁	1	2	yes	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	4	16	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	4	no	—
533	C₂³	—	4	2	no	—
534	C₂³	—	4	2	no	—
535	C₂³	—	4	16	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	8	8	no	—
540	C₂³	—	8	32	no	—
541	C₂³	C₂×D₈×C₄₁	1	8	yes	—
542	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
543	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
544	Q₈	—	4	8	no	—
545	D₄	—	2	4	no	—
546	D₄	—	2	2	no	—
547	D₄	—	2	4	no	—
548	D₄	—	2	2	no	—
549	D₄	—	2	2	no	—
550	D₄	—	2	4	no	—
551	D₄	—	4	16	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	16	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	2	no	—
556	D₄	—	4	8	no	—
557	D₄	—	8	4	no	—
558	D₄	C₂×D₈×C₄₁	1	16	yes	complemented
559	D₄	C₂×#41×C₄₁	1	4	yes	complemented
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	16	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
575	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
579	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
580	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
581	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
582	C₈	—	2	2	no	—
583	C₈	—	2	2	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	16	no	—
586	C₂²	—	2	4	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	4	8	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	8	32	no	—
599	C₂²	—	8	8	no	—
600	C₂²	—	8	4	no	—
601	C₂²	C₄₁×#546	1	2	yes	—
602	C₂²	C₂²×D₈×C₄₁	1	8	yes	—
603	C₂²	C₂²×D₈×C₄₁	1	2	yes	—
604	C₂²	C₄₁×#577	1	2	yes	—
605	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	4	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	4	no	—
611	C₄	C₄₁×#546	1	2	yes	—
612	C₄	C₂²×D₈×C₄₁	1	4	yes	—
613	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
614	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
615	C₂	—	2	8	no	—
616	C₂	—	4	4	no	—
617	C₂	—	8	4	no	—
618	C₂	—	8	2	no	—
619	C₂	C₄₁×#3294	1	4	yes	direct product factor
620	C₂	C₂×C₄₁×#546	1	1	yes	characteristic
621	C₂	C₂³×D₈×C₄₁	1	1	yes	characteristic
622	C₂	C₂×C₄₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×D₄×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄₁×#547	C₂	semidirect
2. C₂×D₄²×C₄₁	C₂	semidirect
3. C₄₁×#3294	C₂	direct (N × H)
4. C₂×C₄₁×#486	C₂	semidirect
5. C₂×C₄₁×#480	C₂	semidirect
6. C₂×C₄₁×#474	C₂	semidirect
7. C₂×C₄₁×#466	C₂	semidirect
8. C₂×C₄₁×#459	C₂	semidirect
9. C₂×C₄₁×#447	C₂	semidirect
10. C₂×C₄×D₈×C₄₁	C₂	semidirect
11. C₂×C₈×D₄×C₄₁	C₂	semidirect
12. C₂²×#102×C₄₁	C₂	semidirect
17. C₂²×D₈×C₄₁	C₂²	semidirect
18. C₂³×C₈×C₄₁	C₂²	semidirect
20. C₄₁×#547	C₂²	semidirect
22. D₄²×C₄₁	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
29. C₂×#121×C₄₁	C₂²	semidirect
33. C₂×C₄×D₄×C₄₁	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₄₁	C₂²	semidirect
35. C₄₁×#486	C₂²	semidirect
36. C₄₁×#480	C₂²	semidirect
37. C₄₁×#474	C₂²	semidirect
38. C₄₁×#466	C₂²	semidirect
39. C₄₁×#459	C₂²	semidirect
40. C₄₁×#447	C₂²	semidirect
41. C₄×D₈×C₄₁	C₂²	semidirect
42. C₈×D₄×C₄₁	C₂²	semidirect
43. C₂×#107×C₄₁	C₂²	semidirect
44. C₂×#107×C₄₁	C₂²	semidirect
45. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
48. C₂×#102×C₄₁	C₂²	semidirect
49. C₂×#102×C₄₁	C₂²	semidirect
50. C₂×#102×C₄₁	C₂²	semidirect
51. C₂×#102×C₄₁	C₂²	semidirect
52. C₂×#102×C₄₁	C₂²	semidirect
53. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
54. C₂×C₄×C₈×C₄₁	C₂²	semidirect
81. C₂×D₈×C₄₁	C₂³	semidirect
86. C₂²×C₈×C₄₁	C₂³	semidirect
88. C₄⋊Q₈×C₄₁	C₂³	semidirect
97. #121×C₄₁	C₂³	semidirect
104. C₄×D₄×C₄₁	C₂³	semidirect
112. C₂×C₄⋊C₄×C₄₁	C₂³	semidirect
119. #107×C₄₁	C₂³	semidirect
120. #107×C₄₁	C₂³	semidirect
121. C₄⋊C₈×C₄₁	C₂³	semidirect
124. #102×C₄₁	C₂³	semidirect
125. #102×C₄₁	C₂³	semidirect
126. #102×C₄₁	C₂³	semidirect
127. #102×C₄₁	C₂³	semidirect
128. C₂²⋊C₈×C₄₁	C₂³	semidirect
129. C₄×C₈×C₄₁	C₂³	semidirect
163. C₂×D₄×C₄₁	D₈	semidirect
165. C₂×D₄×C₄₁	#41	semidirect
246. D₄×C₄₁	C₂×D₈	semidirect
247. D₄×C₄₁	C₂×#41	semidirect
273. C₂×#3294	C₄₁	direct (N × H)
304. C₂×#547	C₂×C₄₁	semidirect
305. C₂×D₄²	C₂×C₄₁	semidirect
306. #3294	C₂×C₄₁	direct (N × H)
307. C₂×#486	C₂×C₄₁	semidirect
308. C₂×#480	C₂×C₄₁	semidirect
309. C₂×#474	C₂×C₄₁	semidirect
310. C₂×#466	C₂×C₄₁	semidirect
311. C₂×#459	C₂×C₄₁	semidirect
312. C₂×#447	C₂×C₄₁	semidirect
313. C₂×C₄×D₈	C₂×C₄₁	semidirect
314. C₂×C₈×D₄	C₂×C₄₁	semidirect
315. C₂²×#102	C₂×C₄₁	semidirect
320. C₂×C₄₁	#3294	direct (N × H)
328. C₂²×D₈	C₂²×C₄₁	semidirect
329. C₂³×C₈	C₂²×C₄₁	semidirect
331. #547	C₂²×C₄₁	semidirect
333. D₄²	C₂²×C₄₁	semidirect
334. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
340. C₂×#121	C₂²×C₄₁	semidirect
344. C₂×C₄×D₄	C₂²×C₄₁	semidirect
345. C₂²×C₄⋊C₄	C₂²×C₄₁	semidirect
346. #486	C₂²×C₄₁	semidirect
347. #480	C₂²×C₄₁	semidirect
348. #474	C₂²×C₄₁	semidirect
349. #466	C₂²×C₄₁	semidirect
350. #459	C₂²×C₄₁	semidirect
351. #447	C₂²×C₄₁	semidirect
352. C₄×D₈	C₂²×C₄₁	semidirect
353. C₈×D₄	C₂²×C₄₁	semidirect
354. C₂×#107	C₂²×C₄₁	semidirect
355. C₂×#107	C₂²×C₄₁	semidirect
356. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
359. C₂×#102	C₂²×C₄₁	semidirect
360. C₂×#102	C₂²×C₄₁	semidirect
361. C₂×#102	C₂²×C₄₁	semidirect
362. C₂×#102	C₂²×C₄₁	semidirect
363. C₂×#102	C₂²×C₄₁	semidirect
364. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
365. C₂×C₄×C₈	C₂²×C₄₁	semidirect
366. C₄₁	C₂×#3294	direct (N × H)
393. C₂×D₈	C₂³×C₄₁	semidirect
398. C₂²×C₈	C₂³×C₄₁	semidirect
400. C₄⋊Q₈	C₂³×C₄₁	semidirect
409. #121	C₂³×C₄₁	semidirect
416. C₄×D₄	C₂³×C₄₁	semidirect
424. C₂×C₄⋊C₄	C₂³×C₄₁	semidirect
431. #107	C₂³×C₄₁	semidirect
432. #107	C₂³×C₄₁	semidirect
433. C₄⋊C₈	C₂³×C₄₁	semidirect
436. #102	C₂³×C₄₁	semidirect
437. #102	C₂³×C₄₁	semidirect
438. #102	C₂³×C₄₁	semidirect
439. #102	C₂³×C₄₁	semidirect
440. C₂²⋊C₈	C₂³×C₄₁	semidirect
441. C₄×C₈	C₂³×C₄₁	semidirect
475. C₂×D₄	D₈×C₄₁	semidirect
477. C₂×D₄	#41×C₄₁	semidirect
558. D₄	C₂×D₈×C₄₁	semidirect
559. D₄	C₂×#41×C₄₁	semidirect
619. C₂	C₄₁×#3294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄₁×#547	2
C₂×D₄²×C₄₁	2
C₄₁×#3294	2
C₂×C₄₁×#486	2
C₂×C₄₁×#480	2
C₂×C₄₁×#474	2
C₂×C₄₁×#466	2
C₂×C₄₁×#459	2
C₂×C₄₁×#447	2
C₂×C₄×D₈×C₄₁	2
C₂×C₈×D₄×C₄₁	2
C₂²×#102×C₄₁	2
C₂×#3294	41

Subgroups — C2×C41×#3294 #23837313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3294 #23837313