Subgroups — C₂×C₄₁×#3303 #23837322

Order 10496.

2210 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3303	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×#465	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#449	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	C₂×C₄₁×#416	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₄₁	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#555	—	2	4	no	—
18	C₄₁×#555	C₂²	1	8	yes	complemented
19	C₄₁×#547	—	2	4	no	—
20	C₄₁×#547	C₂²	1	8	yes	complemented
21	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#124×C₄₁	—	2	2	no	—
24	C₂×#123×C₄₁	—	2	2	no	—
25	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
26	C₂×#121×C₄₁	—	2	1	no	—
27	C₂×#121×C₄₁	—	2	1	no	—
28	C₂×#121×C₄₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#117×C₄₁	—	2	1	no	—
33	C₂×#117×C₄₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
35	C₄₁×#487	C₂²	1	8	yes	complemented
36	C₄₁×#485	C₂²	1	16	yes	complemented
37	C₄₁×#476	C₂²	1	8	yes	complemented
38	C₄₁×#465	C₂²	1	16	yes	complemented
39	C₄₁×#461	C₂²	1	8	yes	complemented
40	C₄₁×#449	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	1	yes	characteristic
44	C₂×#106×C₄₁	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
46	C₄₁×#416	C₂²	1	16	yes	complemented
47	C₂×#103×C₄₁	C₂²	1	2	yes	—
48	C₂×#103×C₄₁	C₂²	1	2	yes	—
49	C₂×#103×C₄₁	C₂²	1	2	yes	—
50	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
51	C₂×#102×C₄₁	C₂²	1	2	yes	—
52	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
55	C₂×#41×C₄₁	—	2	4	no	—
56	C₂×#41×C₄₁	—	2	8	no	—
57	C₂×#41×C₄₁	—	4	4	no	—
58	C₂×#41×C₄₁	C₂³	1	8	yes	—
59	C₂²×Q₈×C₄₁	—	2	1	no	—
60	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
61	C₂²×D₄×C₄₁	—	2	2	no	—
62	C₂²×D₄×C₄₁	—	2	2	no	—
63	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
64	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₁	—	2	1	no	—
67	C₂³×C₄×C₄₁	—	2	2	no	—
68	C₂³×C₄×C₄₁	—	2	1	no	—
69	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
70	C₂×SD₁₆×C₄₁	—	4	4	no	—
71	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
72	C₂×D₈×C₄₁	—	2	4	no	—
73	C₂×D₈×C₄₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₄₁	—	4	4	no	—
75	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
79	C₄⋊Q₈×C₄₁	—	2	2	no	—
80	C₄⋊Q₈×C₄₁	—	2	2	no	—
81	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
82	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
83	#124×C₄₁	—	2	16	no	—
84	#123×C₄₁	—	2	16	no	—
85	C₂²⋊Q₈×C₄₁	—	2	4	no	—
86	C₂²⋊Q₈×C₄₁	C₂³	1	8	yes	—
87	#121×C₄₁	—	2	8	no	—
88	#121×C₄₁	—	2	8	no	—
89	#121×C₄₁	—	2	4	no	—
90	#121×C₄₁	C₂³	1	8	yes	—
91	C₄×D₄×C₄₁	—	2	2	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	D₄	1	4	yes	—
94	C₄×D₄×C₄₁	C₂³	1	4	yes	—
95	C₄×D₄×C₄₁	C₂³	1	4	yes	—
96	C₄×D₄×C₄₁	C₂³	1	4	yes	—
97	#117×C₄₁	—	2	4	no	—
98	#117×C₄₁	—	2	8	no	—
99	#117×C₄₁	C₂³	1	8	yes	—
100	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
105	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
115	C₂×C₄²×C₄₁	—	2	2	no	—
116	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
117	#107×C₄₁	C₂³	1	4	yes	—
118	#106×C₄₁	C₂³	1	8	yes	—
119	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
120	#103×C₄₁	C₂³	1	8	yes	—
121	#103×C₄₁	C₂³	1	8	yes	—
122	#103×C₄₁	C₂³	1	8	yes	—
123	#102×C₄₁	C₂³	1	4	yes	—
124	#102×C₄₁	C₂³	1	8	yes	—
125	#102×C₄₁	C₂³	1	4	yes	—
126	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	—
127	C₄×C₈×C₄₁	C₂³	1	4	yes	—
128	C₂⁴×C₄₁	—	2	1	no	—
129	C₂⁴×C₄₁	—	2	1	no	—
130	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
131	#41×C₄₁	—	2	16	no	—
132	#41×C₄₁	—	4	8	no	—
133	#41×C₄₁	—	4	16	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	—	2	4	no	—
136	C₂×Q₈×C₄₁	—	2	4	no	—
137	C₂×Q₈×C₄₁	—	2	4	no	—
138	C₂×Q₈×C₄₁	—	8	2	no	—
139	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
140	C₂×D₄×C₄₁	—	2	8	no	—
141	C₂×D₄×C₄₁	—	2	8	no	—
142	C₂×D₄×C₄₁	—	2	4	no	—
143	C₂×D₄×C₄₁	—	2	1	no	—
144	C₂×D₄×C₄₁	—	2	1	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	2	no	—
147	C₂×D₄×C₄₁	—	2	2	no	—
148	C₂×D₄×C₄₁	—	4	8	no	—
149	C₂×D₄×C₄₁	—	8	4	no	—
150	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
151	C₂×D₄×C₄₁	#41	1	4	yes	—
152	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
153	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₄₁	—	2	4	no	—
155	C₂²×C₄×C₄₁	—	2	4	no	—
156	C₂²×C₄×C₄₁	—	2	1	no	—
157	C₂²×C₄×C₄₁	—	2	2	no	—
158	C₂²×C₄×C₄₁	—	2	1	no	—
159	C₂²×C₄×C₄₁	—	2	2	no	—
160	C₂²×C₄×C₄₁	—	2	4	no	—
161	C₂²×C₄×C₄₁	—	2	8	no	—
162	C₂²×C₄×C₄₁	—	2	4	no	—
163	C₂²×C₄×C₄₁	—	4	8	no	—
164	C₂²×C₄×C₄₁	—	4	4	no	—
165	C₂²×C₄×C₄₁	—	4	4	no	—
166	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₄₁	—	4	8	no	—
173	D₈×C₄₁	—	2	8	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₂×C₈×C₄₁	—	2	2	no	—
176	C₂×C₈×C₄₁	—	2	4	no	—
177	C₂×C₈×C₄₁	—	4	8	no	—
178	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
179	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
180	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₄₁	—	2	4	no	—
182	C₄⋊C₄×C₄₁	—	2	8	no	—
183	C₄⋊C₄×C₄₁	—	2	4	no	—
184	C₄⋊C₄×C₄₁	—	2	4	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	1	no	—
188	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
190	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
191	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
192	C₂²⋊C₄×C₄₁	—	2	8	no	—
193	C₂²⋊C₄×C₄₁	—	2	8	no	—
194	C₂²⋊C₄×C₄₁	—	2	8	no	—
195	C₂²⋊C₄×C₄₁	—	2	4	no	—
196	C₂²⋊C₄×C₄₁	—	2	4	no	—
197	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
198	C₄²×C₄₁	—	2	8	no	—
199	C₄²×C₄₁	C₂×D₄	1	2	yes	—
200	C₄²×C₄₁	C₂⁴	1	2	yes	—
201	C₂³×C₄₁	—	2	4	no	—
202	C₂³×C₄₁	—	2	4	no	—
203	C₂³×C₄₁	—	2	4	no	—
204	C₂³×C₄₁	—	2	4	no	—
205	C₂³×C₄₁	—	2	2	no	—
206	C₂³×C₄₁	—	2	2	no	—
207	C₂³×C₄₁	—	4	2	no	—
208	C₂³×C₄₁	—	4	2	no	—
209	C₂³×C₄₁	—	4	2	no	—
210	C₂³×C₄₁	—	4	2	no	—
211	C₂³×C₄₁	—	4	2	no	—
212	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
213	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₄₁	—	2	4	no	—
215	Q₈×C₄₁	—	2	4	no	—
216	Q₈×C₄₁	—	4	4	no	—
217	Q₈×C₄₁	—	8	4	no	—
218	D₄×C₄₁	—	2	4	no	—
219	D₄×C₄₁	—	2	8	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	—	2	2	no	—
222	D₄×C₄₁	—	2	2	no	—
223	D₄×C₄₁	—	4	16	no	—
224	D₄×C₄₁	—	4	2	no	—
225	D₄×C₄₁	—	8	8	no	—
226	D₄×C₄₁	C₂×#41	1	4	yes	—
227	C₂×C₄×C₄₁	—	2	4	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	8	no	—
230	C₂×C₄×C₄₁	—	2	4	no	—
231	C₂×C₄×C₄₁	—	2	4	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	4	no	—
234	C₂×C₄×C₄₁	—	2	2	no	—
235	C₂×C₄×C₄₁	—	2	4	no	—
236	C₂×C₄×C₄₁	—	4	16	no	—
237	C₂×C₄×C₄₁	—	4	4	no	—
238	C₂×C₄×C₄₁	—	4	2	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	4	4	no	—
241	C₂×C₄×C₄₁	—	4	8	no	—
242	C₂×C₄×C₄₁	—	4	8	no	—
243	C₂×C₄×C₄₁	#135	1	4	yes	—
244	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
249	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
251	C₈×C₄₁	—	2	2	no	—
252	C₈×C₄₁	—	2	2	no	—
253	C₈×C₄₁	—	2	4	no	—
254	C₂×#3303	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂²×C₄₁	—	2	4	no	—
256	C₂²×C₄₁	—	2	4	no	—
257	C₂²×C₄₁	—	2	4	no	—
258	C₂²×C₄₁	—	2	4	no	—
259	C₂²×C₄₁	—	4	8	no	—
260	C₂²×C₄₁	—	4	8	no	—
261	C₂²×C₄₁	—	4	2	no	—
262	C₂²×C₄₁	—	4	2	no	—
263	C₂²×C₄₁	—	4	2	no	—
264	C₂²×C₄₁	—	4	2	no	—
265	C₂²×C₄₁	—	4	2	no	—
266	C₂²×C₄₁	—	8	4	no	—
267	C₂²×C₄₁	#546	1	2	yes	—
268	C₂²×C₄₁	C₂×#135	1	2	yes	—
269	C₂²×C₄₁	#577	1	2	yes	—
270	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₄₁	—	2	4	no	—
272	C₄×C₄₁	—	2	2	no	—
273	C₄×C₄₁	—	4	4	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	—	4	2	no	—
276	C₄×C₄₁	—	4	4	no	—
277	C₄×C₄₁	#546	1	2	yes	—
278	C₄×C₄₁	C₂×#135	1	4	yes	—
279	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
280	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×#555	C₂×C₄₁	1	1	yes	characteristic, complemented
282	C₂×#547	C₂×C₄₁	1	1	yes	characteristic, complemented
283	#3303	C₂×C₄₁	1	16	yes	direct product factor
284	C₂×#487	C₂×C₄₁	1	1	yes	characteristic, complemented
285	C₂×#485	C₂×C₄₁	1	2	yes	complemented
286	C₂×#476	C₂×C₄₁	1	1	yes	characteristic, complemented
287	C₂×#465	C₂×C₄₁	1	2	yes	complemented
288	C₂×#461	C₂×C₄₁	1	1	yes	characteristic, complemented
289	C₂×#449	C₂×C₄₁	1	2	yes	complemented
290	C₂×C₄×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
291	C₂×C₈×D₄	C₂×C₄₁	1	1	yes	characteristic, complemented
292	C₂×#416	C₂×C₄₁	1	2	yes	complemented
293	C₂×C₄₁	—	4	4	no	—
294	C₂×C₄₁	—	4	4	no	—
295	C₂×C₄₁	—	8	2	no	—
296	C₂×C₄₁	#3303	1	4	yes	direct product factor
297	C₂×C₄₁	C₂×#546	1	1	yes	characteristic
298	C₂×C₄₁	C₂²×#135	1	1	yes	characteristic
299	C₂×C₄₁	C₂×#577	1	1	yes	characteristic
300	C₂²×#41	C₂²×C₄₁	1	2	yes	—
301	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
302	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
303	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
304	#555	—	2	4	no	—
305	#555	C₂²×C₄₁	1	8	yes	complemented
306	#547	—	2	4	no	—
307	#547	C₂²×C₄₁	1	8	yes	complemented
308	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
309	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
310	C₂×#124	—	2	2	no	—
311	C₂×#123	—	2	2	no	—
312	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
313	C₂×#121	—	2	1	no	—
314	C₂×#121	—	2	1	no	—
315	C₂×#121	C₂²×C₄₁	1	2	yes	—
316	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
318	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
319	C₂×#117	—	2	1	no	—
320	C₂×#117	C₂²×C₄₁	1	2	yes	—
321	C₂²×C₄⋊C₄	—	2	1	no	—
322	#487	C₂²×C₄₁	1	8	yes	complemented
323	#485	C₂²×C₄₁	1	16	yes	complemented
324	#476	C₂²×C₄₁	1	8	yes	complemented
325	#465	C₂²×C₄₁	1	16	yes	complemented
326	#461	C₂²×C₄₁	1	8	yes	complemented
327	#449	C₂²×C₄₁	1	16	yes	complemented
328	C₄×D₈	C₂²×C₄₁	1	8	yes	complemented
329	C₈×D₄	C₂²×C₄₁	1	8	yes	complemented
330	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×#106	C₂²×C₄₁	1	2	yes	—
332	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
333	#416	C₂²×C₄₁	1	16	yes	complemented
334	C₂×#103	C₂²×C₄₁	1	2	yes	—
335	C₂×#103	C₂²×C₄₁	1	2	yes	—
336	C₂×#103	C₂²×C₄₁	1	2	yes	—
337	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#102	C₂²×C₄₁	1	2	yes	—
339	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
341	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
342	C₄₁	C₂×#3303	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	—	4	4	no	—
346	C₂×#41	C₂³×C₄₁	1	8	yes	—
347	C₂²×Q₈	—	2	1	no	—
348	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
352	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
353	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	—	2	2	no	—
356	C₂³×C₄	—	2	1	no	—
357	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
358	C₂×SD₁₆	—	4	4	no	—
359	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
360	C₂×D₈	—	2	4	no	—
361	C₂×D₈	C₂³×C₄₁	1	4	yes	—
362	C₂²×C₈	—	4	4	no	—
363	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
365	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
370	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
371	#124	—	2	16	no	—
372	#123	—	2	16	no	—
373	C₂²⋊Q₈	—	2	4	no	—
374	C₂²⋊Q₈	C₂³×C₄₁	1	8	yes	—
375	#121	—	2	8	no	—
376	#121	—	2	8	no	—
377	#121	—	2	4	no	—
378	#121	C₂³×C₄₁	1	8	yes	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	D₄×C₄₁	1	4	yes	—
382	C₄×D₄	C₂³×C₄₁	1	4	yes	—
383	C₄×D₄	C₂³×C₄₁	1	4	yes	—
384	C₄×D₄	C₂³×C₄₁	1	4	yes	—
385	#117	—	2	4	no	—
386	#117	—	2	8	no	—
387	#117	C₂³×C₄₁	1	8	yes	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	8	no	—
393	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
405	#107	C₂³×C₄₁	1	4	yes	—
406	#106	C₂³×C₄₁	1	8	yes	—
407	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
408	#103	C₂³×C₄₁	1	8	yes	—
409	#103	C₂³×C₄₁	1	8	yes	—
410	#103	C₂³×C₄₁	1	8	yes	—
411	#102	C₂³×C₄₁	1	4	yes	—
412	#102	C₂³×C₄₁	1	8	yes	—
413	#102	C₂³×C₄₁	1	4	yes	—
414	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	—
415	C₄×C₈	C₂³×C₄₁	1	4	yes	—
416	C₂⁴	—	2	1	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
419	#41	—	2	16	no	—
420	#41	—	4	8	no	—
421	#41	—	4	16	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	8	2	no	—
427	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	8	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	4	8	no	—
437	C₂×D₄	—	8	4	no	—
438	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
439	C₂×D₄	#41×C₄₁	1	4	yes	—
440	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
441	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	8	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	4	8	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
457	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
458	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
460	SD₁₆	—	4	8	no	—
461	D₈	—	2	8	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	4	no	—
465	C₂×C₈	—	4	8	no	—
466	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
467	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
468	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
478	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
486	C₄²	—	2	8	no	—
487	C₄²	C₂×D₄×C₄₁	1	2	yes	—
488	C₄²	C₂⁴×C₄₁	1	2	yes	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	4	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
501	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
502	Q₈	—	2	4	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	4	4	no	—
505	Q₈	—	8	4	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	8	no	—
508	D₄	—	2	2	no	—
509	D₄	—	2	2	no	—
510	D₄	—	2	2	no	—
511	D₄	—	4	16	no	—
512	D₄	—	4	2	no	—
513	D₄	—	8	8	no	—
514	D₄	C₂×#41×C₄₁	1	4	yes	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	4	16	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	8	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	#135×C₄₁	1	4	yes	—
532	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
533	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
536	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
537	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
538	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
539	C₈	—	2	2	no	—
540	C₈	—	2	2	no	—
541	C₈	—	2	4	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	4	8	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	4	no	—
554	C₂²	C₄₁×#546	1	2	yes	—
555	C₂²	C₂×#135×C₄₁	1	2	yes	—
556	C₂²	C₄₁×#577	1	2	yes	—
557	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
558	C₄	—	2	2	no	—
559	C₄	—	2	4	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	4	no	—
563	C₄	—	4	4	no	—
564	C₄	C₄₁×#546	1	2	yes	—
565	C₄	C₂×#135×C₄₁	1	4	yes	—
566	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
567	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
568	C₂	—	4	4	no	—
569	C₂	—	4	4	no	—
570	C₂	—	8	2	no	—
571	C₂	C₄₁×#3303	1	4	yes	direct product factor
572	C₂	C₂×C₄₁×#546	1	1	yes	characteristic
573	C₂	C₂²×#135×C₄₁	1	1	yes	characteristic
574	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×#555	C₂	semidirect
2. C₂×C₄₁×#547	C₂	semidirect
3. C₄₁×#3303	C₂	direct (N × H)
4. C₂×C₄₁×#487	C₂	semidirect
5. C₂×C₄₁×#485	C₂	semidirect
6. C₂×C₄₁×#476	C₂	semidirect
7. C₂×C₄₁×#465	C₂	semidirect
8. C₂×C₄₁×#461	C₂	semidirect
9. C₂×C₄₁×#449	C₂	semidirect
10. C₂×C₄×D₈×C₄₁	C₂	semidirect
11. C₂×C₈×D₄×C₄₁	C₂	semidirect
12. C₂×C₄₁×#416	C₂	semidirect
18. C₄₁×#555	C₂²	semidirect
20. C₄₁×#547	C₂²	semidirect
35. C₄₁×#487	C₂²	semidirect
36. C₄₁×#485	C₂²	semidirect
37. C₄₁×#476	C₂²	semidirect
38. C₄₁×#465	C₂²	semidirect
39. C₄₁×#461	C₂²	semidirect
40. C₄₁×#449	C₂²	semidirect
41. C₄×D₈×C₄₁	C₂²	semidirect
42. C₈×D₄×C₄₁	C₂²	semidirect
46. C₄₁×#416	C₂²	semidirect
254. C₂×#3303	C₄₁	direct (N × H)
281. C₂×#555	C₂×C₄₁	semidirect
282. C₂×#547	C₂×C₄₁	semidirect
283. #3303	C₂×C₄₁	direct (N × H)
284. C₂×#487	C₂×C₄₁	semidirect
285. C₂×#485	C₂×C₄₁	semidirect
286. C₂×#476	C₂×C₄₁	semidirect
287. C₂×#465	C₂×C₄₁	semidirect
288. C₂×#461	C₂×C₄₁	semidirect
289. C₂×#449	C₂×C₄₁	semidirect
290. C₂×C₄×D₈	C₂×C₄₁	semidirect
291. C₂×C₈×D₄	C₂×C₄₁	semidirect
292. C₂×#416	C₂×C₄₁	semidirect
296. C₂×C₄₁	#3303	direct (N × H)
305. #555	C₂²×C₄₁	semidirect
307. #547	C₂²×C₄₁	semidirect
322. #487	C₂²×C₄₁	semidirect
323. #485	C₂²×C₄₁	semidirect
324. #476	C₂²×C₄₁	semidirect
325. #465	C₂²×C₄₁	semidirect
326. #461	C₂²×C₄₁	semidirect
327. #449	C₂²×C₄₁	semidirect
328. C₄×D₈	C₂²×C₄₁	semidirect
329. C₈×D₄	C₂²×C₄₁	semidirect
333. #416	C₂²×C₄₁	semidirect
342. C₄₁	C₂×#3303	direct (N × H)
571. C₂	C₄₁×#3303	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×#555	2
C₂×C₄₁×#547	2
C₄₁×#3303	2
C₂×C₄₁×#487	2
C₂×C₄₁×#485	2
C₂×C₄₁×#476	2
C₂×C₄₁×#465	2
C₂×C₄₁×#461	2
C₂×C₄₁×#449	2
C₂×C₄×D₈×C₄₁	2
C₂×C₈×D₄×C₄₁	2
C₂×C₄₁×#416	2
C₂×#3303	41

Subgroups — C2×C41×#3303 #23837322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3303 #23837322