Subgroups — C₂×C₄₁×#3096 #23836713

Order 10496.

2362 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#535	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3096	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#481	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#482	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₁×#480	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#102×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#535	—	2	4	no	—
18	C₄₁×#535	C₂²	1	8	yes	—
19	C₂×#127×C₄₁	—	2	1	no	—
20	C₂×#124×C₄₁	—	2	1	no	—
21	C₂×#121×C₄₁	C₂²	1	2	yes	—
22	C₂×#121×C₄₁	C₂²	1	2	yes	—
23	C₂×#120×C₄₁	—	2	2	no	—
24	C₄₁×#520	—	2	4	no	—
25	C₄₁×#520	C₂²	1	8	yes	complemented
26	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
27	C₂×C₄×D₄×C₄₁	—	2	1	no	—
28	C₂×C₄×D₄×C₄₁	—	2	2	no	—
29	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#117×C₄₁	C₂²	1	2	yes	complemented
32	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
34	C₄₁×#482	C₂²	1	16	yes	complemented
35	C₄₁×#482	C₂²	1	16	yes	complemented
36	C₄₁×#481	C₂²	1	16	yes	complemented
37	C₄₁×#480	C₂²	1	16	yes	complemented
38	C₄₁×#428	C₂²	1	8	yes	complemented
39	C₄₁×#427	C₂²	1	8	yes	complemented
40	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×#107×C₄₁	C₂²	1	2	yes	complemented
42	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic, complemented
43	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic, complemented
44	C₂×#106×C₄₁	C₂²	1	2	yes	complemented
45	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic, complemented
46	C₄₁×#418	C₂²	1	8	yes	complemented
47	C₂×#102×C₄₁	C₂²	1	2	yes	—
48	C₂×#102×C₄₁	C₂²	1	2	yes	—
49	C₂×#102×C₄₁	C₂²	1	2	yes	—
50	C₂×#102×C₄₁	C₂²	1	2	yes	—
51	C₂×#102×C₄₁	C₂²	1	8	yes	complemented
52	C₄₁×#408	C₂²	1	8	yes	complemented
53	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	complemented
54	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	complemented
55	C₂⁵×C₄₁	—	2	1	no	—
56	C₂×#41×C₄₁	—	2	4	no	—
57	C₂×#41×C₄₁	—	2	4	no	—
58	C₂×#41×C₄₁	D₄	1	4	yes	—
59	C₂×#41×C₄₁	C₂³	1	4	yes	—
60	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₄₁	—	2	2	no	—
62	C₂²×D₄×C₄₁	—	2	2	no	—
63	C₂²×D₄×C₄₁	—	2	1	no	—
64	C₂²×D₄×C₄₁	—	2	2	no	—
65	C₂²×D₄×C₄₁	—	2	4	no	—
66	C₂²×D₄×C₄₁	—	2	4	no	—
67	C₂²×D₄×C₄₁	D₄	1	2	yes	—
68	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₄₁	—	2	1	no	—
72	C₂³×C₄×C₄₁	—	2	2	no	—
73	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₄₁	—	2	4	no	—
76	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
77	C₂²×C₈×C₄₁	—	2	4	no	—
78	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
81	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
82	#127×C₄₁	—	2	8	no	—
83	#124×C₄₁	—	2	8	no	—
84	#121×C₄₁	—	2	4	no	—
85	#121×C₄₁	—	2	4	no	—
86	#121×C₄₁	C₂³	1	8	yes	—
87	#121×C₄₁	C₂³	1	8	yes	—
88	#120×C₄₁	—	2	16	no	—
89	C₄×Q₈×C₄₁	—	2	8	no	—
90	C₄×D₄×C₄₁	—	2	8	no	—
91	C₄×D₄×C₄₁	—	2	16	no	—
92	#117×C₄₁	—	2	2	no	—
93	#117×C₄₁	—	2	2	no	—
94	#117×C₄₁	—	2	4	no	—
95	#117×C₄₁	C₂³	1	4	yes	—
96	#117×C₄₁	C₂³	1	4	yes	—
97	#117×C₄₁	C₂³	1	8	yes	complemented
98	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
102	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	complemented
109	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂×C₄²×C₄₁	—	2	1	no	—
115	C₂×C₄²×C₄₁	—	2	1	no	—
116	C₂×C₄²×C₄₁	—	2	2	no	—
117	#107×C₄₁	C₂³	1	4	yes	complemented
118	#107×C₄₁	C₂³	1	4	yes	complemented
119	#107×C₄₁	C₂³	1	8	yes	complemented
120	#106×C₄₁	C₂³	1	4	yes	complemented
121	#106×C₄₁	C₂³	1	4	yes	complemented
122	#106×C₄₁	C₂³	1	8	yes	complemented
123	#102×C₄₁	C₂³	1	8	yes	—
124	#102×C₄₁	C₂³	1	8	yes	—
125	#102×C₄₁	C₂³	1	8	yes	—
126	#102×C₄₁	C₂³	1	8	yes	—
127	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	complemented
128	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	complemented
129	C₂⁴×C₄₁	—	2	2	no	—
130	C₂⁴×C₄₁	—	2	4	no	—
131	C₂⁴×C₄₁	—	4	4	no	—
132	C₂⁴×C₄₁	—	8	2	no	—
133	C₂⁴×C₄₁	—	8	2	no	—
134	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
135	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
136	#41×C₄₁	—	2	16	no	—
137	#41×C₄₁	—	2	8	no	—
138	#41×C₄₁	—	2	8	no	—
139	C₂×Q₈×C₄₁	—	2	4	no	—
140	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
141	C₂×D₄×C₄₁	—	2	8	no	—
142	C₂×D₄×C₄₁	—	2	8	no	—
143	C₂×D₄×C₄₁	—	2	8	no	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	4	no	—
147	C₂×D₄×C₄₁	—	2	4	no	—
148	C₂×D₄×C₄₁	—	2	4	no	—
149	C₂×D₄×C₄₁	—	2	8	no	—
150	C₂×D₄×C₄₁	—	2	8	no	—
151	C₂×D₄×C₄₁	—	4	4	no	—
152	C₂×D₄×C₄₁	—	4	8	no	—
153	C₂×D₄×C₄₁	—	8	4	no	—
154	C₂×D₄×C₄₁	—	8	4	no	—
155	C₂×D₄×C₄₁	C₂×D₄	1	8	yes	—
156	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
157	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₄₁	—	2	4	no	—
159	C₂²×C₄×C₄₁	—	2	4	no	—
160	C₂²×C₄×C₄₁	—	2	1	no	—
161	C₂²×C₄×C₄₁	—	2	1	no	—
162	C₂²×C₄×C₄₁	—	2	1	no	—
163	C₂²×C₄×C₄₁	—	2	1	no	—
164	C₂²×C₄×C₄₁	—	2	2	no	—
165	C₂²×C₄×C₄₁	—	2	4	no	—
166	C₂²×C₄×C₄₁	—	2	8	no	—
167	C₂²×C₄×C₄₁	—	8	2	no	—
168	C₂²×C₄×C₄₁	—	8	4	no	—
169	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₄₁	#41	1	4	yes	—
174	C₂²×C₄×C₄₁	#41	1	4	yes	—
175	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
178	M₁₆×C₄₁	—	2	8	no	—
179	C₂×C₈×C₄₁	—	2	2	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₂×C₈×C₄₁	—	2	8	no	—
182	C₂×C₈×C₄₁	—	2	4	no	—
183	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
184	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
185	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	1	no	—
188	C₄⋊C₄×C₄₁	—	2	4	no	—
189	C₄⋊C₄×C₄₁	—	2	8	no	—
190	C₄⋊C₄×C₄₁	—	2	1	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
196	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
198	C₂²⋊C₄×C₄₁	—	2	4	no	—
199	C₂²⋊C₄×C₄₁	—	2	4	no	—
200	C₂²⋊C₄×C₄₁	—	2	8	no	—
201	C₂²⋊C₄×C₄₁	—	2	8	no	—
202	C₂²⋊C₄×C₄₁	—	2	8	no	—
203	C₄²×C₄₁	—	2	4	no	—
204	C₄²×C₄₁	—	2	4	no	—
205	C₄²×C₄₁	—	2	8	no	—
206	C₂³×C₄₁	—	2	2	no	—
207	C₂³×C₄₁	—	2	2	no	—
208	C₂³×C₄₁	—	2	4	no	—
209	C₂³×C₄₁	—	2	4	no	—
210	C₂³×C₄₁	—	2	4	no	—
211	C₂³×C₄₁	—	2	4	no	—
212	C₂³×C₄₁	—	4	8	no	—
213	C₂³×C₄₁	—	4	4	no	—
214	C₂³×C₄₁	—	4	4	no	—
215	C₂³×C₄₁	—	4	4	no	—
216	C₂³×C₄₁	—	8	4	no	—
217	C₂³×C₄₁	—	8	4	no	—
218	C₂³×C₄₁	—	8	16	no	—
219	C₂³×C₄₁	#123	1	4	yes	—
220	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
221	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
222	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₄₁	—	2	8	no	—
224	D₄×C₄₁	—	2	16	no	—
225	D₄×C₄₁	—	2	8	no	—
226	D₄×C₄₁	—	2	4	no	—
227	D₄×C₄₁	—	2	4	no	—
228	D₄×C₄₁	—	4	4	no	—
229	D₄×C₄₁	—	4	8	no	—
230	D₄×C₄₁	—	8	8	no	—
231	D₄×C₄₁	—	8	8	no	—
232	C₂×C₄×C₄₁	—	2	4	no	—
233	C₂×C₄×C₄₁	—	2	8	no	—
234	C₂×C₄×C₄₁	—	2	4	no	—
235	C₂×C₄×C₄₁	—	2	4	no	—
236	C₂×C₄×C₄₁	—	2	2	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	2	no	—
240	C₂×C₄×C₄₁	—	2	4	no	—
241	C₂×C₄×C₄₁	—	4	2	no	—
242	C₂×C₄×C₄₁	—	4	4	no	—
243	C₂×C₄×C₄₁	—	4	2	no	—
244	C₂×C₄×C₄₁	—	4	2	no	—
245	C₂×C₄×C₄₁	—	4	2	no	—
246	C₂×C₄×C₄₁	—	8	4	no	—
247	C₂×C₄×C₄₁	—	8	8	no	—
248	C₂×C₄×C₄₁	#123	1	4	yes	—
249	C₂×C₄×C₄₁	#123	1	8	yes	—
250	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
251	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
254	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
255	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
256	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
257	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
258	C₈×C₄₁	—	2	2	no	—
259	C₈×C₄₁	—	2	2	no	—
260	C₈×C₄₁	—	2	4	no	—
261	C₂×#3096	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂²×C₄₁	—	2	8	no	—
263	C₂²×C₄₁	—	2	2	no	—
264	C₂²×C₄₁	—	2	2	no	—
265	C₂²×C₄₁	—	2	4	no	—
266	C₂²×C₄₁	—	2	4	no	—
267	C₂²×C₄₁	—	2	4	no	—
268	C₂²×C₄₁	—	4	8	no	—
269	C₂²×C₄₁	—	4	4	no	—
270	C₂²×C₄₁	—	4	4	no	—
271	C₂²×C₄₁	—	4	4	no	—
272	C₂²×C₄₁	—	8	16	no	—
273	C₂²×C₄₁	—	8	8	no	—
274	C₂²×C₄₁	C₂×#123	1	4	yes	—
275	C₂²×C₄₁	C₂×#123	1	2	yes	—
276	C₂²×C₄₁	#576	1	2	yes	—
277	C₂²×C₄₁	#577	1	2	yes	—
278	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₄₁	—	2	4	no	—
280	C₄×C₄₁	—	4	2	no	—
281	C₄×C₄₁	—	4	4	no	—
282	C₄×C₄₁	—	4	2	no	—
283	C₄×C₄₁	—	4	2	no	—
284	C₄×C₄₁	—	4	2	no	—
285	C₄×C₄₁	C₂×#123	1	2	yes	—
286	C₄×C₄₁	C₂×#123	1	4	yes	—
287	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
288	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
289	C₂×#535	C₂×C₄₁	1	1	yes	characteristic
290	C₂×#520	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#3096	C₂×C₄₁	1	16	yes	direct product factor
292	C₂×#481	C₂×C₄₁	1	2	yes	complemented
293	C₂×#482	C₂×C₄₁	1	2	yes	complemented
294	C₂×#482	C₂×C₄₁	1	2	yes	complemented
295	C₂×#480	C₂×C₄₁	1	2	yes	complemented
296	C₂×#428	C₂×C₄₁	1	1	yes	characteristic, complemented
297	C₂×#427	C₂×C₄₁	1	1	yes	characteristic, complemented
298	C₂×#418	C₂×C₄₁	1	1	yes	characteristic, complemented
299	C₂²×#102	C₂×C₄₁	1	1	yes	characteristic, complemented
300	C₂×#408	C₂×C₄₁	1	1	yes	characteristic, complemented
301	C₂×C₄₁	—	2	4	no	—
302	C₂×C₄₁	—	4	4	no	—
303	C₂×C₄₁	—	8	4	no	—
304	C₂×C₄₁	#3096	1	4	yes	direct product factor
305	C₂×C₄₁	C₂²×#123	1	1	yes	characteristic
306	C₂×C₄₁	C₂×#576	1	1	yes	characteristic
307	C₂×C₄₁	C₂×#577	1	1	yes	characteristic
308	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
309	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
310	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
311	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
312	#535	—	2	4	no	—
313	#535	C₂²×C₄₁	1	8	yes	—
314	C₂×#127	—	2	1	no	—
315	C₂×#124	—	2	1	no	—
316	C₂×#121	C₂²×C₄₁	1	2	yes	—
317	C₂×#121	C₂²×C₄₁	1	2	yes	—
318	C₂×#120	—	2	2	no	—
319	#520	—	2	4	no	—
320	#520	C₂²×C₄₁	1	8	yes	complemented
321	C₂×C₄×Q₈	—	2	1	no	—
322	C₂×C₄×D₄	—	2	1	no	—
323	C₂×C₄×D₄	—	2	2	no	—
324	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
325	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
326	C₂×#117	C₂²×C₄₁	1	2	yes	complemented
327	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
328	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
329	#482	C₂²×C₄₁	1	16	yes	complemented
330	#482	C₂²×C₄₁	1	16	yes	complemented
331	#481	C₂²×C₄₁	1	16	yes	complemented
332	#480	C₂²×C₄₁	1	16	yes	complemented
333	#428	C₂²×C₄₁	1	8	yes	complemented
334	#427	C₂²×C₄₁	1	8	yes	complemented
335	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic, complemented
336	C₂×#107	C₂²×C₄₁	1	2	yes	complemented
337	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic, complemented
338	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic, complemented
339	C₂×#106	C₂²×C₄₁	1	2	yes	complemented
340	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic, complemented
341	#418	C₂²×C₄₁	1	8	yes	complemented
342	C₂×#102	C₂²×C₄₁	1	2	yes	—
343	C₂×#102	C₂²×C₄₁	1	2	yes	—
344	C₂×#102	C₂²×C₄₁	1	2	yes	—
345	C₂×#102	C₂²×C₄₁	1	2	yes	—
346	C₂×#102	C₂²×C₄₁	1	8	yes	complemented
347	#408	C₂²×C₄₁	1	8	yes	complemented
348	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	complemented
349	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	complemented
350	C₄₁	C₂×#3096	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁵	—	2	1	no	—
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	4	no	—
354	C₂×#41	D₄×C₄₁	1	4	yes	—
355	C₂×#41	C₂³×C₄₁	1	4	yes	—
356	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
357	C₂²×D₄	—	2	2	no	—
358	C₂²×D₄	—	2	2	no	—
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	4	no	—
362	C₂²×D₄	—	2	4	no	—
363	C₂²×D₄	D₄×C₄₁	1	2	yes	—
364	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
365	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
366	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	—	2	2	no	—
369	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
370	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
371	C₂×M₁₆	—	2	4	no	—
372	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
373	C₂²×C₈	—	2	4	no	—
374	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
377	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
378	#127	—	2	8	no	—
379	#124	—	2	8	no	—
380	#121	—	2	4	no	—
381	#121	—	2	4	no	—
382	#121	C₂³×C₄₁	1	8	yes	—
383	#121	C₂³×C₄₁	1	8	yes	—
384	#120	—	2	16	no	—
385	C₄×Q₈	—	2	8	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	16	no	—
388	#117	—	2	2	no	—
389	#117	—	2	2	no	—
390	#117	—	2	4	no	—
391	#117	C₂³×C₄₁	1	4	yes	—
392	#117	C₂³×C₄₁	1	4	yes	—
393	#117	C₂³×C₄₁	1	8	yes	complemented
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	2	no	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	4	no	—
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	complemented
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	2	2	no	—
413	#107	C₂³×C₄₁	1	4	yes	complemented
414	#107	C₂³×C₄₁	1	4	yes	complemented
415	#107	C₂³×C₄₁	1	8	yes	complemented
416	#106	C₂³×C₄₁	1	4	yes	complemented
417	#106	C₂³×C₄₁	1	4	yes	complemented
418	#106	C₂³×C₄₁	1	8	yes	complemented
419	#102	C₂³×C₄₁	1	8	yes	—
420	#102	C₂³×C₄₁	1	8	yes	—
421	#102	C₂³×C₄₁	1	8	yes	—
422	#102	C₂³×C₄₁	1	8	yes	—
423	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	complemented
424	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	complemented
425	C₂⁴	—	2	2	no	—
426	C₂⁴	—	2	4	no	—
427	C₂⁴	—	4	4	no	—
428	C₂⁴	—	8	2	no	—
429	C₂⁴	—	8	2	no	—
430	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
431	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
432	#41	—	2	16	no	—
433	#41	—	2	8	no	—
434	#41	—	2	8	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
437	C₂×D₄	—	2	8	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	8	4	no	—
450	C₂×D₄	—	8	4	no	—
451	C₂×D₄	C₂×D₄×C₄₁	1	8	yes	—
452	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
453	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	8	no	—
463	C₂²×C₄	—	8	2	no	—
464	C₂²×C₄	—	8	4	no	—
465	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
467	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
469	C₂²×C₄	#41×C₄₁	1	4	yes	—
470	C₂²×C₄	#41×C₄₁	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
474	M₁₆	—	2	8	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	8	no	—
478	C₂×C₈	—	2	4	no	—
479	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
480	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
481	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
490	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
491	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
492	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
493	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₄²	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	8	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	4	8	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	4	no	—
512	C₂³	—	8	4	no	—
513	C₂³	—	8	4	no	—
514	C₂³	—	8	16	no	—
515	C₂³	#123×C₄₁	1	4	yes	—
516	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
517	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
518	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
519	Q₈	—	2	8	no	—
520	D₄	—	2	16	no	—
521	D₄	—	2	8	no	—
522	D₄	—	2	4	no	—
523	D₄	—	2	4	no	—
524	D₄	—	4	4	no	—
525	D₄	—	4	8	no	—
526	D₄	—	8	8	no	—
527	D₄	—	8	8	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	—	8	8	no	—
544	C₂×C₄	#123×C₄₁	1	4	yes	—
545	C₂×C₄	#123×C₄₁	1	8	yes	—
546	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
547	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
549	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
550	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
551	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
552	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
553	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	2	4	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	8	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	8	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	8	16	no	—
568	C₂²	—	8	8	no	—
569	C₂²	C₂×#123×C₄₁	1	4	yes	—
570	C₂²	C₂×#123×C₄₁	1	2	yes	—
571	C₂²	C₄₁×#576	1	2	yes	—
572	C₂²	C₄₁×#577	1	2	yes	—
573	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
574	C₄	—	2	4	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	2	no	—
580	C₄	C₂×#123×C₄₁	1	2	yes	—
581	C₄	C₂×#123×C₄₁	1	4	yes	—
582	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
583	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
584	C₂	—	2	4	no	—
585	C₂	—	4	4	no	—
586	C₂	—	8	4	no	—
587	C₂	C₄₁×#3096	1	4	yes	direct product factor
588	C₂	C₂²×#123×C₄₁	1	1	yes	characteristic
589	C₂	C₂×C₄₁×#576	1	1	yes	characteristic
590	C₂	C₂×C₄₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂×C₄₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#520	C₂	semidirect
3. C₄₁×#3096	C₂	direct (N × H)
4. C₂×C₄₁×#481	C₂	semidirect
5. C₂×C₄₁×#482	C₂	semidirect
6. C₂×C₄₁×#482	C₂	semidirect
7. C₂×C₄₁×#480	C₂	semidirect
8. C₂×C₄₁×#428	C₂	semidirect
9. C₂×C₄₁×#427	C₂	semidirect
10. C₂×C₄₁×#418	C₂	semidirect
11. C₂²×#102×C₄₁	C₂	semidirect
12. C₂×C₄₁×#408	C₂	semidirect
25. C₄₁×#520	C₂²	semidirect
31. C₂×#117×C₄₁	C₂²	semidirect
33. C₂²×C₄⋊C₄×C₄₁	C₂²	semidirect
34. C₄₁×#482	C₂²	semidirect
35. C₄₁×#482	C₂²	semidirect
36. C₄₁×#481	C₂²	semidirect
37. C₄₁×#480	C₂²	semidirect
38. C₄₁×#428	C₂²	semidirect
39. C₄₁×#427	C₂²	semidirect
40. C₂×#107×C₄₁	C₂²	semidirect
41. C₂×#107×C₄₁	C₂²	semidirect
42. C₂×#107×C₄₁	C₂²	semidirect
43. C₂×#106×C₄₁	C₂²	semidirect
44. C₂×#106×C₄₁	C₂²	semidirect
45. C₂×#106×C₄₁	C₂²	semidirect
46. C₄₁×#418	C₂²	semidirect
51. C₂×#102×C₄₁	C₂²	semidirect
52. C₄₁×#408	C₂²	semidirect
53. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
54. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
97. #117×C₄₁	C₂³	semidirect
108. C₂×C₄⋊C₄×C₄₁	C₂³	semidirect
117. #107×C₄₁	C₂³	semidirect
118. #107×C₄₁	C₂³	semidirect
119. #107×C₄₁	C₂³	semidirect
120. #106×C₄₁	C₂³	semidirect
121. #106×C₄₁	C₂³	semidirect
122. #106×C₄₁	C₂³	semidirect
127. C₂²⋊C₈×C₄₁	C₂³	semidirect
128. C₂²⋊C₈×C₄₁	C₂³	semidirect
261. C₂×#3096	C₄₁	direct (N × H)
290. C₂×#520	C₂×C₄₁	semidirect
291. #3096	C₂×C₄₁	direct (N × H)
292. C₂×#481	C₂×C₄₁	semidirect
293. C₂×#482	C₂×C₄₁	semidirect
294. C₂×#482	C₂×C₄₁	semidirect
295. C₂×#480	C₂×C₄₁	semidirect
296. C₂×#428	C₂×C₄₁	semidirect
297. C₂×#427	C₂×C₄₁	semidirect
298. C₂×#418	C₂×C₄₁	semidirect
299. C₂²×#102	C₂×C₄₁	semidirect
300. C₂×#408	C₂×C₄₁	semidirect
304. C₂×C₄₁	#3096	direct (N × H)
320. #520	C₂²×C₄₁	semidirect
326. C₂×#117	C₂²×C₄₁	semidirect
328. C₂²×C₄⋊C₄	C₂²×C₄₁	semidirect
329. #482	C₂²×C₄₁	semidirect
330. #482	C₂²×C₄₁	semidirect
331. #481	C₂²×C₄₁	semidirect
332. #480	C₂²×C₄₁	semidirect
333. #428	C₂²×C₄₁	semidirect
334. #427	C₂²×C₄₁	semidirect
335. C₂×#107	C₂²×C₄₁	semidirect
336. C₂×#107	C₂²×C₄₁	semidirect
337. C₂×#107	C₂²×C₄₁	semidirect
338. C₂×#106	C₂²×C₄₁	semidirect
339. C₂×#106	C₂²×C₄₁	semidirect
340. C₂×#106	C₂²×C₄₁	semidirect
341. #418	C₂²×C₄₁	semidirect
346. C₂×#102	C₂²×C₄₁	semidirect
347. #408	C₂²×C₄₁	semidirect
348. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
349. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
350. C₄₁	C₂×#3096	direct (N × H)
393. #117	C₂³×C₄₁	semidirect
404. C₂×C₄⋊C₄	C₂³×C₄₁	semidirect
413. #107	C₂³×C₄₁	semidirect
414. #107	C₂³×C₄₁	semidirect
415. #107	C₂³×C₄₁	semidirect
416. #106	C₂³×C₄₁	semidirect
417. #106	C₂³×C₄₁	semidirect
418. #106	C₂³×C₄₁	semidirect
423. C₂²⋊C₈	C₂³×C₄₁	semidirect
424. C₂²⋊C₈	C₂³×C₄₁	semidirect
587. C₂	C₄₁×#3096	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×#535	2
C₂×C₄₁×#520	2
C₄₁×#3096	2
C₂×C₄₁×#481	2
C₂×C₄₁×#482	2
C₂×C₄₁×#480	2
C₂×C₄₁×#428	2
C₂×C₄₁×#427	2
C₂×C₄₁×#418	2
C₂²×#102×C₄₁	2
C₂×C₄₁×#408	2
C₂×#3096	41

Subgroups — C2×C41×#3096 #23836713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3096 #23836713