Subgroups — C₂×C₄₁×#2634 #23810801

Order 10496.

2286 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2634	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#400	C₂	1	2	yes	maximal
5	C₂×C₄₁×#400	C₂	1	2	yes	maximal
6	C₂×C₄₁×#398	C₂	1	2	yes	maximal
7	C₂×C₄₁×#395	C₂	1	2	yes	maximal
8	C₂×C₄₁×#387	C₂	1	4	yes	maximal
9	C₂×C₄₁×#383	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#125×C₄₁	—	2	8	no	—
12	C₂×#125×C₄₁	C₂²	1	8	yes	—
13	C₂×C₄×Q₈×C₄₁	—	2	8	no	—
14	C₂×C₄×Q₈×C₄₁	C₂²	1	8	yes	—
15	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
17	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#400	C₂²	1	16	yes	—
25	C₄₁×#400	C₂²	1	16	yes	—
26	C₄₁×#398	C₂²	1	16	yes	—
27	C₄₁×#395	C₂²	1	16	yes	—
28	C₄₁×#387	C₂²	1	32	yes	—
29	C₄₁×#383	C₂²	1	8	yes	—
30	C₂×#95×C₄₁	C₂²	1	2	yes	—
31	C₂×#95×C₄₁	C₂²	1	2	yes	—
32	C₂×#95×C₄₁	C₂²	1	2	yes	—
33	C₂×#95×C₄₁	C₂²	1	4	yes	—
34	C₂×#95×C₄₁	C₂²	1	4	yes	—
35	C₂×#95×C₄₁	C₂²	1	4	yes	—
36	C₂²×Q₈×C₄₁	—	2	4	no	—
37	C₂²×Q₈×C₄₁	—	2	8	no	—
38	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
40	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
47	#125×C₄₁	—	2	32	no	—
48	C₄×Q₈×C₄₁	—	2	32	no	—
49	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
50	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
52	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
54	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
55	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
56	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
59	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₄₁	C₂³	1	16	yes	—
63	C₂×C₄²×C₄₁	—	2	8	no	—
64	C₂×C₄²×C₄₁	D₄	1	8	yes	—
65	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
66	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
67	#95×C₄₁	C₂³	1	8	yes	—
68	#95×C₄₁	C₂³	1	8	yes	—
69	#95×C₄₁	C₂³	1	8	yes	—
70	#95×C₄₁	C₂³	1	16	yes	—
71	#95×C₄₁	C₂³	1	16	yes	—
72	#95×C₄₁	C₂³	1	16	yes	—
73	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
74	C₂×Q₈×C₄₁	—	2	32	no	—
75	C₂×Q₈×C₄₁	—	2	8	no	—
76	C₂×Q₈×C₄₁	—	2	16	no	—
77	C₂²×C₄×C₄₁	—	2	4	no	—
78	C₂²×C₄×C₄₁	—	2	4	no	—
79	C₂²×C₄×C₄₁	—	2	4	no	—
80	C₂²×C₄×C₄₁	—	2	4	no	—
81	C₂²×C₄×C₄₁	—	2	4	no	—
82	C₂²×C₄×C₄₁	—	2	4	no	—
83	C₂²×C₄×C₄₁	—	2	4	no	—
84	C₂²×C₄×C₄₁	—	2	4	no	—
85	C₂²×C₄×C₄₁	—	2	8	no	—
86	C₂²×C₄×C₄₁	—	2	8	no	—
87	C₂²×C₄×C₄₁	—	2	8	no	—
88	C₂²×C₄×C₄₁	—	2	8	no	—
89	C₂²×C₄×C₄₁	—	2	8	no	—
90	C₂²×C₄×C₄₁	—	2	8	no	—
91	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₄₁	#41	1	8	yes	—
95	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
96	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
98	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
103	C₄⋊C₄×C₄₁	—	2	16	no	—
104	C₄⋊C₄×C₄₁	—	2	8	no	—
105	C₄⋊C₄×C₄₁	—	2	16	no	—
106	C₄⋊C₄×C₄₁	—	2	8	no	—
107	C₄⋊C₄×C₄₁	—	2	8	no	—
108	C₄⋊C₄×C₄₁	—	2	16	no	—
109	C₄⋊C₄×C₄₁	—	2	32	no	—
110	C₄²×C₄₁	—	2	16	no	—
111	C₄²×C₄₁	C₂×D₄	1	16	yes	—
112	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
113	C₂³×C₄₁	C₂×#41	1	4	yes	—
114	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
115	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
116	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
117	Q₈×C₄₁	—	2	32	no	—
118	C₂×C₄×C₄₁	—	2	8	no	—
119	C₂×C₄×C₄₁	—	2	16	no	—
120	C₂×C₄×C₄₁	—	2	8	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	2	4	no	—
127	C₂×C₄×C₄₁	—	2	4	no	—
128	C₂×C₄×C₄₁	—	2	4	no	—
129	C₂×C₄×C₄₁	—	2	4	no	—
130	C₂×C₄×C₄₁	—	2	8	no	—
131	C₂×C₄×C₄₁	—	2	8	no	—
132	C₂×C₄×C₄₁	—	2	8	no	—
133	C₂×C₄×C₄₁	—	2	8	no	—
134	C₂×C₄×C₄₁	—	2	8	no	—
135	C₂×C₄×C₄₁	—	2	8	no	—
136	C₂×C₄×C₄₁	—	4	8	no	—
137	C₂×C₄×C₄₁	—	4	8	no	—
138	C₂×C₄×C₄₁	—	4	8	no	—
139	C₂×C₄×C₄₁	—	4	16	no	—
140	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
142	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
143	C₂×C₄×C₄₁	C₂×#41	1	8	yes	—
144	C₂×#2634	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₄₁	#525	1	4	yes	—
146	C₂²×C₄₁	#536	1	4	yes	—
147	C₂²×C₄₁	#536	1	4	yes	—
148	C₂²×C₄₁	#541	1	16	yes	—
149	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
150	C₂²×C₄₁	C₂²×#41	1	2	yes	—
151	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	2	yes	—
152	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	2	yes	—
153	C₄×C₄₁	—	2	4	no	—
154	C₄×C₄₁	—	2	8	no	—
155	C₄×C₄₁	—	2	4	no	—
156	C₄×C₄₁	—	2	4	no	—
157	C₄×C₄₁	—	4	4	no	—
158	C₄×C₄₁	—	4	4	no	—
159	C₄×C₄₁	—	4	4	no	—
160	C₄×C₄₁	—	4	8	no	—
161	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic
162	C₂²×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic
163	#2634	C₂×C₄₁	1	16	yes	direct product factor
164	C₂×#400	C₂×C₄₁	1	2	yes	—
165	C₂×#400	C₂×C₄₁	1	2	yes	—
166	C₂×#398	C₂×C₄₁	1	2	yes	—
167	C₂×#395	C₂×C₄₁	1	2	yes	—
168	C₂×#387	C₂×C₄₁	1	4	yes	—
169	C₂×#383	C₂×C₄₁	1	1	yes	characteristic
170	C₂×C₄₁	#2634	1	8	yes	direct product factor
171	C₂×C₄₁	C₂×#525	1	1	yes	characteristic
172	C₂×C₄₁	C₂×#536	1	1	yes	characteristic
173	C₂×C₄₁	C₂×#536	1	1	yes	characteristic
174	C₂×C₄₁	C₂×#541	1	4	yes	—
175	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×#125	—	2	8	no	—
177	C₂×#125	C₂²×C₄₁	1	8	yes	—
178	C₂×C₄×Q₈	—	2	8	no	—
179	C₂×C₄×Q₈	C₂²×C₄₁	1	8	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
182	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
183	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
186	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
187	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
188	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
189	#400	C₂²×C₄₁	1	16	yes	—
190	#400	C₂²×C₄₁	1	16	yes	—
191	#398	C₂²×C₄₁	1	16	yes	—
192	#395	C₂²×C₄₁	1	16	yes	—
193	#387	C₂²×C₄₁	1	32	yes	—
194	#383	C₂²×C₄₁	1	8	yes	—
195	C₂×#95	C₂²×C₄₁	1	2	yes	—
196	C₂×#95	C₂²×C₄₁	1	4	yes	—
197	C₂×#95	C₂²×C₄₁	1	2	yes	—
198	C₂×#95	C₂²×C₄₁	1	4	yes	—
199	C₂×#95	C₂²×C₄₁	1	4	yes	—
200	C₂×#95	C₂²×C₄₁	1	2	yes	—
201	C₄₁	C₂×#2634	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	8	no	—
204	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
205	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
206	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
207	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
208	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
209	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
210	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
212	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
213	#125	—	2	32	no	—
214	C₄×Q₈	—	2	32	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	8	no	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	8	no	—
221	C₂×C₄⋊C₄	—	2	16	no	—
222	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
223	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
224	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
225	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
226	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
227	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
228	C₂×C₄⋊C₄	C₂³×C₄₁	1	16	yes	—
229	C₂×C₄²	—	2	8	no	—
230	C₂×C₄²	D₄×C₄₁	1	8	yes	—
231	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
232	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
233	#95	C₂³×C₄₁	1	8	yes	—
234	#95	C₂³×C₄₁	1	8	yes	—
235	#95	C₂³×C₄₁	1	8	yes	—
236	#95	C₂³×C₄₁	1	16	yes	—
237	#95	C₂³×C₄₁	1	16	yes	—
238	#95	C₂³×C₄₁	1	16	yes	—
239	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
240	C₂×Q₈	—	2	32	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	16	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
258	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
259	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
260	C₂²×C₄	#41×C₄₁	1	8	yes	—
261	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
262	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
263	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
264	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
265	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
267	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
268	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
269	C₄⋊C₄	—	2	16	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	16	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	16	no	—
275	C₄⋊C₄	—	2	32	no	—
276	C₄²	—	2	16	no	—
277	C₄²	C₂×D₄×C₄₁	1	16	yes	—
278	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
279	C₂³	C₂×#41×C₄₁	1	4	yes	—
280	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
281	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
282	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
283	Q₈	—	2	32	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	16	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	16	no	—
306	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
307	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
309	C₂×C₄	C₂×#41×C₄₁	1	8	yes	—
310	C₂²	C₄₁×#525	1	4	yes	—
311	C₂²	C₄₁×#536	1	4	yes	—
312	C₂²	C₄₁×#536	1	4	yes	—
313	C₂²	C₄₁×#541	1	16	yes	—
314	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
315	C₂²	C₂²×#41×C₄₁	1	2	yes	—
316	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	2	yes	—
317	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	8	no	—
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	8	no	—
326	C₂	C₄₁×#2634	1	8	yes	direct product factor
327	C₂	C₂×C₄₁×#525	1	1	yes	characteristic
328	C₂	C₂×C₄₁×#536	1	1	yes	characteristic
329	C₂	C₂×C₄₁×#536	1	1	yes	characteristic
330	C₂	C₂×C₄₁×#541	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2634	C₂	direct (N × H)
144. C₂×#2634	C₄₁	direct (N × H)
163. #2634	C₂×C₄₁	direct (N × H)
170. C₂×C₄₁	#2634	direct (N × H)
201. C₄₁	C₂×#2634	direct (N × H)
326. C₂	C₄₁×#2634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₁	2
C₂²×C₄×Q₈×C₄₁	2
C₄₁×#2634	2
C₂×C₄₁×#400	2
C₂×C₄₁×#398	2
C₂×C₄₁×#395	2
C₂×C₄₁×#387	2
C₂×C₄₁×#383	2
C₂×#2634	41

Subgroups — C2×C41×#2634 #23810801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2634 #23810801