Subgroups — C₄₁×#34637 #23810838

Order 10496.

2390 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2642	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2641	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2640	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2639	C₂	1	8	yes	maximal, complemented
7	C₄₁×#2398	C₂	1	8	yes	maximal, complemented
8	C₄₁×#2297	C₂	1	4	yes	maximal, complemented
9	C₂×C₄₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₄₁	C₂²	1	2	yes	—
11	C₄₁×#533	—	2	8	no	—
12	C₄₁×#532	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#125×C₄₁	C₂²	1	2	yes	—
17	C₂×#124×C₄₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	4	yes	—
19	C₂×#121×C₄₁	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
22	C₂×#117×C₄₁	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
24	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#400	C₂²	1	16	yes	—
26	C₄₁×#397	C₂²	1	16	yes	—
27	C₄₁×#394	C₂²	1	16	yes	—
28	C₄₁×#393	C₂²	1	16	yes	—
29	C₄₁×#391	C₂²	1	8	yes	—
30	C₄₁×#390	C₂²	1	8	yes	—
31	C₄₁×#386	C₂²	1	16	yes	—
32	C₄×C₂²⋊C₄×C₄₁	C₂²	1	16	yes	—
33	C₄₁×#376	C₂²	1	2	yes	—
34	C₄₁×#376	C₂²	1	2	yes	—
35	C₄₁×#376	C₂²	1	4	yes	—
36	C₂×#95×C₄₁	C₂²	1	4	yes	—
37	C₂×#41×C₄₁	—	2	8	no	—
38	C₂×#41×C₄₁	—	2	16	no	—
39	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
40	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
41	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
42	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
43	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
44	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
45	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
46	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₁	—	2	4	no	—
49	C₄⋊Q₈×C₄₁	—	2	8	no	—
50	#127×C₄₁	—	2	4	no	—
51	#125×C₄₁	—	2	8	no	—
52	#124×C₄₁	—	2	8	no	—
53	C₂²⋊Q₈×C₄₁	—	2	16	no	—
54	#121×C₄₁	—	2	16	no	—
55	C₄×Q₈×C₄₁	—	2	16	no	—
56	C₄×D₄×C₄₁	—	2	16	no	—
57	#117×C₄₁	—	2	8	no	—
58	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
64	C₂×C₂²⋊C₄×C₄₁	C₂³	1	32	yes	—
65	C₂×C₄²×C₄₁	—	2	8	no	—
66	C₂×C₄²×C₄₁	D₄	1	8	yes	—
67	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
68	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
69	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
70	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
73	#95×C₄₁	C₂³	1	8	yes	—
74	#95×C₄₁	C₂³	1	8	yes	—
75	#95×C₄₁	C₂³	1	32	yes	—
76	C₂⁴×C₄₁	C₂⁴	1	4	yes	—
77	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
78	#41×C₄₁	—	4	32	no	—
79	C₂×Q₈×C₄₁	—	2	8	no	—
80	C₂×Q₈×C₄₁	—	2	4	no	—
81	C₂×Q₈×C₄₁	—	2	8	no	—
82	C₂×Q₈×C₄₁	—	4	8	no	—
83	C₂×D₄×C₄₁	—	2	8	no	—
84	C₂×D₄×C₄₁	—	2	16	no	—
85	C₂×D₄×C₄₁	—	2	8	no	—
86	C₂×D₄×C₄₁	—	2	4	no	—
87	C₂×D₄×C₄₁	—	2	8	no	—
88	C₂×D₄×C₄₁	—	4	8	no	—
89	C₂²×C₄×C₄₁	—	2	8	no	—
90	C₂²×C₄×C₄₁	—	2	8	no	—
91	C₂²×C₄×C₄₁	—	2	16	no	—
92	C₂²×C₄×C₄₁	—	4	4	no	—
93	C₂²×C₄×C₄₁	—	4	8	no	—
94	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
95	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₄₁	#41	1	8	yes	—
97	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
103	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₁	—	2	8	no	—
105	C₄⋊C₄×C₄₁	—	2	8	no	—
106	C₄⋊C₄×C₄₁	—	2	16	no	—
107	C₄⋊C₄×C₄₁	—	2	8	no	—
108	C₂²⋊C₄×C₄₁	—	2	8	no	—
109	C₂²⋊C₄×C₄₁	—	2	16	no	—
110	C₂²⋊C₄×C₄₁	—	4	32	no	—
111	C₄²×C₄₁	—	2	8	no	—
112	C₄²×C₄₁	—	2	8	no	—
113	C₄²×C₄₁	—	2	8	no	—
114	C₄²×C₄₁	C₂×D₄	1	8	yes	—
115	C₄²×C₄₁	C₂×D₄	1	4	yes	—
116	C₄²×C₄₁	C₂×D₄	1	4	yes	—
117	C₂³×C₄₁	—	2	4	no	—
118	C₂³×C₄₁	—	2	4	no	—
119	C₂³×C₄₁	—	2	8	no	—
120	C₂³×C₄₁	—	2	16	no	—
121	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
122	C₂³×C₄₁	C₂×#41	1	4	yes	—
123	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
124	Q₈×C₄₁	—	4	16	no	—
125	Q₈×C₄₁	—	4	8	no	—
126	D₄×C₄₁	—	4	16	no	—
127	D₄×C₄₁	—	4	16	no	—
128	D₄×C₄₁	—	4	8	no	—
129	C₂×C₄×C₄₁	—	2	4	no	—
130	C₂×C₄×C₄₁	—	2	4	no	—
131	C₂×C₄×C₄₁	—	2	8	no	—
132	C₂×C₄×C₄₁	—	2	4	no	—
133	C₂×C₄×C₄₁	—	2	8	no	—
134	C₂×C₄×C₄₁	—	2	8	no	—
135	C₂×C₄×C₄₁	—	2	16	no	—
136	C₂×C₄×C₄₁	—	2	16	no	—
137	C₂×C₄×C₄₁	—	4	8	no	—
138	C₂×C₄×C₄₁	—	4	16	no	—
139	C₂×C₄×C₄₁	—	4	16	no	—
140	C₂×C₄×C₄₁	#124	1	16	yes	—
141	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
142	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
143	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
146	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
147	#34637	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₁	—	2	4	no	—
149	C₂²×C₄₁	—	2	8	no	—
150	C₂²×C₄₁	—	4	4	no	—
151	C₂²×C₄₁	—	4	8	no	—
152	C₂²×C₄₁	—	4	16	no	—
153	C₂²×C₄₁	C₂×#124	1	4	yes	—
154	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
155	C₂²×C₄₁	C₂²×#41	1	2	yes	—
156	C₂²×C₄₁	#585	1	4	yes	—
157	C₄×C₄₁	—	2	4	no	—
158	C₄×C₄₁	—	2	4	no	—
159	C₄×C₄₁	—	4	4	no	—
160	C₄×C₄₁	—	4	8	no	—
161	C₄×C₄₁	—	4	8	no	—
162	C₄×C₄₁	C₂×#124	1	4	yes	—
163	C₄×C₄₁	C₂×#124	1	4	yes	—
164	C₂×#533	C₂×C₄₁	1	1	yes	characteristic, complemented
165	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
166	#2642	C₂×C₄₁	1	2	yes	complemented
167	#2641	C₂×C₄₁	1	2	yes	complemented
168	#2640	C₂×C₄₁	1	4	yes	complemented
169	#2639	C₂×C₄₁	1	8	yes	complemented
170	#2398	C₂×C₄₁	1	8	yes	complemented
171	#2297	C₂×C₄₁	1	4	yes	complemented
172	C₂×#376	C₂×C₄₁	1	1	yes	characteristic, complemented
173	C₂×C₄₁	—	2	4	no	—
174	C₂×C₄₁	—	8	4	no	—
175	C₂×C₄₁	C₂²×#124	1	1	yes	characteristic
176	C₂×C₄₁	#3449	1	2	yes	—
177	C₂×C₄₁	#3455	1	4	yes	—
178	C₂²×#41	C₂²×C₄₁	1	2	yes	—
179	#533	—	2	8	no	—
180	#532	—	2	8	no	—
181	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
182	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
183	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
184	C₂×#125	C₂²×C₄₁	1	2	yes	—
185	C₂×#124	C₂²×C₄₁	1	2	yes	—
186	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	4	yes	—
187	C₂×#121	C₂²×C₄₁	1	4	yes	—
188	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
189	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
190	C₂×#117	C₂²×C₄₁	1	2	yes	—
191	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
192	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
193	#400	C₂²×C₄₁	1	16	yes	—
194	#397	C₂²×C₄₁	1	16	yes	—
195	#394	C₂²×C₄₁	1	16	yes	—
196	#393	C₂²×C₄₁	1	16	yes	—
197	#391	C₂²×C₄₁	1	8	yes	—
198	#390	C₂²×C₄₁	1	8	yes	—
199	#386	C₂²×C₄₁	1	16	yes	—
200	C₄×C₂²⋊C₄	C₂²×C₄₁	1	16	yes	—
201	#376	C₂²×C₄₁	1	2	yes	—
202	#376	C₂²×C₄₁	1	2	yes	—
203	#376	C₂²×C₄₁	1	4	yes	—
204	C₂×#95	C₂²×C₄₁	1	4	yes	—
205	C₄₁	#34637	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×#41	—	2	8	no	—
207	C₂×#41	—	2	16	no	—
208	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
209	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
210	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
211	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
212	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
213	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
214	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
215	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
216	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	2	8	no	—
219	#127	—	2	4	no	—
220	#125	—	2	8	no	—
221	#124	—	2	8	no	—
222	C₂²⋊Q₈	—	2	16	no	—
223	#121	—	2	16	no	—
224	C₄×Q₈	—	2	16	no	—
225	C₄×D₄	—	2	16	no	—
226	#117	—	2	8	no	—
227	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
228	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
229	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
230	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
231	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
232	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
233	C₂×C₂²⋊C₄	C₂³×C₄₁	1	32	yes	—
234	C₂×C₄²	—	2	8	no	—
235	C₂×C₄²	D₄×C₄₁	1	8	yes	—
236	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
237	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
238	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
239	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
240	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
242	#95	C₂³×C₄₁	1	8	yes	—
243	#95	C₂³×C₄₁	1	8	yes	—
244	#95	C₂³×C₄₁	1	32	yes	—
245	C₂⁴	C₂⁴×C₄₁	1	4	yes	—
246	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
247	#41	—	4	32	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	8	no	—
251	C₂×Q₈	—	4	8	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	16	no	—
254	C₂×D₄	—	2	8	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	8	no	—
257	C₂×D₄	—	4	8	no	—
258	C₂²×C₄	—	2	8	no	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	16	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	8	no	—
263	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
264	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
265	C₂²×C₄	#41×C₄₁	1	8	yes	—
266	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
267	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
268	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
269	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
272	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	2	16	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	2	16	no	—
279	C₂²⋊C₄	—	4	32	no	—
280	C₄²	—	2	8	no	—
281	C₄²	—	2	8	no	—
282	C₄²	—	2	8	no	—
283	C₄²	C₂×D₄×C₄₁	1	8	yes	—
284	C₄²	C₂×D₄×C₄₁	1	4	yes	—
285	C₄²	C₂×D₄×C₄₁	1	4	yes	—
286	C₂³	—	2	4	no	—
287	C₂³	—	2	4	no	—
288	C₂³	—	2	8	no	—
289	C₂³	—	2	16	no	—
290	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
291	C₂³	C₂×#41×C₄₁	1	4	yes	—
292	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
293	Q₈	—	4	16	no	—
294	Q₈	—	4	8	no	—
295	D₄	—	4	16	no	—
296	D₄	—	4	16	no	—
297	D₄	—	4	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	16	no	—
305	C₂×C₄	—	2	16	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	16	no	—
309	C₂×C₄	#124×C₄₁	1	16	yes	—
310	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
312	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
313	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
314	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
315	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	8	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	16	no	—
321	C₂²	C₂×#124×C₄₁	1	4	yes	—
322	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
323	C₂²	C₂²×#41×C₄₁	1	2	yes	—
324	C₂²	C₄₁×#585	1	4	yes	—
325	C₄	—	2	4	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	4	no	—
328	C₄	—	4	8	no	—
329	C₄	—	4	8	no	—
330	C₄	C₂×#124×C₄₁	1	4	yes	—
331	C₄	C₂×#124×C₄₁	1	4	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂²×#124×C₄₁	1	1	yes	characteristic
335	C₂	C₄₁×#3449	1	2	yes	—
336	C₂	C₄₁×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
socle	C₂³×C₄₁	C₂⁵
prime radical	C₂×C₄₁×#532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#533	C₂	semidirect
3. C₄₁×#2642	C₂	semidirect
4. C₄₁×#2641	C₂	semidirect
5. C₄₁×#2640	C₂	semidirect
6. C₄₁×#2639	C₂	semidirect
7. C₄₁×#2398	C₂	semidirect
8. C₄₁×#2297	C₂	semidirect
9. C₂×C₄₁×#376	C₂	semidirect
147. #34637	C₄₁	direct (N × H)
164. C₂×#533	C₂×C₄₁	semidirect
166. #2642	C₂×C₄₁	semidirect
167. #2641	C₂×C₄₁	semidirect
168. #2640	C₂×C₄₁	semidirect
169. #2639	C₂×C₄₁	semidirect
170. #2398	C₂×C₄₁	semidirect
171. #2297	C₂×C₄₁	semidirect
172. C₂×#376	C₂×C₄₁	semidirect
205. C₄₁	#34637	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×#533	2
C₂×C₄₁×#532	2
C₄₁×#2642	2
C₄₁×#2641	2
C₄₁×#2640	2
C₄₁×#2639	2
C₄₁×#2398	2
C₄₁×#2297	2
C₂×C₄₁×#376	2
#34637	41

Subgroups — C41×#34637 #23810838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34637 #23810838