Subgroups — C₄₁×#32668 #23808869

Order 10496.

920 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2548	C₂	1	2	yes	maximal, complemented
2	C₄₁×#2054	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2011	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1970	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1925	C₂	1	4	yes	maximal, complemented
7	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
8	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
9	C₂×#127×C₄₁	C₂²	1	2	yes	—
10	C₂×#124×C₄₁	—	2	2	no	—
11	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#102×C₄₁	C₂²	1	4	yes	—
15	C₂×#102×C₄₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#394	—	2	2	no	—
18	C₄₁×#394	—	2	2	no	—
19	C₄₁×#390	—	2	2	no	—
20	C₄₁×#390	C₂²	1	4	yes	—
21	C₄₁×#390	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#340	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₄₁	—	2	2	no	—
27	C₂²×D₄×C₄₁	—	2	2	no	—
28	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
29	C₂×D₈×C₄₁	—	2	4	no	—
30	C₂×D₈×C₄₁	—	4	2	no	—
31	C₂×D₈×C₄₁	—	4	2	no	—
32	C₂×D₈×C₄₁	—	4	4	no	—
33	C₂×D₈×C₄₁	—	4	2	no	—
34	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
36	#127×C₄₁	—	4	2	no	—
37	#127×C₄₁	—	4	2	no	—
38	#124×C₄₁	—	4	8	no	—
39	C₄×Q₈×C₄₁	—	4	1	no	—
40	C₄×Q₈×C₄₁	—	4	1	no	—
41	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	2	2	no	—
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	#107×C₄₁	—	4	2	no	—
52	C₄⋊C₈×C₄₁	—	4	1	no	—
53	#102×C₄₁	—	2	8	no	—
54	#102×C₄₁	—	4	4	no	—
55	C₄×C₈×C₄₁	—	4	1	no	—
56	#95×C₄₁	—	2	2	no	—
57	C₂⁴×C₄₁	—	2	4	no	—
58	C₂×Q₈×C₄₁	—	4	1	no	—
59	C₂×Q₈×C₄₁	—	4	1	no	—
60	C₂×Q₈×C₄₁	D₈	1	4	yes	complemented
61	C₂×D₄×C₄₁	—	2	4	no	—
62	C₂×D₄×C₄₁	—	2	4	no	—
63	C₂×D₄×C₄₁	—	2	4	no	—
64	C₂×D₄×C₄₁	—	2	4	no	—
65	C₂×D₄×C₄₁	—	2	4	no	—
66	C₂×D₄×C₄₁	—	2	4	no	—
67	C₂×D₄×C₄₁	—	4	4	no	—
68	C₂×D₄×C₄₁	—	4	4	no	—
69	C₂×D₄×C₄₁	—	4	4	no	—
70	C₂×D₄×C₄₁	—	4	4	no	—
71	C₂×D₄×C₄₁	—	8	4	no	—
72	C₂²×C₄×C₄₁	—	2	4	no	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
78	D₈×C₄₁	—	4	8	no	—
79	D₈×C₄₁	—	4	8	no	—
80	C₂×C₈×C₄₁	—	2	2	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	4	no	—
83	C₂×C₈×C₄₁	—	2	2	no	—
84	C₂×C₈×C₄₁	#41	1	4	yes	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	—	4	2	no	—
88	C₄⋊C₄×C₄₁	—	4	1	no	—
89	C₂²⋊C₄×C₄₁	—	4	8	no	—
90	C₂²⋊C₄×C₄₁	—	4	8	no	—
91	C₂²⋊C₄×C₄₁	—	8	4	no	—
92	C₂²⋊C₄×C₄₁	—	8	4	no	—
93	C₄²×C₄₁	—	2	1	no	—
94	C₄²×C₄₁	—	2	1	no	—
95	C₄²×C₄₁	—	2	2	no	—
96	C₄²×C₄₁	—	2	2	no	—
97	C₄²×C₄₁	—	4	4	no	—
98	C₂³×C₄₁	—	4	4	no	—
99	C₂³×C₄₁	—	4	4	no	—
100	C₂³×C₄₁	—	4	4	no	—
101	C₂³×C₄₁	—	4	4	no	—
102	C₂³×C₄₁	—	4	4	no	—
103	C₂³×C₄₁	—	4	4	no	—
104	C₂³×C₄₁	—	4	4	no	—
105	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₄₁	—	4	4	no	—
107	D₄×C₄₁	—	4	4	no	—
108	D₄×C₄₁	—	4	4	no	—
109	D₄×C₄₁	—	4	4	no	—
110	D₄×C₄₁	—	4	4	no	—
111	D₄×C₄₁	—	8	8	no	—
112	D₄×C₄₁	—	8	8	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	1	no	—
117	C₂×C₄×C₄₁	—	2	1	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
125	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
127	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₄₁	—	2	4	no	—
131	C₈×C₄₁	—	4	2	no	—
132	#32668	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₄₁	—	8	4	no	—
134	C₂²×C₄₁	—	8	4	no	—
135	C₂²×C₄₁	—	8	4	no	—
136	C₂²×C₄₁	—	8	4	no	—
137	C₂²×C₄₁	—	8	4	no	—
138	C₂²×C₄₁	—	8	4	no	—
139	C₂²×C₄₁	—	8	4	no	—
140	C₂²×C₄₁	#534	1	1	yes	characteristic
141	C₂²×C₄₁	#550	1	2	yes	—
142	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
143	C₂²×C₄₁	#575	1	1	yes	characteristic
144	C₂²×C₄₁	#576	1	2	yes	—
145	C₄×C₄₁	—	2	4	no	—
146	C₄×C₄₁	—	2	2	no	—
147	C₄×C₄₁	—	4	1	no	—
148	C₄×C₄₁	—	4	2	no	—
149	C₄×C₄₁	—	8	4	no	—
150	C₄×C₄₁	#538	1	2	yes	—
151	C₄×C₄₁	#555	1	1	yes	characteristic
152	C₄×C₄₁	H(3,𝔽₄)	1	1	yes	characteristic
153	#2548	C₂×C₄₁	1	2	yes	complemented
154	#2054	C₂×C₄₁	1	2	yes	complemented
155	#2011	C₂×C₄₁	1	4	yes	complemented
156	#1984	C₂×C₄₁	1	1	yes	characteristic, complemented
157	#1970	C₂×C₄₁	1	2	yes	complemented
158	#1925	C₂×C₄₁	1	4	yes	complemented
159	C₂×C₄₁	—	16	4	no	—
160	C₂×C₄₁	#2733	1	1	yes	characteristic
161	C₂×C₄₁	#3186	1	1	yes	characteristic
162	C₂×C₄₁	#3199	1	1	yes	characteristic
163	C₂×C₄₁	#3391	1	2	yes	—
164	C₂×C₄₁	#3403	1	2	yes	—
165	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
166	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
167	C₂×#127	C₂²×C₄₁	1	2	yes	—
168	C₂×#124	—	2	2	no	—
169	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×#107	C₂²×C₄₁	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
172	C₂×#102	C₂²×C₄₁	1	4	yes	—
173	C₂×#102	C₂²×C₄₁	1	4	yes	—
174	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
175	#394	—	2	2	no	—
176	#394	—	2	2	no	—
177	#390	—	2	2	no	—
178	#390	C₂²×C₄₁	1	4	yes	—
179	#390	C₂²×C₄₁	1	4	yes	—
180	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
181	#340	C₂²×C₄₁	1	2	yes	—
182	#336	C₂²×C₄₁	1	4	yes	—
183	C₄₁	#32668	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	2	2	no	—
187	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
188	C₂×D₈	—	2	4	no	—
189	C₂×D₈	—	4	2	no	—
190	C₂×D₈	—	4	2	no	—
191	C₂×D₈	—	4	4	no	—
192	C₂×D₈	—	4	2	no	—
193	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
195	#127	—	4	2	no	—
196	#127	—	4	2	no	—
197	#124	—	4	8	no	—
198	C₄×Q₈	—	4	1	no	—
199	C₄×Q₈	—	4	1	no	—
200	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
201	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	—	2	4	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
209	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
210	#107	—	4	2	no	—
211	C₄⋊C₈	—	4	1	no	—
212	#102	—	2	8	no	—
213	#102	—	4	4	no	—
214	C₄×C₈	—	4	1	no	—
215	#95	—	2	2	no	—
216	C₂⁴	—	2	4	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	D₈×C₄₁	1	4	yes	complemented
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	8	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
237	D₈	—	4	8	no	—
238	D₈	—	4	8	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	#41×C₄₁	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₂²⋊C₄	—	4	8	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₂²⋊C₄	—	8	4	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
284	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
285	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
286	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
287	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
288	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	C₄₁×#534	1	1	yes	characteristic
299	C₂²	C₄₁×#550	1	2	yes	—
300	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
301	C₂²	C₄₁×#575	1	1	yes	characteristic
302	C₂²	C₄₁×#576	1	2	yes	—
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	8	4	no	—
308	C₄	C₄₁×#538	1	2	yes	—
309	C₄	C₄₁×#555	1	1	yes	characteristic
310	C₄	C₄₁×H(3,𝔽₄)	1	1	yes	characteristic
311	C₂	—	16	4	no	—
312	C₂	C₄₁×#2733	1	1	yes	characteristic
313	C₂	C₄₁×#3186	1	1	yes	characteristic
314	C₂	C₄₁×#3199	1	1	yes	characteristic
315	C₂	C₄₁×#3391	1	2	yes	—
316	C₂	C₄₁×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#2548	C₂	semidirect
2. C₄₁×#2054	C₂	semidirect
3. C₄₁×#2011	C₂	semidirect
4. C₄₁×#1984	C₂	semidirect
5. C₄₁×#1970	C₂	semidirect
6. C₄₁×#1925	C₂	semidirect
60. C₂×Q₈×C₄₁	D₈	semidirect
132. #32668	C₄₁	direct (N × H)
153. #2548	C₂×C₄₁	semidirect
154. #2054	C₂×C₄₁	semidirect
155. #2011	C₂×C₄₁	semidirect
156. #1984	C₂×C₄₁	semidirect
157. #1970	C₂×C₄₁	semidirect
158. #1925	C₂×C₄₁	semidirect
183. C₄₁	#32668	direct (N × H)
219. C₂×Q₈	D₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#2548	2
C₄₁×#2054	2
C₄₁×#2011	2
C₄₁×#1984	2
C₄₁×#1970	2
C₄₁×#1925	2
#32668	41

Subgroups — C41×#32668 #23808869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#32668 #23808869