Subgroups — C₄₁×#21779 #23797980

Order 10496.

1270 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3485	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2888	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1915	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1891	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1829	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1785	C₂	1	2	yes	maximal, complemented
7	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#568	C₂²	1	4	yes	—
10	C₄₁×#547	—	2	4	no	—
11	C₄₁×#546	—	2	4	no	—
12	C₄₁×#536	C₂²	1	2	yes	—
13	C₄₁×#534	C₄	1	4	yes	—
14	C₄₁×#525	C₂²	1	2	yes	—
15	C₂×#123×C₄₁	C₄	1	4	yes	—
16	C₄₁×#518	C₂²	1	1	yes	characteristic
17	C₄₁×#413	—	2	4	no	—
18	C₄₁×#413	C₂²	1	4	yes	—
19	C₂×#101×C₄₁	C₂²	1	4	yes	complemented
20	C₂×#100×C₄₁	—	2	4	no	—
21	C₂×#100×C₄₁	C₂²	1	4	yes	—
22	C₄₁×#407	C₂²	1	4	yes	complemented
23	C₄₁×#344	C₂²	1	8	yes	complemented
24	H(5,𝔽₂)×C₄₁	—	4	4	no	—
25	C₂×#41×C₄₁	—	2	4	no	—
26	C₂×#41×C₄₁	—	4	4	no	—
27	C₂×#41×C₄₁	D₄	1	4	yes	—
28	C₂×#41×C₄₁	C₂³	1	2	yes	—
29	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
31	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
32	C₂²×D₄×C₄₁	D₄	1	4	yes	—
33	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	#131×C₄₁	—	4	8	no	—
37	C₂×M₁₆×C₄₁	—	2	8	no	—
38	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
39	C₄⋊Q₈×C₄₁	—	2	2	no	—
40	#124×C₄₁	—	2	2	no	—
41	#123×C₄₁	—	2	8	no	—
42	#123×C₄₁	—	2	4	no	—
43	#123×C₄₁	C₂×C₄	1	4	yes	—
44	#123×C₄₁	C₂×C₄	1	8	yes	—
45	C₂²⋊Q₈×C₄₁	—	2	2	no	—
46	C₂²⋊Q₈×C₄₁	—	2	4	no	—
47	#121×C₄₁	—	2	8	no	—
48	#121×C₄₁	—	2	2	no	—
49	#120×C₄₁	—	2	4	no	—
50	#120×C₄₁	C₂×C₄	1	4	yes	—
51	C₄×D₄×C₄₁	—	2	4	no	—
52	C₄×D₄×C₄₁	—	2	4	no	—
53	#117×C₄₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
57	#101×C₄₁	—	2	4	no	—
58	#101×C₄₁	—	4	4	no	—
59	#100×C₄₁	—	2	4	no	—
60	#100×C₄₁	—	2	8	no	—
61	C₂²⋊C₈×C₄₁	—	4	4	no	—
62	C₂⁴×C₄₁	C₂²⋊C₄	1	4	yes	—
63	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
64	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
65	#41×C₄₁	—	2	8	no	—
66	#41×C₄₁	—	4	8	no	—
67	#41×C₄₁	—	4	16	no	—
68	C₂×Q₈×C₄₁	—	2	2	no	—
69	C₂×Q₈×C₄₁	—	2	4	no	—
70	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
71	C₂×D₄×C₄₁	—	2	4	no	—
72	C₂×D₄×C₄₁	—	2	4	no	—
73	C₂×D₄×C₄₁	—	2	4	no	—
74	C₂×D₄×C₄₁	—	4	8	no	—
75	C₂×D₄×C₄₁	—	4	4	no	—
76	C₂×D₄×C₄₁	—	4	8	no	—
77	C₂×D₄×C₄₁	C₂²⋊C₄	1	8	yes	—
78	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
79	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
80	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	—	2	4	no	—
82	C₂²×C₄×C₄₁	—	2	4	no	—
83	C₂²×C₄×C₄₁	—	2	2	no	—
84	C₂²×C₄×C₄₁	—	2	4	no	—
85	C₂²×C₄×C₄₁	—	4	4	no	—
86	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
87	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₄₁	—	2	8	no	—
92	M₁₆×C₄₁	—	2	8	no	—
93	M₁₆×C₄₁	—	4	8	no	—
94	C₂×C₈×C₄₁	—	2	8	no	—
95	C₂×C₈×C₄₁	—	4	8	no	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	2	8	no	—
98	C₄⋊C₄×C₄₁	—	2	2	no	—
99	C₂²⋊C₄×C₄₁	—	2	4	no	—
100	C₂²⋊C₄×C₄₁	—	2	8	no	—
101	C₂²⋊C₄×C₄₁	—	2	4	no	—
102	C₂²⋊C₄×C₄₁	—	2	4	no	—
103	C₂²⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₂³×C₄₁	—	2	4	no	—
106	C₂³×C₄₁	—	2	1	no	—
107	C₂³×C₄₁	—	2	2	no	—
108	C₂³×C₄₁	—	4	4	no	—
109	C₂³×C₄₁	—	8	1	no	—
110	C₂³×C₄₁	—	8	2	no	—
111	C₂³×C₄₁	—	8	4	no	—
112	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₄₁	—	2	4	no	—
116	Q₈×C₄₁	—	4	4	no	—
117	D₄×C₄₁	—	2	8	no	—
118	D₄×C₄₁	—	2	8	no	—
119	D₄×C₄₁	—	2	4	no	—
120	D₄×C₄₁	—	4	8	no	—
121	D₄×C₄₁	—	4	8	no	—
122	D₄×C₄₁	—	8	4	no	—
123	C₂×C₄×C₄₁	—	2	8	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	1	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	2	4	no	—
129	C₂×C₄×C₄₁	—	4	4	no	—
130	C₂×C₄×C₄₁	—	4	8	no	—
131	C₂×C₄×C₄₁	—	4	4	no	—
132	C₂×C₄×C₄₁	—	8	4	no	—
133	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
136	C₈×C₄₁	—	4	8	no	—
137	#21779	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₁	—	2	4	no	—
139	C₂²×C₄₁	—	2	4	no	—
140	C₂²×C₄₁	—	4	1	no	—
141	C₂²×C₄₁	—	4	4	no	—
142	C₂²×C₄₁	—	4	4	no	—
143	C₂²×C₄₁	—	4	2	no	—
144	C₂²×C₄₁	—	4	4	no	—
145	C₂²×C₄₁	—	4	1	no	—
146	C₂²×C₄₁	—	8	4	no	—
147	C₂²×C₄₁	—	16	4	no	—
148	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₁	#518	1	2	yes	—
150	C₄×C₄₁	—	2	4	no	—
151	C₄×C₄₁	—	4	4	no	—
152	C₄×C₄₁	—	4	1	no	—
153	C₄×C₄₁	—	4	2	no	—
154	C₄×C₄₁	#534	1	4	yes	—
155	#3485	C₂×C₄₁	1	1	yes	characteristic
156	#2888	C₂×C₄₁	1	2	yes	complemented
157	#1915	C₂×C₄₁	1	4	yes	complemented
158	#1891	C₂×C₄₁	1	4	yes	complemented
159	#1829	C₂×C₄₁	1	2	yes	complemented
160	#1785	C₂×C₄₁	1	2	yes	complemented
161	C₂×C₄₁	—	2	1	no	—
162	C₂×C₄₁	—	2	2	no	—
163	C₂×C₄₁	—	4	4	no	—
164	C₂×C₄₁	—	4	4	no	—
165	C₂×C₄₁	—	8	1	no	—
166	C₂×C₄₁	#2308	1	1	yes	characteristic
167	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
168	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
169	#568	C₂²×C₄₁	1	4	yes	—
170	#547	—	2	4	no	—
171	#546	—	2	4	no	—
172	#536	C₂²×C₄₁	1	2	yes	—
173	#534	C₄×C₄₁	1	4	yes	—
174	#525	C₂²×C₄₁	1	2	yes	—
175	C₂×#123	C₄×C₄₁	1	4	yes	—
176	#518	C₂²×C₄₁	1	1	yes	characteristic
177	#413	—	2	4	no	—
178	#413	C₂²×C₄₁	1	4	yes	—
179	C₂×#101	C₂²×C₄₁	1	4	yes	complemented
180	C₂×#100	—	2	4	no	—
181	C₂×#100	C₂²×C₄₁	1	4	yes	—
182	#407	C₂²×C₄₁	1	4	yes	complemented
183	#344	C₂²×C₄₁	1	8	yes	complemented
184	C₄₁	#21779	1	1	yes	characteristic, direct product factor, Sylow
185	H(5,𝔽₂)	—	4	4	no	—
186	C₂×#41	—	2	4	no	—
187	C₂×#41	—	4	4	no	—
188	C₂×#41	D₄×C₄₁	1	4	yes	—
189	C₂×#41	C₂³×C₄₁	1	2	yes	—
190	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
191	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
192	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
193	C₂²×D₄	D₄×C₄₁	1	4	yes	—
194	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
195	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
196	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
197	#131	—	4	8	no	—
198	C₂×M₁₆	—	2	8	no	—
199	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
200	C₄⋊Q₈	—	2	2	no	—
201	#124	—	2	2	no	—
202	#123	—	2	8	no	—
203	#123	—	2	4	no	—
204	#123	C₂×C₄×C₄₁	1	4	yes	—
205	#123	C₂×C₄×C₄₁	1	8	yes	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	4	no	—
208	#121	—	2	8	no	—
209	#121	—	2	2	no	—
210	#120	—	2	4	no	—
211	#120	C₂×C₄×C₄₁	1	4	yes	—
212	C₄×D₄	—	2	4	no	—
213	C₄×D₄	—	2	4	no	—
214	#117	C₂³×C₄₁	1	2	yes	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	—	2	4	no	—
217	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
218	#101	—	2	4	no	—
219	#101	—	4	4	no	—
220	#100	—	2	4	no	—
221	#100	—	2	8	no	—
222	C₂²⋊C₈	—	4	4	no	—
223	C₂⁴	C₂²⋊C₄×C₄₁	1	4	yes	—
224	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
225	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
226	#41	—	2	8	no	—
227	#41	—	4	8	no	—
228	#41	—	4	16	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	C₂²⋊C₄×C₄₁	1	8	yes	—
239	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
240	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
241	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
251	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
252	M₁₆	—	2	8	no	—
253	M₁₆	—	2	8	no	—
254	M₁₆	—	4	8	no	—
255	C₂×C₈	—	2	8	no	—
256	C₂×C₈	—	4	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
265	C₄²	—	2	2	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	4	no	—
270	C₂³	—	8	1	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	4	no	—
273	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
274	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
275	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
276	Q₈	—	2	4	no	—
277	Q₈	—	4	4	no	—
278	D₄	—	2	8	no	—
279	D₄	—	2	8	no	—
280	D₄	—	2	4	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
296	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
297	C₈	—	4	8	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	8	4	no	—
307	C₂²	—	16	4	no	—
308	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
309	C₂²	C₄₁×#518	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	C₄₁×#534	1	4	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	4	4	no	—
318	C₂	—	4	4	no	—
319	C₂	—	8	1	no	—
320	C₂	C₄₁×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#2308
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2888	C₂	semidirect
3. C₄₁×#1915	C₂	semidirect
4. C₄₁×#1891	C₂	semidirect
5. C₄₁×#1829	C₂	semidirect
6. C₄₁×#1785	C₂	semidirect
19. C₂×#101×C₄₁	C₂²	semidirect
22. C₄₁×#407	C₂²	semidirect
23. C₄₁×#344	C₂²	semidirect
137. #21779	C₄₁	direct (N × H)
156. #2888	C₂×C₄₁	semidirect
157. #1915	C₂×C₄₁	semidirect
158. #1891	C₂×C₄₁	semidirect
159. #1829	C₂×C₄₁	semidirect
160. #1785	C₂×C₄₁	semidirect
179. C₂×#101	C₂²×C₄₁	semidirect
182. #407	C₂²×C₄₁	semidirect
183. #344	C₂²×C₄₁	semidirect
184. C₄₁	#21779	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3485	2
C₄₁×#2888	2
C₄₁×#1915	2
C₄₁×#1891	2
C₄₁×#1829	2
C₄₁×#1785	2
#21779	41

Subgroups — C41×#21779 #23797980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21779 #23797980