Subgroups — C₄₁×#13776 #23789977

Order 10496.

2318 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#381	C₂	1	2	yes	maximal
3	C₂×C₄×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal
4	C₄₁×#1436	C₂	1	8	yes	maximal, complemented
5	C₄₁×C₂²⋊#95	C₂	1	2	yes	maximal
6	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₄₁	—	2	8	no	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	8	yes	complemented
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
14	C₄₁×#381	C₄	1	16	yes	complemented
15	C₄×C₂²⋊C₄×C₄₁	—	2	8	no	—
16	C₂×#95×C₄₁	C₂²	1	4	yes	—
17	C₂×#95×C₄₁	C₂²	1	8	yes	—
18	C₂×#95×C₄₁	C₂²	1	8	yes	—
19	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₄₁	—	2	4	no	—
21	C₂³×C₄×C₄₁	—	2	4	no	—
22	C₂³×C₄×C₄₁	—	2	4	no	—
23	C₂³×C₄×C₄₁	—	4	4	no	—
24	C₂³×C₄×C₄₁	—	4	4	no	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₁	D₄	1	8	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
31	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₁	—	2	32	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
38	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
40	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
41	C₂×C₄²×C₄₁	—	2	8	no	—
42	C₂×C₄²×C₄₁	—	4	8	no	—
43	#95×C₄₁	—	2	8	no	—
44	#95×C₄₁	—	4	8	no	—
45	#95×C₄₁	C₂×C₄	1	32	yes	complemented
46	C₂⁴×C₄₁	—	2	4	no	—
47	C₂⁴×C₄₁	—	2	4	no	—
48	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
49	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₁	#41	1	4	yes	—
52	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	8	no	—
54	C₂²×C₄×C₄₁	—	2	8	no	—
55	C₂²×C₄×C₄₁	—	2	32	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	8	no	—
59	C₂²×C₄×C₄₁	—	2	8	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	16	no	—
64	C₂²×C₄×C₄₁	—	2	8	no	—
65	C₂²×C₄×C₄₁	—	2	16	no	—
66	C₂²×C₄×C₄₁	—	4	8	no	—
67	C₂²×C₄×C₄₁	—	4	8	no	—
68	C₂²×C₄×C₄₁	—	4	16	no	—
69	C₂²×C₄×C₄₁	—	4	16	no	—
70	C₂²×C₄×C₄₁	C₄²	1	16	yes	complemented
71	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	16	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
79	C₄⋊C₄×C₄₁	—	2	32	no	—
80	C₂²⋊C₄×C₄₁	—	2	32	no	—
81	C₄²×C₄₁	—	4	16	no	—
82	C₂³×C₄₁	—	2	4	no	—
83	C₂³×C₄₁	—	2	4	no	—
84	C₂³×C₄₁	—	2	16	no	—
85	C₂³×C₄₁	—	2	8	no	—
86	C₂³×C₄₁	—	2	8	no	—
87	C₂³×C₄₁	—	2	16	no	—
88	C₂³×C₄₁	C₂×C₄²	1	4	yes	—
89	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
90	C₂³×C₄₁	#117	1	4	yes	—
91	C₂³×C₄₁	C₄×D₄	1	16	yes	—
92	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
93	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
94	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
95	C₂³×C₄₁	C₂×#41	1	2	yes	—
96	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
97	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	2	16	no	—
101	C₂×C₄×C₄₁	—	2	8	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	32	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	16	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	2	32	no	—
109	C₂×C₄×C₄₁	—	4	16	no	—
110	C₂×C₄×C₄₁	—	4	16	no	—
111	C₂×C₄×C₄₁	—	4	8	no	—
112	C₂×C₄×C₄₁	—	4	8	no	—
113	C₂×C₄×C₄₁	—	4	8	no	—
114	C₂×C₄×C₄₁	—	4	16	no	—
115	C₂×C₄×C₄₁	C₂×C₄²	1	8	yes	—
116	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
117	#13776	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	2	4	no	—
119	C₂²×C₄₁	—	2	4	no	—
120	C₂²×C₄₁	—	2	8	no	—
121	C₂²×C₄₁	—	2	32	no	—
122	C₂²×C₄₁	—	2	8	no	—
123	C₂²×C₄₁	C₄×C₂²⋊C₄	1	8	yes	—
124	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
125	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
127	C₂²×C₄₁	C₂×C₄×D₄	1	4	yes	—
128	C₂²×C₄₁	#518	1	2	yes	—
129	C₂²×C₄₁	#519	1	2	yes	—
130	C₂²×C₄₁	#520	1	8	yes	—
131	C₂²×C₄₁	#534	1	4	yes	—
132	C₂²×C₄₁	#536	1	4	yes	—
133	C₂²×C₄₁	#539	1	8	yes	—
134	C₄×C₄₁	—	2	8	no	—
135	C₄×C₄₁	—	2	8	no	—
136	C₄×C₄₁	—	4	4	no	—
137	C₄×C₄₁	—	4	16	no	—
138	C₂³×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
139	C₂×#381	C₂×C₄₁	1	2	yes	—
140	C₂×C₄×C₂²⋊C₄	C₂×C₄₁	1	2	yes	—
141	#1436	C₂×C₄₁	1	8	yes	complemented
142	C₂²⋊#95	C₂×C₄₁	1	2	yes	—
143	C₂×C₄₁	—	2	8	no	—
144	C₂×C₄₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄₁	#2274	1	2	yes	—
146	C₂×C₄₁	#2308	1	1	yes	characteristic
147	C₂×C₄₁	#2310	1	1	yes	characteristic
148	C₂×C₄₁	#2313	1	2	yes	—
149	C₂×C₄₁	#2322	1	8	yes	—
150	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
151	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
152	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	2	8	no	—
154	C₂²×C₄⋊C₄	C₄×C₄₁	1	8	yes	complemented
155	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
156	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
157	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
158	#381	C₄×C₄₁	1	16	yes	complemented
159	C₄×C₂²⋊C₄	—	2	8	no	—
160	C₂×#95	C₂²×C₄₁	1	4	yes	—
161	C₂×#95	C₂²×C₄₁	1	8	yes	—
162	C₂×#95	C₂²×C₄₁	1	8	yes	—
163	C₄₁	#13776	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
165	C₂³×C₄	—	2	4	no	—
166	C₂³×C₄	—	2	4	no	—
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	4	4	no	—
169	C₂³×C₄	—	4	4	no	—
170	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
171	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
172	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
173	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
174	C₂³×C₄	D₄×C₄₁	1	8	yes	—
175	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
176	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
177	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
178	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
179	C₂×C₄⋊C₄	—	2	32	no	—
180	C₂×C₄⋊C₄	—	2	8	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
183	C₂×C₂²⋊C₄	—	2	8	no	—
184	C₂×C₂²⋊C₄	—	2	8	no	—
185	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	4	8	no	—
188	#95	—	2	8	no	—
189	#95	—	4	8	no	—
190	#95	C₂×C₄×C₄₁	1	32	yes	complemented
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	2	4	no	—
193	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
194	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
195	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
196	C₂⁴	#41×C₄₁	1	4	yes	—
197	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	32	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	16	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	—	4	8	no	—
213	C₂²×C₄	—	4	16	no	—
214	C₂²×C₄	—	4	16	no	—
215	C₂²×C₄	C₄²×C₄₁	1	16	yes	complemented
216	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
217	C₂²×C₄	C₂²×C₄×C₄₁	1	16	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
222	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
224	C₄⋊C₄	—	2	32	no	—
225	C₂²⋊C₄	—	2	32	no	—
226	C₄²	—	4	16	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	16	no	—
230	C₂³	—	2	8	no	—
231	C₂³	—	2	8	no	—
232	C₂³	—	2	16	no	—
233	C₂³	C₂×C₄²×C₄₁	1	4	yes	—
234	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
235	C₂³	#117×C₄₁	1	4	yes	—
236	C₂³	C₄×D₄×C₄₁	1	16	yes	—
237	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
238	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
239	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
240	C₂³	C₂×#41×C₄₁	1	2	yes	—
241	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
242	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	16	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	32	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	16	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	32	no	—
254	C₂×C₄	—	4	16	no	—
255	C₂×C₄	—	4	16	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	16	no	—
260	C₂×C₄	C₂×C₄²×C₄₁	1	8	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	32	no	—
267	C₂²	C₄×C₂²⋊C₄×C₄₁	1	8	yes	—
268	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
270	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
271	C₂²	C₂×C₄×D₄×C₄₁	1	4	yes	—
272	C₂²	C₄₁×#518	1	2	yes	—
273	C₂²	C₄₁×#519	1	2	yes	—
274	C₂²	C₄₁×#520	1	8	yes	—
275	C₂²	C₄₁×#534	1	4	yes	—
276	C₂²	C₄₁×#536	1	4	yes	—
277	C₂²	C₄₁×#539	1	8	yes	—
278	C₄	—	2	8	no	—
279	C₄	—	2	8	no	—
280	C₄	—	4	4	no	—
281	C₄	—	4	16	no	—
282	C₂	—	2	8	no	—
283	C₂	C₂×C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
284	C₂	C₄₁×#2274	1	2	yes	—
285	C₂	C₄₁×#2308	1	1	yes	characteristic
286	C₂	C₄₁×#2310	1	1	yes	characteristic
287	C₂	C₄₁×#2313	1	2	yes	—
288	C₂	C₄₁×#2322	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1436	C₂	semidirect
10. C₂²×C₄⋊C₄×C₄₁	C₄	semidirect
14. C₄₁×#381	C₄	semidirect
45. #95×C₄₁	C₂×C₄	semidirect
70. C₂²×C₄×C₄₁	C₄²	semidirect
117. #13776	C₄₁	direct (N × H)
141. #1436	C₂×C₄₁	semidirect
154. C₂²×C₄⋊C₄	C₄×C₄₁	semidirect
158. #381	C₄×C₄₁	semidirect
163. C₄₁	#13776	direct (N × H)
190. #95	C₂×C₄×C₄₁	semidirect
215. C₂²×C₄	C₄²×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₄₁	2
C₂×C₄₁×#381	2
C₂×C₄×C₂²⋊C₄×C₄₁	2
C₄₁×#1436	2
C₄₁×C₂²⋊#95	2
#13776	41

Subgroups — C41×#13776 #23789977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13776 #23789977