Subgroups — C₄₁×#34364 #23810565

Order 10496.

3390 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	C₂²×#127×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2380	C₂	1	4	yes	maximal, complemented
3	C₄₁×#2366	C₂	1	4	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₄₁	C₂	1	8	yes	maximal, complemented
5	C₄₁×#2328	C₂	1	8	yes	maximal, complemented
6	C₄₁×#2308	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2283	C₂	1	2	yes	maximal, complemented
8	C₂³×D₄×C₄₁	C₂²	1	2	yes	—
9	C₂³×D₄×C₄₁	C₂²	1	4	yes	—
10	C₂×#127×C₄₁	C₄	1	16	yes	complemented
11	C₂×#127×C₄₁	C₂²	1	4	yes	—
12	C₂×#127×C₄₁	C₂²	1	2	yes	—
13	C₂×#127×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₄₁	C₂²	1	16	yes	complemented
15	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#390	C₂²	1	16	yes	complemented
18	C₄₁×#386	C₂²	1	16	yes	complemented
19	C₄₁×#386	C₂²	1	16	yes	complemented
20	C₄₁×#386	C₂²	1	32	yes	complemented
21	C₄₁×#383	C₂²	1	4	yes	complemented
22	C₄₁×#382	C₂²	1	4	yes	complemented
23	C₄₁×#381	C₂²	1	4	yes	—
24	C₄₁×#380	C₂²	1	4	yes	—
25	C₄₁×#379	C₂²	1	8	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
27	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
28	C₄₁×#376	C₂²	1	4	yes	complemented
29	C₂⁵×C₄₁	C₂³	1	4	yes	—
30	C₂²×D₄×C₄₁	—	4	16	no	—
31	C₂²×D₄×C₄₁	—	4	8	no	—
32	C₂²×D₄×C₄₁	C₂×C₄	1	32	yes	complemented
33	C₂²×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
34	C₂²×D₄×C₄₁	C₂³	1	8	yes	—
35	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
36	C₂²×D₄×C₄₁	C₂³	1	8	yes	—
37	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
38	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
39	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₁	C₂³	1	16	yes	—
41	#127×C₄₁	C₂×C₄	1	16	yes	complemented
42	#127×C₄₁	C₂×C₄	1	32	yes	complemented
43	#127×C₄₁	C₂×C₄	1	16	yes	complemented
44	C₄×D₄×C₄₁	—	4	32	no	—
45	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₄₁	—	4	16	no	—
47	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
48	C₂×C₂²⋊C₄×C₄₁	C₂³	1	16	yes	—
49	C₂×C₂²⋊C₄×C₄₁	C₂³	1	32	yes	—
50	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	complemented
51	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
55	#95×C₄₁	C₂³	1	8	yes	—
56	#95×C₄₁	C₂³	1	8	yes	—
57	#95×C₄₁	C₂³	1	8	yes	—
58	#95×C₄₁	C₂³	1	8	yes	—
59	C₂⁴×C₄₁	—	2	16	no	—
60	C₂⁴×C₄₁	—	4	16	no	—
61	C₂⁴×C₄₁	C₂²×C₄	1	16	yes	—
62	C₂⁴×C₄₁	C₂⁴	1	8	yes	—
63	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
64	C₂×D₄×C₄₁	—	2	32	no	—
65	C₂×D₄×C₄₁	—	4	64	no	—
66	C₂×D₄×C₄₁	—	4	32	no	—
67	C₂×D₄×C₄₁	—	4	4	no	—
68	C₂×D₄×C₄₁	—	4	8	no	—
69	C₂×D₄×C₄₁	—	4	16	no	—
70	C₂×D₄×C₄₁	—	4	4	no	—
71	C₂×D₄×C₄₁	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₄₁	C₂²×C₄	1	32	yes	—
73	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
74	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
75	C₂²×C₄×C₄₁	—	2	4	no	—
76	C₂²×C₄×C₄₁	—	2	8	no	—
77	C₂²×C₄×C₄₁	—	4	16	no	—
78	C₂²×C₄×C₄₁	—	4	32	no	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
86	C₄⋊C₄×C₄₁	—	2	16	no	—
87	C₂²⋊C₄×C₄₁	—	2	16	no	—
88	C₂²⋊C₄×C₄₁	—	4	16	no	—
89	C₂²⋊C₄×C₄₁	—	4	32	no	—
90	C₄²×C₄₁	—	2	16	no	—
91	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
92	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
93	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
94	C₂³×C₄₁	—	2	4	no	—
95	C₂³×C₄₁	—	2	8	no	—
96	C₂³×C₄₁	—	2	32	no	—
97	C₂³×C₄₁	—	2	8	no	—
98	C₂³×C₄₁	—	2	32	no	—
99	C₂³×C₄₁	—	4	32	no	—
100	C₂³×C₄₁	—	8	32	no	—
101	C₂³×C₄₁	C₂³×C₄	1	16	yes	—
102	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
103	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
104	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
105	D₄×C₄₁	—	2	64	no	—
106	D₄×C₄₁	—	4	32	no	—
107	D₄×C₄₁	—	4	16	no	—
108	D₄×C₄₁	—	4	16	no	—
109	C₂×C₄×C₄₁	—	2	8	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	16	no	—
114	C₂×C₄×C₄₁	—	2	16	no	—
115	C₂×C₄×C₄₁	—	2	16	no	—
116	C₂×C₄×C₄₁	—	2	16	no	—
117	C₂×C₄×C₄₁	—	8	32	no	—
118	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
121	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₄₁	H(5,𝔽₂)	1	8	yes	—
123	#34364	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	2	4	no	—
125	C₂²×C₄₁	—	2	8	no	—
126	C₂²×C₄₁	—	2	32	no	—
127	C₂²×C₄₁	—	2	16	no	—
128	C₂²×C₄₁	—	4	16	no	—
129	C₂²×C₄₁	—	4	16	no	—
130	C₂²×C₄₁	—	8	32	no	—
131	C₂²×C₄₁	#518	1	4	yes	—
132	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
133	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	4	yes	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	2	2	no	—
137	C₄×C₄₁	—	2	4	no	—
138	C₄×C₄₁	—	4	16	no	—
139	C₄×C₄₁	#518	1	8	yes	—
140	C₂²×#127	C₂×C₄₁	1	1	yes	characteristic
141	#2380	C₂×C₄₁	1	4	yes	complemented
142	#2366	C₂×C₄₁	1	4	yes	complemented
143	D₄×C₂²⋊C₄	C₂×C₄₁	1	8	yes	complemented
144	#2328	C₂×C₄₁	1	8	yes	complemented
145	#2308	C₂×C₄₁	1	4	yes	complemented
146	#2283	C₂×C₄₁	1	2	yes	complemented
147	C₂×C₄₁	—	2	4	no	—
148	C₂×C₄₁	—	4	16	no	—
149	C₂×C₄₁	C₂×#518	1	1	yes	characteristic
150	C₂×C₄₁	C₂×#518	1	2	yes	—
151	C₂×C₄₁	#3543	1	4	yes	—
152	C₂³×D₄	C₂²×C₄₁	1	2	yes	—
153	C₂³×D₄	C₂²×C₄₁	1	4	yes	—
154	C₂×#127	C₄×C₄₁	1	16	yes	complemented
155	C₂×#127	C₂²×C₄₁	1	4	yes	—
156	C₂×#127	C₂²×C₄₁	1	2	yes	—
157	C₂×#127	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄×D₄	C₂²×C₄₁	1	16	yes	complemented
159	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
160	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
161	#390	C₂²×C₄₁	1	16	yes	complemented
162	#386	C₂²×C₄₁	1	16	yes	complemented
163	#386	C₂²×C₄₁	1	16	yes	complemented
164	#386	C₂²×C₄₁	1	32	yes	complemented
165	#383	C₂²×C₄₁	1	4	yes	complemented
166	#382	C₂²×C₄₁	1	4	yes	complemented
167	#381	C₂²×C₄₁	1	4	yes	—
168	#380	C₂²×C₄₁	1	4	yes	—
169	#379	C₂²×C₄₁	1	8	yes	—
170	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
171	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
172	#376	C₂²×C₄₁	1	4	yes	complemented
173	C₄₁	#34364	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁵	C₂³×C₄₁	1	4	yes	—
175	C₂²×D₄	—	4	16	no	—
176	C₂²×D₄	—	4	8	no	—
177	C₂²×D₄	C₂×C₄×C₄₁	1	32	yes	complemented
178	C₂²×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
179	C₂²×D₄	C₂³×C₄₁	1	8	yes	—
180	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
181	C₂²×D₄	C₂³×C₄₁	1	8	yes	—
182	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
183	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
184	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₄₁	1	16	yes	—
186	#127	C₂×C₄×C₄₁	1	16	yes	complemented
187	#127	C₂×C₄×C₄₁	1	32	yes	complemented
188	#127	C₂×C₄×C₄₁	1	16	yes	complemented
189	C₄×D₄	—	4	32	no	—
190	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
191	C₂×C₂²⋊C₄	—	4	16	no	—
192	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₄₁	1	16	yes	—
194	C₂×C₂²⋊C₄	C₂³×C₄₁	1	32	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	complemented
196	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
197	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
200	#95	C₂³×C₄₁	1	8	yes	—
201	#95	C₂³×C₄₁	1	8	yes	—
202	#95	C₂³×C₄₁	1	8	yes	—
203	#95	C₂³×C₄₁	1	8	yes	—
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂²×C₄×C₄₁	1	16	yes	—
207	C₂⁴	C₂⁴×C₄₁	1	8	yes	—
208	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	4	64	no	—
211	C₂×D₄	—	4	32	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	16	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	C₂²×C₄×C₄₁	1	32	yes	—
217	C₂×D₄	C₂²×C₄×C₄₁	1	32	yes	—
218	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
219	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	—	4	32	no	—
224	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
226	C₂²×C₄	C₂⁴×C₄₁	1	16	yes	—
227	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
231	C₄⋊C₄	—	2	16	no	—
232	C₂²⋊C₄	—	2	16	no	—
233	C₂²⋊C₄	—	4	16	no	—
234	C₂²⋊C₄	—	4	32	no	—
235	C₄²	—	2	16	no	—
236	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
237	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
238	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	32	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	32	no	—
244	C₂³	—	4	32	no	—
245	C₂³	—	8	32	no	—
246	C₂³	C₂³×C₄×C₄₁	1	16	yes	—
247	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
248	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
249	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
250	D₄	—	2	64	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	16	no	—
253	D₄	—	4	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	8	32	no	—
263	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
266	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
267	C₂×C₄	H(5,𝔽₂)×C₄₁	1	8	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	—	2	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	8	32	no	—
275	C₂²	C₄₁×#518	1	4	yes	—
276	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
277	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
278	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	4	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	16	no	—
283	C₄	C₄₁×#518	1	8	yes	—
284	C₂	—	2	4	no	—
285	C₂	—	4	16	no	—
286	C₂	C₂×C₄₁×#518	1	1	yes	characteristic
287	C₂	C₂×C₄₁×#518	1	2	yes	—
288	C₂	C₄₁×#3543	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂²×#127×C₄₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2380	C₂	semidirect
3. C₄₁×#2366	C₂	semidirect
4. D₄×C₂²⋊C₄×C₄₁	C₂	semidirect
5. C₄₁×#2328	C₂	semidirect
6. C₄₁×#2308	C₂	semidirect
7. C₄₁×#2283	C₂	semidirect
10. C₂×#127×C₄₁	C₄	semidirect
14. C₂×C₄×D₄×C₄₁	C₂²	semidirect
17. C₄₁×#390	C₂²	semidirect
18. C₄₁×#386	C₂²	semidirect
19. C₄₁×#386	C₂²	semidirect
20. C₄₁×#386	C₂²	semidirect
21. C₄₁×#383	C₂²	semidirect
22. C₄₁×#382	C₂²	semidirect
26. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
28. C₄₁×#376	C₂²	semidirect
32. C₂²×D₄×C₄₁	C₂×C₄	semidirect
33. C₂²×D₄×C₄₁	C₂×C₄	semidirect
41. #127×C₄₁	C₂×C₄	semidirect
42. #127×C₄₁	C₂×C₄	semidirect
43. #127×C₄₁	C₂×C₄	semidirect
50. C₂×C₄²×C₄₁	C₂×C₄	semidirect
123. #34364	C₄₁	direct (N × H)
141. #2380	C₂×C₄₁	semidirect
142. #2366	C₂×C₄₁	semidirect
143. D₄×C₂²⋊C₄	C₂×C₄₁	semidirect
144. #2328	C₂×C₄₁	semidirect
145. #2308	C₂×C₄₁	semidirect
146. #2283	C₂×C₄₁	semidirect
154. C₂×#127	C₄×C₄₁	semidirect
158. C₂×C₄×D₄	C₂²×C₄₁	semidirect
161. #390	C₂²×C₄₁	semidirect
162. #386	C₂²×C₄₁	semidirect
163. #386	C₂²×C₄₁	semidirect
164. #386	C₂²×C₄₁	semidirect
165. #383	C₂²×C₄₁	semidirect
166. #382	C₂²×C₄₁	semidirect
170. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
172. #376	C₂²×C₄₁	semidirect
173. C₄₁	#34364	direct (N × H)
177. C₂²×D₄	C₂×C₄×C₄₁	semidirect
178. C₂²×D₄	C₂×C₄×C₄₁	semidirect
186. #127	C₂×C₄×C₄₁	semidirect
187. #127	C₂×C₄×C₄₁	semidirect
188. #127	C₂×C₄×C₄₁	semidirect
195. C₂×C₄²	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₄₁	2
C₄₁×#2380	2
C₄₁×#2366	2
D₄×C₂²⋊C₄×C₄₁	2
C₄₁×#2328	2
C₄₁×#2308	2
C₄₁×#2283	2
#34364	41

Subgroups — C41×#34364 #23810565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34364 #23810565