Subgroups — C₄₁×#11385 #23787586

Order 10496.

1038 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#340	C₂	1	4	yes	maximal
3	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic
4	C₄₁×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
6	C₂×#125×C₄₁	—	2	8	no	—
7	C₂×#125×C₄₁	—	2	2	no	—
8	C₂×#125×C₄₁	C₄	1	4	yes	complemented
9	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#95×C₄₁	—	2	2	no	—
14	C₄₁×#340	—	2	8	no	—
15	C₄₁×#336	—	2	2	no	—
16	C₂³×C₄×C₄₁	—	2	2	no	—
17	C₂³×C₄×C₄₁	—	2	2	no	—
18	C₂³×C₄×C₄₁	D₄	1	2	yes	—
19	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₄₁	—	2	2	no	—
21	C₂²×C₈×C₄₁	—	2	4	no	—
22	C₂²×C₈×C₄₁	—	2	2	no	—
23	C₂²×C₈×C₄₁	—	2	4	no	—
24	C₂²×C₈×C₄₁	Q₈	1	4	yes	—
25	#125×C₄₁	—	2	16	no	—
26	#125×C₄₁	—	2	16	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₁	D₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₄₁	Q₈	1	4	yes	—
37	C₂×C₄²×C₄₁	—	2	4	no	—
38	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₁	D₄	1	2	yes	—
40	#95×C₄₁	—	4	4	no	—
41	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₁	—	2	4	no	—
43	C₂²×C₄×C₄₁	—	2	4	no	—
44	C₂²×C₄×C₄₁	—	2	8	no	—
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	—	2	2	no	—
47	C₂²×C₄×C₄₁	—	2	2	no	—
48	C₂²×C₄×C₄₁	—	2	4	no	—
49	C₂²×C₄×C₄₁	—	4	4	no	—
50	C₂²×C₄×C₄₁	—	4	2	no	—
51	C₂²×C₄×C₄₁	—	4	2	no	—
52	C₂²×C₄×C₄₁	—	4	2	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	—	4	4	no	—
55	C₂²×C₄×C₄₁	—	4	4	no	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₁	#41	1	2	yes	—
63	C₂²×C₄×C₄₁	#41	1	2	yes	—
64	C₂×C₈×C₄₁	—	2	8	no	—
65	C₂×C₈×C₄₁	—	2	4	no	—
66	C₂×C₈×C₄₁	—	4	2	no	—
67	C₂×C₈×C₄₁	—	4	4	no	—
68	C₂×C₈×C₄₁	—	4	2	no	—
69	C₄⋊C₄×C₄₁	—	2	32	no	—
70	C₄⋊C₄×C₄₁	—	2	4	no	—
71	C₄⋊C₄×C₄₁	—	2	8	no	—
72	C₄⋊C₄×C₄₁	—	2	8	no	—
73	C₄⋊C₄×C₄₁	—	4	4	no	—
74	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
75	C₄²×C₄₁	—	2	4	no	—
76	C₄²×C₄₁	—	2	4	no	—
77	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₁	#120	1	2	yes	—
79	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₄₁	#124	1	1	yes	characteristic
81	C₂³×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂³×C₄₁	#135	1	4	yes	—
83	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
84	C₂³×C₄₁	#137	1	2	yes	—
85	C₂×C₄×C₄₁	—	2	4	no	—
86	C₂×C₄×C₄₁	—	2	8	no	—
87	C₂×C₄×C₄₁	—	2	4	no	—
88	C₂×C₄×C₄₁	—	2	4	no	—
89	C₂×C₄×C₄₁	—	4	4	no	—
90	C₂×C₄×C₄₁	—	4	4	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	8	no	—
93	C₂×C₄×C₄₁	—	4	4	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	—	4	4	no	—
96	C₂×C₄×C₄₁	—	4	8	no	—
97	C₂×C₄×C₄₁	—	8	4	no	—
98	C₂×C₄×C₄₁	—	8	2	no	—
99	C₂×C₄×C₄₁	—	8	2	no	—
100	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₄₁	#120	1	2	yes	—
104	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₄₁	#124	1	4	yes	—
108	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₄₁	—	4	4	no	—
112	#11385	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₄₁	#379	1	1	yes	characteristic
114	C₂²×C₄₁	#391	1	1	yes	characteristic
115	C₂²×C₄₁	#391	1	1	yes	characteristic
116	C₂²×C₄₁	#393	1	2	yes	—
117	C₂²×C₄₁	#395	1	2	yes	—
118	C₂²×C₄₁	#416	1	2	yes	—
119	C₂²×C₄₁	#417	1	2	yes	—
120	C₂²×C₄₁	#449	1	4	yes	—
121	C₂²×C₄₁	#452	1	4	yes	—
122	C₂²×C₄₁	#478	1	4	yes	—
123	C₂²×C₄₁	#479	1	4	yes	—
124	C₂²×C₄₁	#488	1	2	yes	—
125	C₂²×C₄₁	#490	1	1	yes	characteristic
126	C₂²×C₄₁	#491	1	1	yes	characteristic
127	C₂²×C₄₁	#499	1	2	yes	—
128	C₂²×C₄₁	#501	1	2	yes	—
129	C₄×C₄₁	—	4	4	no	—
130	C₄×C₄₁	—	4	8	no	—
131	C₄×C₄₁	—	8	4	no	—
132	C₄×C₄₁	#391	1	4	yes	—
133	C₄×C₄₁	#393	1	2	yes	—
134	C₄×C₄₁	#395	1	2	yes	—
135	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic
136	C₂×#340	C₂×C₄₁	1	4	yes	—
137	C₂×#336	C₂×C₄₁	1	1	yes	characteristic
138	C₄⋊#95	C₂×C₄₁	1	1	yes	characteristic
139	C₂×C₄₁	#1446	1	1	yes	characteristic
140	C₂×C₄₁	#1788	1	2	yes	—
141	C₂×C₄₁	#1989	1	2	yes	—
142	C₂×C₄₁	#1991	1	1	yes	characteristic
143	C₂×C₄₁	#1992	1	1	yes	characteristic
144	C₂×C₄₁	#2025	1	2	yes	—
145	C₂×C₄₁	#2026	1	2	yes	—
146	C₂×C₄₁	#2059	1	4	yes	—
147	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
148	C₂×#125	—	2	8	no	—
149	C₂×#125	—	2	2	no	—
150	C₂×#125	C₄×C₄₁	1	4	yes	complemented
151	C₂²×C₄⋊C₄	—	2	2	no	—
152	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
154	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
155	C₂×#95	—	2	2	no	—
156	#340	—	2	8	no	—
157	#336	—	2	2	no	—
158	C₄₁	#11385	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₄₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	4	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	Q₈×C₄₁	1	4	yes	—
168	#125	—	2	16	no	—
169	#125	—	2	16	no	—
170	C₂×C₄⋊C₄	—	2	16	no	—
171	C₂×C₄⋊C₄	—	2	8	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
178	C₂×C₄⋊C₄	D₄×C₄₁	1	8	yes	—
179	C₂×C₄⋊C₄	Q₈×C₄₁	1	4	yes	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
182	C₂×C₄²	D₄×C₄₁	1	2	yes	—
183	#95	—	4	4	no	—
184	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
202	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
205	C₂²×C₄	#41×C₄₁	1	2	yes	—
206	C₂²×C₄	#41×C₄₁	1	2	yes	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₄⋊C₄	—	2	32	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
221	C₂³	#120×C₄₁	1	2	yes	—
222	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
223	C₂³	#124×C₄₁	1	1	yes	characteristic
224	C₂³	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
225	C₂³	#135×C₄₁	1	4	yes	—
226	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
227	C₂³	#137×C₄₁	1	2	yes	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
244	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
245	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
246	C₂×C₄	#120×C₄₁	1	2	yes	—
247	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
248	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
249	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
250	C₂×C₄	#124×C₄₁	1	4	yes	—
251	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
252	C₂×C₄	C₄⋊Q₈×C₄₁	1	4	yes	—
253	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
254	C₈	—	4	4	no	—
255	C₂²	C₄₁×#379	1	1	yes	characteristic
256	C₂²	C₄₁×#391	1	1	yes	characteristic
257	C₂²	C₄₁×#391	1	1	yes	characteristic
258	C₂²	C₄₁×#393	1	2	yes	—
259	C₂²	C₄₁×#395	1	2	yes	—
260	C₂²	C₄₁×#416	1	2	yes	—
261	C₂²	C₄₁×#417	1	2	yes	—
262	C₂²	C₄₁×#449	1	4	yes	—
263	C₂²	C₄₁×#452	1	4	yes	—
264	C₂²	C₄₁×#478	1	4	yes	—
265	C₂²	C₄₁×#479	1	4	yes	—
266	C₂²	C₄₁×#488	1	2	yes	—
267	C₂²	C₄₁×#490	1	1	yes	characteristic
268	C₂²	C₄₁×#491	1	1	yes	characteristic
269	C₂²	C₄₁×#499	1	2	yes	—
270	C₂²	C₄₁×#501	1	2	yes	—
271	C₄	—	4	4	no	—
272	C₄	—	4	8	no	—
273	C₄	—	8	4	no	—
274	C₄	C₄₁×#391	1	4	yes	—
275	C₄	C₄₁×#393	1	2	yes	—
276	C₄	C₄₁×#395	1	2	yes	—
277	C₂	C₄₁×#1446	1	1	yes	characteristic
278	C₂	C₄₁×#1788	1	2	yes	—
279	C₂	C₄₁×#1989	1	2	yes	—
280	C₂	C₄₁×#1991	1	1	yes	characteristic
281	C₂	C₄₁×#1992	1	1	yes	characteristic
282	C₂	C₄₁×#2025	1	2	yes	—
283	C₂	C₄₁×#2026	1	2	yes	—
284	C₂	C₄₁×#2059	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×#125×C₄₁	C₄	semidirect
112. #11385	C₄₁	direct (N × H)
150. C₂×#125	C₄×C₄₁	semidirect
158. C₄₁	#11385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₁	2
C₂×C₄₁×#340	2
C₂×C₄₁×#336	2
C₄₁×C₄⋊#95	2
#11385	41

Subgroups — C41×#11385 #23787586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11385 #23787586