Subgroups — C₄₁×#20006 #23796207

Order 10496.

586 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3380	C₂	1	2	yes	maximal
2	C₄₁×#3338	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1676	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1675	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1632	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#554	C₂²	1	4	yes	—
10	C₄₁×#500	C₂²	1	2	yes	—
11	C₄₁×#500	C₂²	1	2	yes	—
12	C₄₁×#499	—	2	2	no	—
13	C₄₁×#494	C₂²	1	2	yes	—
14	C₄₁×#484	—	2	4	no	—
15	C₄₁×#478	—	2	4	no	—
16	C₄₁×#477	C₂²	1	4	yes	—
17	C₄₁×#476	C₂²	1	4	yes	—
18	C₄₁×#467	—	2	2	no	—
19	C₈×Q₈×C₄₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
21	C₄×D₈×C₄₁	—	2	2	no	—
22	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₄₁	C₂²	1	2	yes	—
24	C₂×#107×C₄₁	—	2	2	no	—
25	C₄₁×#335	C₂²	1	1	yes	characteristic
26	C₄₁×#335	C₂²	1	1	yes	characteristic
27	C₄₁×#333	C₂²	1	4	yes	—
28	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₄₁	—	4	2	no	—
32	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
33	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₁	—	2	2	no	—
35	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
36	#125×C₄₁	—	2	4	no	—
37	C₂²⋊Q₈×C₄₁	—	2	4	no	—
38	C₂²⋊Q₈×C₄₁	—	4	4	no	—
39	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
40	C₄×D₄×C₄₁	—	2	4	no	—
41	C₄×D₄×C₄₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
43	#107×C₄₁	—	2	2	no	—
44	#107×C₄₁	—	2	2	no	—
45	#107×C₄₁	—	2	2	no	—
46	#107×C₄₁	—	2	2	no	—
47	#107×C₄₁	—	2	4	no	—
48	#107×C₄₁	D₄	1	1	yes	characteristic
49	#107×C₄₁	D₄	1	1	yes	characteristic
50	#106×C₄₁	—	2	4	no	—
51	C₄⋊C₈×C₄₁	—	2	2	no	—
52	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
53	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
54	#103×C₄₁	—	2	4	no	—
55	#103×C₄₁	D₄	1	2	yes	—
56	#102×C₄₁	—	2	4	no	—
57	#102×C₄₁	D₄	1	2	yes	—
58	C₂²⋊C₈×C₄₁	—	2	2	no	—
59	C₄×C₈×C₄₁	—	2	2	no	—
60	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₄₁	—	2	4	no	—
64	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₁	—	2	2	no	—
67	C₂²×C₄×C₄₁	—	4	4	no	—
68	Q₁₆×C₄₁	—	2	2	no	—
69	Q₁₆×C₄₁	—	8	4	no	—
70	D₈×C₄₁	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₄₁	—	2	1	no	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	4	4	no	—
75	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₄₁	—	2	2	no	—
80	C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	4	4	no	—
83	C₄⋊C₄×C₄₁	—	4	4	no	—
84	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄×C₄₁	—	2	2	no	—
87	C₂²⋊C₄×C₄₁	—	4	4	no	—
88	C₄²×C₄₁	—	2	2	no	—
89	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
90	C₂³×C₄₁	—	2	2	no	—
91	Q₈×C₄₁	—	2	2	no	—
92	Q₈×C₄₁	—	2	2	no	—
93	Q₈×C₄₁	—	8	4	no	—
94	D₄×C₄₁	—	2	2	no	—
95	D₄×C₄₁	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	4	2	no	—
100	C₂×C₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	16	4	no	—
103	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₈×C₄₁	—	2	2	no	—
107	C₈×C₄₁	—	2	1	no	—
108	C₈×C₄₁	—	2	1	no	—
109	C₈×C₄₁	—	4	2	no	—
110	C₈×C₄₁	—	4	2	no	—
111	C₈×C₄₁	C₂×Q₁₆	1	2	yes	—
112	C₈×C₄₁	#135	1	2	yes	—
113	#20006	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₄₁	—	2	4	no	—
115	C₂²×C₄₁	—	4	2	no	—
116	C₂²×C₄₁	—	4	2	no	—
117	C₂²×C₄₁	D₄²	1	1	yes	characteristic
118	C₄×C₄₁	—	2	1	no	—
119	C₄×C₄₁	—	4	2	no	—
120	C₄×C₄₁	—	4	2	no	—
121	C₄×C₄₁	—	4	2	no	—
122	C₄×C₄₁	—	8	4	no	—
123	C₄×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
125	C₄×C₄₁	#578	1	1	yes	characteristic
126	C₄×C₄₁	#578	1	1	yes	characteristic
127	#3380	C₂×C₄₁	1	2	yes	—
128	#3338	C₂×C₄₁	1	2	yes	complemented
129	#1713	C₂×C₄₁	1	1	yes	characteristic, complemented
130	#1676	C₂×C₄₁	1	2	yes	complemented
131	#1675	C₂×C₄₁	1	2	yes	complemented
132	#1632	C₂×C₄₁	1	4	yes	complemented
133	C₈×Q₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
134	C₈×D₈	C₂×C₄₁	1	1	yes	characteristic
135	C₂×C₄₁	—	4	4	no	—
136	C₂×C₄₁	#3283	1	1	yes	characteristic
137	C₂×C₄₁	D₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₄₁	#3292	1	1	yes	characteristic
139	#554	C₂²×C₄₁	1	4	yes	—
140	#500	C₂²×C₄₁	1	2	yes	—
141	#500	C₂²×C₄₁	1	2	yes	—
142	#499	—	2	2	no	—
143	#494	C₂²×C₄₁	1	2	yes	—
144	#484	—	2	4	no	—
145	#478	—	2	4	no	—
146	#477	C₂²×C₄₁	1	4	yes	—
147	#476	C₂²×C₄₁	1	4	yes	—
148	#467	—	2	2	no	—
149	C₈×Q₈	C₂²×C₄₁	1	2	yes	—
150	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
151	C₄×D₈	—	2	2	no	—
152	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
153	C₈×D₄	C₂²×C₄₁	1	2	yes	—
154	C₂×#107	—	2	2	no	—
155	#335	C₂²×C₄₁	1	1	yes	characteristic
156	#335	C₂²×C₄₁	1	1	yes	characteristic
157	#333	C₂²×C₄₁	1	4	yes	—
158	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
159	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
160	C₈²	C₂²×C₄₁	1	1	yes	characteristic
161	C₄₁	#20006	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
164	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
165	C₂²×C₈	—	2	2	no	—
166	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
167	#125	—	2	4	no	—
168	C₂²⋊Q₈	—	2	4	no	—
169	C₂²⋊Q₈	—	4	4	no	—
170	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
171	C₄×D₄	—	2	4	no	—
172	C₄×D₄	C₂³×C₄₁	1	2	yes	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	#107	—	2	2	no	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#107	—	2	4	no	—
179	#107	D₄×C₄₁	1	1	yes	characteristic
180	#107	D₄×C₄₁	1	1	yes	characteristic
181	#106	—	2	4	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
184	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
185	#103	—	2	4	no	—
186	#103	D₄×C₄₁	1	2	yes	—
187	#102	—	2	4	no	—
188	#102	D₄×C₄₁	1	2	yes	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
196	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	4	no	—
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	8	4	no	—
201	D₈	Q₁₆×C₄₁	1	4	yes	complemented
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
216	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	8	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂×Q₁₆×C₄₁	1	4	yes	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
236	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	C₂×Q₁₆×C₄₁	1	2	yes	—
243	C₈	#135×C₄₁	1	2	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	D₄²×C₄₁	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	4	no	—
253	C₄	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
254	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
255	C₄	C₄₁×#578	1	1	yes	characteristic
256	C₄	C₄₁×#578	1	1	yes	characteristic
257	C₂	—	4	4	no	—
258	C₂	C₄₁×#3283	1	1	yes	characteristic
259	C₂	D₄×Q₁₆×C₄₁	1	1	yes	characteristic
260	C₂	C₄₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×D₈×C₄₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3338	C₂	semidirect
3. C₄₁×#1713	C₂	semidirect
4. C₄₁×#1676	C₂	semidirect
5. C₄₁×#1675	C₂	semidirect
6. C₄₁×#1632	C₂	semidirect
7. C₈×Q₁₆×C₄₁	C₂	semidirect
70. D₈×C₄₁	Q₁₆	semidirect
113. #20006	C₄₁	direct (N × H)
128. #3338	C₂×C₄₁	semidirect
129. #1713	C₂×C₄₁	semidirect
130. #1676	C₂×C₄₁	semidirect
131. #1675	C₂×C₄₁	semidirect
132. #1632	C₂×C₄₁	semidirect
133. C₈×Q₁₆	C₂×C₄₁	semidirect
161. C₄₁	#20006	direct (N × H)
201. D₈	Q₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3380	2
C₄₁×#3338	2
C₄₁×#1713	2
C₄₁×#1676	2
C₄₁×#1675	2
C₄₁×#1632	2
C₈×Q₁₆×C₄₁	2
C₈×D₈×C₄₁	2
#20006	41

Subgroups — C41×#20006 #23796207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20006 #23796207