Subgroups — C₄₁×#20764 #23796965

Order 10496.

1014 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1836	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1834	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₄₁	C₄	1	8	yes	complemented
13	C₂×#97×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#384	—	2	2	no	—
16	C₄₁×#384	C₂²	1	2	yes	—
17	C₄₁×#384	C₂²	1	2	yes	—
18	C₄₁×#384	C₂²	1	2	yes	—
19	C₄₁×#382	C₂²	1	2	yes	—
20	C₄₁×#380	—	2	2	no	—
21	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
22	C₄₁×#340	C₂²	1	16	yes	—
23	C₂³×C₄×C₄₁	—	2	2	no	—
24	C₂³×C₄×C₄₁	D₄	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₁	—	2	4	no	—
27	C₂×M₁₆×C₄₁	—	2	4	no	—
28	C₂×M₁₆×C₄₁	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
30	#117×C₄₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	—	4	8	no	—
38	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄²×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₁	D₄	1	2	yes	—
44	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
45	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	#97×C₄₁	C₂×C₄	1	8	yes	complemented
48	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	complemented
49	#95×C₄₁	—	2	4	no	—
50	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	—	4	8	no	—
56	C₂²×C₄×C₄₁	—	4	4	no	—
57	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₁	#41	1	4	yes	—
64	C₂²×C₄×C₄₁	#41	1	4	yes	—
65	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₄₁	—	2	16	no	—
67	C₂×C₈×C₄₁	—	2	4	no	—
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
70	C₄⋊C₄×C₄₁	—	2	8	no	—
71	C₄⋊C₄×C₄₁	—	2	8	no	—
72	C₄⋊C₄×C₄₁	—	4	8	no	—
73	C₄⋊C₄×C₄₁	—	4	4	no	—
74	C₄⋊C₄×C₄₁	—	4	4	no	—
75	C₄⋊C₄×C₄₁	—	4	8	no	—
76	C₄⋊C₄×C₄₁	—	4	4	no	—
77	C₂²⋊C₄×C₄₁	—	4	4	no	—
78	C₄²×C₄₁	—	4	4	no	—
79	C₄²×C₄₁	C₄⋊C₄	1	8	yes	—
80	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
84	C₂³×C₄₁	—	2	2	no	—
85	C₂³×C₄₁	—	2	2	no	—
86	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₄₁	#124	1	2	yes	—
88	C₂³×C₄₁	#124	1	2	yes	—
89	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₁	—	2	2	no	—
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	16	no	—
93	C₂×C₄×C₄₁	—	4	4	no	—
94	C₂×C₄×C₄₁	—	4	4	no	—
95	C₂×C₄×C₄₁	—	4	8	no	—
96	C₂×C₄×C₄₁	—	4	4	no	—
97	C₂×C₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	—	8	8	no	—
99	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₁	#117	1	8	yes	—
103	C₂×C₄×C₄₁	#124	1	2	yes	—
104	C₂×C₄×C₄₁	#124	1	2	yes	—
105	C₂×C₄×C₄₁	#125	1	8	yes	—
106	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₄₁	#137	1	4	yes	—
108	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
111	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
112	C₈×C₄₁	—	2	8	no	—
113	#20764	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₄₁	—	2	2	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	2	4	no	—
117	C₂²×C₄₁	—	2	4	no	—
118	C₂²×C₄₁	#382	1	4	yes	—
119	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
121	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
122	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₄₁	C₂×#137	1	2	yes	—
124	C₄×C₄₁	—	2	4	no	—
125	C₄×C₄₁	—	8	4	no	—
126	C₄×C₄₁	—	8	4	no	—
127	C₄×C₄₁	#382	1	4	yes	—
128	C₄×C₄₁	#428	1	8	yes	—
129	C₄×C₄₁	C₂×#117	1	2	yes	—
130	C₄×C₄₁	C₂×#125	1	2	yes	—
131	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic
132	#2285	C₂×C₄₁	1	1	yes	characteristic
133	C₂×#384	C₂×C₄₁	1	1	yes	characteristic
134	#1836	C₂×C₄₁	1	4	yes	complemented
135	#1834	C₂×C₄₁	1	4	yes	complemented
136	#1736	C₂×C₄₁	1	4	yes	—
137	C₂×C₄₁	—	2	4	no	—
138	C₂×C₄₁	C₂×#382	1	1	yes	characteristic
139	C₂×C₄₁	C₂×#428	1	2	yes	—
140	C₂×C₄₁	#3138	1	1	yes	characteristic
141	C₂×C₄₁	#3139	1	1	yes	characteristic
142	C₂×C₄₁	#3141	1	2	yes	—
143	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
144	C₂×#117	C₂²×C₄₁	1	2	yes	—
145	C₂²×C₄⋊C₄	—	2	2	no	—
146	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
147	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
148	C₄×M₁₆	C₄×C₄₁	1	8	yes	complemented
149	C₂×#97	C₂²×C₄₁	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
151	#384	—	2	2	no	—
152	#384	C₂²×C₄₁	1	2	yes	—
153	#384	C₂²×C₄₁	1	2	yes	—
154	#384	C₂²×C₄₁	1	2	yes	—
155	#382	C₂²×C₄₁	1	2	yes	—
156	#380	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	#340	C₂²×C₄₁	1	16	yes	—
159	C₄₁	#20764	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₄₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×C₄×C₄₁	1	8	yes	—
166	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
167	#117	—	2	8	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	4	8	no	—
175	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
176	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
180	C₂×C₄²	D₄×C₄₁	1	2	yes	—
181	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
182	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
184	#97	C₂×C₄×C₄₁	1	8	yes	complemented
185	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	complemented
186	#95	—	2	4	no	—
187	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
195	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
200	C₂²×C₄	#41×C₄₁	1	4	yes	—
201	C₂²×C₄	#41×C₄₁	1	4	yes	—
202	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
203	M₁₆	—	2	16	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	8	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₄⋊C₄×C₄₁	1	8	yes	—
217	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
218	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
219	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
220	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
224	C₂³	#124×C₄₁	1	2	yes	—
225	C₂³	#124×C₄₁	1	2	yes	—
226	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	8	no	—
236	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
239	C₂×C₄	#117×C₄₁	1	8	yes	—
240	C₂×C₄	#124×C₄₁	1	2	yes	—
241	C₂×C₄	#124×C₄₁	1	2	yes	—
242	C₂×C₄	#125×C₄₁	1	8	yes	—
243	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
244	C₂×C₄	#137×C₄₁	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
247	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
248	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₄₁×#382	1	4	yes	—
255	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
256	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
257	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
258	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
259	C₂²	C₂×#137×C₄₁	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	8	4	no	—
262	C₄	—	8	4	no	—
263	C₄	C₄₁×#382	1	4	yes	—
264	C₄	C₄₁×#428	1	8	yes	—
265	C₄	C₂×#117×C₄₁	1	2	yes	—
266	C₄	C₂×#125×C₄₁	1	2	yes	—
267	C₂	—	2	4	no	—
268	C₂	C₂×C₄₁×#382	1	1	yes	characteristic
269	C₂	C₂×C₄₁×#428	1	2	yes	—
270	C₂	C₄₁×#3138	1	1	yes	characteristic
271	C₂	C₄₁×#3139	1	1	yes	characteristic
272	C₂	C₄₁×#3141	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1836	C₂	semidirect
5. C₄₁×#1834	C₂	semidirect
12. C₄×M₁₆×C₄₁	C₄	semidirect
47. #97×C₄₁	C₂×C₄	semidirect
48. C₄×C₈×C₄₁	C₂×C₄	semidirect
113. #20764	C₄₁	direct (N × H)
134. #1836	C₂×C₄₁	semidirect
135. #1834	C₂×C₄₁	semidirect
148. C₄×M₁₆	C₄×C₄₁	semidirect
159. C₄₁	#20764	direct (N × H)
184. #97	C₂×C₄×C₄₁	semidirect
185. C₄×C₈	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
C₂×C₄×M₁₆×C₄₁	2
C₄₁×#2285	2
C₂×C₄₁×#384	2
C₄₁×#1836	2
C₄₁×#1834	2
C₄₁×#1736	2
#20764	41

Subgroups — C41×#20764 #23796965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20764 #23796965