Subgroups — #35225175

Order 20864.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782286	C₂	1	2	yes	maximal, complemented
2	#23782070	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782049	C₂	1	1	yes	maximal, characteristic, complemented
4	#23782025	C₂	1	2	yes	maximal, complemented
5	#23781911	C₂	1	2	yes	maximal, complemented
6	#23781906	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781901	C₂	1	1	yes	maximal, characteristic
8	#23781767	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₆₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₃₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₆₅₂	C₂²	1	1	yes	characteristic
13	C₂×#14579688	C₂²	1	2	yes	—
14	C₂×D₁₃₀₄	C₂²	1	1	yes	characteristic
15	#19104735	C₂²	1	2	yes	—
16	#19104730	—	2	2	no	—
17	#19104723	C₂²	1	2	yes	—
18	#19104721	—	2	2	no	—
19	#19104717	—	2	2	no	—
20	#19104714	C₂²	1	2	yes	—
21	#19104713	—	2	2	no	—
22	#19104709	C₂²	1	2	yes	—
23	C₄×Dic₃₂₆	C₂²	1	2	yes	—
24	C₄×D₆₅₂	C₂²	1	2	yes	—
25	#19104669	C₂²	1	1	yes	characteristic
26	#19104669	C₂²	1	1	yes	characteristic
27	#19104669	C₂²	1	2	yes	—
28	#19104660	C₂²	1	2	yes	—
29	#19104660	C₂²	1	2	yes	—
30	#19104657	C₂²	1	2	yes	—
31	#19104657	C₂²	1	2	yes	—
32	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
34	#117×C₁₆₃	C₂²	1	1	yes	characteristic
35	#97×C₁₆₃	C₂²	1	1	yes	characteristic
36	#97×C₁₆₃	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₂₆	—	2	2	no	—
40	C₂×Dic₃₂₆	C₂³	1	2	yes	—
41	C₂×D₆₅₂	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₁₆₃	—	2	2	no	—
43	C₂×C₄×D₁₆₃	—	2	2	no	—
44	Dic₆₅₂	—	2	2	no	—
45	#14579688	—	2	4	no	—
46	D₁₃₀₄	—	2	2	no	—
47	C₄⋊Dic₁₆₃	—	2	2	no	—
48	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
49	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
50	Dic₁₆₃⋊C₄	—	2	2	no	—
51	Dic₁₆₃⋊C₄	—	2	2	no	—
52	C₂²⋊Dic₁₆₃	—	2	2	no	—
53	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
54	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
55	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
56	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
57	C₄×Dic₁₆₃	—	2	2	no	—
58	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₆₃	—	2	2	no	—
60	C₂×C₈×C₁₆₃	—	2	2	no	—
61	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
69	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
70	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
71	C₂²×D₁₆₃	—	2	2	no	—
72	Dic₃₂₆	—	2	2	no	—
73	Dic₃₂₆	—	2	2	no	—
74	Dic₃₂₆	—	8	2	no	—
75	D₆₅₂	—	2	2	no	—
76	D₆₅₂	—	2	2	no	—
77	C₂²⋊D₁₆₃	—	8	2	no	—
78	C₂×Dic₁₆₃	—	2	2	no	—
79	C₂×Dic₁₆₃	—	2	2	no	—
80	C₂×Dic₁₆₃	—	2	2	no	—
81	C₄×D₁₆₃	—	8	2	no	—
82	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₆₃	—	2	1	no	—
84	C₂×C₄×C₁₆₃	—	2	1	no	—
85	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₆₃	—	2	1	no	—
92	C₈×C₁₆₃	—	2	1	no	—
93	C₈×C₁₆₃	#41	1	2	yes	—
94	C₈×C₁₆₃	#41	1	2	yes	—
95	C₂²×C₁₆₃	—	2	1	no	—
96	C₂²×C₁₆₃	—	2	1	no	—
97	C₂²×C₁₆₃	—	2	1	no	—
98	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₁₆₃	—	4	2	no	—
100	C₂×D₁₆₃	—	4	2	no	—
101	C₂×D₁₆₃	—	4	2	no	—
102	C₄×C₁₆₃	—	2	1	no	—
103	C₄×C₁₆₃	—	2	1	no	—
104	C₄×C₁₆₃	—	2	1	no	—
105	C₄×C₁₆₃	—	2	1	no	—
106	C₄×C₁₆₃	—	2	1	no	—
107	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
109	Dic₁₆₃	—	4	2	no	—
110	Dic₁₆₃	—	4	2	no	—
111	Dic₁₆₃	—	4	2	no	—
112	C₂×C₁₆₃	—	4	1	no	—
113	C₂×C₁₆₃	#532	1	1	yes	characteristic
114	C₂×C₁₆₃	#576	1	1	yes	characteristic
115	C₂×C₁₆₃	#578	1	1	yes	characteristic
116	D₁₆₃	—	8	2	no	—
117	C₁₆₃	#3175	1	1	yes	characteristic, complemented, Sylow
118	#3175	—	163	1	no	maximal, Sylow
119	#542	—	163	2	no	—
120	#500	—	163	1	no	—
121	#495	—	163	1	no	—
122	#489	—	163	2	no	—
123	#468	—	163	2	no	—
124	#467	—	163	1	no	—
125	#466	—	163	1	no	—
126	#440	—	163	2	no	—
127	C₄×Q₁₆	—	163	1	no	—
128	C₄×D₈	—	163	1	no	—
129	#405	D₁₆₃	1	1	yes	characteristic, complemented
130	C₂×Q₁₆	—	163	1	no	—
131	C₂×SD₁₆	—	163	2	no	—
132	C₂×D₈	—	163	1	no	—
133	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	163	2	no	—
136	#126	—	326	2	no	—
137	#124	—	326	2	no	—
138	#124	—	163	2	no	—
139	#123	—	326	2	no	—
140	C₂²⋊Q₈	—	326	2	no	—
141	C₂²⋊Q₈	—	163	2	no	—
142	#121	—	163	2	no	—
143	C₄×Q₈	—	163	2	no	—
144	C₄×D₄	—	163	2	no	—
145	#117	C₂×D₁₆₃	1	1	yes	characteristic
146	#107	—	163	1	no	—
147	#107	—	163	1	no	—
148	#107	—	163	2	no	—
149	#103	—	163	2	no	—
150	#103	—	163	2	no	—
151	#102	—	163	2	no	—
152	#102	—	163	2	no	—
153	#97	C₂×D₁₆₃	1	1	yes	characteristic
154	#97	C₂×D₁₆₃	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
157	C₂×Q₈	—	326	2	no	—
158	C₂×Q₈	—	163	2	no	—
159	C₂×D₄	—	326	2	no	—
160	C₂×D₄	—	163	2	no	—
161	C₂²×C₄	—	326	2	no	—
162	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
163	Q₁₆	—	326	2	no	—
164	SD₁₆	—	326	4	no	—
165	D₈	—	326	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
172	C₄⋊C₄	—	326	2	no	—
173	C₄⋊C₄	—	326	2	no	—
174	C₄⋊C₄	—	326	2	no	—
175	C₄⋊C₄	—	163	2	no	—
176	C₄⋊C₄	—	163	2	no	—
177	C₄⋊C₄	D₆₅₂	1	1	yes	characteristic
178	C₄⋊C₄	D₆₅₂	1	1	yes	characteristic
179	C₂²⋊C₄	—	326	2	no	—
180	C₂²⋊C₄	—	326	2	no	—
181	C₂²⋊C₄	—	326	2	no	—
182	C₂²⋊C₄	—	326	2	no	—
183	C₂²⋊C₄	—	326	2	no	—
184	C₂²⋊C₄	D₆₅₂	1	1	yes	characteristic
185	C₂²⋊C₄	D₆₅₂	1	1	yes	characteristic
186	C₄²	—	326	2	no	—
187	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
189	C₂³	—	326	2	no	—
190	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
191	Q₈	—	1304	2	no	—
192	Q₈	—	326	2	no	—
193	Q₈	—	326	2	no	—
194	D₄	—	1304	2	no	—
195	D₄	—	326	2	no	—
196	D₄	—	326	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	1304	2	no	—
200	C₂×C₄	—	326	2	no	—
201	C₂×C₄	—	326	2	no	—
202	C₂×C₄	—	326	2	no	—
203	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
205	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
206	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
207	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#14579699	1	2	yes	—
212	C₈	#14579699	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	652	2	no	—
217	C₂²	—	652	2	no	—
218	C₂²	—	652	2	no	—
219	C₂²	C₂²×D₆₅₂	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	652	2	no	—
226	C₄	—	652	2	no	—
227	C₄	—	652	2	no	—
228	C₄	C₂×#14579699	1	1	yes	characteristic
229	C₄	C₂×#14579699	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	1304	2	no	—
232	C₂	#23782215	1	1	yes	characteristic
233	C₂	#23782470	1	1	yes	characteristic
234	C₂	#23782482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₅₂
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#405×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	#23781901	C₂
hypercenter	#405	D₁₆₃
hypocenter	C₁₆₃	#3175
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #23782286	C₂	semidirect
2. #23782070	C₂	semidirect
3. #23782049	C₂	semidirect
4. #23782025	C₂	semidirect
5. #23781911	C₂	semidirect
6. #23781906	C₂	semidirect
8. #23781767	C₂	semidirect
9. C₄×Dic₆₅₂	C₂	semidirect
10. C₄×D₁₃₀₄	C₂	semidirect
11. #405×C₁₆₃	C₂	semidirect
117. C₁₆₃	#3175	semidirect
129. #405	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782286	2
#23782070	2
#23782049	2
#23782025	2
#23781911	2
#23781906	2
#23781901	2
#23781767	2
C₄×Dic₆₅₂	2
C₄×D₁₃₀₄	2
#405×C₁₆₃	2
#3175	163