Subgroups — #35223231

Order 20864.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104742	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781737	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆₃⋊C₈	C₂	1	4	yes	maximal, complemented
5	#23781692	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781632	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781629	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781621	C₂	1	1	yes	maximal, characteristic, complemented
9	#525×C₁₆₃	C₂	1	1	yes	maximal, characteristic
10	#19104742	—	2	4	no	—
11	C₂×#14579683	C₂²	1	1	yes	characteristic
12	C₂×#14579683	C₂²	1	1	yes	characteristic
13	C₂×#14579683	C₂²	1	2	yes	complemented
14	C₂²×C₁₆₃⋊C₈	—	2	4	no	—
15	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	complemented
20	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	complemented
21	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	—
22	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	complemented
23	#19104646	C₂²	1	2	yes	complemented
24	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	complemented
25	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
27	#123×C₁₆₃	C₄	1	2	yes	—
28	C₂²⋊Q₈×C₁₆₃	C₄	1	2	yes	—
29	#121×C₁₆₃	C₄	1	2	yes	—
30	#120×C₁₆₃	C₄	1	2	yes	—
31	C₄×D₄×C₁₆₃	C₂²	1	4	yes	—
32	#117×C₁₆₃	C₂²	1	1	yes	characteristic
33	#14579683	—	2	4	no	—
34	#14579683	—	2	2	no	—
35	#14579683	—	2	2	no	—
36	C₂×C₁₆₃⋊C₈	—	2	4	no	—
37	C₂×C₁₆₃⋊C₈	—	2	4	no	—
38	C₂×C₁₆₃⋊C₈	—	2	4	no	—
39	C₂×C₁₆₃⋊C₈	—	2	2	no	—
40	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	complemented
41	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
42	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
44	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
46	C₂⁴×C₁₆₃	C₂×C₄	1	1	yes	characteristic
47	#41×C₁₆₃	—	2	2	no	—
48	#41×C₁₆₃	—	2	2	no	—
49	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₁₆₃	C₂×C₄	1	4	yes	—
51	C₂×D₄×C₁₆₃	C₂×C₄	1	2	yes	—
52	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₆₃	—	2	2	no	—
54	C₂²×C₄×C₁₆₃	—	2	2	no	—
55	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
65	C₄²×C₁₆₃	C₂³	1	2	yes	—
66	C₁₆₃⋊C₈	—	2	2	no	—
67	C₁₆₃⋊C₈	—	2	2	no	—
68	C₁₆₃⋊C₈	—	2	2	no	—
69	C₁₆₃⋊C₈	C₂×D₄	1	2	yes	—
70	C₁₆₃⋊C₈	C₂×D₄	1	2	yes	—
71	C₂³×C₁₆₃	—	2	2	no	—
72	C₂³×C₁₆₃	—	2	4	no	—
73	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
76	Q₈×C₁₆₃	—	2	2	no	—
77	D₄×C₁₆₃	—	2	4	no	—
78	D₄×C₁₆₃	—	2	8	no	—
79	D₄×C₁₆₃	—	2	2	no	—
80	C₂×C₄×C₁₆₃	—	2	2	no	—
81	C₂×C₄×C₁₆₃	—	2	4	no	—
82	C₂×C₄×C₁₆₃	—	2	2	no	—
83	C₂×C₄×C₁₆₃	—	2	2	no	—
84	C₂×C₄×C₁₆₃	—	2	2	no	—
85	C₂×C₄×C₁₆₃	—	2	2	no	—
86	C₂×C₄×C₁₆₃	C₂²×C₄	1	4	yes	—
87	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₆₃	#41	1	2	yes	—
92	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
93	C₂²×C₁₆₃	—	2	8	no	—
94	C₂²×C₁₆₃	—	2	2	no	—
95	C₂²×C₁₆₃	—	2	2	no	—
96	C₂²×C₁₆₃	—	2	2	no	—
97	C₂²×C₁₆₃	—	2	2	no	—
98	C₂²×C₁₆₃	—	2	4	no	—
99	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
100	C₂²×C₁₆₃	#131	1	4	yes	complemented
101	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₆₃	—	2	2	no	—
103	C₄×C₁₆₃	—	2	2	no	—
104	C₄×C₁₆₃	—	2	4	no	—
105	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
106	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₁₆₃	—	2	2	no	—
109	C₂×C₁₆₃	—	2	4	no	—
110	C₂×C₁₆₃	—	4	2	no	—
111	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₁₆₃	C₂×#131	1	2	yes	—
113	C₁₆₃	#2929	1	1	yes	characteristic, complemented, Sylow
114	#2929	—	163	1	no	maximal, Sylow
115	C₂×#131	—	163	1	no	—
116	C₂³×C₈	—	163	1	no	—
117	#525	D₁₆₃	1	1	yes	characteristic
118	#435	—	163	4	no	—
119	C₈×D₄	—	163	4	no	—
120	#424	—	163	1	no	—
121	#408	—	163	1	no	—
122	#407	—	163	1	no	—
123	#405	—	163	1	no	—
124	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
126	#131	—	326	4	no	—
127	C₂×M₁₆	—	163	1	no	—
128	C₂×M₁₆	—	163	2	no	—
129	C₂×M₁₆	—	163	1	no	—
130	C₂²×C₈	—	326	4	no	—
131	C₂²×C₈	—	163	1	no	—
132	C₂²×C₈	—	163	2	no	—
133	C₂²×C₈	—	163	2	no	—
134	C₂²×C₈	—	163	2	no	—
135	C₂²×C₈	—	163	1	no	—
136	#123	Dic₁₆₃	1	2	yes	—
137	C₂²⋊Q₈	Dic₁₆₃	1	2	yes	—
138	#121	Dic₁₆₃	1	2	yes	—
139	#120	Dic₁₆₃	1	2	yes	—
140	C₄×D₄	C₂×D₁₆₃	1	4	yes	—
141	#117	C₂×D₁₆₃	1	1	yes	characteristic
142	C₄⋊C₈	—	163	4	no	—
143	C₂²⋊C₈	—	163	4	no	—
144	C₂²⋊C₈	—	163	4	no	—
145	#97	—	163	2	no	—
146	C₄×C₈	—	163	2	no	—
147	C₂⁴	C₂×Dic₁₆₃	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₁₆₃	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₁₆₃	1	4	yes	—
152	C₂×D₄	C₂×Dic₁₆₃	1	2	yes	—
153	C₂×D₄	C₂×Dic₁₆₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
157	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
161	M₁₆	—	326	2	no	—
162	M₁₆	—	326	2	no	—
163	M₁₆	—	326	4	no	—
164	C₂×C₈	—	326	2	no	—
165	C₂×C₈	—	326	4	no	—
166	C₂×C₈	—	326	4	no	—
167	C₂×C₈	—	326	4	no	—
168	C₂×C₈	—	163	2	no	—
169	C₂×C₈	—	163	2	no	—
170	C₂×C₈	—	163	2	no	—
171	C₂×C₈	—	163	4	no	—
172	C₂×C₈	—	163	1	no	—
173	C₂×C₈	—	163	1	no	—
174	C₄⋊C₄	C₂×Dic₁₆₃	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₁₆₃	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
179	C₄²	C₂²×D₁₆₃	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×Dic₁₆₃	1	2	yes	—
183	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
184	C₂³	C₂²×Dic₁₆₃	1	2	yes	—
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	8	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂²×Dic₁₆₃	1	4	yes	—
196	C₂×C₄	C₂²×Dic₁₆₃	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₆₃	1	2	yes	—
198	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
199	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
200	C₂×C₄	#14579698	1	2	yes	—
201	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
202	C₈	—	326	2	no	—
203	C₈	—	326	2	no	—
204	C₈	—	326	2	no	—
205	C₈	—	163	2	no	—
206	C₈	—	163	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	D₄×Dic₁₆₃	1	2	yes	—
214	C₂²	#19104742	1	4	yes	complemented
215	C₂²	C₂³×Dic₁₆₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	D₄×Dic₁₆₃	1	2	yes	—
220	C₄	C₂×D₄×D₁₆₃	1	1	yes	characteristic
221	C₄	C₂×#14579698	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₁₆₃	1	1	yes	characteristic
226	C₂	C₂×#19104742	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting, prime radical	#525×C₁₆₃	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#525	D₁₆₃
hypocenter	C₁₆₃	#2929
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#19104742	C₂	semidirect
2. C₂³×C₁₆₃⋊C₈	C₂	semidirect
3. #23781737	C₂	semidirect
4. D₄×C₁₆₃⋊C₈	C₂	semidirect
5. #23781692	C₂	semidirect
6. #23781632	C₂	semidirect
7. #23781629	C₂	semidirect
8. #23781621	C₂	semidirect
13. C₂×#14579683	C₂²	semidirect
19. C₂²×C₁₆₃⋊C₈	C₂²	semidirect
20. C₄⋊(C₁₆₃⋊C₈)	C₂²	semidirect
22. C₂²⋊(C₁₆₃⋊C₈)	C₂²	semidirect
23. #19104646	C₂²	semidirect
24. C₄×C₁₆₃⋊C₈	C₂²	semidirect
40. C₂×C₁₆₃⋊C₈	D₄	semidirect
100. C₂²×C₁₆₃	#131	semidirect
113. C₁₆₃	#2929	semidirect
214. C₂²	#19104742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104742	2
C₂³×C₁₆₃⋊C₈	2
#23781737	2
D₄×C₁₆₃⋊C₈	2
#23781692	2
#23781632	2
#23781629	2
#23781621	2
#525×C₁₆₃	2
#2929	163