Subgroups — #35223703

Order 20864.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782199	C₂	1	1	yes	maximal, characteristic
2	#23781794	C₂	1	2	yes	maximal, complemented
3	#23781741	C₂	1	2	yes	maximal, complemented
4	#23781735	C₂	1	2	yes	maximal, complemented
5	#23781734	C₂	1	2	yes	maximal, complemented
6	#23781725	C₂	1	2	yes	maximal, complemented
7	#23781722	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781690	C₂	1	2	yes	maximal, complemented
9	#423×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579682	C₂²	1	2	yes	—
11	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
12	#19104729	C₄	1	2	yes	—
13	#19104727	C₄	1	1	yes	characteristic
14	#19104723	C₄	1	1	yes	characteristic
15	#19104718	C₄	1	2	yes	—
16	#19104714	C₄	1	1	yes	characteristic
17	#19104709	C₄	1	1	yes	characteristic
18	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
19	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
21	#19104692	C₂²	1	2	yes	—
22	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
25	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
26	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
27	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
29	#19104646	C₂²	1	2	yes	—
30	#19104645	C₂²	1	2	yes	—
31	C₈×Dic₁₆₃	C₂²	1	2	yes	—
32	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
36	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
37	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
38	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
41	#14579682	—	4	2	no	—
42	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
44	C₈×D₁₆₃	—	4	2	no	—
45	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
47	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
52	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₆₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
60	M₁₆×C₁₆₃	—	4	1	no	—
61	M₁₆×C₁₆₃	—	4	1	no	—
62	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
63	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
64	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
67	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
68	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
69	Dic₃₂₆	—	4	1	no	—
70	D₆₅₂	—	4	1	no	—
71	C₂²⋊D₁₆₃	—	4	2	no	—
72	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
75	C₄×D₁₆₃	—	2	2	no	—
76	C₄×D₁₆₃	—	2	2	no	—
77	C₁₆₃⋊C₈	—	2	2	no	—
78	C₁₆₃⋊C₈	—	2	2	no	—
79	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₆₃	—	2	1	no	—
81	C₂×C₄×C₁₆₃	—	2	1	no	—
82	C₂×C₄×C₁₆₃	—	4	1	no	—
83	C₂×C₄×C₁₆₃	—	4	1	no	—
84	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₆₃	—	2	2	no	—
91	C₈×C₁₆₃	—	2	2	no	—
92	C₂²×C₁₆₃	—	2	1	no	—
93	C₂²×C₁₆₃	—	2	1	no	—
94	C₂²×C₁₆₃	—	2	1	no	—
95	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₆₃	—	2	2	no	—
97	C₂×D₁₆₃	—	2	2	no	—
98	C₂×D₁₆₃	—	2	2	no	—
99	C₄×C₁₆₃	—	2	1	no	—
100	C₄×C₁₆₃	—	2	1	no	—
101	C₄×C₁₆₃	—	2	1	no	—
102	C₄×C₁₆₃	—	2	1	no	—
103	C₄×C₁₆₃	—	2	1	no	—
104	C₄×C₁₆₃	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₁₆₃	—	2	2	no	—
107	Dic₁₆₃	—	2	2	no	—
108	Dic₁₆₃	—	2	2	no	—
109	C₂×C₁₆₃	—	4	1	no	—
110	C₂×C₁₆₃	#520	1	1	yes	characteristic
111	C₂×C₁₆₃	#568	1	1	yes	characteristic
112	C₂×C₁₆₃	#568	1	1	yes	characteristic
113	D₁₆₃	—	4	2	no	—
114	C₁₆₃	#2995	1	1	yes	characteristic, complemented, Sylow
115	#2995	—	163	1	no	maximal, Sylow
116	#529	—	163	1	no	—
117	#446	—	163	2	no	—
118	#436	—	163	2	no	—
119	#435	—	163	2	no	—
120	#435	—	163	2	no	—
121	#433	—	163	2	no	—
122	#432	—	163	1	no	—
123	#424	—	163	2	no	—
124	#423	D₁₆₃	1	1	yes	characteristic, complemented
125	C₂×M₁₆	—	163	2	no	—
126	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
127	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
128	C₂²×C₈	—	163	2	no	—
129	#126	—	163	2	no	—
130	#125	—	163	1	no	—
131	#124	—	163	1	no	—
132	#123	—	163	2	no	—
133	C₂²⋊Q₈	—	163	1	no	—
134	#121	—	163	1	no	—
135	C₄×Q₈	—	163	1	no	—
136	C₄×D₄	—	163	2	no	—
137	C₄×D₄	—	163	1	no	—
138	#117	—	163	2	no	—
139	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
140	C₄⋊C₈	—	163	2	no	—
141	C₄⋊C₈	—	163	2	no	—
142	C₄⋊C₈	—	163	1	no	—
143	C₄⋊C₈	—	163	1	no	—
144	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
145	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
146	C₂²⋊C₈	—	163	2	no	—
147	C₂²⋊C₈	—	163	2	no	—
148	C₂²⋊C₈	—	163	2	no	—
149	#97	—	163	2	no	—
150	#97	—	163	2	no	—
151	C₄×C₈	—	163	2	no	—
152	C₂×Q₈	—	163	1	no	—
153	C₂×D₄	—	163	2	no	—
154	C₂×D₄	—	163	1	no	—
155	C₂²×C₄	—	163	2	no	—
156	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	652	2	no	—
162	C₂×C₈	—	652	2	no	—
163	C₂×C₈	—	163	2	no	—
164	C₂×C₈	—	163	2	no	—
165	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
166	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
167	C₄⋊C₄	—	163	2	no	—
168	C₄⋊C₄	—	163	2	no	—
169	C₄⋊C₄	—	163	1	no	—
170	C₄⋊C₄	—	163	2	no	—
171	C₄⋊C₄	—	163	1	no	—
172	C₄⋊C₄	—	163	2	no	—
173	C₂²⋊C₄	—	163	2	no	—
174	C₂²⋊C₄	—	163	2	no	—
175	C₂²⋊C₄	—	163	2	no	—
176	C₂²⋊C₄	—	163	2	no	—
177	C₂²⋊C₄	—	163	2	no	—
178	C₄²	—	163	2	no	—
179	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
180	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
181	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
182	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₂³	—	163	2	no	—
184	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
185	Q₈	—	652	1	no	—
186	D₄	—	652	1	no	—
187	D₄	—	652	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	326	2	no	—
193	C₂×C₄	—	326	2	no	—
194	C₂×C₄	—	163	2	no	—
195	C₂×C₄	—	163	2	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	326	2	no	—
206	C₈	—	326	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	326	2	no	—
211	C₂²	—	326	2	no	—
212	C₂²	—	326	2	no	—
213	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	326	2	no	—
220	C₄	—	326	2	no	—
221	C₄	—	326	2	no	—
222	C₄	#19104780	1	1	yes	characteristic
223	C₄	#19104783	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	652	2	no	—
226	C₂	#23782146	1	1	yes	characteristic
227	C₂	#23782431	1	1	yes	characteristic
228	C₂	#23782431	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#423×C₁₆₃	C₂
prime radical	#19104709	C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#423	D₁₆₃
hypocenter	C₁₆₃	#2995
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #23781794	C₂	semidirect
3. #23781741	C₂	semidirect
4. #23781735	C₂	semidirect
5. #23781734	C₂	semidirect
6. #23781725	C₂	semidirect
7. #23781722	C₂	semidirect
8. #23781690	C₂	semidirect
9. #423×C₁₆₃	C₂	semidirect
114. C₁₆₃	#2995	semidirect
124. #423	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782199	2
#23781794	2
#23781741	2
#23781735	2
#23781734	2
#23781725	2
#23781722	2
#23781690	2
#423×C₁₆₃	2
#2995	163