Subgroups — #35223177

Order 20864.

400 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	#23782133	C₂	1	1	yes	maximal, characteristic
2	#23781727	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781726	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781724	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781717	C₂	1	2	yes	maximal, complemented
6	#23781715	C₂	1	2	yes	maximal, complemented
7	#23781690	C₂	1	2	yes	maximal, complemented
8	#23781630	C₂	1	2	yes	maximal
9	#23781617	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	C₂×#14579682	C₂²	1	2	yes	—
14	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
15	#19104700	C₄	1	2	yes	—
16	#19104697	C₄	1	2	yes	—
17	(C₂²⋊D₁₆₃)⋊C₄	C₄	1	2	yes	—
18	(C₂²⋊D₁₆₃)⋊C₄	C₄	1	2	yes	—
19	#19104692	C₂²	1	1	yes	characteristic
20	#19104692	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
22	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
23	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
25	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
26	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
30	#19104646	C₂²	1	1	yes	characteristic
31	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×Dic₁₆₃	C₂²	1	2	yes	—
33	C₈×Dic₁₆₃	C₂²	1	2	yes	—
34	#117×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
37	#97×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
39	#14579699	—	2	4	no	—
40	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
49	#14579682	—	2	4	no	—
50	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
51	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
52	C₈×D₁₆₃	—	2	4	no	—
53	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
64	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
67	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
68	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
71	Dic₃₂₆	—	2	2	no	—
72	D₆₅₂	—	2	2	no	—
73	C₂²⋊D₁₆₃	—	2	2	no	—
74	C₂²⋊D₁₆₃	—	2	2	no	—
75	C₂×Dic₁₆₃	—	2	4	no	—
76	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
80	C₄×D₁₆₃	—	2	2	no	—
81	C₄×D₁₆₃	—	2	4	no	—
82	C₄×D₁₆₃	—	2	4	no	—
83	C₄×D₁₆₃	#41	1	4	yes	—
84	C₁₆₃⋊C₈	—	2	2	no	—
85	C₁₆₃⋊C₈	—	2	2	no	—
86	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₃	—	2	2	no	—
88	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₆₃	—	2	2	no	—
93	C₈×C₁₆₃	—	2	2	no	—
94	C₂²×C₁₆₃	—	2	1	no	—
95	C₂²×C₁₆₃	—	2	2	no	—
96	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₆₃	—	2	2	no	—
98	C₂×D₁₆₃	—	2	1	no	—
99	C₂×D₁₆₃	—	2	2	no	—
100	C₂×D₁₆₃	#117	1	2	yes	—
101	C₄×C₁₆₃	—	2	1	no	—
102	C₄×C₁₆₃	—	2	2	no	—
103	C₄×C₁₆₃	—	2	2	no	—
104	C₄×C₁₆₃	C₂×#41	1	2	yes	—
105	Dic₁₆₃	—	2	1	no	—
106	Dic₁₆₃	—	2	2	no	—
107	Dic₁₆₃	#117	1	2	yes	—
108	Dic₁₆₃	#131	1	4	yes	—
109	C₂×C₁₆₃	—	4	1	no	—
110	C₂×C₁₆₃	C₂×#117	1	1	yes	characteristic
111	C₂×C₁₆₃	C₂×#131	1	2	yes	—
112	D₁₆₃	—	2	2	no	—
113	D₁₆₃	—	4	1	no	—
114	C₁₆₃	#2922	1	1	yes	characteristic, complemented, Sylow
115	#2922	—	163	1	no	maximal, Sylow
116	C₄×#41	—	163	1	no	—
117	#433	—	163	1	no	—
118	#433	—	163	1	no	—
119	#433	—	163	1	no	—
120	#433	D₁₆₃	1	1	yes	characteristic, complemented
121	#431	—	163	2	no	—
122	#431	—	163	2	no	—
123	#424	—	163	2	no	—
124	#408	—	163	2	no	—
125	C₄×M₁₆	—	163	1	no	—
126	C₂×C₄×C₈	—	163	1	no	—
127	C₂×#41	—	163	1	no	—
128	C₂×M₁₆	—	163	2	no	—
129	C₂²×C₈	—	163	2	no	—
130	C₄×Q₈	—	163	2	no	—
131	C₄×D₄	—	163	2	no	—
132	C₄×D₄	—	163	2	no	—
133	C₄×D₄	—	163	2	no	—
134	#117	—	163	1	no	—
135	#117	—	163	1	no	—
136	#117	C₂×D₁₆₃	1	1	yes	characteristic
137	C₂×C₄²	—	163	1	no	—
138	C₂×C₄²	—	163	1	no	—
139	C₂×C₄²	—	163	1	no	—
140	C₄⋊C₈	—	163	2	no	—
141	C₄⋊C₈	—	163	2	no	—
142	C₄⋊C₈	—	163	2	no	—
143	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
144	C₂²⋊C₈	—	163	2	no	—
145	C₂²⋊C₈	—	163	2	no	—
146	C₂²⋊C₈	—	163	2	no	—
147	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
148	#97	—	163	1	no	—
149	#97	C₂×D₁₆₃	1	1	yes	characteristic
150	C₄×C₈	—	163	2	no	—
151	C₄×C₈	—	163	2	no	—
152	C₄×C₈	—	163	1	no	—
153	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
154	#41	—	326	4	no	—
155	C₂×Q₈	—	163	1	no	—
156	C₂×D₄	—	163	1	no	—
157	C₂×D₄	—	163	1	no	—
158	C₂×D₄	—	163	1	no	—
159	C₂²×C₄	—	163	2	no	—
160	C₂²×C₄	—	163	2	no	—
161	C₂²×C₄	—	163	2	no	—
162	C₂²×C₄	—	163	1	no	—
163	C₂²×C₄	—	163	1	no	—
164	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
165	M₁₆	—	326	4	no	—
166	C₂×C₈	—	326	4	no	—
167	C₂×C₈	—	163	2	no	—
168	C₂×C₈	—	163	2	no	—
169	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
170	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
171	C₄⋊C₄	—	163	2	no	—
172	C₄⋊C₄	—	163	2	no	—
173	C₄⋊C₄	C₄×D₁₆₃	1	2	yes	—
174	C₂²⋊C₄	—	163	2	no	—
175	C₂²⋊C₄	—	163	2	no	—
176	C₂²⋊C₄	C₄×D₁₆₃	1	2	yes	—
177	C₄²	—	163	2	no	—
178	C₄²	—	163	2	no	—
179	C₄²	—	163	2	no	—
180	C₄²	—	163	1	no	—
181	C₄²	—	163	1	no	—
182	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
184	C₂³	—	163	1	no	—
185	C₂³	—	163	1	no	—
186	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
187	Q₈	—	326	2	no	—
188	D₄	—	326	2	no	—
189	D₄	—	326	2	no	—
190	D₄	—	326	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	326	4	no	—
193	C₂×C₄	—	326	4	no	—
194	C₂×C₄	—	326	4	no	—
195	C₂×C₄	—	326	2	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	—	163	1	no	—
198	C₂×C₄	—	163	2	no	—
199	C₂×C₄	—	163	1	no	—
200	C₂×C₄	—	163	4	no	—
201	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
204	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	326	2	no	—
208	C₈	—	326	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	326	2	no	—
212	C₂²	—	326	2	no	—
213	C₂²	—	326	1	no	—
214	C₂²	—	163	2	no	—
215	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	326	2	no	—
220	C₄	—	326	1	no	—
221	C₄	—	163	4	no	—
222	C₄	—	163	2	no	—
223	C₄	#41×D₁₆₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	326	2	no	—
226	C₂	—	652	1	no	—
227	C₂	#117×D₁₆₃	1	1	yes	characteristic
228	C₂	#131×D₁₆₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#433×C₁₆₃	C₂
prime radical	C₂×#14579699	C₂²
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#433	D₁₆₃
hypocenter	C₁₆₃	#2922
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #23781727	C₂	semidirect
3. #23781726	C₂	semidirect
4. #23781724	C₂	semidirect
5. #23781717	C₂	semidirect
6. #23781715	C₂	semidirect
7. #23781690	C₂	semidirect
9. #23781617	C₂	semidirect
10. C₄×C₈×D₁₆₃	C₂	semidirect
11. #433×C₁₆₃	C₂	semidirect
114. C₁₆₃	#2922	semidirect
120. #433	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782133	2
#23781727	2
#23781726	2
#23781724	2
#23781717	2
#23781715	2
#23781690	2
#23781630	2
#23781617	2
C₄×C₈×D₁₆₃	2
#433×C₁₆₃	2
#2922	163