Subgroups — #35223480

Order 20864.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782218	C₂	1	1	yes	maximal, characteristic
2	#23781795	C₂	1	2	yes	maximal, complemented
3	#23781793	C₂	1	2	yes	maximal, complemented
4	#23781793	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₃₂₆	C₂	1	2	yes	maximal, complemented
6	#23781692	C₂	1	1	yes	maximal, characteristic
7	#23781691	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#23781623	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
11	#405×C₁₆₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579683	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
14	#19104734	C₄	1	1	yes	characteristic
15	#19104734	C₄	1	1	yes	characteristic
16	#19104726	C₄	1	1	yes	characteristic
17	#19104726	C₄	1	1	yes	characteristic
18	#19104712	C₄	1	2	yes	—
19	#19104712	C₄	1	2	yes	—
20	C₄×Dic₃₂₆	C₂²	1	2	yes	—
21	C₄×Dic₃₂₆	C₂²	1	2	yes	—
22	#19104694	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
25	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
26	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
29	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
30	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
31	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
32	#19104645	C₂²	1	2	yes	—
33	#19104645	C₂²	1	2	yes	—
34	C₈×Dic₁₆₃	C₂²	1	2	yes	—
35	C₈×Dic₁₆₃	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
38	#117×C₁₆₃	C₂²	1	1	yes	characteristic
39	#97×C₁₆₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₆₃	C₂²	1	2	yes	—
41	C₂×Dic₃₂₆	C₂×C₄	1	2	yes	—
42	C₂×Dic₃₂₆	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
45	#14579683	—	2	2	no	—
46	C₂×C₁₆₃⋊C₈	—	2	2	no	—
47	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
50	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
54	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₆₃	C₂³	1	2	yes	—
63	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
65	M₁₆×C₁₆₃	Q₈	1	4	yes	complemented
66	C₂×C₈×C₁₆₃	—	2	2	no	—
67	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
73	C₄²×C₁₆₃	C₂³	1	2	yes	—
74	Dic₃₂₆	—	2	4	no	—
75	Dic₃₂₆	—	4	2	no	—
76	C₂×Dic₁₆₃	—	2	2	no	—
77	C₂×Dic₁₆₃	—	4	1	no	—
78	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₆₃⋊C₈	—	2	1	no	—
85	C₁₆₃⋊C₈	—	2	1	no	—
86	C₁₆₃⋊C₈	—	2	2	no	—
87	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₆₃	#41	1	2	yes	—
93	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₆₃	—	2	1	no	—
95	C₈×C₁₆₃	—	2	1	no	—
96	C₈×C₁₆₃	C₂×Q₈	1	4	yes	—
97	C₂²×C₁₆₃	—	2	1	no	—
98	C₂²×C₁₆₃	—	2	1	no	—
99	C₂²×C₁₆₃	C₄×Q₈	1	2	yes	—
100	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₆₃	—	2	2	no	—
102	C₄×C₁₆₃	—	2	2	no	—
103	C₄×C₁₆₃	C₄×Q₈	1	2	yes	—
104	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
106	Dic₁₆₃	—	2	1	no	—
107	Dic₁₆₃	—	2	2	no	—
108	Dic₁₆₃	—	2	1	no	—
109	Dic₁₆₃	—	2	1	no	—
110	Dic₁₆₃	—	2	2	no	—
111	Dic₁₆₃	#131	1	2	yes	—
112	C₂×C₁₆₃	—	2	2	no	—
113	C₂×C₁₆₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
115	C₂×C₁₆₃	#568	1	1	yes	characteristic
116	C₁₆₃	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	163	1	no	maximal, Sylow
118	#533	—	163	1	no	—
119	#446	—	163	2	no	—
120	#446	—	163	2	no	—
121	#446	—	163	2	no	—
122	C₈×Q₈	—	163	2	no	—
123	#424	—	163	1	no	—
124	#424	—	163	1	no	—
125	C₂×C₄⋊C₈	—	163	1	no	—
126	#405	—	163	1	no	—
127	#405	D₁₆₃	1	1	yes	characteristic
128	C₄×M₁₆	—	163	1	no	—
129	C₂×M₁₆	—	163	1	no	—
130	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
131	C₂²×C₈	—	163	1	no	—
132	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	163	1	no	—
134	C₄⋊Q₈	—	163	1	no	—
135	#125	—	163	1	no	—
136	#125	—	163	1	no	—
137	C₂²⋊Q₈	—	163	2	no	—
138	C₂²⋊Q₈	—	163	2	no	—
139	C₄×Q₈	—	163	2	no	—
140	C₄×Q₈	—	163	2	no	—
141	#117	—	163	1	no	—
142	#117	C₂×D₁₆₃	1	1	yes	characteristic
143	C₂×C₄²	—	163	1	no	—
144	C₄⋊C₈	—	163	1	no	—
145	C₄⋊C₈	—	163	1	no	—
146	C₄⋊C₈	—	163	1	no	—
147	C₄⋊C₈	—	163	1	no	—
148	C₄⋊C₈	—	163	2	no	—
149	C₄⋊C₈	—	163	2	no	—
150	C₄⋊C₈	—	163	2	no	—
151	C₄⋊C₈	—	163	2	no	—
152	#97	—	163	2	no	—
153	#97	—	163	2	no	—
154	#97	C₂×D₁₆₃	1	2	yes	—
155	C₄×C₈	—	163	2	no	—
156	C₄×C₈	—	163	2	no	—
157	C₄×C₈	C₂×D₁₆₃	1	2	yes	—
158	C₂×Q₈	—	163	2	no	—
159	C₂×Q₈	—	163	2	no	—
160	C₂²×C₄	—	163	1	no	—
161	C₂²×C₄	—	163	1	no	—
162	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
163	M₁₆	—	326	2	no	—
164	M₁₆	Dic₃₂₆	1	4	yes	complemented
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	326	2	no	—
167	C₂×C₈	—	163	1	no	—
168	C₂×C₈	—	163	2	no	—
169	C₂×C₈	—	163	1	no	—
170	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
173	C₄⋊C₄	—	163	1	no	—
174	C₄⋊C₄	—	163	1	no	—
175	C₄⋊C₄	—	163	2	no	—
176	C₄⋊C₄	—	163	2	no	—
177	C₄⋊C₄	—	163	2	no	—
178	C₄⋊C₄	—	163	2	no	—
179	C₄⋊C₄	—	163	2	no	—
180	C₄⋊C₄	—	163	2	no	—
181	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
183	C₂²⋊C₄	—	163	2	no	—
184	C₂²⋊C₄	C₄×D₁₆₃	1	2	yes	—
185	C₄²	—	163	1	no	—
186	C₄²	—	163	1	no	—
187	C₄²	—	163	1	no	—
188	C₄²	—	163	2	no	—
189	C₄²	—	163	1	no	—
190	C₄²	C₂²×D₁₆₃	1	2	yes	—
191	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
192	Q₈	—	326	4	no	—
193	Q₈	—	652	2	no	—
194	C₂×C₄	—	326	2	no	—
195	C₂×C₄	—	652	1	no	—
196	C₂×C₄	—	163	1	no	—
197	C₂×C₄	—	163	1	no	—
198	C₂×C₄	—	163	1	no	—
199	C₂×C₄	—	163	2	no	—
200	C₂×C₄	—	163	2	no	—
201	C₂×C₄	—	163	1	no	—
202	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
205	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
206	C₂×C₄	#14579699	1	2	yes	—
207	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	326	1	no	—
211	C₈	—	326	1	no	—
212	C₈	—	326	2	no	—
213	C₈	C₂×Dic₃₂₆	1	4	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₄×Dic₃₂₆	1	2	yes	—
217	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	326	1	no	—
221	C₄	—	326	2	no	—
222	C₄	—	326	1	no	—
223	C₄	—	326	1	no	—
224	C₄	—	326	2	no	—
225	C₄	—	163	2	no	—
226	C₄	C₄×Dic₃₂₆	1	2	yes	—
227	C₄	C₂²×Dic₃₂₆	1	1	yes	characteristic
228	C₄	C₂×#14579699	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×C₄×Dic₃₂₆	1	1	yes	characteristic
231	C₂	#131×D₁₆₃	1	1	yes	characteristic
232	C₂	#23782430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#405×C₁₆₃	C₂
prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#405	D₁₆₃
hypocenter	C₁₆₃	#2962
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #23781795	C₂	semidirect
3. #23781793	C₂	semidirect
4. #23781793	C₂	semidirect
5. C₈×Dic₃₂₆	C₂	semidirect
65. M₁₆×C₁₆₃	Q₈	semidirect
116. C₁₆₃	#2962	semidirect
164. M₁₆	Dic₃₂₆	semidirect

Maximal subgroups

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

subgroup	index
#23782218	2
#23781795	2
#23781793	2
C₈×Dic₃₂₆	2
#23781692	2
#23781691	2
C₂×Dic₁₆₃⋊C₈	2
#23781623	2
M₁₆×Dic₁₆₃	2
#405×C₁₆₃	2
#2962	163