Subgroups — #35224318

Order 20864.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579699	C₂	1	1	yes	maximal, characteristic
2	#23781969	C₂	1	2	yes	maximal, complemented
3	#23781962	C₂	1	2	yes	maximal, complemented
4	#23781957	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781952	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781947	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781942	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781701	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₁₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#23781659	C₂	1	2	yes	maximal
11	#533×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
14	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
16	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
17	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
19	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
20	#19104694	C₂²	1	1	yes	characteristic
21	#19104692	—	2	1	no	—
22	C₄²×D₁₆₃	—	2	1	no	—
23	#19104668	C₂²	1	2	yes	—
24	#19104668	C₂²	1	2	yes	—
25	#19104666	C₂²	1	2	yes	—
26	#19104666	C₂²	1	2	yes	—
27	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
30	#19104658	C₂²	1	2	yes	—
31	#19104658	C₂²	1	2	yes	—
32	#19104655	C₂²	1	2	yes	—
33	#19104655	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
36	#125×C₁₆₃	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₁₆₃	—	2	1	no	—
38	C₂²⋊Q₈×C₁₆₃	—	2	1	no	—
39	C₄×Q₈×C₁₆₃	—	2	1	no	—
40	C₄×Q₈×C₁₆₃	—	2	1	no	—
41	#117×C₁₆₃	C₂²	1	2	yes	—
42	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
43	#14579699	—	2	2	no	—
44	#14579699	Q₈	1	4	yes	complemented
45	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
46	C₂×D₆₅₂	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
48	C₂×C₄×D₁₆₃	—	2	1	no	—
49	C₂×C₄×D₁₆₃	—	2	1	no	—
50	C₂×C₄×D₁₆₃	—	2	1	no	—
51	C₂×C₁₆₃⋊C₈	—	2	4	no	—
52	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
54	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
56	Dic₁₆₃⋊C₄	—	2	1	no	—
57	Dic₁₆₃⋊C₄	—	2	1	no	—
58	C₂²⋊Dic₁₆₃	—	2	1	no	—
59	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
60	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
61	C₄×Dic₁₆₃	—	2	1	no	—
62	C₄×Dic₁₆₃	—	2	1	no	—
63	C₄×Dic₁₆₃	—	2	1	no	—
64	C₂×Q₈×C₁₆₃	—	2	1	no	—
65	C₂×Q₈×C₁₆₃	—	2	1	no	—
66	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₆₃	—	2	2	no	—
70	C₄⋊C₄×C₁₆₃	—	2	2	no	—
71	C₄⋊C₄×C₁₆₃	—	2	1	no	—
72	C₄⋊C₄×C₁₆₃	—	2	1	no	—
73	C₄⋊C₄×C₁₆₃	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₆₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
76	C₄²×C₁₆₃	—	2	2	no	—
77	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
78	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
79	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
80	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
81	C₂²×D₁₆₃	—	2	1	no	—
82	Dic₃₂₆	—	2	1	no	—
83	Dic₃₂₆	C₂×Q₈	1	2	yes	—
84	D₆₅₂	—	2	1	no	—
85	D₆₅₂	C₂×Q₈	1	2	yes	—
86	C₂²⋊D₁₆₃	—	2	4	no	—
87	C₂×Dic₁₆₃	—	2	1	no	—
88	C₂×Dic₁₆₃	—	2	1	no	—
89	C₂×Dic₁₆₃	—	2	1	no	—
90	C₄×D₁₆₃	—	2	4	no	—
91	C₄×D₁₆₃	—	4	2	no	—
92	C₄×D₁₆₃	—	4	2	no	—
93	C₁₆₃⋊C₈	—	2	2	no	—
94	C₁₆₃⋊C₈	—	2	2	no	—
95	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₁₆₃	—	4	2	no	—
97	Q₈×C₁₆₃	—	4	2	no	—
98	C₂×C₄×C₁₆₃	—	2	2	no	—
99	C₂×C₄×C₁₆₃	—	2	2	no	—
100	C₂×C₄×C₁₆₃	—	2	2	no	—
101	C₂×C₄×C₁₆₃	—	4	1	no	—
102	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₁₆₃	#41	1	2	yes	—
109	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
110	C₂²×C₁₆₃	—	2	1	no	—
111	C₂²×C₁₆₃	—	2	1	no	—
112	C₂²×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₆₃	—	2	2	no	—
115	C₂×D₁₆₃	—	4	1	no	—
116	C₂×D₁₆₃	—	4	1	no	—
117	C₄×C₁₆₃	—	2	1	no	—
118	C₄×C₁₆₃	—	2	1	no	—
119	C₄×C₁₆₃	—	2	1	no	—
120	C₄×C₁₆₃	—	4	2	no	—
121	C₄×C₁₆₃	—	4	2	no	—
122	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₁₆₃	#135	1	2	yes	—
124	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
126	Dic₁₆₃	—	2	2	no	—
127	Dic₁₆₃	—	4	1	no	—
128	Dic₁₆₃	—	4	1	no	—
129	C₂×C₁₆₃	—	2	2	no	—
130	C₂×C₁₆₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₆₃	C₂×#135	1	1	yes	characteristic
132	C₂×C₁₆₃	#575	1	1	yes	characteristic
133	D₁₆₃	—	4	2	no	—
134	C₁₆₃	#3076	1	1	yes	characteristic, complemented, Sylow
135	#3076	—	163	1	no	maximal, Sylow
136	#533	D₁₆₃	1	1	yes	characteristic, complemented
137	C₄×#41	—	163	1	no	—
138	#479	—	163	2	no	—
139	#478	—	163	2	no	—
140	#477	—	163	1	no	—
141	#476	—	163	1	no	—
142	#475	—	163	1	no	—
143	#474	—	163	1	no	—
144	#427	—	163	2	no	—
145	C₂×C₄⋊C₈	—	163	1	no	—
146	#416	—	163	2	no	—
147	C₂×#41	—	163	1	no	—
148	C₂²×C₈	—	163	2	no	—
149	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
151	#125	C₂×D₁₆₃	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	—	163	1	no	—
157	C₄×Q₈	—	163	1	no	—
158	C₄×D₄	—	326	1	no	—
159	C₄×D₄	—	326	1	no	—
160	C₄×D₄	—	163	1	no	—
161	C₄×D₄	—	163	1	no	—
162	#117	—	326	1	no	—
163	#117	—	163	1	no	—
164	#117	C₂×D₁₆₃	1	2	yes	—
165	C₂×C₄²	—	326	1	no	—
166	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
167	#107	—	163	2	no	—
168	#107	—	163	2	no	—
169	#106	—	163	2	no	—
170	#106	—	163	2	no	—
171	C₄⋊C₈	—	163	1	no	—
172	C₄⋊C₈	—	163	1	no	—
173	C₄⋊C₈	—	163	2	no	—
174	#103	—	163	2	no	—
175	#103	—	163	2	no	—
176	#102	—	163	2	no	—
177	#102	—	163	2	no	—
178	#41	—	326	2	no	—
179	#41	—	163	4	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	163	1	no	—
183	C₂×D₄	—	326	1	no	—
184	C₂×D₄	—	163	1	no	—
185	C₂²×C₄	—	326	1	no	—
186	C₂²×C₄	—	326	1	no	—
187	C₂²×C₄	—	326	1	no	—
188	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
191	C₂×C₈	—	326	4	no	—
192	C₂×C₈	—	163	2	no	—
193	C₂×C₈	—	163	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	326	1	no	—
199	C₄⋊C₄	—	326	1	no	—
200	C₄⋊C₄	—	163	1	no	—
201	C₄⋊C₄	—	163	1	no	—
202	C₄⋊C₄	C₂²×D₁₆₃	1	2	yes	—
203	C₄⋊C₄	C₂²×D₁₆₃	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	326	1	no	—
206	C₂²⋊C₄	—	326	1	no	—
207	C₂²⋊C₄	—	326	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	326	1	no	—
210	C₄²	—	326	1	no	—
211	C₄²	—	326	1	no	—
212	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
213	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
214	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
215	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
216	C₂³	—	326	1	no	—
217	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	326	1	no	—
221	Q₈	—	163	2	no	—
222	D₄	—	326	4	no	—
223	D₄	—	326	1	no	—
224	D₄	—	163	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	326	1	no	—
230	C₂×C₄	—	326	4	no	—
231	C₂×C₄	—	326	1	no	—
232	C₂×C₄	—	326	1	no	—
233	C₂×C₄	—	652	2	no	—
234	C₂×C₄	—	652	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
241	C₂×C₄	#14579700	1	2	yes	—
242	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
243	C₈	—	326	2	no	—
244	C₈	—	326	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	326	2	no	—
248	C₂²	—	652	1	no	—
249	C₂²	—	652	1	no	—
250	C₂²	#19104711	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	326	2	no	—
258	C₄	—	652	1	no	—
259	C₄	—	652	1	no	—
260	C₄	#19104711	1	2	yes	—
261	C₄	#19104755	1	2	yes	—
262	C₄	C₂×Q₈×D₁₆₃	1	1	yes	characteristic
263	C₄	C₂×#14579700	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	652	2	no	—
266	C₂	C₂×#19104711	1	1	yes	characteristic
267	C₂	C₂×#19104755	1	1	yes	characteristic
268	C₂	#23782463	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₆₃
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#533×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#14579699	C₂²
hypercenter	#533	D₁₆₃
hypocenter	C₁₆₃	#3076
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #23781969	C₂	semidirect
3. #23781962	C₂	semidirect
4. #23781957	C₂	semidirect
5. #23781952	C₂	semidirect
6. #23781947	C₂	semidirect
7. #23781942	C₂	semidirect
8. #23781701	C₂	semidirect
9. C₂×C₄⋊(C₁₆₃⋊C₈)	C₂	semidirect
11. #533×C₁₆₃	C₂	semidirect
44. #14579699	Q₈	semidirect
134. C₁₆₃	#3076	semidirect
136. #533	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579699	2
#23781969	2
#23781962	2
#23781957	2
#23781952	2
#23781947	2
#23781942	2
#23781701	2
C₂×C₄⋊(C₁₆₃⋊C₈)	2
#23781659	2
#533×C₁₆₃	2
#3076	163