Subgroups — #35227869

Order 20864.

1556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579699	C₂	1	2	yes	maximal, complemented
2	#23782246	C₂	1	4	yes	maximal, complemented
3	#23782241	C₂	1	4	yes	maximal, complemented
4	#23782234	C₂	1	4	yes	maximal, complemented
5	#23782227	C₂	1	4	yes	maximal, complemented
6	#23782215	C₂	1	4	yes	maximal, complemented
7	#23782177	C₂	1	8	yes	maximal, complemented
8	C₂×#117×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579699	—	2	16	no	—
10	C₂×#14579699	C₂²	1	4	yes	—
11	C₂×#14579699	C₂²	1	4	yes	—
12	C₂×#14579699	C₂²	1	4	yes	complemented
13	C₂×#14579699	C₂²	1	4	yes	complemented
14	C₂²×Dic₃₂₆	C₂²	1	2	yes	complemented
15	C₂²×D₆₅₂	C₂²	1	2	yes	complemented
16	C₂²×C₂²⋊D₁₆₃	C₂²	1	4	yes	—
17	C₂²×C₄×D₁₆₃	C₂²	1	4	yes	complemented
18	#19104735	C₂²	1	4	yes	complemented
19	#19104732	C₂²	1	4	yes	complemented
20	#19104723	C₂²	1	8	yes	complemented
21	#19104719	C₂²	1	16	yes	complemented
22	#19104715	C₂²	1	16	yes	complemented
23	#19104710	C₂²	1	16	yes	complemented
24	#19104709	C₂²	1	16	yes	complemented
25	#19104705	C₂²	1	16	yes	complemented
26	C₄×D₆₅₂	C₂²	1	16	yes	complemented
27	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
28	#117×C₁₆₃	C₂²	1	4	yes	complemented
29	#117×C₁₆₃	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	complemented
31	C₂×C₂²⋊C₄×C₁₆₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	complemented
33	C₂³×D₁₆₃	C₂³	1	4	yes	—
34	#14579699	—	2	16	no	—
35	#14579699	—	2	16	no	—
36	#14579699	—	2	16	no	—
37	#14579699	—	2	16	no	—
38	C₂×Dic₃₂₆	—	2	8	no	—
39	C₂×Dic₃₂₆	C₂³	1	4	yes	—
40	C₂×Dic₃₂₆	C₂³	1	4	yes	—
41	C₂×D₆₅₂	—	2	8	no	—
42	C₂×D₆₅₂	C₂³	1	4	yes	—
43	C₂×D₆₅₂	C₂³	1	4	yes	—
44	C₂×D₆₅₂	C₂³	1	16	yes	—
45	C₂×C₂²⋊D₁₆₃	—	2	16	no	—
46	C₂×C₂²⋊D₁₆₃	C₂³	1	8	yes	—
47	C₂×C₂²⋊D₁₆₃	C₂³	1	8	yes	—
48	C₂²×Dic₁₆₃	C₂³	1	4	yes	—
49	C₂×C₄×D₁₆₃	—	2	16	no	—
50	C₂×C₄×D₁₆₃	C₂³	1	8	yes	—
51	C₂×C₄×D₁₆₃	C₂³	1	8	yes	—
52	C₄⋊Dic₁₆₃	C₂³	1	16	yes	—
53	(C₂×D₁₆₃)⋊C₄	C₂³	1	32	yes	—
54	C₂⁴×C₁₆₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₆₃	D₄	1	8	yes	complemented
56	C₂²×C₄×C₁₆₃	C₂³	1	4	yes	—
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	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₆₃	C₂³	1	8	yes	—
62	C₂²⋊C₄×C₁₆₃	C₂³	1	8	yes	—
63	C₄²×C₁₆₃	C₂³	1	4	yes	—
64	C₄²×C₁₆₃	C₂³	1	4	yes	—
65	C₂²×D₁₆₃	—	2	8	no	—
66	C₂²×D₁₆₃	—	4	8	no	—
67	C₂²×D₁₆₃	C₂⁴	1	8	yes	—
68	Dic₃₂₆	—	2	8	no	—
69	Dic₃₂₆	—	2	8	no	—
70	D₆₅₂	—	2	8	no	—
71	D₆₅₂	—	2	8	no	—
72	D₆₅₂	—	4	16	no	—
73	C₂²⋊D₁₆₃	—	2	16	no	—
74	C₂²⋊D₁₆₃	—	2	16	no	—
75	C₂×Dic₁₆₃	—	2	8	no	—
76	C₂×Dic₁₆₃	C₂⁴	1	8	yes	—
77	C₄×D₁₆₃	—	2	16	no	—
78	C₄×D₁₆₃	—	2	16	no	—
79	C₂³×C₁₆₃	—	4	2	no	—
80	C₂³×C₁₆₃	C₂×D₄	1	4	yes	—
81	C₂³×C₁₆₃	C₂⁴	1	2	yes	—
82	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₆₃	—	4	4	no	—
84	C₂×C₄×C₁₆₃	C₂×D₄	1	8	yes	—
85	C₂×C₄×C₁₆₃	C₂×D₄	1	8	yes	—
86	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₆₃	C₂⁴	1	8	yes	—
89	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
92	C₂²×C₁₆₃	—	2	2	no	—
93	C₂²×C₁₆₃	—	2	4	no	—
94	C₂²×C₁₆₃	—	4	4	no	—
95	C₂²×C₁₆₃	C₂²×D₄	1	4	yes	—
96	C₂²×C₁₆₃	C₂²×D₄	1	2	yes	—
97	C₂²×C₁₆₃	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₆₃	—	2	8	no	—
99	C₂×D₁₆₃	—	2	16	no	—
100	C₂×D₁₆₃	—	4	16	no	—
101	C₄×C₁₆₃	—	2	8	no	—
102	C₄×C₁₆₃	C₂²×D₄	1	2	yes	—
103	C₄×C₁₆₃	C₂²×D₄	1	4	yes	—
104	C₄×C₁₆₃	C₂²×D₄	1	2	yes	—
105	Dic₁₆₃	—	2	8	no	—
106	C₂×C₁₆₃	—	2	4	no	—
107	C₂×C₁₆₃	—	2	2	no	—
108	C₂×C₁₆₃	C₂³×D₄	1	1	yes	characteristic
109	C₂×C₁₆₃	#585	1	2	yes	—
110	D₁₆₃	—	4	8	no	—
111	C₁₆₃	#3449	1	1	yes	characteristic, complemented, Sylow
112	#3449	—	163	1	no	maximal, Sylow
113	C₂²×#41	—	163	2	no	—
114	#537	—	163	4	no	—
115	#536	—	163	4	no	—
116	#535	—	163	4	no	—
117	#534	—	163	4	no	—
118	#532	—	163	4	no	—
119	#525	—	163	8	no	—
120	C₂×#117	D₁₆₃	1	1	yes	characteristic, complemented
121	C₂×#41	—	326	16	no	—
122	C₂×#41	—	163	4	no	—
123	C₂×#41	—	163	4	no	—
124	C₂×#41	—	163	4	no	—
125	C₂×#41	—	163	4	no	—
126	C₂²×Q₈	—	163	2	no	—
127	C₂²×D₄	—	163	4	no	—
128	C₂²×D₄	—	163	2	no	—
129	C₂³×C₄	—	163	4	no	—
130	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	163	4	no	—
132	#127	—	163	4	no	—
133	#124	—	163	8	no	—
134	#123	—	163	16	no	—
135	C₂²⋊Q₈	—	163	16	no	—
136	#121	—	163	16	no	—
137	#121	—	163	16	no	—
138	#120	—	163	16	no	—
139	C₄×D₄	—	163	16	no	—
140	#117	C₂×D₁₆₃	1	4	yes	complemented
141	#117	C₂×D₁₆₃	1	4	yes	complemented
142	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	complemented
143	C₂×C₂²⋊C₄	C₂×D₁₆₃	1	2	yes	—
144	C₂×C₄²	C₂×D₁₆₃	1	2	yes	complemented
145	C₂⁴	—	163	4	no	—
146	C₂⁴	C₂²×D₁₆₃	1	1	yes	characteristic
147	#41	—	326	16	no	—
148	#41	—	326	16	no	—
149	#41	—	326	16	no	—
150	#41	—	326	16	no	—
151	C₂×Q₈	—	326	8	no	—
152	C₂×Q₈	—	163	4	no	—
153	C₂×Q₈	—	163	4	no	—
154	C₂×D₄	—	326	16	no	—
155	C₂×D₄	—	326	8	no	—
156	C₂×D₄	—	163	8	no	—
157	C₂×D₄	—	163	8	no	—
158	C₂×D₄	—	163	4	no	—
159	C₂×D₄	—	163	4	no	—
160	C₂×D₄	—	163	16	no	—
161	C₂²×C₄	—	326	16	no	—
162	C₂²×C₄	—	163	4	no	—
163	C₂²×C₄	—	163	8	no	—
164	C₂²×C₄	—	163	8	no	—
165	C₂²×C₄	D₆₅₂	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₁₆₃	1	4	yes	—
167	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
171	C₄⋊C₄	—	163	16	no	—
172	C₄⋊C₄	C₂²×D₁₆₃	1	8	yes	—
173	C₂²⋊C₄	—	163	32	no	—
174	C₂²⋊C₄	C₂²×D₁₆₃	1	8	yes	—
175	C₄²	C₂²×D₁₆₃	1	4	yes	—
176	C₄²	C₂²×D₁₆₃	1	4	yes	—
177	C₂³	—	4	2	no	—
178	C₂³	—	326	8	no	—
179	C₂³	—	652	8	no	—
180	C₂³	—	163	8	no	—
181	C₂³	C₂×D₆₅₂	1	4	yes	—
182	C₂³	C₂³×D₁₆₃	1	2	yes	—
183	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
184	Q₈	—	326	8	no	—
185	Q₈	—	326	8	no	—
186	D₄	—	326	16	no	—
187	D₄	—	326	16	no	—
188	D₄	—	326	8	no	—
189	D₄	—	326	8	no	—
190	D₄	—	652	16	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	326	8	no	—
193	C₂×C₄	—	326	16	no	—
194	C₂×C₄	—	326	16	no	—
195	C₂×C₄	—	163	8	no	—
196	C₂×C₄	C₂×D₆₅₂	1	8	yes	—
197	C₂×C₄	C₂×D₆₅₂	1	8	yes	—
198	C₂×C₄	C₂×D₆₅₂	1	4	yes	—
199	C₂×C₄	C₂×D₆₅₂	1	4	yes	—
200	C₂×C₄	C₂³×D₁₆₃	1	8	yes	—
201	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
203	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	326	8	no	—
208	C₂²	—	326	16	no	—
209	C₂²	—	652	16	no	—
210	C₂²	C₂²×D₆₅₂	1	4	yes	—
211	C₂²	C₂²×D₆₅₂	1	2	yes	—
212	C₂²	C₂⁴×D₁₆₃	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	326	8	no	—
215	C₄	C₂²×D₆₅₂	1	2	yes	—
216	C₄	C₂²×D₆₅₂	1	4	yes	—
217	C₄	C₂²×D₆₅₂	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	—	2	2	no	—
220	C₂	—	652	8	no	—
221	C₂	C₂³×D₆₅₂	1	1	yes	characteristic
222	C₂	#23782501	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₆₃
commutator, socle, cosocle	C₂²×C₁₆₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₆₃
Fitting	C₂×#117×C₁₆₃	C₂
hypercenter	C₂×#117	D₁₆₃
hypocenter	C₁₆₃	#3449
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579699	C₂	semidirect
2. #23782246	C₂	semidirect
3. #23782241	C₂	semidirect
4. #23782234	C₂	semidirect
5. #23782227	C₂	semidirect
6. #23782215	C₂	semidirect
7. #23782177	C₂	semidirect
8. C₂×#117×C₁₆₃	C₂	semidirect
12. C₂×#14579699	C₂²	semidirect
13. C₂×#14579699	C₂²	semidirect
14. C₂²×Dic₃₂₆	C₂²	semidirect
15. C₂²×D₆₅₂	C₂²	semidirect
17. C₂²×C₄×D₁₆₃	C₂²	semidirect
18. #19104735	C₂²	semidirect
19. #19104732	C₂²	semidirect
20. #19104723	C₂²	semidirect
21. #19104719	C₂²	semidirect
22. #19104715	C₂²	semidirect
23. #19104710	C₂²	semidirect
24. #19104709	C₂²	semidirect
25. #19104705	C₂²	semidirect
26. C₄×D₆₅₂	C₂²	semidirect
28. #117×C₁₆₃	C₂²	semidirect
29. #117×C₁₆₃	C₂²	semidirect
30. C₂×C₄⋊C₄×C₁₆₃	C₂²	semidirect
32. C₂×C₄²×C₁₆₃	C₂²	semidirect
55. C₂²×C₄×C₁₆₃	D₄	semidirect
111. C₁₆₃	#3449	semidirect
120. C₂×#117	D₁₆₃	semidirect
140. #117	C₂×D₁₆₃	semidirect
141. #117	C₂×D₁₆₃	semidirect
142. C₂×C₄⋊C₄	C₂×D₁₆₃	semidirect
144. C₂×C₄²	C₂×D₁₆₃	semidirect
165. C₂²×C₄	D₆₅₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579699	2
#23782246	2
#23782241	2
#23782234	2
#23782227	2
#23782215	2
#23782177	2
C₂×#117×C₁₆₃	2
#3449	163