Finite Group Databasebeta

Subgroups — #22573348

Order 8576.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864201	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863832	C₂	1	3	yes	maximal, complemented
3	#16863831	C₂	1	3	yes	maximal, complemented
4	#16863826	C₂	1	3	yes	maximal, complemented
5	C₂×#14523984	C₂	1	3	yes	maximal, complemented
6	#16863813	C₂	1	1	yes	maximal, characteristic, complemented
7	#576×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524089	—	2	1	no	—
9	#14524071	—	2	3	no	—
10	#14524065	C₂²	1	3	yes	complemented
11	C₂×D₅₃₆	C₂²	1	3	yes	complemented
12	#14523989	C₂²	1	1	yes	characteristic, complemented
13	#14523987	C₂²	1	3	yes	complemented
14	#14523986	C₂²	1	3	yes	complemented
15	#14523984	C₂²	1	3	yes	complemented
16	#14523984	C₂²	1	6	yes	complemented
17	#14523983	C₂²	1	3	yes	complemented
18	C₂×C₆₇⋊C₁₆	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₆₇	—	2	1	no	—
20	AGL(1,ℤ/8)×C₆₇	—	2	3	no	—
21	#135×C₆₇	C₂²	1	3	yes	complemented
22	C₂×D₈×C₆₇	C₂²	1	3	yes	complemented
23	#131×C₆₇	C₂²	1	1	yes	characteristic
24	#11761629	—	4	1	no	—
25	#11761628	—	2	3	no	—
26	C₂×D₂₆₈	—	2	3	no	—
27	D₄×D₆₇	—	4	3	no	—
28	Dic₂₆₈	C₂³	1	1	yes	characteristic
29	#11761617	—	2	3	no	—
30	D₅₃₆	—	2	3	no	—
31	D₅₃₆	C₂³	1	3	yes	—
32	C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
33	C₆₇⋊C₁₆	C₂³	1	3	yes	—
34	#41×C₆₇	—	2	3	no	—
35	#41×C₆₇	—	4	1	no	—
36	#41×C₆₇	D₄	1	1	yes	characteristic
37	C₂×D₄×C₆₇	—	2	3	no	—
38	C₂×D₄×C₆₇	—	4	3	no	—
39	Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
40	SD₁₆×C₆₇	—	2	3	no	—
41	D₈×C₆₇	—	2	3	no	—
42	D₈×C₆₇	C₂³	1	3	yes	—
43	M₁₆×C₆₇	D₄	1	3	yes	—
44	C₂×C₈×C₆₇	C₂³	1	3	yes	—
45	C₂³×C₆₇	—	4	3	no	—
46	C₂²×D₆₇	—	4	3	no	—
47	Q₈×C₆₇	—	2	1	no	—
48	Q₈×C₆₇	D₈	1	1	yes	characteristic
49	D₄×C₆₇	—	2	3	no	—
50	D₄×C₆₇	—	4	3	no	—
51	D₄×C₆₇	—	4	3	no	—
52	D₄×C₆₇	D₈	1	3	yes	—
53	C₂×C₄×C₆₇	—	4	3	no	—
54	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
55	C₂²⋊D₆₇	—	4	3	no	—
56	D₂₆₈	—	2	3	no	—
57	D₂₆₈	—	4	3	no	—
58	C₄×D₆₇	—	4	3	no	—
59	Dic₁₃₄	—	2	1	no	—
60	C₈×C₆₇	C₂×D₄	1	3	yes	—
61	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₇	—	4	3	no	—
63	C₂²×C₆₇	—	16	3	no	—
64	C₂²×C₆₇	C₂×D₈	1	3	yes	—
65	C₂×D₆₇	—	4	3	no	—
66	C₂×D₆₇	—	16	3	no	—
67	C₄×C₆₇	—	4	1	no	—
68	C₄×C₆₇	C₂×D₈	1	3	yes	—
69	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₆₇	—	4	1	no	—
71	C₂×C₆₇	—	2	3	no	—
72	C₂×C₆₇	—	8	3	no	—
73	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
74	D₆₇	—	8	3	no	—
75	#3415	—	67	1	no	maximal, Sylow
76	C₆₇	#3415	1	1	yes	characteristic, complemented, Sylow
77	#576	—	67	1	no	—
78	#576	D₆₇	1	1	yes	characteristic, complemented
79	#509	—	67	3	no	—
80	#509	—	67	3	no	—
81	#508	—	67	3	no	—
82	C₂×D₁₆	—	67	3	no	—
83	#504	—	67	1	no	—
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	134	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	134	3	no	—
88	#135	—	67	3	no	—
89	#135	C₂×D₆₇	1	3	yes	complemented
90	C₂×D₈	—	67	3	no	—
91	C₂×D₈	C₂×D₆₇	1	3	yes	complemented
92	#131	C₂×D₆₇	1	1	yes	characteristic
93	Q₃₂	—	67	1	no	—
94	SD₃₂	—	67	3	no	—
95	SD₃₂	—	67	3	no	—
96	D₁₆	—	67	3	no	—
97	D₁₆	—	67	6	no	—
98	M₃₂	—	67	3	no	—
99	C₂×C₁₆	—	67	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	268	1	no	—
103	#41	—	134	3	no	—
104	#41	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	268	3	no	—
108	C₂×D₄	—	134	3	no	—
109	Q₁₆	—	67	1	no	—
110	Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	134	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	67	3	no	—
115	D₈	—	134	3	no	—
116	D₈	C₂²×D₆₇	1	3	yes	—
117	M₁₆	C₂²⋊D₆₇	1	3	yes	—
118	C₂×C₈	C₂²×D₆₇	1	3	yes	—
119	C₁₆	—	67	1	no	—
120	C₁₆	—	67	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	268	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	134	1	no	—
125	Q₈	#11761613	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	268	3	no	—
130	D₄	—	268	3	no	—
131	D₄	—	134	3	no	—
132	D₄	#11761613	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	268	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
136	C₈	C₂×C₂²⋊D₆₇	1	3	yes	—
137	C₈	C₂³×D₆₇	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	268	3	no	—
140	C₂²	—	16	3	no	—
141	C₂²	—	1072	3	no	—
142	C₂²	C₂×#11761613	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	268	1	no	—
145	C₄	C₂×#11761613	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	536	3	no	—
150	C₂	C₂²×#11761613	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#11761613
commutator	C₈×C₆₇	C₂⁴
Frattini	C₈	C₂³×D₆₇
Fitting	#576×C₆₇	C₂
socle	C₂×C₆₇	C₂²×D₈
hypercenter	#576	D₆₇
hypocenter	C₆₇	#3415
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #16864201	C₂	semidirect
2. #16863832	C₂	semidirect
3. #16863831	C₂	semidirect
4. #16863826	C₂	semidirect
5. C₂×#14523984	C₂	semidirect
6. #16863813	C₂	semidirect
7. #576×C₆₇	C₂	semidirect
10. #14524065	C₂²	semidirect
11. C₂×D₅₃₆	C₂²	semidirect
12. #14523989	C₂²	semidirect
13. #14523987	C₂²	semidirect
14. #14523986	C₂²	semidirect
15. #14523984	C₂²	semidirect
16. #14523984	C₂²	semidirect
17. #14523983	C₂²	semidirect
18. C₂×C₆₇⋊C₁₆	C₂²	semidirect
21. #135×C₆₇	C₂²	semidirect
22. C₂×D₈×C₆₇	C₂²	semidirect
76. C₆₇	#3415	semidirect
78. #576	D₆₇	semidirect
89. #135	C₂×D₆₇	semidirect
91. C₂×D₈	C₂×D₆₇	semidirect