Finite Group Databasebeta

Subgroups — #22573357

Order 8576.

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

normal subgroups only

This group has 360 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	#16864213	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864201	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863839	C₂	1	3	yes	maximal, complemented
4	#16863834	C₂	1	3	yes	maximal, complemented
5	#16863828	C₂	1	3	yes	maximal, complemented
6	C₂×#14523988	C₂	1	3	yes	maximal, complemented
7	#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524090	—	2	1	no	—
9	#14524089	—	2	1	no	—
10	#14524076	—	2	3	no	—
11	#14524071	—	2	3	no	—
12	#14524065	C₂²	1	3	yes	—
13	#14524065	C₂²	1	3	yes	complemented
14	C₂×Dic₂₆₈	C₂²	1	3	yes	complemented
15	C₂×D₅₃₆	C₂²	1	3	yes	—
16	Dic₅₃₆	C₂²	1	3	yes	complemented
17	#14523988	C₂²	1	3	yes	complemented
18	#14523988	C₂²	1	6	yes	complemented
19	#14523988	C₂²	1	1	yes	characteristic, complemented
20	D₁₀₇₂	C₂²	1	3	yes	complemented
21	#131×C₆₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₆₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₆₇	C₂²	1	3	yes	complemented
24	#11761629	—	4	1	no	—
25	#11761628	—	2	3	no	—
26	#11761628	—	2	3	no	—
27	#11761627	—	4	3	no	—
28	C₂×Dic₁₃₄	—	2	3	no	—
29	Q₈×D₆₇	—	4	1	no	—
30	C₂×D₂₆₈	—	2	3	no	—
31	D₄×D₆₇	—	4	3	no	—
32	Dic₂₆₈	—	2	3	no	—
33	Dic₂₆₈	C₂³	1	3	yes	—
34	Dic₂₆₈	C₂³	1	1	yes	characteristic
35	#11761617	—	2	3	no	—
36	#11761617	—	2	3	no	—
37	D₅₃₆	—	2	3	no	—
38	D₅₃₆	C₂³	1	1	yes	characteristic
39	D₅₃₆	C₂³	1	3	yes	—
40	#41×C₆₇	D₄	1	1	yes	characteristic
41	M₁₆×C₆₇	D₄	1	3	yes	—
42	C₂×C₈×C₆₇	C₂³	1	3	yes	—
43	C₁₆×C₆₇	C₂³	1	1	yes	characteristic
44	C₁₆×C₆₇	C₂³	1	3	yes	—
45	C₂²×D₆₇	—	4	3	no	—
46	Q₈×C₆₇	D₈	1	1	yes	characteristic
47	D₄×C₆₇	D₈	1	3	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
49	C₂²⋊D₆₇	—	4	3	no	—
50	C₂²⋊D₆₇	—	4	3	no	—
51	C₂×Dic₆₇	—	4	3	no	—
52	D₂₆₈	—	2	3	no	—
53	D₂₆₈	—	2	1	no	—
54	D₂₆₈	—	4	3	no	—
55	C₄×D₆₇	—	4	3	no	—
56	C₄×D₆₇	—	4	3	no	—
57	Dic₁₃₄	—	2	1	no	—
58	Dic₁₃₄	—	2	3	no	—
59	Dic₁₃₄	—	4	3	no	—
60	C₈×C₆₇	C₂×D₄	1	3	yes	—
61	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×D₈	1	3	yes	—
63	C₂×D₆₇	—	4	1	no	—
64	C₂×D₆₇	—	4	3	no	—
65	C₂×D₆₇	—	16	3	no	—
66	C₄×C₆₇	C₂×D₈	1	3	yes	—
67	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
68	Dic₆₇	—	4	3	no	—
69	Dic₆₇	—	4	1	no	—
70	C₂×C₆₇	—	2	3	no	—
71	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
72	D₆₇	—	8	1	no	—
73	D₆₇	—	8	3	no	—
74	#3416	—	67	1	no	maximal, Sylow
75	C₆₇	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	67	1	no	—
77	#576	—	67	1	no	—
78	#510	—	67	3	no	—
79	#509	—	67	3	no	—
80	#508	—	67	3	no	—
81	C₂×SD₃₂	—	67	3	no	—
82	#504	D₆₇	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	134	1	no	—
84	H(5,𝔽₂)	—	134	1	no	—
85	#137	—	134	3	no	—
86	AGL(1,ℤ/8)	—	134	3	no	—
87	#135	—	67	3	no	—
88	#135	—	67	3	no	—
89	C₂×Q₁₆	—	67	3	no	—
90	C₂×D₈	—	67	3	no	—
91	#131	C₂×D₆₇	1	1	yes	characteristic
92	Q₃₂	—	67	3	no	—
93	SD₃₂	—	67	3	no	—
94	SD₃₂	—	67	6	no	—
95	SD₃₂	—	67	1	no	—
96	D₁₆	—	67	3	no	—
97	M₃₂	C₂×D₆₇	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₆₇	1	3	yes	complemented
99	#41	—	268	1	no	—
100	#41	—	268	3	no	—
101	#41	—	134	3	no	—
102	#41	—	134	3	no	—
103	#41	D₂₆₈	1	1	yes	characteristic
104	C₂×Q₈	—	268	1	no	—
105	C₂×Q₈	—	134	3	no	—
106	C₂×D₄	—	268	3	no	—
107	C₂×D₄	—	134	3	no	—
108	Q₁₆	—	67	3	no	—
109	Q₁₆	—	67	1	no	—
110	Q₁₆	—	134	3	no	—
111	SD₁₆	—	134	3	no	—
112	SD₁₆	—	134	3	no	—
113	D₈	—	67	1	no	—
114	D₈	—	67	3	no	—
115	D₈	—	134	3	no	—
116	M₁₆	D₂₆₈	1	3	yes	—
117	C₂×C₈	C₂²×D₆₇	1	3	yes	—
118	C₁₆	C₂²×D₆₇	1	1	yes	characteristic
119	C₁₆	C₂²×D₆₇	1	3	yes	—
120	C₂³	—	268	3	no	—
121	Q₈	—	268	3	no	—
122	Q₈	—	134	1	no	—
123	Q₈	—	134	3	no	—
124	Q₈	D₅₃₆	1	1	yes	characteristic
125	D₄	—	268	3	no	—
126	D₄	—	268	3	no	—
127	D₄	—	268	3	no	—
128	D₄	—	134	3	no	—
129	D₄	—	134	1	no	—
130	D₄	D₅₃₆	1	3	yes	—
131	C₂×C₄	—	268	3	no	—
132	C₂×C₄	—	268	3	no	—
133	C₂×C₄	—	268	3	no	—
134	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
135	C₈	C₂×D₂₆₈	1	3	yes	—
136	C₈	C₂³×D₆₇	1	1	yes	characteristic
137	C₂²	—	268	1	no	—
138	C₂²	—	268	3	no	—
139	C₂²	—	1072	3	no	—
140	C₂²	C₂×D₅₃₆	1	3	yes	—
141	C₄	—	268	1	no	—
142	C₄	—	268	3	no	—
143	C₄	C₂×D₅₃₆	1	3	yes	—
144	C₄	C₂²×D₂₆₈	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	536	3	no	—
147	C₂	—	536	1	no	—
148	C₂	C₂²×D₅₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₅₃₆
commutator	C₈×C₆₇	C₂⁴
Frattini	C₈	C₂³×D₆₇
Fitting	#504×C₆₇	C₂
socle	C₂×C₆₇	C₂²×D₈
hypercenter	#504	D₆₇
hypocenter	C₆₇	#3416
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #16864213	C₂	semidirect
2. #16864201	C₂	semidirect
3. #16863839	C₂	semidirect
4. #16863834	C₂	semidirect
5. #16863828	C₂	semidirect
6. C₂×#14523988	C₂	semidirect
7. #504×C₆₇	C₂	semidirect
13. #14524065	C₂²	semidirect
14. C₂×Dic₂₆₈	C₂²	semidirect
16. Dic₅₃₆	C₂²	semidirect
17. #14523988	C₂²	semidirect
18. #14523988	C₂²	semidirect
19. #14523988	C₂²	semidirect
20. D₁₀₇₂	C₂²	semidirect
22. M₃₂×C₆₇	C₂²	semidirect
23. C₂×C₁₆×C₆₇	C₂²	semidirect
75. C₆₇	#3416	semidirect
82. #504	D₆₇	semidirect
97. M₃₂	C₂×D₆₇	semidirect
98. C₂×C₁₆	C₂×D₆₇	semidirect