Finite Group Databasebeta

Subgroups — #22601488

Order 9088.

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	#17986816	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986804	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986442	C₂	1	3	yes	maximal, complemented
4	#17986437	C₂	1	3	yes	maximal, complemented
5	#17986431	C₂	1	3	yes	maximal, complemented
6	C₂×#14556073	C₂	1	3	yes	maximal, complemented
7	#504×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556175	—	2	1	no	—
9	#14556174	—	2	1	no	—
10	#14556161	—	2	3	no	—
11	#14556156	—	2	3	no	—
12	#14556150	C₂²	1	3	yes	—
13	#14556150	C₂²	1	3	yes	complemented
14	C₂×Dic₂₈₄	C₂²	1	3	yes	complemented
15	C₂×D₅₆₈	C₂²	1	3	yes	—
16	Dic₅₆₈	C₂²	1	3	yes	complemented
17	#14556073	C₂²	1	3	yes	complemented
18	#14556073	C₂²	1	6	yes	complemented
19	#14556073	C₂²	1	1	yes	characteristic, complemented
20	D₁₁₃₆	C₂²	1	3	yes	complemented
21	#131×C₇₁	C₂²	1	1	yes	characteristic
22	M₃₂×C₇₁	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₇₁	C₂²	1	3	yes	complemented
24	#11765866	—	4	1	no	—
25	#11765865	—	2	3	no	—
26	#11765865	—	2	3	no	—
27	#11765864	—	4	3	no	—
28	C₂×Dic₁₄₂	—	2	3	no	—
29	Q₈×D₇₁	—	4	1	no	—
30	C₂×D₂₈₄	—	2	3	no	—
31	D₄×D₇₁	—	4	3	no	—
32	Dic₂₈₄	—	2	3	no	—
33	Dic₂₈₄	C₂³	1	3	yes	—
34	Dic₂₈₄	C₂³	1	1	yes	characteristic
35	#11765854	—	2	3	no	—
36	#11765854	—	2	3	no	—
37	D₅₆₈	—	2	3	no	—
38	D₅₆₈	C₂³	1	1	yes	characteristic
39	D₅₆₈	C₂³	1	3	yes	—
40	#41×C₇₁	D₄	1	1	yes	characteristic
41	M₁₆×C₇₁	D₄	1	3	yes	—
42	C₂×C₈×C₇₁	C₂³	1	3	yes	—
43	C₁₆×C₇₁	C₂³	1	1	yes	characteristic
44	C₁₆×C₇₁	C₂³	1	3	yes	—
45	C₂²×D₇₁	—	4	3	no	—
46	Q₈×C₇₁	D₈	1	1	yes	characteristic
47	D₄×C₇₁	D₈	1	3	yes	—
48	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
49	C₂²⋊D₇₁	—	4	3	no	—
50	C₂²⋊D₇₁	—	4	3	no	—
51	C₂×Dic₇₁	—	4	3	no	—
52	D₂₈₄	—	2	3	no	—
53	D₂₈₄	—	2	1	no	—
54	D₂₈₄	—	4	3	no	—
55	C₄×D₇₁	—	4	3	no	—
56	C₄×D₇₁	—	4	3	no	—
57	Dic₁₄₂	—	2	1	no	—
58	Dic₁₄₂	—	2	3	no	—
59	Dic₁₄₂	—	4	3	no	—
60	C₈×C₇₁	C₂×D₄	1	3	yes	—
61	C₈×C₇₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₇₁	C₂×D₈	1	3	yes	—
63	C₂×D₇₁	—	4	1	no	—
64	C₂×D₇₁	—	4	3	no	—
65	C₂×D₇₁	—	16	3	no	—
66	C₄×C₇₁	C₂×D₈	1	3	yes	—
67	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₇₁	—	4	3	no	—
69	Dic₇₁	—	4	1	no	—
70	C₂×C₇₁	—	2	3	no	—
71	C₂×C₇₁	C₂²×D₈	1	1	yes	characteristic
72	D₇₁	—	8	1	no	—
73	D₇₁	—	8	3	no	—
74	#3416	—	71	1	no	maximal, Sylow
75	C₇₁	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	71	1	no	—
77	#576	—	71	1	no	—
78	#510	—	71	3	no	—
79	#509	—	71	3	no	—
80	#508	—	71	3	no	—
81	C₂×SD₃₂	—	71	3	no	—
82	#504	D₇₁	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	142	1	no	—
84	H(5,𝔽₂)	—	142	1	no	—
85	#137	—	142	3	no	—
86	AGL(1,ℤ/8)	—	142	3	no	—
87	#135	—	71	3	no	—
88	#135	—	71	3	no	—
89	C₂×Q₁₆	—	71	3	no	—
90	C₂×D₈	—	71	3	no	—
91	#131	C₂×D₇₁	1	1	yes	characteristic
92	Q₃₂	—	71	3	no	—
93	SD₃₂	—	71	3	no	—
94	SD₃₂	—	71	6	no	—
95	SD₃₂	—	71	1	no	—
96	D₁₆	—	71	3	no	—
97	M₃₂	C₂×D₇₁	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₇₁	1	3	yes	complemented
99	#41	—	284	1	no	—
100	#41	—	284	3	no	—
101	#41	—	142	3	no	—
102	#41	—	142	3	no	—
103	#41	D₂₈₄	1	1	yes	characteristic
104	C₂×Q₈	—	284	1	no	—
105	C₂×Q₈	—	142	3	no	—
106	C₂×D₄	—	284	3	no	—
107	C₂×D₄	—	142	3	no	—
108	Q₁₆	—	71	3	no	—
109	Q₁₆	—	71	1	no	—
110	Q₁₆	—	142	3	no	—
111	SD₁₆	—	142	3	no	—
112	SD₁₆	—	142	3	no	—
113	D₈	—	71	1	no	—
114	D₈	—	71	3	no	—
115	D₈	—	142	3	no	—
116	M₁₆	D₂₈₄	1	3	yes	—
117	C₂×C₈	C₂²×D₇₁	1	3	yes	—
118	C₁₆	C₂²×D₇₁	1	1	yes	characteristic
119	C₁₆	C₂²×D₇₁	1	3	yes	—
120	C₂³	—	284	3	no	—
121	Q₈	—	284	3	no	—
122	Q₈	—	142	1	no	—
123	Q₈	—	142	3	no	—
124	Q₈	D₅₆₈	1	1	yes	characteristic
125	D₄	—	284	3	no	—
126	D₄	—	284	3	no	—
127	D₄	—	284	3	no	—
128	D₄	—	142	3	no	—
129	D₄	—	142	1	no	—
130	D₄	D₅₆₈	1	3	yes	—
131	C₂×C₄	—	284	3	no	—
132	C₂×C₄	—	284	3	no	—
133	C₂×C₄	—	284	3	no	—
134	C₂×C₄	C₂×D₂₈₄	1	3	yes	—
135	C₈	C₂×D₂₈₄	1	3	yes	—
136	C₈	C₂³×D₇₁	1	1	yes	characteristic
137	C₂²	—	284	1	no	—
138	C₂²	—	284	3	no	—
139	C₂²	—	1136	3	no	—
140	C₂²	C₂×D₅₆₈	1	3	yes	—
141	C₄	—	284	1	no	—
142	C₄	—	284	3	no	—
143	C₄	C₂×D₅₆₈	1	3	yes	—
144	C₄	C₂²×D₂₈₄	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	568	3	no	—
147	C₂	—	568	1	no	—
148	C₂	C₂²×D₅₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₅₆₈
commutator	C₈×C₇₁	C₂⁴
Frattini	C₈	C₂³×D₇₁
Fitting	#504×C₇₁	C₂
socle	C₂×C₇₁	C₂²×D₈
hypercenter	#504	D₇₁
hypocenter	C₇₁	#3416
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #17986816	C₂	semidirect
2. #17986804	C₂	semidirect
3. #17986442	C₂	semidirect
4. #17986437	C₂	semidirect
5. #17986431	C₂	semidirect
6. C₂×#14556073	C₂	semidirect
7. #504×C₇₁	C₂	semidirect
13. #14556150	C₂²	semidirect
14. C₂×Dic₂₈₄	C₂²	semidirect
16. Dic₅₆₈	C₂²	semidirect
17. #14556073	C₂²	semidirect
18. #14556073	C₂²	semidirect
19. #14556073	C₂²	semidirect
20. D₁₁₃₆	C₂²	semidirect
22. M₃₂×C₇₁	C₂²	semidirect
23. C₂×C₁₆×C₇₁	C₂²	semidirect
75. C₇₁	#3416	semidirect
82. #504	D₇₁	semidirect
97. M₃₂	C₂×D₇₁	semidirect
98. C₂×C₁₆	C₂×D₇₁	semidirect