Finite Group Databasebeta

Subgroups — #24920611

Order 10624.

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	#19128518	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128506	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128144	C₂	1	3	yes	maximal, complemented
4	#19128139	C₂	1	3	yes	maximal, complemented
5	#19128133	C₂	1	3	yes	maximal, complemented
6	C₂×#14581737	C₂	1	3	yes	maximal, complemented
7	#504×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581839	—	2	1	no	—
9	#14581838	—	2	1	no	—
10	#14581825	—	2	3	no	—
11	#14581820	—	2	3	no	—
12	#14581814	C₂²	1	3	yes	—
13	#14581814	C₂²	1	3	yes	complemented
14	C₂×Dic₃₃₂	C₂²	1	3	yes	complemented
15	C₂×D₆₆₄	C₂²	1	3	yes	—
16	Dic₆₆₄	C₂²	1	3	yes	complemented
17	#14581737	C₂²	1	3	yes	complemented
18	#14581737	C₂²	1	6	yes	complemented
19	#14581737	C₂²	1	1	yes	characteristic, complemented
20	D₁₃₂₈	C₂²	1	3	yes	complemented
21	#131×C₈₃	C₂²	1	1	yes	characteristic
22	M₃₂×C₈₃	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₈₃	C₂²	1	3	yes	complemented
24	#13050591	—	4	1	no	—
25	#13050590	—	2	3	no	—
26	#13050590	—	2	3	no	—
27	#13050589	—	4	3	no	—
28	C₂×Dic₁₆₆	—	2	3	no	—
29	Q₈×D₈₃	—	4	1	no	—
30	C₂×D₃₃₂	—	2	3	no	—
31	D₄×D₈₃	—	4	3	no	—
32	Dic₃₃₂	—	2	3	no	—
33	Dic₃₃₂	C₂³	1	3	yes	—
34	Dic₃₃₂	C₂³	1	1	yes	characteristic
35	#13050579	—	2	3	no	—
36	#13050579	—	2	3	no	—
37	D₆₆₄	—	2	3	no	—
38	D₆₆₄	C₂³	1	3	yes	—
39	D₆₆₄	C₂³	1	1	yes	characteristic
40	#41×C₈₃	D₄	1	1	yes	characteristic
41	M₁₆×C₈₃	D₄	1	3	yes	—
42	C₂×C₈×C₈₃	C₂³	1	3	yes	—
43	C₁₆×C₈₃	C₂³	1	1	yes	characteristic
44	C₁₆×C₈₃	C₂³	1	3	yes	—
45	C₂²×D₈₃	—	4	3	no	—
46	Q₈×C₈₃	D₈	1	1	yes	characteristic
47	D₄×C₈₃	D₈	1	3	yes	—
48	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
49	C₂²⋊D₈₃	—	4	3	no	—
50	C₂²⋊D₈₃	—	4	3	no	—
51	C₂×Dic₈₃	—	4	3	no	—
52	D₃₃₂	—	2	3	no	—
53	D₃₃₂	—	2	1	no	—
54	D₃₃₂	—	4	3	no	—
55	C₄×D₈₃	—	4	3	no	—
56	C₄×D₈₃	—	4	3	no	—
57	Dic₁₆₆	—	2	1	no	—
58	Dic₁₆₆	—	2	3	no	—
59	Dic₁₆₆	—	4	3	no	—
60	C₈×C₈₃	C₂×D₄	1	3	yes	—
61	C₈×C₈₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₈₃	C₂×D₈	1	3	yes	—
63	C₂×D₈₃	—	4	1	no	—
64	C₂×D₈₃	—	4	3	no	—
65	C₂×D₈₃	—	16	3	no	—
66	C₄×C₈₃	C₂×D₈	1	3	yes	—
67	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
68	Dic₈₃	—	4	3	no	—
69	Dic₈₃	—	4	1	no	—
70	C₂×C₈₃	—	2	3	no	—
71	C₂×C₈₃	C₂²×D₈	1	1	yes	characteristic
72	D₈₃	—	8	1	no	—
73	D₈₃	—	8	3	no	—
74	#3416	—	83	1	no	maximal, Sylow
75	C₈₃	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	83	1	no	—
77	#576	—	83	1	no	—
78	#510	—	83	3	no	—
79	#509	—	83	3	no	—
80	#508	—	83	3	no	—
81	C₂×SD₃₂	—	83	3	no	—
82	#504	D₈₃	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	166	1	no	—
84	H(5,𝔽₂)	—	166	1	no	—
85	#137	—	166	3	no	—
86	AGL(1,ℤ/8)	—	166	3	no	—
87	#135	—	83	3	no	—
88	#135	—	83	3	no	—
89	C₂×Q₁₆	—	83	3	no	—
90	C₂×D₈	—	83	3	no	—
91	#131	C₂×D₈₃	1	1	yes	characteristic
92	Q₃₂	—	83	3	no	—
93	SD₃₂	—	83	3	no	—
94	SD₃₂	—	83	6	no	—
95	SD₃₂	—	83	1	no	—
96	D₁₆	—	83	3	no	—
97	M₃₂	C₂×D₈₃	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₈₃	1	3	yes	complemented
99	#41	—	332	1	no	—
100	#41	—	332	3	no	—
101	#41	—	166	3	no	—
102	#41	—	166	3	no	—
103	#41	D₃₃₂	1	1	yes	characteristic
104	C₂×Q₈	—	332	1	no	—
105	C₂×Q₈	—	166	3	no	—
106	C₂×D₄	—	332	3	no	—
107	C₂×D₄	—	166	3	no	—
108	Q₁₆	—	83	3	no	—
109	Q₁₆	—	83	1	no	—
110	Q₁₆	—	166	3	no	—
111	SD₁₆	—	166	3	no	—
112	SD₁₆	—	166	3	no	—
113	D₈	—	83	1	no	—
114	D₈	—	83	3	no	—
115	D₈	—	166	3	no	—
116	M₁₆	D₃₃₂	1	3	yes	—
117	C₂×C₈	C₂²×D₈₃	1	3	yes	—
118	C₁₆	C₂²×D₈₃	1	1	yes	characteristic
119	C₁₆	C₂²×D₈₃	1	3	yes	—
120	C₂³	—	332	3	no	—
121	Q₈	—	332	3	no	—
122	Q₈	—	166	1	no	—
123	Q₈	—	166	3	no	—
124	Q₈	D₆₆₄	1	1	yes	characteristic
125	D₄	—	332	3	no	—
126	D₄	—	332	3	no	—
127	D₄	—	332	3	no	—
128	D₄	—	166	3	no	—
129	D₄	—	166	1	no	—
130	D₄	D₆₆₄	1	3	yes	—
131	C₂×C₄	—	332	3	no	—
132	C₂×C₄	—	332	3	no	—
133	C₂×C₄	—	332	3	no	—
134	C₂×C₄	C₂×D₃₃₂	1	3	yes	—
135	C₈	C₂×D₃₃₂	1	3	yes	—
136	C₈	C₂³×D₈₃	1	1	yes	characteristic
137	C₂²	—	1328	3	no	—
138	C₂²	—	332	1	no	—
139	C₂²	—	332	3	no	—
140	C₂²	C₂×D₆₆₄	1	3	yes	—
141	C₄	—	332	1	no	—
142	C₄	—	332	3	no	—
143	C₄	C₂×D₆₆₄	1	3	yes	—
144	C₄	C₂²×D₃₃₂	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	664	3	no	—
147	C₂	—	664	1	no	—
148	C₂	C₂²×D₆₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₆₄
commutator, cosocle	C₈×C₈₃	C₂⁴
Frattini	C₈	C₂³×D₈₃
Fitting	#504×C₈₃	C₂
socle	C₂×C₈₃	C₂²×D₈
hypercenter	#504	D₈₃
hypocenter	C₈₃	#3416
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #19128518	C₂	semidirect
2. #19128506	C₂	semidirect
3. #19128144	C₂	semidirect
4. #19128139	C₂	semidirect
5. #19128133	C₂	semidirect
6. C₂×#14581737	C₂	semidirect
7. #504×C₈₃	C₂	semidirect
13. #14581814	C₂²	semidirect
14. C₂×Dic₃₃₂	C₂²	semidirect
16. Dic₆₆₄	C₂²	semidirect
17. #14581737	C₂²	semidirect
18. #14581737	C₂²	semidirect
19. #14581737	C₂²	semidirect
20. D₁₃₂₈	C₂²	semidirect
22. M₃₂×C₈₃	C₂²	semidirect
23. C₂×C₁₆×C₈₃	C₂²	semidirect
75. C₈₃	#3416	semidirect
82. #504	D₈₃	semidirect
97. M₃₂	C₂×D₈₃	semidirect
98. C₂×C₁₆	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128518	2
#19128506	2
#19128144	2
#19128139	2
#19128133	2
C₂×#14581737	2
#504×C₈₃	2
#3416	83