Subgroups — #24909714

Order 10624.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581738	C₂	1	2	yes	maximal
3	#19128079	C₂	1	2	yes	maximal
4	C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	C₂²	1	2	yes	—
7	C₂×#13050580	—	2	2	no	—
8	#14581790	—	2	2	no	—
9	Dic₆₆₄	—	2	2	no	—
10	#14581738	—	2	4	no	—
11	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
14	Q₃₂×C₈₃	—	2	2	no	—
15	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₆₆	—	2	2	no	—
17	Dic₃₃₂	—	2	2	no	—
18	Dic₃₃₂	—	2	2	no	—
19	#13050580	—	4	4	no	—
20	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
21	Dic₈₃⋊C₄	—	4	4	no	—
22	C₄×Dic₈₃	D₄	1	1	yes	characteristic
23	C₈₃⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₈₃	—	2	2	no	—
25	Q₁₆×C₈₃	—	2	2	no	—
26	Q₁₆×C₈₃	—	2	2	no	—
27	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
28	C₁₆×C₈₃	D₄	1	2	yes	—
29	Q₈×C₈₃	—	4	2	no	—
30	Q₈×C₈₃	—	4	2	no	—
31	C₂×C₄×C₈₃	—	4	2	no	—
32	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₈₃	—	4	2	no	—
34	C₂×Dic₈₃	D₈	1	2	yes	—
35	Dic₁₆₆	—	4	2	no	—
36	Dic₁₆₆	—	4	2	no	—
37	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₈₃⋊C₈	D₈	1	2	yes	—
40	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₈₃	—	8	2	no	—
42	C₄×C₈₃	#127	1	1	yes	characteristic
43	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
44	Dic₈₃	—	8	2	no	—
45	Dic₈₃	Q₃₂	1	4	yes	complemented
46	C₂×C₈₃	#493	1	1	yes	characteristic
47	C₂×C₈₃	C₂×Q₃₂	1	2	yes	—
48	#2247	—	83	1	no	maximal, Sylow
49	C₈₃	#2247	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	83	2	no	—
51	C₂×Q₃₂	—	83	1	no	—
52	C₂×Q₃₂	D₈₃	1	1	yes	characteristic
53	#494	—	83	2	no	—
54	C₄×C₁₆	—	83	1	no	—
55	C₂×Q₁₆	—	83	2	no	—
56	C₂×Q₁₆	—	166	2	no	—
57	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
58	C₄⋊Q₈	—	166	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	166	4	no	—
61	Q₃₂	—	166	2	no	—
62	C₂×C₁₆	—	83	1	no	—
63	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
64	C₄×C₈	—	83	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	166	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	332	4	no	—
70	Q₁₆	—	166	2	no	—
71	Q₁₆	—	166	2	no	—
72	C₂×C₈	—	83	1	no	—
73	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
74	C₄⋊C₄	—	332	4	no	—
75	C₄²	—	83	1	no	—
76	C₁₆	—	83	2	no	—
77	C₁₆	C₂²⋊D₈₃	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	332	2	no	—
81	Q₈	—	332	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	332	2	no	—
84	C₂×C₄	—	83	2	no	—
85	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
86	C₈	—	83	2	no	—
87	C₈	D₄×D₈₃	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
89	C₂²	D₈×D₈₃	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	83	4	no	—
92	C₄	—	664	2	no	—
93	C₄	#14581788	1	1	yes	characteristic
94	C₄	D₈×D₈₃	1	1	yes	characteristic
95	C₂	#19128076	1	1	yes	characteristic
96	C₂	Q₃₂×D₈₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	C₂×Q₃₂×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₈₃
hypocenter	C₈₃	#2247
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₈₃	Q₃₂	semidirect
49. C₈₃	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₆₄	2
C₂×#14581738	2
#19128079	2
C₁₆×Dic₈₃	2
C₂×Q₃₂×C₈₃	2
#2247	83