Group #24908264 — subgroups

Subgroups — #24908264

Order 10624.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581792	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581715	C₂	1	4	yes	maximal
3	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#384×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	#14581792	—	2	2	no	—
8	#14581792	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₈₃	—	2	2	no	—
10	#14581715	—	2	8	no	—
11	C₄×C₈₃⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₆₆	—	2	4	no	—
16	C₂×Dic₁₆₆	—	2	4	no	—
17	C₂×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₁₆₆	C₂×C₄	1	4	yes	—
19	C₂²×Dic₈₃	—	2	2	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₈₃	—	2	8	no	—
24	C₂²×C₄×C₈₃	—	2	2	no	—
25	C₂²×C₄×C₈₃	D₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	2	4	no	—
28	C₄⋊C₄×C₈₃	—	4	4	no	—
29	C₄²×C₈₃	C₂×C₄	1	2	yes	—
30	C₄²×C₈₃	D₄	1	2	yes	—
31	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	—	4	4	no	—
34	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₈₃	SD₁₆	1	4	yes	—
36	C₂×C₄×C₈₃	Q₁₆	1	4	yes	—
37	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	#41	1	2	yes	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	4	2	no	—
43	C₂×Dic₈₃	—	4	2	no	—
44	Dic₁₆₆	—	2	8	no	—
45	Dic₁₆₆	—	4	4	no	—
46	C₈₃⋊C₈	—	2	4	no	—
47	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈₃	#124	1	1	yes	characteristic
49	C₂²×C₈₃	#127	1	1	yes	characteristic
50	C₂²×C₈₃	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₈₃	C₂×Q₁₆	1	2	yes	—
52	C₄×C₈₃	—	8	2	no	—
53	C₄×C₈₃	#103	1	8	yes	complemented
54	C₄×C₈₃	C₄×D₄	1	2	yes	—
55	C₄×C₈₃	#121	1	2	yes	—
56	Dic₈₃	—	4	4	no	—
57	C₂×C₈₃	#390	1	1	yes	characteristic
58	C₂×C₈₃	C₂×#103	1	2	yes	—
59	C₂×C₈₃	#487	1	2	yes	—
60	C₂×C₈₃	#492	1	1	yes	characteristic
61	C₂×C₈₃	#494	1	1	yes	characteristic
62	#1956	—	83	1	no	maximal, Sylow
63	C₈₃	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	83	1	no	—
65	C₂×#103	—	83	4	no	—
66	C₂×C₄×C₈	—	83	1	no	—
67	#384	D₈₃	1	1	yes	characteristic
68	C₂²×Q₈	—	83	2	no	—
69	C₂²×C₈	—	83	2	no	—
70	C₄⋊Q₈	—	83	4	no	—
71	C₄⋊Q₈	—	166	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	166	2	no	—
74	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
75	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
76	#103	—	166	8	no	—
77	C₄×C₈	—	166	2	no	—
78	C₂×Q₈	—	83	4	no	—
79	C₂×Q₈	—	166	4	no	—
80	C₂×Q₈	—	166	4	no	—
81	C₂×Q₈	—	166	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	166	2	no	—
84	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
86	C₂×C₈	—	83	4	no	—
87	C₂×C₈	—	166	2	no	—
88	C₂×C₈	—	166	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	166	8	no	—
92	C₄²	C₄×D₈₃	1	2	yes	—
93	C₄²	D₃₃₂	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
95	Q₈	—	332	4	no	—
96	Q₈	—	166	8	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	332	2	no	—
100	C₂×C₄	—	332	2	no	—
101	C₂×C₄	—	166	4	no	—
102	C₂×C₄	(C₂×D₈₃)⋊C₄	1	4	yes	—
103	C₂×C₄	#13050578	1	4	yes	—
104	C₂×C₄	#13050580	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₈₃	1	2	yes	—
107	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
108	C₂×C₄	#13050591	1	2	yes	—
109	C₈	—	166	4	no	—
110	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
111	C₂²	#14581781	1	1	yes	characteristic
112	C₂²	#14581788	1	1	yes	characteristic
113	C₂²	C₂×#13050578	1	2	yes	—
114	C₂²	C₂×#13050580	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	332	4	no	—
117	C₄	#14581715	1	8	yes	complemented
118	C₄	#14581754	1	2	yes	—
119	C₄	#14581767	1	2	yes	—
120	C₂	#19127602	1	1	yes	characteristic
121	C₂	C₂×#14581715	1	2	yes	—
122	C₂	#19128053	1	2	yes	—
123	C₂	#19128073	1	1	yes	characteristic
124	C₂	#19128080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

C₂×#14581715

C₂

yes

maximal

C₂×C₄×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#384×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581792

—

#14581792

C₄

yes

complemented

C₂×C₄⋊Dic₈₃

—

#14581715

—

C₄×C₈₃⋊C₈

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

Q₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₂×SD₁₆

yes

—

C₂²×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

—

C₄×C₈₃

#103

yes

complemented

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

#121

yes

—

Dic₈₃

—

C₂×C₈₃

#390

yes

characteristic

C₂×C₈₃

C₂×#103

yes

—

C₂×C₈₃

#487

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

166

—

C₄×C₈

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

C₄×D₈₃

yes

—

C₄²

D₃₃₂

yes

—

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

100

C₂×C₄

—

332

—

101

C₂×C₄

—

166

—

102

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

103

C₂×C₄

#13050578

yes

—

104

C₂×C₄

#13050580

yes

—

105

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

106

C₂×C₄

D₄×D₈₃

yes

—

107

C₂×C₄

C₂×D₃₃₂

yes

characteristic

108

C₂×C₄

#13050591

yes

—

109

C₈

—

166

—

110

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

111

C₂²

#14581781

yes

characteristic

112

C₂²

#14581788

yes

characteristic

113

C₂²

C₂×#13050578

yes

—

114

C₂²

C₂×#13050580

yes

—

115

C₄

—

116

C₄

—

332

—

117

C₄

#14581715

yes

complemented

118

C₄

#14581754

yes

—

119

C₄

#14581767

yes

—

120

C₂

#19127602

yes

characteristic

121

C₂

C₂×#14581715

yes

—

122

C₂

#19128053

yes

—

123

C₂

#19128073

yes

characteristic

124

C₂

#19128080

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#384×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#384	D₈₃
hypocenter	C₈₃	#1956
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#384×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #14581792	C₄	semidirect
53. C₄×C₈₃	#103	semidirect
63. C₈₃	#1956	semidirect
117. C₄	#14581715	semidirect

H ≅ G/N

type

8. #14581792

C₄

semidirect

53. C₄×C₈₃

#103

semidirect

63. C₈₃

#1956

semidirect

117. C₄

#14581715

semidirect

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

subgroup	index
C₂×#14581792	2
C₂×#14581715	2
C₂×C₄×C₈₃⋊C₈	2
#384×C₈₃	2
#1956	83

subgroup

index

C₂×#14581792

C₂×#14581715

C₂×C₄×C₈₃⋊C₈

#384×C₈₃

#1956

Subgroups — #24908264

Conjugacy classes of subgroups

Complements

Maximal subgroups