Group #24907618 — subgroups

Subgroups — #24907618

Order 10624.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581724	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581723	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127570	C₂	1	2	yes	maximal
5	C₂×#106×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	C₂×Dic₈₃⋊C₄	—	2	4	no	—
9	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
10	#14581724	—	2	2	no	—
11	#14581723	—	2	4	no	—
12	C₈×Dic₈₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
15	#106×C₈₃	—	2	2	no	—
16	C₂²×Dic₈₃	—	2	2	no	—
17	C₂²×Dic₈₃	D₄	1	2	yes	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
20	C₂×C₈₃⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₈₃	—	2	2	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	Dic₈₃⋊C₄	—	4	8	no	—
24	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₈₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₈₃	—	2	2	no	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₃	D₄	1	2	yes	—
30	C₂×C₈×C₈₃	Q₈	1	2	yes	—
31	C₄⋊C₄×C₈₃	—	2	2	no	—
32	C₄⋊C₄×C₈₃	—	2	2	no	—
33	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	—	4	2	no	—
35	C₂×C₄×C₈₃	—	4	4	no	—
36	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₈₃	—	4	2	no	—
43	C₂×Dic₈₃	—	4	4	no	—
44	C₂×Dic₈₃	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₈₃	SD₁₆	1	8	yes	—
46	C₈×C₈₃	C₄⋊C₄	1	4	yes	complemented
47	C₈₃⋊C₈	C₄⋊C₄	1	4	yes	complemented
48	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈₃	#127	1	1	yes	characteristic
50	C₂²×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₈₃	C₂×SD₁₆	1	4	yes	—
52	C₄×C₈₃	—	8	2	no	—
53	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₈₃	—	8	2	no	—
58	Dic₈₃	#106	1	8	yes	complemented
59	C₂×C₈₃	#384	1	1	yes	characteristic
60	C₂×C₈₃	C₂×#106	1	2	yes	—
61	C₂×C₈₃	#492	1	2	yes	—
62	C₂×C₈₃	#498	1	2	yes	—
63	#1844	—	83	1	no	maximal, Sylow
64	C₈₃	#1844	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	—	83	1	no	—
66	C₂×#106	—	83	2	no	—
67	C₂×#106	D₈₃	1	1	yes	characteristic
68	C₂×C₄×C₈	—	83	1	no	—
69	#384	—	83	2	no	—
70	C₂²×C₈	—	83	1	no	—
71	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	83	2	no	—
73	C₂×C₄⋊C₄	—	166	4	no	—
74	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
75	C₂×C₄²	—	83	1	no	—
76	#106	—	2	2	no	—
77	#106	—	166	4	no	—
78	#106	—	166	2	no	—
79	C₄×C₈	—	83	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	83	2	no	—
82	C₂²×C₄	—	166	2	no	—
83	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
84	C₂×C₈	—	83	2	no	—
85	C₂×C₈	—	83	2	no	—
86	C₂×C₈	—	83	2	no	—
87	C₂×C₈	Dic₁₆₆	1	2	yes	—
88	C₂×C₈	C₄×D₈₃	1	2	yes	—
89	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	332	8	no	—
93	C₄⋊C₄	—	166	2	no	—
94	C₄⋊C₄	—	166	2	no	—
95	C₄²	—	83	2	no	—
96	C₄²	—	83	2	no	—
97	C₂³	D₄×D₈₃	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	332	2	no	—
101	C₂×C₄	—	332	4	no	—
102	C₂×C₄	—	83	8	no	—
103	C₂×C₄	—	83	4	no	—
104	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
110	C₈	—	83	4	no	—
111	C₈	Dic₈₃⋊C₄	1	4	yes	complemented
112	C₂²	C₄⋊C₄×D₈₃	1	1	yes	characteristic
113	C₂²	#14581788	1	1	yes	characteristic
114	C₂²	#14581791	1	1	yes	characteristic
115	C₂²	SD₁₆×D₈₃	1	4	yes	—
116	C₄	—	8	2	no	—
117	C₄	—	83	8	no	—
118	C₄	—	664	2	no	—
119	C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
120	C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
121	C₄	#14581790	1	1	yes	characteristic
122	C₄	#14581791	1	1	yes	characteristic
123	C₂	#19127570	1	1	yes	characteristic
124	C₂	#106×D₈₃	1	2	yes	—
125	C₂	#19128072	1	2	yes	—
126	C₂	#19128098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581724

C₂

yes

maximal, characteristic

C₂×#14581723

C₂

yes

maximal

C₂×C₈×Dic₈₃

C₂

yes

maximal, characteristic

#19127570

C₂

yes

maximal

C₂×#106×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581724

—

#14581723

—

C₈×Dic₈₃

C₄

yes

complemented

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

#106×C₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

D₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

Q₈

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

Q₈

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

C₂×Dic₈₃

SD₁₆

yes

—

C₈×C₈₃

C₄⋊C₄

yes

complemented

C₈₃⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

#127

yes

characteristic

C₂²×C₈₃

C₄⋊Q₈

yes

characteristic

C₂²×C₈₃

C₂×SD₁₆

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#492

yes

—

C₂×C₈₃

#498

yes

—

#1844

—

maximal, Sylow

C₈₃

#1844

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

—

C₂×#106

D₈₃

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

—

#106

—

#106

—

166

—

#106

—

166

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₆₆

yes

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

C₄²

—

C₂³

D₄×D₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

332

—

101

C₂×C₄

—

332

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

105

C₂×C₄

D₄×D₈₃

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

107

C₂×C₄

Q₈×D₈₃

yes

characteristic

108

C₂×C₄

Q₈×D₈₃

yes

characteristic

109

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

110

C₈

—

111

C₈

Dic₈₃⋊C₄

yes

complemented

112

C₂²

C₄⋊C₄×D₈₃

yes

characteristic

113

C₂²

#14581788

yes

characteristic

114

C₂²

#14581791

yes

characteristic

115

C₂²

SD₁₆×D₈₃

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

664

—

119

C₄

C₄⋊C₄×D₈₃

yes

characteristic

120

C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

121

C₄

#14581790

yes

characteristic

122

C₄

#14581791

yes

characteristic

123

C₂

#19127570

yes

characteristic

124

C₂

#106×D₈₃

yes

—

125

C₂

#19128072

yes

—

126

C₂

#19128098

yes

—

	subgroup	G/H
center	C₂³	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×#106×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×#106	D₈₃
hypocenter	C₈₃	#1844
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×#106×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 ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₈×Dic₈₃	C₄	semidirect
46. C₈×C₈₃	C₄⋊C₄	semidirect
47. C₈₃⋊C₈	C₄⋊C₄	semidirect
58. Dic₈₃	#106	semidirect
64. C₈₃	#1844	semidirect
111. C₈	Dic₈₃⋊C₄	semidirect

H ≅ G/N

type

12. C₈×Dic₈₃

C₄

semidirect

46. C₈×C₈₃

C₄⋊C₄

semidirect

47. C₈₃⋊C₈

C₄⋊C₄

semidirect

58. Dic₈₃

#106

semidirect

64. C₈₃

#1844

semidirect

111. C₈

Dic₈₃⋊C₄

semidirect

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

subgroup	index
C₂×#14581724	2
C₂×#14581723	2
C₂×C₈×Dic₈₃	2
#19127570	2
C₂×#106×C₈₃	2
#1844	83

subgroup

index

C₂×C₈×Dic₈₃

C₂×#106×C₈₃

Subgroups — #24907618

Conjugacy classes of subgroups

Complements

Maximal subgroups