Group #24912492 — subgroups

Subgroups — #24912492

Order 10624.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14581791	C₂	1	4	yes	maximal
2	#19127621	C₂	1	4	yes	maximal
3	#19127597	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₈₃	C₂	1	2	yes	maximal
5	#384×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	4	yes	—
7	#14581791	—	2	16	no	—
8	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	C₂×Dic₈₃⋊C₄	C₂²	1	8	yes	—
11	C₂×C₄×Dic₈₃	C₂²	1	4	yes	—
12	C₈₃⋊#95	C₂²	1	4	yes	—
13	C₈₃⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₆₆	—	2	16	no	—
18	C₂×Dic₁₆₆	—	4	4	no	—
19	C₂²×Dic₈₃	C₂³	1	4	yes	—
20	C₂²×Dic₈₃	C₂³	1	4	yes	—
21	C₄⋊Dic₈₃	D₄	1	8	yes	—
22	C₄⋊Dic₈₃	Q₈	1	8	yes	complemented
23	Dic₈₃⋊C₄	—	2	16	no	—
24	C₄×Dic₈₃	—	2	8	no	—
25	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
28	C₄⋊C₄×C₈₃	—	2	8	no	—
29	C₄⋊C₄×C₈₃	—	4	2	no	—
30	C₄²×C₈₃	Q₈	1	4	yes	complemented
31	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	1	no	—
33	C₂×C₄×C₈₃	—	2	1	no	—
34	C₂×C₄×C₈₃	—	2	8	no	—
35	C₂×C₄×C₈₃	—	2	4	no	—
36	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
38	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
40	C₂×Dic₈₃	—	2	4	no	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	2	4	no	—
43	C₂×Dic₈₃	—	2	4	no	—
44	C₂×Dic₈₃	C₂×D₄	1	8	yes	—
45	C₂×Dic₈₃	C₂×Q₈	1	8	yes	—
46	Dic₁₆₆	—	4	16	no	—
47	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
48	C₂²×C₈₃	C₂²×Q₈	1	2	yes	—
49	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₈₃	—	2	2	no	—
53	C₄×C₈₃	—	4	4	no	—
54	C₄×C₈₃	C₄⋊Q₈	1	8	yes	complemented
55	Dic₈₃	—	2	8	no	—
56	Dic₈₃	—	2	8	no	—
57	C₂×C₈₃	C₂×C₄⋊Q₈	1	2	yes	—
58	C₂×C₈₃	#544	1	1	yes	characteristic
59	C₂×C₈₃	D₄²	1	1	yes	characteristic
60	C₂×C₈₃	D₄×Q₈	1	2	yes	—
61	C₂×C₈₃	Q₈²	1	1	yes	characteristic
62	#2551	—	83	1	no	maximal, Sylow
63	C₈₃	#2551	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	83	4	no	—
65	#395	—	83	4	no	—
66	#389	—	83	4	no	—
67	#384	D₈₃	1	1	yes	characteristic
68	C₄×C₄⋊C₄	—	83	2	no	—
69	C₂²×Q₈	—	83	4	no	—
70	C₄⋊Q₈	—	166	16	no	—
71	C₂×C₄⋊C₄	—	83	2	no	—
72	C₂×C₄⋊C₄	—	83	2	no	—
73	C₂×C₄⋊C₄	—	83	8	no	—
74	C₂×C₄⋊C₄	C₂×D₈₃	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
76	C₂×C₄²	—	83	4	no	—
77	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
78	#95	—	83	4	no	—
79	#95	—	83	4	no	—
80	C₂×Q₈	—	332	4	no	—
81	C₂×Q₈	—	166	16	no	—
82	C₂²×C₄	—	83	4	no	—
83	C₂²×C₄	—	83	4	no	—
84	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	83	8	no	—
90	C₄⋊C₄	—	83	8	no	—
91	C₄⋊C₄	—	166	16	no	—
92	C₄²	—	166	8	no	—
93	C₄²	Dic₁₆₆	1	4	yes	complemented
94	C₂³	C₂³×D₈₃	1	1	yes	characteristic
95	Q₈	—	332	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	83	8	no	—
101	C₂×C₄	—	83	8	no	—
102	C₂×C₄	—	166	4	no	—
103	C₂×C₄	—	166	4	no	—
104	C₂×C₄	—	166	4	no	—
105	C₂×C₄	—	166	4	no	—
106	C₂×C₄	D₄×D₈₃	1	4	yes	—
107	C₂×C₄	Q₈×D₈₃	1	4	yes	—
108	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₆₆	1	4	yes	—
110	C₂²	C₂×D₄×D₈₃	1	2	yes	—
111	C₂²	C₂×Q₈×D₈₃	1	2	yes	—
112	C₂²	C₂²×Dic₁₆₆	1	1	yes	characteristic
113	C₂²	#14581837	1	1	yes	characteristic
114	C₂²	#14581840	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	166	8	no	—
118	C₄	—	166	8	no	—
119	C₄	#14581791	1	8	yes	complemented
120	C₂	C₂×#14581791	1	2	yes	—
121	C₂	#19128332	1	1	yes	characteristic
122	C₂	#19128336	1	1	yes	characteristic
123	C₂	#19128366	1	2	yes	—
124	C₂	#19128431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581791

C₂

yes

maximal

#19127621

C₂

yes

maximal

#19127597

C₂

yes

maximal

C₄×C₄⋊Dic₈₃

C₂

yes

maximal

#384×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

#14581791

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

Q₈

yes

complemented

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

Q₈

yes

complemented

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×Q₈

yes

—

Dic₁₆₆

—

C₂²×C₈₃

C₂²×D₄

yes

—

C₂²×C₈₃

C₂²×Q₈

yes

—

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₃

2₋¹⁺⁴

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄⋊Q₈

yes

complemented

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₂×C₄⋊Q₈

yes

—

C₂×C₈₃

#544

yes

characteristic

C₂×C₈₃

D₄²

yes

characteristic

C₂×C₈₃

D₄×Q₈

yes

—

C₂×C₈₃

Q₈²

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₈₃

yes

characteristic

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

332

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

Dic₁₆₆

yes

complemented

C₂³

C₂³×D₈₃

yes

characteristic

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

166

—

103

C₂×C₄

—

166

—

104

C₂×C₄

—

166

—

105

C₂×C₄

—

166

—

106

C₂×C₄

D₄×D₈₃

yes

—

107

C₂×C₄

Q₈×D₈₃

yes

—

108

C₂×C₄

C₂×Dic₁₆₆

yes

—

109

C₂×C₄

C₂×Dic₁₆₆

yes

—

110

C₂²

C₂×D₄×D₈₃

yes

—

111

C₂²

C₂×Q₈×D₈₃

yes

—

112

C₂²

C₂²×Dic₁₆₆

yes

characteristic

113

C₂²

#14581837

yes

characteristic

114

C₂²

#14581840

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

166

—

118

C₄

—

166

—

119

C₄

#14581791

yes

complemented

120

C₂

C₂×#14581791

yes

—

121

C₂

#19128332

yes

characteristic

122

C₂

#19128336

yes

characteristic

123

C₂

#19128366

yes

—

124

C₂

#19128431

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	#384×C₈₃	C₂
hypercenter	#384	D₈₃
hypocenter	C₈₃	#2551
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

#384×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
22. C₄⋊Dic₈₃	Q₈	semidirect
30. C₄²×C₈₃	Q₈	semidirect
54. C₄×C₈₃	C₄⋊Q₈	semidirect
63. C₈₃	#2551	semidirect
93. C₄²	Dic₁₆₆	semidirect
119. C₄	#14581791	semidirect

H ≅ G/N

type

22. C₄⋊Dic₈₃

Q₈

semidirect

30. C₄²×C₈₃

Q₈

semidirect

54. C₄×C₈₃

C₄⋊Q₈

semidirect

63. C₈₃

#2551

semidirect

93. C₄²

Dic₁₆₆

semidirect

119. C₄

#14581791

semidirect

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

subgroup	index
C₂×#14581791	2
#19127621	2
#19127597	2
C₄×C₄⋊Dic₈₃	2
#384×C₈₃	2
#2551	83

subgroup

index

C₂×#14581791

#19127621

#19127597

C₄×C₄⋊Dic₈₃

#384×C₈₃

#2551

Subgroups — #24912492

Conjugacy classes of subgroups

Complements

Maximal subgroups