Group #24917799 — subgroups

Subgroups — C₂×#19128067 #24917799

Order 10624.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581789	C₂	1	1	yes	maximal, characteristic
2	#19128067	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14581712	C₂	1	4	yes	maximal, complemented
4	C₂×#14581703	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₃₂	—	2	2	no	—
7	C₂²×D₃₃₂	C₂²	1	2	yes	—
8	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#14581789	—	2	2	no	—
10	#14581789	C₂²	1	4	yes	—
11	#14581712	C₂²	1	16	yes	complemented
12	#14581703	C₂²	1	4	yes	complemented
13	#125×C₈₃	—	2	2	no	—
14	#125×C₈₃	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
16	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
18	C₂³×D₈₃	—	2	2	no	—
19	C₂×D₃₃₂	—	2	8	no	—
20	C₂×D₃₃₂	—	2	2	no	—
21	C₂×D₃₃₂	—	2	8	no	—
22	C₂×D₃₃₂	—	4	8	no	—
23	C₂×D₃₃₂	C₂³	1	4	yes	—
24	C₂×C₈₃⋊C₈	—	2	4	no	—
25	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₄×C₈₃	—	2	2	no	—
27	C₂²×C₄×C₈₃	D₄	1	2	yes	—
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₃	—	2	8	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
32	C₄²×C₈₃	D₄	1	2	yes	—
33	C₄²×C₈₃	C₂³	1	2	yes	—
34	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₈₃	—	2	4	no	—
36	C₂²×D₈₃	—	4	4	no	—
37	C₂²×D₈₃	—	4	8	no	—
38	C₂×C₄×C₈₃	—	2	4	no	—
39	C₂×C₄×C₈₃	—	2	4	no	—
40	C₂×C₄×C₈₃	—	4	4	no	—
41	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#41	1	4	yes	—
44	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
45	D₃₃₂	—	2	8	no	—
46	D₃₃₂	—	4	16	no	—
47	D₃₃₂	—	4	4	no	—
48	C₈₃⋊C₈	—	2	4	no	—
49	C₂²×C₈₃	#124	1	2	yes	—
50	C₂²×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₈₃	—	4	4	no	—
53	C₂×D₈₃	—	4	8	no	—
54	C₂×D₈₃	—	8	8	no	—
55	C₄×C₈₃	—	2	4	no	—
56	C₄×C₈₃	—	4	4	no	—
57	C₄×C₈₃	#124	1	2	yes	—
58	C₄×C₈₃	C₂×#41	1	2	yes	—
59	C₂×C₈₃	#490	1	4	yes	complemented
60	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
61	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
62	D₈₃	—	8	4	no	—
63	C₂×#490	—	83	1	no	maximal, Sylow
64	C₈₃	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	—	83	1	no	—
66	C₂×#125	D₈₃	1	1	yes	characteristic, complemented
67	#490	—	83	8	no	—
68	C₂×#102	—	83	4	no	—
69	C₂×#97	—	83	1	no	—
70	C₂²×D₄	—	83	2	no	—
71	C₂²×D₄	—	166	2	no	—
72	C₂²×C₈	—	83	2	no	—
73	#127	—	83	4	no	—
74	#127	—	166	2	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₈₃	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
79	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
80	#102	—	83	16	no	—
81	#97	—	83	4	no	—
82	C₂⁴	—	166	2	no	—
83	C₂×D₄	—	332	8	no	—
84	C₂×D₄	—	83	4	no	—
85	C₂×D₄	—	166	8	no	—
86	C₂×D₄	—	166	2	no	—
87	C₂×D₄	—	166	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
91	C₂×C₈	—	83	4	no	—
92	C₂×C₈	—	166	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
96	C₄²	C₂²⋊D₈₃	1	2	yes	—
97	C₄²	C₂²×D₈₃	1	2	yes	—
98	C₂³	—	332	4	no	—
99	C₂³	—	332	8	no	—
100	C₂³	—	166	4	no	—
101	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
102	D₄	—	332	16	no	—
103	D₄	—	332	4	no	—
104	D₄	—	166	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₈₃	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
110	C₂×C₄	#13050591	1	4	yes	—
111	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
112	C₈	—	166	4	no	—
113	C₂²	—	332	4	no	—
114	C₂²	—	332	8	no	—
115	C₂²	—	664	8	no	—
116	C₂²	#14581781	1	2	yes	—
117	C₂²	#14581817	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#14581781	1	2	yes	—
122	C₄	C₂×#13050591	1	2	yes	—
123	C₂	—	664	4	no	—
124	C₂	#19128067	1	4	yes	direct product factor
125	C₂	C₂×#14581781	1	1	yes	characteristic
126	C₂	C₂×#14581817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581789

C₂

yes

maximal, characteristic

#19128067

C₂

yes

maximal, direct product factor

C₂×#14581712

C₂

yes

maximal, complemented

C₂×#14581703

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₃₂

—

C₂²×D₃₃₂

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581789

—

#14581789

C₂²

yes

—

#14581712

C₂²

yes

complemented

#14581703

C₂²

yes

complemented

#125×C₈₃

—

#125×C₈₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂³×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

C₂³

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₈₃⋊C₈

—

C₂²×C₈₃

#124

yes

—

C₂²×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

C₄×C₈₃

C₂×#41

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

C₂×#490

—

maximal, Sylow

C₈₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#125

D₈₃

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₂²×D₄

—

C₂²×D₄

—

166

—

C₂²×C₈

—

#127

—

#127

—

166

—

#125

—

#125

C₂×D₈₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#102

—

#97

—

C₂⁴

—

166

—

C₂×D₄

—

332

—

C₂×D₄

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₈₃

yes

—

C₄²

C₂²⋊D₈₃

yes

—

C₄²

C₂²×D₈₃

yes

—

C₂³

—

332

—

C₂³

—

332

—

100

C₂³

—

166

—

101

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

102

D₄

—

332

—

103

D₄

—

332

—

104

D₄

—

166

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

109

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

110

C₂×C₄

#13050591

yes

—

111

C₂×C₄

C₂³×D₈₃

yes

characteristic

112

C₈

—

166

—

113

C₂²

—

332

—

114

C₂²

—

332

—

115

C₂²

—

664

—

116

C₂²

#14581781

yes

—

117

C₂²

#14581817

yes

—

118

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#14581781

yes

—

122

C₄

C₂×#13050591

yes

—

123

C₂

—

664

—

124

C₂

#19128067

yes

direct product factor

125

C₂

C₂×#14581781

yes

characteristic

126

C₂

C₂×#14581817

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂×#125×C₈₃	C₂
socle	C₂³×C₈₃	C₂×D₄
prime radical	C₂×#14581789	C₂
hypercenter	C₂×#125	D₈₃
hypocenter	C₈₃	C₂×#490
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₂×#125×C₈₃

C₂

socle

C₂³×C₈₃

C₂×D₄

C₂

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19128067	C₂	direct (N × H)
3. C₂×#14581712	C₂	semidirect
4. C₂×#14581703	C₂	semidirect
5. C₂×#125×C₈₃	C₂	semidirect
11. #14581712	C₂²	semidirect
12. #14581703	C₂²	semidirect
14. #125×C₈₃	C₂²	semidirect
59. C₂×C₈₃	#490	semidirect
64. C₈₃	C₂×#490	semidirect
66. C₂×#125	D₈₃	semidirect
76. #125	C₂×D₈₃	semidirect
124. C₂	#19128067	direct (N × H)

H ≅ G/N

type

2. #19128067

C₂

direct (N × H)

3. C₂×#14581712

C₂

semidirect

4. C₂×#14581703

C₂

semidirect

5. C₂×#125×C₈₃

C₂

semidirect

11. #14581712

C₂²

semidirect

12. #14581703

C₂²

semidirect

14. #125×C₈₃

C₂²

semidirect

59. C₂×C₈₃

#490

semidirect

64. C₈₃

C₂×#490

semidirect

66. C₂×#125

D₈₃

semidirect

76. #125

C₂×D₈₃

semidirect

124. C₂

#19128067

direct (N × H)

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

subgroup	index
C₂×#14581789	2
#19128067	2
C₂×#14581712	2
C₂×#14581703	2
C₂×#125×C₈₃	2
C₂×#490	83

subgroup

index

C₂×#125×C₈₃

Subgroups — C₂×#19128067 #24917799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19128067 #24917799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128067 #24917799