Group #24917804 — subgroups

Subgroups — C₂×#19128071 #24917804

Order 10624.

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

normal subgroups only

This group has 384 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×#14581792	C₂	1	1	yes	maximal, characteristic
2	#19128071	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14581715	C₂	1	4	yes	maximal
4	C₂×#14581703	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
8	#14581792	—	2	2	no	—
9	#14581792	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₈₃	—	2	2	no	—
11	#14581715	C₂²	1	16	yes	—
12	#14581703	C₂²	1	4	yes	—
13	#125×C₈₃	—	2	2	no	—
14	#125×C₈₃	C₂²	1	4	yes	—
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	C₂×Dic₁₆₆	—	2	8	no	—
19	C₂×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₁₆₆	C₂³	1	4	yes	—
21	C₂²×Dic₈₃	—	2	2	no	—
22	C₂×C₈₃⋊C₈	—	2	4	no	—
23	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
24	C₄⋊Dic₈₃	—	2	8	no	—
25	C₂²×C₄×C₈₃	—	2	2	no	—
26	C₂²×C₄×C₈₃	D₄	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₃	—	2	8	no	—
29	C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
31	C₄²×C₈₃	D₄	1	2	yes	—
32	C₄²×C₈₃	C₂³	1	2	yes	—
33	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	—	2	4	no	—
35	C₂×C₄×C₈₃	—	2	4	no	—
36	C₂×C₄×C₈₃	—	4	4	no	—
37	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	#41	1	4	yes	—
40	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	4	4	no	—
43	Dic₁₆₆	—	2	8	no	—
44	Dic₁₆₆	—	4	4	no	—
45	C₈₃⋊C₈	—	2	4	no	—
46	C₂²×C₈₃	#124	1	2	yes	—
47	C₂²×C₈₃	#137	1	4	yes	—
48	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₈₃	—	2	4	no	—
50	C₄×C₈₃	—	4	4	no	—
51	C₄×C₈₃	#124	1	2	yes	—
52	C₄×C₈₃	C₂×#41	1	2	yes	—
53	Dic₈₃	—	4	4	no	—
54	C₂×C₈₃	#491	1	4	yes	complemented
55	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
56	C₂×C₈₃	C₂×#137	1	2	yes	—
57	C₂×#491	—	83	1	no	maximal, Sylow
58	C₈₃	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	83	1	no	—
60	C₂×#125	D₈₃	1	1	yes	characteristic
61	#491	—	83	8	no	—
62	C₂×#103	—	83	4	no	—
63	C₂×#97	—	83	1	no	—
64	C₂²×Q₈	—	83	2	no	—
65	C₂²×C₈	—	83	2	no	—
66	C₄⋊Q₈	—	83	4	no	—
67	C₄⋊Q₈	—	166	2	no	—
68	#125	—	2	2	no	—
69	#125	C₂×D₈₃	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	166	2	no	—
72	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
73	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
74	#103	—	83	16	no	—
75	#97	—	83	4	no	—
76	C₂×Q₈	—	83	4	no	—
77	C₂×Q₈	—	166	8	no	—
78	C₂×Q₈	—	166	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	166	2	no	—
81	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
82	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
83	C₂×C₈	—	83	4	no	—
84	C₂×C₈	—	166	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	166	8	no	—
88	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
89	C₄²	C₂²⋊D₈₃	1	2	yes	—
90	C₄²	C₂²×D₈₃	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
92	Q₈	—	332	4	no	—
93	Q₈	—	166	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	332	4	no	—
98	C₂×C₄	—	166	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₈₃	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
101	C₂×C₄	#13050591	1	4	yes	—
102	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
103	C₈	—	166	4	no	—
104	C₂²	#14581781	1	2	yes	—
105	C₂²	#14581822	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	332	4	no	—
110	C₄	#14581781	1	2	yes	—
111	C₄	C₂×#13050591	1	2	yes	—
112	C₂	#19128071	1	4	yes	direct product factor
113	C₂	C₂×#14581781	1	1	yes	characteristic
114	C₂	C₂×#14581822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

#19128071

C₂

yes

maximal, direct product factor

C₂×#14581715

C₂

yes

maximal

C₂×#14581703

C₂

yes

maximal, characteristic

C₂×#125×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581792

—

#14581792

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

#14581715

C₂²

yes

—

#14581703

C₂²

yes

—

#125×C₈₃

—

#125×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

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

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

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

#124

yes

—

C₂²×C₈₃

#137

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

C₄×C₈₃

C₂×#41

yes

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×#491

—

maximal, Sylow

C₈₃

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

D₈₃

yes

characteristic

#491

—

C₂×#103

—

C₂×#97

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

#125

—

#125

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

#97

—

C₂×Q₈

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

C₂²×D₈₃

yes

—

C₄²

C₂²⋊D₈₃

yes

—

C₄²

C₂²×D₈₃

yes

—

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

101

C₂×C₄

#13050591

yes

—

102

C₂×C₄

C₂³×D₈₃

yes

characteristic

103

C₈

—

166

—

104

C₂²

#14581781

yes

—

105

C₂²

#14581822

yes

—

106

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

332

—

110

C₄

#14581781

yes

—

111

C₄

C₂×#13050591

yes

—

112

C₂

#19128071

yes

direct product factor

113

C₂

C₂×#14581781

yes

characteristic

114

C₂

C₂×#14581822

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, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	C₂×#125	D₈₃
hypocenter	C₈₃	C₂×#491
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, prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19128071	C₂	direct (N × H)
54. C₂×C₈₃	#491	semidirect
58. C₈₃	C₂×#491	semidirect
112. C₂	#19128071	direct (N × H)

H ≅ G/N

type

2. #19128071

C₂

direct (N × H)

54. C₂×C₈₃

#491

semidirect

58. C₈₃

C₂×#491

semidirect

112. C₂

#19128071

direct (N × H)

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

subgroup	index
C₂×#14581792	2
#19128071	2
C₂×#14581715	2
C₂×#14581703	2
C₂×#125×C₈₃	2
C₂×#491	83

subgroup

index

C₂×#125×C₈₃

Subgroups — C₂×#19128071 #24917804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19128071 #24917804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128071 #24917804