Group #24915821 — subgroups

Subgroups — #24915821

Order 10624.

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

normal subgroups only

This group has 752 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×#14581789	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581781	C₂	1	4	yes	maximal, complemented
3	#19127602	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₃₂	C₂²	1	4	yes	—
7	C₂²×D₃₃₂	C₂²	1	2	yes	—
8	#14581789	—	2	4	no	—
9	#14581781	—	2	16	no	—
10	C₂×(C₂×D₈₃)⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₈₃	C₂²	1	4	yes	—
12	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
13	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₈₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂³×D₈₃	C₂³	1	4	yes	—
18	C₂×D₃₃₂	—	2	8	no	—
19	C₂×D₃₃₂	—	2	8	no	—
20	C₂×D₃₃₂	—	2	4	no	—
21	C₂×D₃₃₂	—	2	16	no	—
22	C₂²×Dic₈₃	C₂³	1	4	yes	—
23	(C₂×D₈₃)⋊C₄	—	2	32	no	—
24	C₄×Dic₈₃	—	2	8	no	—
25	C₄×Dic₈₃	D₄	1	8	yes	complemented
26	C₂×Q₈×C₈₃	—	2	8	no	—
27	C₂×Q₈×C₈₃	—	2	4	no	—
28	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
31	C₄⋊C₄×C₈₃	—	2	8	no	—
32	C₄²×C₈₃	D₄	1	4	yes	—
33	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
34	C₂²×D₈₃	—	2	8	no	—
35	C₂²×D₈₃	—	2	4	no	—
36	C₂²×D₈₃	—	2	16	no	—
37	Q₈×C₈₃	—	4	8	no	—
38	C₂×C₄×C₈₃	—	2	4	no	—
39	C₂×C₄×C₈₃	—	2	8	no	—
40	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₈₃	#41	1	8	yes	—
44	C₂×Dic₈₃	—	2	8	no	—
45	C₂×Dic₈₃	C₂×D₄	1	8	yes	—
46	D₃₃₂	—	4	16	no	—
47	D₃₃₂	—	4	8	no	—
48	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
49	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₈₃	C₂×#41	1	4	yes	—
51	C₂×D₈₃	—	4	8	no	—
52	C₂×D₈₃	—	4	4	no	—
53	C₂×D₈₃	—	4	16	no	—
54	C₄×C₈₃	—	4	4	no	—
55	C₄×C₈₃	#124	1	8	yes	complemented
56	C₄×C₈₃	#127	1	4	yes	—
57	Dic₈₃	—	2	8	no	—
58	C₂×C₈₃	C₂×#124	1	2	yes	—
59	C₂×C₈₃	C₂×#127	1	1	yes	characteristic
60	C₂×C₈₃	#532	1	4	yes	—
61	D₈₃	—	8	4	no	—
62	#2858	—	83	1	no	maximal, Sylow
63	C₈₃	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	D₈₃	1	1	yes	characteristic, complemented
65	C₂×#127	—	83	1	no	—
66	C₂×#124	—	83	4	no	—
67	#390	—	83	8	no	—
68	C₄³	—	83	1	no	—
69	C₂²×Q₈	C₂×D₈₃	1	2	yes	—
70	C₂²×D₄	—	83	2	no	—
71	C₂²×D₄	—	83	4	no	—
72	C₄⋊Q₈	—	2	4	no	—
73	#127	—	166	4	no	—
74	#124	—	166	16	no	—
75	C₂×C₄⋊C₄	C₂×D₈₃	1	4	yes	—
76	C₂×C₂²⋊C₄	—	83	16	no	—
77	C₂×C₄²	—	83	2	no	—
78	C₂×C₄²	—	83	4	no	—
79	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
80	C₂⁴	—	83	4	no	—
81	C₂×Q₈	—	2	8	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×D₄	—	166	8	no	—
84	C₂×D₄	—	166	8	no	—
85	C₂×D₄	—	166	4	no	—
86	C₂×D₄	—	166	16	no	—
87	C₂²×C₄	—	83	4	no	—
88	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
89	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
91	C₄⋊C₄	—	2	8	no	—
92	C₂²⋊C₄	—	166	32	no	—
93	C₄²	—	83	8	no	—
94	C₄²	—	166	8	no	—
95	C₄²	C₂²⋊D₈₃	1	4	yes	—
96	C₂³	—	166	8	no	—
97	C₂³	—	166	4	no	—
98	C₂³	—	166	16	no	—
99	C₂³	C₂³×D₈₃	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	332	16	no	—
102	D₄	—	332	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	83	8	no	—
106	C₂×C₄	—	166	8	no	—
107	C₂×C₄	D₄×D₈₃	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₈₃	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
110	C₂×C₄	#13050591	1	8	yes	—
111	C₂²	—	332	8	no	—
112	C₂²	—	332	4	no	—
113	C₂²	—	332	16	no	—
114	C₂²	C₂×D₄×D₈₃	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
116	C₂²	C₂×#13050591	1	4	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	166	8	no	—
119	C₄	#14581781	1	8	yes	complemented
120	C₄	#14581788	1	4	yes	—
121	C₂	—	664	4	no	—
122	C₂	C₂×#14581781	1	2	yes	—
123	C₂	C₂×#14581788	1	1	yes	characteristic
124	C₂	#19128252	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581789

C₂

yes

maximal, characteristic

C₂×#14581781

C₂

yes

maximal, complemented

#19127602

C₂

yes

maximal, complemented

C₄²×Dic₈₃

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₃₂

C₂²

yes

—

C₂²×D₃₃₂

C₂²

yes

—

#14581789

—

#14581781

—

C₂×(C₂×D₈₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂²×Q₈×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂³×D₈₃

C₂³

yes

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂²×Dic₈₃

C₂³

yes

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

complemented

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

D₃₃₂

—

D₃₃₂

—

C₂²×C₈₃

C₂²×D₄

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

complemented

C₄×C₈₃

#127

yes

—

Dic₈₃

—

C₂×C₈₃

C₂×#124

yes

—

C₂×C₈₃

C₂×#127

yes

characteristic

C₂×C₈₃

#532

yes

—

D₈₃

—

#2858

—

maximal, Sylow

C₈₃

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₃

yes

characteristic, complemented

C₂×#127

—

C₂×#124

—

#390

—

C₄³

—

C₂²×Q₈

C₂×D₈₃

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

166

—

#124

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

166

—

C₄²

C₂²⋊D₈₃

yes

—

C₂³

—

166

—

C₂³

—

166

—

C₂³

—

166

—

C₂³

C₂³×D₈₃

yes

characteristic

100

Q₈

—

101

D₄

—

332

—

102

D₄

—

332

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

166

—

107

C₂×C₄

D₄×D₈₃

yes

—

108

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

109

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

110

C₂×C₄

#13050591

yes

—

111

C₂²

—

332

—

112

C₂²

—

332

—

113

C₂²

—

332

—

114

C₂²

C₂×D₄×D₈₃

yes

—

115

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

116

C₂²

C₂×#13050591

yes

—

117

C₄

—

118

C₄

—

166

—

119

C₄

#14581781

yes

complemented

120

C₄

#14581788

yes

—

121

C₂

—

664

—

122

C₂

C₂×#14581781

yes

—

123

C₂

C₂×#14581788

yes

characteristic

124

C₂

#19128252

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, cosocle	C₂³×C₈₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₈₃	C₂
prime radical	C₂×#14581789	C₂
hypercenter	C₂×C₄⋊Q₈	D₈₃
hypocenter	C₈₃	#2858
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, cosocle

C₂³×C₈₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₈₃

C₂

prime radical

C₂×#14581789

C₂

hypercenter

C₂×C₄⋊Q₈

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#14581781	C₂	semidirect
3. #19127602	C₂	semidirect
4. C₄²×Dic₈₃	C₂	semidirect
5. C₂×C₄⋊Q₈×C₈₃	C₂	semidirect
25. C₄×Dic₈₃	D₄	semidirect
55. C₄×C₈₃	#124	semidirect
63. C₈₃	#2858	semidirect
64. C₂×C₄⋊Q₈	D₈₃	semidirect
119. C₄	#14581781	semidirect

H ≅ G/N

type

2. C₂×#14581781

C₂

semidirect

3. #19127602

C₂

semidirect

4. C₄²×Dic₈₃

C₂

semidirect

5. C₂×C₄⋊Q₈×C₈₃

C₂

semidirect

25. C₄×Dic₈₃

D₄

semidirect

55. C₄×C₈₃

#124

semidirect

63. C₈₃

#2858

semidirect

64. C₂×C₄⋊Q₈

D₈₃

semidirect

119. C₄

#14581781

semidirect

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

subgroup	index
C₂×#14581789	2
C₂×#14581781	2
#19127602	2
C₄²×Dic₈₃	2
C₂×C₄⋊Q₈×C₈₃	2
#2858	83

subgroup

index

C₂×#14581789

C₂×#14581781

#19127602

C₄²×Dic₈₃

C₂×C₄⋊Q₈×C₈₃

#2858

Subgroups — #24915821

Conjugacy classes of subgroups

Complements

Maximal subgroups