Group #24917986 — subgroups

Subgroups — #24917986

Order 10624.

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

normal subgroups only

This group has 348 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	#19128254	C₂	1	2	yes	maximal
2	#19128109	C₂	1	4	yes	maximal, complemented
3	#19128108	C₂	1	4	yes	maximal, complemented
4	#19127739	C₂	1	1	yes	maximal, characteristic
5	#19127737	C₂	1	2	yes	maximal
6	C₂×#14581702	C₂	1	1	yes	maximal, characteristic
7	#427×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14581791	C₂²	1	4	yes	—
10	#14581783	C₂²	1	4	yes	—
11	#14581769	—	2	4	no	—
12	C₄×Dic₁₆₆	—	2	4	no	—
13	#14581751	C₂²	1	2	yes	—
14	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
15	#14581726	C₂²	1	2	yes	—
16	#14581725	C₂²	1	4	yes	—
17	#14581724	C₂²	1	2	yes	—
18	#14581723	C₂²	1	4	yes	—
19	#14581702	C₂²	1	2	yes	—
20	#14581702	C₂²	1	2	yes	—
21	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#117×C₈₃	C₂²	1	2	yes	—
23	#107×C₈₃	C₂²	1	2	yes	—
24	#106×C₈₃	C₂²	1	2	yes	—
25	C₂×Dic₁₆₆	—	2	4	no	—
26	C₂²×Dic₈₃	D₄	1	2	yes	—
27	C₂×C₈₃⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₈₃	—	2	4	no	—
30	C₄⋊Dic₈₃	C₂³	1	4	yes	—
31	Dic₈₃⋊C₄	—	2	4	no	—
32	Dic₈₃⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₈₃	—	2	2	no	—
34	C₄×Dic₈₃	—	2	2	no	—
35	C₄×Dic₈₃	D₄	1	2	yes	—
36	C₄×Dic₈₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₈₃	Q₈	1	4	yes	complemented
39	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₈₃	—	2	2	no	—
43	C₄²×C₈₃	—	2	2	no	—
44	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	—	2	4	no	—
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	4	no	—
51	C₂×Dic₈₃	—	4	2	no	—
52	C₂×Dic₈₃	C₂×D₄	1	4	yes	—
53	Dic₁₆₆	—	4	8	no	—
54	C₈×C₈₃	C₂×Q₈	1	2	yes	—
55	C₈×C₈₃	C₂×Q₈	1	2	yes	—
56	C₈₃⋊C₈	C₂×Q₈	1	4	yes	—
57	C₂²×C₈₃	—	2	2	no	—
58	C₂²×C₈₃	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₈₃	—	4	4	no	—
61	C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
62	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
64	Dic₈₃	—	2	4	no	—
65	Dic₈₃	—	4	4	no	—
66	C₂×C₈₃	—	2	2	no	—
67	C₂×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₈₃	#575	1	2	yes	—
69	#3162	—	83	1	no	maximal, Sylow
70	C₈₃	#3162	1	1	yes	characteristic, complemented, Sylow
71	#533	—	83	2	no	—
72	#501	—	83	4	no	—
73	#501	—	83	4	no	—
74	#427	—	83	1	no	—
75	#427	—	83	2	no	—
76	#427	D₈₃	1	1	yes	characteristic
77	C₂×#97	—	83	1	no	—
78	C₂²×C₈	—	83	1	no	—
79	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
80	C₄⋊Q₈	—	83	4	no	—
81	#125	—	83	4	no	—
82	C₂²⋊Q₈	—	166	4	no	—
83	C₄×Q₈	—	166	4	no	—
84	#117	—	83	2	no	—
85	#117	C₂×D₈₃	1	2	yes	—
86	C₂×C₄²	—	83	1	no	—
87	#107	—	83	2	no	—
88	#107	—	83	4	no	—
89	#107	C₂×D₈₃	1	2	yes	—
90	#106	—	83	2	no	—
91	#106	—	83	4	no	—
92	#106	C₂×D₈₃	1	2	yes	—
93	#97	—	83	2	no	—
94	#97	—	83	2	no	—
95	C₂×Q₈	—	166	4	no	—
96	C₂²×C₄	—	83	2	no	—
97	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
98	C₂×C₈	—	83	4	no	—
99	C₂×C₈	—	83	2	no	—
100	C₂×C₈	Dic₁₆₆	1	4	yes	complemented
101	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
103	C₄⋊C₄	—	83	4	no	—
104	C₄⋊C₄	—	166	4	no	—
105	C₄⋊C₄	—	166	4	no	—
106	C₄⋊C₄	—	166	4	no	—
107	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	166	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	83	2	no	—
112	C₄²	—	83	2	no	—
113	C₄²	—	166	2	no	—
114	C₂³	D₄×D₈₃	1	1	yes	characteristic
115	Q₈	—	332	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	332	2	no	—
118	C₂×C₄	—	83	4	no	—
119	C₂×C₄	—	166	4	no	—
120	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₈₃	1	2	yes	—
122	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
123	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
124	C₈	—	83	4	no	—
125	C₈	C₂×Dic₁₆₆	1	2	yes	—
126	C₈	C₂×Dic₁₆₆	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#14581791	1	2	yes	—
129	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	332	4	no	—
132	C₄	—	166	4	no	—
133	C₄	#14581791	1	2	yes	—
134	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
135	C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#14581791	1	1	yes	characteristic
138	C₂	#19128498	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128254

C₂

yes

maximal

#19128109

C₂

yes

maximal, complemented

#19128108

C₂

yes

maximal, complemented

#19127739

C₂

yes

maximal, characteristic

#19127737

C₂

yes

maximal

C₂×#14581702

C₂

yes

maximal, characteristic

#427×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581791

C₂²

yes

—

#14581783

C₂²

yes

—

#14581769

—

C₄×Dic₁₆₆

—

#14581751

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581726

C₂²

yes

—

#14581725

C₂²

yes

—

#14581724

C₂²

yes

—

#14581723

C₂²

yes

—

#14581702

C₂²

yes

—

#14581702

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#107×C₈₃

C₂²

yes

—

#106×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

D₄

yes

—

C₂×C₈₃⋊C₈

Q₈

yes

complemented

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

complemented

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

C₂³

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

Dic₁₆₆

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈₃⋊C₈

C₂×Q₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄⋊Q₈

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₄⋊Q₈

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

C₂²×Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₃

#575

yes

—

#3162

—

maximal, Sylow

C₈₃

#3162

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₃

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

#117

C₂×D₈₃

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₈₃

yes

—

#106

—

#106

—

#106

C₂×D₈₃

yes

—

#97

—

#97

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₁₆₆

yes

complemented

101

C₂×C₈

C₂²×D₈₃

yes

characteristic

102

C₂×C₈

C₂²×D₈₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

C₂²×D₈₃

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

166

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

166

—

114

C₂³

D₄×D₈₃

yes

characteristic

115

Q₈

—

332

—

116

C₂×C₄

—

117

C₂×C₄

—

332

—

118

C₂×C₄

—

119

C₂×C₄

—

166

—

120

C₂×C₄

D₄×D₈₃

yes

characteristic

121

C₂×C₄

Q₈×D₈₃

yes

—

122

C₂×C₄

C₂×Dic₁₆₆

yes

—

123

C₂×C₄

C₂³×D₈₃

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₁₆₆

yes

—

126

C₈

C₂×Dic₁₆₆

yes

—

127

C₂²

—

128

C₂²

#14581791

yes

—

129

C₂²

C₂×D₄×D₈₃

yes

characteristic

130

C₄

—

131

C₄

—

332

—

132

C₄

—

166

—

133

C₄

#14581791

yes

—

134

C₄

C₂×Q₈×D₈₃

yes

characteristic

135

C₄

C₂²×Dic₁₆₆

yes

characteristic

136

C₂

—

137

C₂

C₂×#14581791

yes

characteristic

138

C₂

#19128498

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#427×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#427	D₈₃
hypocenter	C₈₃	#3162
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#427×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #19128109	C₂	semidirect
3. #19128108	C₂	semidirect
27. C₂×C₈₃⋊C₈	Q₈	semidirect
38. C₂×C₈×C₈₃	Q₈	semidirect
70. C₈₃	#3162	semidirect
100. C₂×C₈	Dic₁₆₆	semidirect

H ≅ G/N

type

2. #19128109

C₂

semidirect

3. #19128108

C₂

semidirect

27. C₂×C₈₃⋊C₈

Q₈

semidirect

38. C₂×C₈×C₈₃

Q₈

semidirect

70. C₈₃

#3162

semidirect

100. C₂×C₈

Dic₁₆₆

semidirect

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

subgroup	index
#19128254	2
#19128109	2
#19128108	2
#19127739	2
#19127737	2
C₂×#14581702	2
#427×C₈₃	2
#3162	83

subgroup

index

#427×C₈₃

Subgroups — #24917986

Conjugacy classes of subgroups

Complements

Maximal subgroups