Group #24905936 — subgroups

Subgroups — #24905936

Order 10624.

202 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 81 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	#19127758	C₂	1	1	yes	maximal, characteristic
2	#19127754	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₈₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#533×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
9	#14581703	—	2	1	no	—
10	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₈₃	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₈₃	C₄	1	1	yes	characteristic
13	#125×C₈₃	—	2	1	no	—
14	C₂²⋊Q₈×C₈₃	—	2	1	no	—
15	C₂²⋊Q₈×C₈₃	C₄	1	2	yes	—
16	C₄×Q₈×C₈₃	—	2	1	no	—
17	C₄×Q₈×C₈₃	C₂²	1	2	yes	—
18	#117×C₈₃	—	2	1	no	—
19	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×Q₈×C₈₃	—	2	1	no	—
23	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	2	1	no	—
28	C₄⋊C₄×C₈₃	—	2	1	no	—
29	C₄⋊C₄×C₈₃	—	2	1	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₄⋊C₄×C₈₃	—	2	2	no	—
32	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
33	C₂²⋊C₄×C₈₃	—	2	2	no	—
34	C₄²×C₈₃	—	2	2	no	—
35	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₈₃	—	2	4	no	—
41	Q₈×C₈₃	—	4	2	no	—
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	2	2	no	—
45	C₂×C₄×C₈₃	—	2	2	no	—
46	C₂×C₄×C₈₃	—	4	1	no	—
47	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₈₃⋊C₈	—	4	2	no	—
54	C₈₃⋊C₈	—	4	2	no	—
55	C₂²×C₈₃	—	2	1	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₈₃	—	2	1	no	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	2	1	no	—
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	—	4	2	no	—
64	C₄×C₈₃	—	4	2	no	—
65	C₄×C₈₃	#131	1	2	yes	—
66	C₄×C₈₃	#137	1	2	yes	—
67	C₂×C₈₃	—	4	1	no	—
68	C₂×C₈₃	#408	1	1	yes	characteristic
69	C₂×C₈₃	#419	1	1	yes	characteristic
70	C₂×C₈₃	#421	1	1	yes	characteristic
71	#1497	—	83	1	no	maximal, Sylow
72	C₈₃	#1497	1	1	yes	characteristic, complemented, Sylow
73	#533	D₈₃	1	1	yes	characteristic
74	#432	—	83	1	no	—
75	#431	—	83	1	no	—
76	Q₈⋊C₈	—	83	4	no	—
77	C₄⋊Q₈	Dic₈₃	1	1	yes	characteristic
78	C₄⋊Q₈	Dic₈₃	1	1	yes	characteristic
79	#125	—	2	1	no	—
80	C₂²⋊Q₈	—	2	1	no	—
81	C₂²⋊Q₈	Dic₈₃	1	2	yes	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	C₂×D₈₃	1	2	yes	—
84	#117	—	2	1	no	—
85	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
86	C₄⋊C₈	—	83	2	no	—
87	C₄⋊C₈	—	166	1	no	—
88	C₂²⋊C₈	—	166	1	no	—
89	C₂²⋊C₈	—	166	1	no	—
90	#97	—	166	1	no	—
91	C₄×C₈	—	83	2	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	C₂×Dic₈₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
97	C₂×C₈	—	166	2	no	—
98	C₂×C₈	—	166	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
105	C₂²⋊C₄	—	2	2	no	—
106	C₄²	—	2	2	no	—
107	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
108	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
110	C₄²	C₂²×D₈₃	1	1	yes	characteristic
111	C₂³	C₂²⋊Dic₈₃	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
125	C₈	—	332	2	no	—
126	C₈	—	332	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	#14581799	1	2	yes	—
138	C₄	#14581823	1	2	yes	—
139	C₂	—	4	1	no	—
140	C₂	#19127668	1	1	yes	characteristic
141	C₂	#19127710	1	1	yes	characteristic
142	C₂	#19127718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127758

C₂

yes

maximal, characteristic

#19127754

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, complemented

#533×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₄

yes

characteristic

C₄⋊Q₈×C₈₃

C₄

yes

characteristic

#125×C₈₃

—

C₂²⋊Q₈×C₈₃

—

C₂²⋊Q₈×C₈₃

C₄

yes

—

C₄×Q₈×C₈₃

—

C₄×Q₈×C₈₃

C₂²

yes

—

#117×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#131

yes

—

C₄×C₈₃

#137

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#533

D₈₃

yes

characteristic

#432

—

#431

—

Q₈⋊C₈

—

C₄⋊Q₈

Dic₈₃

yes

characteristic

C₄⋊Q₈

Dic₈₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₈₃

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₃

yes

—

#117

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

#97

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂×Dic₈₃

yes

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

C₂×Dic₈₃

yes

characteristic

108

C₄²

C₂²⋊D₈₃

yes

characteristic

109

C₄²

C₂²⋊D₈₃

yes

characteristic

110

C₄²

C₂²×D₈₃

yes

characteristic

111

C₂³

C₂²⋊Dic₈₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

122

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

125

C₈

—

332

—

126

C₈

—

332

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#14581799

yes

—

138

C₄

#14581823

yes

—

139

C₂

—

140

C₂

#19127668

yes

characteristic

141

C₂

#19127710

yes

characteristic

142

C₂

#19127718

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#533×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#533	D₈₃
hypocenter	C₈₃	#1497
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#533×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊(C₈₃⋊C₈)	C₂	semidirect
72. C₈₃	#1497	semidirect

H ≅ G/N

type

3. Q₈⋊(C₈₃⋊C₈)

C₂

semidirect

72. C₈₃

#1497

semidirect

Subgroups — #24905936

Conjugacy classes of subgroups

Complements