Group #24917845 — subgroups

Subgroups — #24917845

Order 10624.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	4	yes	maximal, complemented
3	#19128053	C₂	1	4	yes	maximal, complemented
4	#19127709	C₂	1	4	yes	maximal
5	C₄×#13050574	C₂	1	1	yes	maximal, characteristic
6	#533×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
8	C₂×#13050574	C₂²	1	2	yes	—
9	#14581792	—	2	2	no	—
10	#14581792	C₂²	1	2	yes	—
11	#14581792	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₃	—	2	2	no	—
13	#14581715	C₂²	1	16	yes	—
14	#14581703	C₂²	1	2	yes	—
15	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
17	#125×C₈₃	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₈₃	—	2	2	no	—
19	C₄×Q₈×C₈₃	—	2	2	no	—
20	#117×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₆₆	—	2	8	no	—
23	C₂×Dic₁₆₆	—	2	2	no	—
24	C₂×Dic₁₆₆	C₂³	1	4	yes	—
25	C₂²×Dic₈₃	—	2	2	no	—
26	#13050574	—	2	4	no	—
27	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₈₃	—	2	4	no	—
29	C₄⋊Dic₈₃	—	2	4	no	—
30	C₂×Q₈×C₈₃	—	2	2	no	—
31	C₂²×C₄×C₈₃	D₄	1	2	yes	—
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	—	2	4	no	—
34	C₄⋊C₄×C₈₃	—	2	2	no	—
35	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₈₃	—	2	2	no	—
37	C₄²×C₈₃	—	2	2	no	—
38	C₄²×C₈₃	D₄	1	2	yes	—
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₈₃	—	4	4	no	—
43	C₂×C₄×C₈₃	—	2	4	no	—
44	C₂×C₄×C₈₃	—	2	4	no	—
45	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₈₃	#41	1	4	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	4	no	—
51	C₂×Dic₈₃	—	4	4	no	—
52	Dic₁₆₆	—	2	8	no	—
53	Dic₁₆₆	—	4	4	no	—
54	C₈₃⋊C₈	—	2	4	no	—
55	C₂²×C₈₃	—	2	2	no	—
56	C₂²×C₈₃	#124	1	2	yes	—
57	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₈₃	—	2	2	no	—
59	C₄×C₈₃	—	4	4	no	—
60	C₄×C₈₃	#124	1	2	yes	—
61	C₄×C₈₃	#137	1	4	yes	—
62	C₄×C₈₃	C₂×#41	1	2	yes	—
63	Dic₈₃	—	4	4	no	—
64	C₂×C₈₃	—	2	2	no	—
65	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
66	C₂×C₈₃	C₂×#137	1	2	yes	—
67	#3139	—	83	1	no	maximal, Sylow
68	C₈₃	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	D₈₃	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	83	1	no	—
71	#491	—	83	4	no	—
72	#487	—	83	4	no	—
73	#419	—	83	4	no	—
74	C₄×M₁₆	—	83	1	no	—
75	C₂²×Q₈	—	83	2	no	—
76	C₂×M₁₆	—	83	2	no	—
77	C₄⋊Q₈	—	83	2	no	—
78	C₄⋊Q₈	—	83	2	no	—
79	C₄⋊Q₈	—	166	2	no	—
80	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
81	#125	C₂×D₈₃	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×D₈₃	1	2	yes	—
85	C₂×C₄⋊C₄	—	166	2	no	—
86	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
87	#103	—	83	16	no	—
88	#97	—	83	2	no	—
89	C₄×C₈	—	83	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	83	4	no	—
92	C₂×Q₈	—	166	8	no	—
93	C₂×Q₈	—	166	2	no	—
94	C₂²×C₄	—	166	2	no	—
95	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
97	M₁₆	—	166	4	no	—
98	C₂×C₈	—	83	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	166	4	no	—
102	C₄⋊C₄	—	166	4	no	—
103	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂²⋊D₈₃	1	2	yes	—
107	C₄²	C₂²×D₈₃	1	1	yes	characteristic
108	C₄²	C₂²×D₈₃	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	332	4	no	—
112	Q₈	—	166	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	332	4	no	—
116	C₂×C₄	—	166	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
120	C₂×C₄	#13050591	1	4	yes	—
121	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
122	C₈	—	166	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14581781	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	332	4	no	—
129	C₄	#14581781	1	2	yes	—
130	C₄	#14581822	1	4	yes	—
131	C₄	C₂×#13050591	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#14581781	1	1	yes	characteristic
134	C₂	C₂×#14581822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

#19128071

C₂

yes

maximal, complemented

#19128053

C₂

yes

maximal, complemented

#19127709

C₂

yes

maximal

C₄×#13050574

C₂

yes

maximal, characteristic

#533×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂×#13050574

C₂²

yes

—

#14581792

—

#14581792

C₂²

yes

—

#14581792

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

#14581715

C₂²

yes

—

#14581703

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

#125×C₈₃

C₂²

yes

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#124

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

C₄×C₈₃

#137

yes

—

C₄×C₈₃

C₂×#41

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×#124

yes

characteristic

C₂×C₈₃

C₂×#137

yes

—

#3139

—

maximal, Sylow

C₈₃

#3139

yes

characteristic, complemented, Sylow

#533

D₈₃

yes

characteristic

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

C₂×D₈₃

yes

—

#125

C₂×D₈₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

166

—

102

C₄⋊C₄

—

166

—

103

C₄⋊C₄

C₂²×D₈₃

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₈₃

yes

—

107

C₄²

C₂²×D₈₃

yes

characteristic

108

C₄²

C₂²×D₈₃

yes

characteristic

109

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

110

Q₈

—

111

Q₈

—

332

—

112

Q₈

—

166

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

332

—

116

C₂×C₄

—

166

—

117

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

118

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

120

C₂×C₄

#13050591

yes

—

121

C₂×C₄

C₂³×D₈₃

yes

characteristic

122

C₈

—

166

—

123

C₂²

—

124

C₂²

#14581781

yes

—

125

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

332

—

129

C₄

#14581781

yes

—

130

C₄

#14581822

yes

—

131

C₄

C₂×#13050591

yes

—

132

C₂

—

133

C₂

C₂×#14581781

yes

characteristic

134

C₂

C₂×#14581822

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#533×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#533	D₈₃
hypocenter	C₈₃	#3139
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#533×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19128071	C₂	semidirect
3. #19128053	C₂	semidirect
68. C₈₃	#3139	semidirect

H ≅ G/N

type

2. #19128071

C₂

semidirect

3. #19128053

C₂

semidirect

68. C₈₃

#3139

semidirect

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

subgroup	index
C₂×#14581792	2
#19128071	2
#19128053	2
#19127709	2
C₄×#13050574	2
#533×C₈₃	2
#3139	83

subgroup

index

#533×C₈₃

Subgroups — #24917845

Conjugacy classes of subgroups

Complements

Maximal subgroups