Group #35319171 — subgroups

Subgroups — #35319171

Order 21248.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	#24909715	C₂	1	2	yes	maximal
2	#24909692	C₂	1	2	yes	maximal
3	#24907081	C₂	1	1	yes	maximal, characteristic
4	#24905381	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1707	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581738	—	2	4	no	—
7	#19128106	—	2	1	no	—
8	#19128106	—	2	1	no	—
9	#19128081	C₂²	1	2	yes	—
10	#19127504	—	2	4	no	—
11	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
12	#500×C₈₃	C₂²	1	1	yes	characteristic
13	#498×C₈₃	—	2	1	no	—
14	#494×C₈₃	C₂²	1	1	yes	characteristic
15	#487×C₈₃	—	2	1	no	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	#14581792	—	2	2	no	—
20	#14581738	—	4	8	no	—
21	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
22	#14581726	—	4	2	no	—
23	#14581726	—	4	2	no	—
24	C₂×Q₁₆×C₈₃	—	2	2	no	—
25	C₄⋊Q₈×C₈₃	—	2	1	no	—
26	C₄⋊Q₈×C₈₃	—	2	1	no	—
27	#107×C₈₃	—	2	2	no	—
28	#106×C₈₃	—	4	2	no	—
29	#103×C₈₃	—	4	2	no	—
30	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₆₆	—	4	2	no	—
34	Dic₃₃₂	—	4	4	no	—
35	Dic₃₃₂	—	4	4	no	—
36	C₄⋊Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₈₃⋊C₁₆	—	2	4	no	—
39	C₂×Q₈×C₈₃	—	4	1	no	—
40	C₂×Q₈×C₈₃	—	4	1	no	—
41	Q₁₆×C₈₃	—	4	4	no	—
42	C₂×C₈×C₈₃	D₈	1	2	yes	—
43	C₂×C₈×C₈₃	D₈	1	2	yes	—
44	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₈₃	—	4	1	no	—
46	C₄⋊C₄×C₈₃	—	4	1	no	—
47	C₄⋊C₄×C₈₃	—	4	1	no	—
48	C₄⋊C₄×C₈₃	—	4	1	no	—
49	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₈₃	—	8	2	no	—
51	Q₈×C₈₃	—	8	2	no	—
52	C₂×C₄×C₈₃	—	8	1	no	—
53	C₂×C₄×C₈₃	—	8	1	no	—
54	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₈₃	—	8	2	no	—
58	Dic₁₆₆	—	8	2	no	—
59	Dic₁₆₆	—	8	2	no	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	Q₃₂	1	4	yes	complemented
62	C₈×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₈₃	#493	1	1	yes	characteristic
64	C₄×C₈₃	—	16	1	no	—
65	C₄×C₈₃	—	16	1	no	—
66	C₄×C₈₃	#496	1	2	yes	—
67	C₄×C₈₃	C₂×Q₃₂	1	2	yes	—
68	C₄×C₈₃	#510	1	2	yes	—
69	Dic₈₃	—	16	2	no	—
70	#12528	—	83	1	no	maximal, Sylow
71	C₂×C₈₃	#1721	1	1	yes	characteristic
72	C₂×C₈₃	#2247	1	1	yes	characteristic
73	C₂×C₈₃	#2251	1	1	yes	characteristic
74	#2247	—	83	2	no	—
75	#2241	—	83	2	no	—
76	#1713	—	83	1	no	—
77	#1707	D₈₃	1	1	yes	characteristic
78	#1312	—	83	1	no	—
79	C₈₃	#12528	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₃₂	—	166	4	no	—
81	#500	—	166	1	no	—
82	#500	—	166	1	no	—
83	#500	C₂×D₈₃	1	1	yes	characteristic
84	#498	—	2	1	no	—
85	#494	—	83	2	no	—
86	#494	C₂×D₈₃	1	1	yes	characteristic
87	#487	—	2	1	no	—
88	#358	—	166	4	no	—
89	C₄×C₁₆	—	83	2	no	—
90	C₈²	C₂×D₈₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	166	2	no	—
93	C₂×Q₁₆	—	166	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	166	2	no	—
97	Q₃₂	—	332	8	no	—
98	C₂×C₁₆	—	83	4	no	—
99	#107	—	2	2	no	—
100	#107	—	332	2	no	—
101	#107	—	332	2	no	—
102	#106	—	4	2	no	—
103	#103	—	4	2	no	—
104	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	332	2	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	332	4	no	—
112	Q₁₆	—	332	4	no	—
113	C₂×C₈	#13050575	1	2	yes	—
114	C₂×C₈	#13050575	1	2	yes	—
115	C₂×C₈	D₄×D₈₃	1	2	yes	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	332	2	no	—
121	C₄⋊C₄	—	332	2	no	—
122	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
123	C₁₆	—	166	4	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	—	664	2	no	—
127	Q₈	—	664	2	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	664	2	no	—
131	C₂×C₄	#14581788	1	1	yes	characteristic
132	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
133	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	#14581738	1	4	yes	complemented
136	C₈	#14581819	1	4	yes	—
137	C₂²	#19128077	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	1328	2	no	—
141	C₄	#19128087	1	2	yes	—
142	C₄	C₂×#14581738	1	2	yes	—
143	C₄	#19128141	1	2	yes	—
144	C₂	#24907127	1	1	yes	characteristic
145	C₂	#24909715	1	1	yes	characteristic
146	C₂	#24909735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909715

C₂

yes

maximal

#24909692

C₂

yes

maximal

#24907081

C₂

yes

maximal, characteristic

#24905381

C₂

yes

maximal, characteristic

C₈₃×#1707

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#19127504

—

C₄×C₈₃⋊C₁₆

C₂²

yes

—

#500×C₈₃

C₂²

yes

characteristic

#498×C₈₃

—

#494×C₈₃

C₂²

yes

characteristic

#487×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

#14581738

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

#14581726

—

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

#107×C₈₃

—

#106×C₈₃

—

#103×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

Q₃₂

yes

complemented

C₈×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#496

yes

—

C₄×C₈₃

C₂×Q₃₂

yes

—

C₄×C₈₃

#510

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

—

D₈₃

yes

characteristic

#1312

—

C₈₃

#12528

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

166

—

#500

—

166

—

#500

—

166

—

#500

C₂×D₈₃

yes

—

—

C₂×D₈₃

yes

characteristic

#487

—

#358

—

166

—

C₄×C₁₆

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

Q₃₂

—

332

—

C₂×C₁₆

—

#107

—

100

#107

—

332

—

101

#107

—

332

—

102

#106

—

103

#103

—

104

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

105

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

106

C₄×C₈

C₂²×D₈₃

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

332

—

110

Q₁₆

—

111

Q₁₆

—

332

—

112

Q₁₆

—

332

—

113

C₂×C₈

#13050575

yes

—

114

C₂×C₈

#13050575

yes

—

115

C₂×C₈

D₄×D₈₃

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

332

—

121

C₄⋊C₄

—

332

—

122

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

123

C₁₆

—

166

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

664

—

127

Q₈

—

664

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

664

—

131

C₂×C₄

#14581788

yes

characteristic

132

C₂×C₄

C₂×#13050575

yes

characteristic

133

C₂×C₄

C₂×#13050575

yes

characteristic

134

C₈

—

135

C₈

#14581738

yes

complemented

136

C₈

#14581819

yes

—

137

C₂²

#19128077

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1328

—

141

C₄

#19128087

yes

—

142

C₄

C₂×#14581738

yes

—

143

C₄

#19128141

yes

—

144

C₂

#24907127

yes

characteristic

145

C₂

#24909715

yes

characteristic

146

C₂

#24909735

yes

characteristic

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1707	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1707	D₈₃
hypocenter	C₈₃	#12528
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#19128077

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1707

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

N	H ≅ G/N	type
61. C₈×C₈₃	Q₃₂	semidirect
79. C₈₃	#12528	semidirect
135. C₈	#14581738	semidirect

H ≅ G/N

type

61. C₈×C₈₃

Q₃₂

semidirect

79. C₈₃

#12528

semidirect

135. C₈

#14581738

semidirect

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

subgroup	index
#24909715	2
#24909692	2
#24907081	2
#24905381	2
C₈₃×#1707	2
#12528	83

subgroup

index

C₈₃×#1707

Subgroups — #35319171

Conjugacy classes of subgroups

Complements

Maximal subgroups