Group #16832166 — subgroups

Subgroups — #16832166

Order 3712.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₅₈	C₂	1	1	yes	maximal, characteristic
2	#14271513	C₂	1	3	yes	maximal
3	#14271458	C₂	1	3	yes	maximal
4	#14271370	C₂	1	3	yes	maximal
5	Q₈×#6597	C₂	1	1	yes	maximal, characteristic
6	C₄×#90785	C₂	1	3	yes	maximal
7	C₂₉×#557	C₂	1	1	yes	maximal, characteristic
8	C₂₉×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₂₉×#125	—	2	3	no	—
10	C₄×Q₈×C₂₉	—	2	1	no	—
11	C₄×Q₈×C₂₉	C₂²	1	3	yes	—
12	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
13	Q₈×Dic₂₉	—	2	1	no	—
14	C₂×#90785	C₂²	1	3	yes	—
15	#11745250	—	2	3	no	—
16	#11745231	C₂²	1	3	yes	—
17	C₄×Dic₅₈	C₂²	1	3	yes	—
18	#11745183	C₂²	1	3	yes	—
19	#11745183	C₂²	1	6	yes	—
20	#11745158	C₂²	1	1	yes	characteristic
21	#11745155	C₂²	1	3	yes	—
22	C₄⋊#6597	C₂²	1	3	yes	—
23	C₄×#6597	C₂²	1	3	yes	—
24	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₂₉	C₂³	1	3	yes	—
26	C₂×Dic₅₈	—	2	3	no	—
27	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₉	—	2	3	no	—
29	C₄⋊C₄×C₂₉	—	2	3	no	—
30	C₄⋊C₄×C₂₉	—	2	3	no	—
31	C₄⋊C₄×C₂₉	D₄	1	3	yes	—
32	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
33	C₄²×C₂₉	—	2	3	no	—
34	C₄²×C₂₉	C₂³	1	3	yes	—
35	#90785	—	2	6	no	—
36	C₄⋊Dic₂₉	—	2	3	no	—
37	C₄⋊Dic₂₉	C₂³	1	3	yes	—
38	#90779	—	2	3	no	—
39	C₄×Dic₂₉	—	2	3	no	—
40	C₂×#6597	C₂³	1	3	yes	—
41	C₂×#6597	C₂³	1	1	yes	characteristic
42	Q₈×C₂₉	—	2	3	no	—
43	Q₈×C₂₉	—	2	3	no	—
44	Q₈×C₂₉	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₂₉	—	2	1	no	—
46	C₂×C₄×C₂₉	—	2	3	no	—
47	C₂×C₄×C₂₉	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₂₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₉	—	2	1	no	—
51	C₂×Dic₂₉	—	2	3	no	—
52	Dic₅₈	—	2	1	no	—
53	Dic₅₈	—	8	3	no	—
54	Dic₅₈	#41	1	2	yes	—
55	#6597	—	2	3	no	—
56	#6597	—	2	1	no	—
57	#3363	—	29	1	no	maximal, Sylow
58	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₉	—	2	3	no	—
60	C₄×C₂₉	—	4	3	no	—
61	C₄×C₂₉	—	4	1	no	—
62	C₄×C₂₉	C₂×Q₁₆	1	6	yes	—
63	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
64	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
65	Dic₂₉	—	2	2	no	—
66	Dic₂₉	—	4	3	no	—
67	Q₈²	—	29	1	no	—
68	#557	D₂₉	1	1	yes	characteristic
69	#487	—	29	3	no	—
70	#477	—	29	3	no	—
71	#462	—	29	3	no	—
72	C₈×Q₈	—	29	1	no	—
73	C₄×Q₁₆	—	29	3	no	—
74	C₂×C₂₉	#548	1	1	yes	characteristic
75	C₂×C₂₉	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₂₉	#577	1	1	yes	characteristic
77	C₂×Q₁₆	—	29	3	no	—
78	C₄⋊Q₈	—	29	3	no	—
79	C₄⋊Q₈	—	58	3	no	—
80	C₄⋊Q₈	C₂×D₂₉	1	3	yes	—
81	#125	—	2	3	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	29	3	no	—
84	C₄×Q₈	—	58	1	no	—
85	C₄×Q₈	C₂×D₂₉	1	3	yes	—
86	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
87	#107	—	29	3	no	—
88	C₄⋊C₈	—	29	3	no	—
89	#103	—	29	3	no	—
90	#103	—	29	6	no	—
91	#103	—	29	1	no	—
92	C₄×C₈	—	29	3	no	—
93	C₂₉	#3363	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	29	1	no	—
95	C₂×Q₈	—	58	3	no	—
96	C₂×Q₈	C₂²⋊D₂₉	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₂₉	1	3	yes	—
98	Q₁₆	—	58	6	no	—
99	C₂×C₈	—	29	1	no	—
100	C₂×C₈	—	29	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	29	3	no	—
105	C₄⋊C₄	—	58	3	no	—
106	C₄⋊C₄	—	58	3	no	—
107	C₄⋊C₄	C₂²⋊D₂₉	1	3	yes	—
108	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	58	3	no	—
111	C₄²	C₂²×D₂₉	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	29	2	no	—
115	Q₈	—	58	1	no	—
116	Q₈	—	232	3	no	—
117	Q₈	#90785	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	58	1	no	—
121	C₂×C₄	—	58	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₂₉	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₉	1	3	yes	—
124	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
125	C₈	—	58	3	no	—
126	C₈	—	58	1	no	—
127	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	58	2	no	—
132	C₄	—	116	3	no	—
133	C₄	C₂×#90785	1	6	yes	—
134	C₄	#41×D₂₉	1	1	yes	characteristic
135	C₄	#11745365	1	1	yes	characteristic
136	C₂	#14271820	1	1	yes	characteristic
137	C₂	C₂²×#90785	1	1	yes	characteristic
138	C₂	#14271971	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₅₈

C₂

yes

maximal, characteristic

#14271513

C₂

yes

maximal

#14271458

C₂

yes

maximal

#14271370

C₂

yes

maximal

Q₈×#6597

C₂

yes

maximal, characteristic

C₄×#90785

C₂

yes

maximal

C₂₉×#557

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₂²

yes

—

C₂₉×#125

—

C₄×Q₈×C₂₉

—

C₄×Q₈×C₂₉

C₂²

yes

—

C₄×Q₈×C₂₉

C₂²

yes

characteristic

Q₈×Dic₂₉

—

C₂×#90785

C₂²

yes

—

#11745250

—

#11745231

C₂²

yes

—

C₄×Dic₅₈

C₂²

yes

—

#11745183

C₂²

yes

—

#11745183

C₂²

yes

—

#11745158

C₂²

yes

characteristic

#11745155

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₄×#6597

C₂²

yes

—

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×Q₈×C₂₉

C₂³

yes

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₄⋊C₄×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

—

C₄²×C₂₉

C₂³

yes

—

#90785

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

C₄×Dic₂₉

—

C₂×#6597

C₂³

yes

—

C₂×#6597

C₂³

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

Q₁₆

yes

complemented

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

#41

yes

—

—

—

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×Q₁₆

yes

—

C₄×C₂₉

C₂×#41

yes

characteristic

C₄×C₂₉

2₋¹⁺⁴

yes

characteristic

Dic₂₉

—

Dic₂₉

—

Q₈²

—

#557

D₂₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₂₉

#548

yes

characteristic

C₂×C₂₉

C₂²×Q₁₆

yes

characteristic

C₂×C₂₉

#577

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₉

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₉

yes

—

C₄×Q₈

C₂×D₂₉

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂₉

#3363

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₉

yes

characteristic

C₂×Q₈

C₂²×D₂₉

yes

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²⋊D₂₉

yes

—

108

C₄⋊C₄

C₂²×D₂₉

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₂₉

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

232

—

117

Q₈

#90785

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

124

C₂×C₄

C₂³×D₂₉

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₂₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

116

—

133

C₄

C₂×#90785

yes

—

134

C₄

#41×D₂₉

yes

characteristic

135

C₄

#11745365

yes

characteristic

136

C₂

#14271820

yes

characteristic

137

C₂

C₂²×#90785

yes

characteristic

138

C₂

#14271971

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#557	C₂
socle, prime radical	C₂²×C₂₉	C₂²×D₄
hypercenter	#557	D₂₉
hypocenter	C₂₉	#3363
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₉

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂₉×#557

C₂

socle, prime radical

C₂²×C₂₉

C₂²×D₄

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₂₉	Q₁₆	semidirect
93. C₂₉	#3363	semidirect
117. Q₈	#90785	semidirect

H ≅ G/N

type

44. Q₈×C₂₉

Q₁₆

semidirect

93. C₂₉

#3363

semidirect

117. Q₈

#90785

semidirect

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

subgroup	index
Q₈×Dic₅₈	2
#14271513	2
#14271458	2
#14271370	2
Q₈×#6597	2
C₄×#90785	2
C₂₉×#557	2
#3363	29

subgroup

index

Q₈×Dic₅₈

C₂₉×#557

Subgroups — #16832166

Conjugacy classes of subgroups

Complements

Maximal subgroups