Group #16828426 — subgroups

Subgroups — #16828426

Order 3712.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#14271720	C₂	1	1	yes	maximal, characteristic
2	#14271293	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₅₈	C₂	1	4	yes	maximal, complemented
4	#14271189	C₂	1	2	yes	maximal
5	#14271187	C₂	1	1	yes	maximal, characteristic
6	#14271118	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745288	C₂²	1	2	yes	—
12	#11745286	C₄	1	4	yes	—
13	C₂×#90777	C₂²	1	2	yes	—
14	#11745232	C₄	1	2	yes	—
15	#11745231	C₄	1	2	yes	—
16	C₄×Dic₅₈	C₂²	1	4	yes	—
17	C₄×C₈×C₂₉	C₂²	1	2	yes	—
18	C₄×C₈×C₂₉	C₂²	1	2	yes	—
19	#11745163	C₂²	1	4	yes	—
20	#11745162	C₂²	1	8	yes	—
21	C₈×Dic₂₉	C₂²	1	4	yes	—
22	C₄⋊#6597	C₂²	1	4	yes	—
23	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₂₉	—	2	2	no	—
27	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
28	C₂×C₈×C₂₉	C₂³	1	2	yes	—
29	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
31	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₂₉	C₂³	1	2	yes	—
34	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
35	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
37	#90779	C₂×C₄	1	8	yes	—
38	C₄×Dic₂₉	C₂³	1	4	yes	—
39	#90777	—	2	4	no	—
40	C₂×#6597	C₂³	1	4	yes	—
41	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	2	4	no	—
43	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₉	#41	1	2	yes	—
48	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₂₉	C₂²×C₄	1	8	yes	—
50	Dic₅₈	—	2	8	no	—
51	C₈×C₂₉	—	2	2	no	—
52	C₈×C₂₉	C₂×Q₈	1	2	yes	—
53	C₈×C₂₉	C₂×Q₈	1	2	yes	—
54	#6597	—	2	4	no	—
55	#2960	—	29	1	no	maximal, Sylow
56	C₂²×C₂₉	—	2	2	no	—
57	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
58	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₂₉	—	2	2	no	—
60	C₄×C₂₉	C₄×Q₈	1	2	yes	—
61	C₄×C₂₉	#131	1	4	yes	—
62	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
64	Dic₂₉	—	2	8	no	—
65	#533	—	29	1	no	—
66	#446	—	29	4	no	—
67	C₈×Q₈	—	29	4	no	—
68	#424	—	29	2	no	—
69	#423	—	29	1	no	—
70	#405	—	29	2	no	—
71	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
72	C₂×C₂₉	—	2	2	no	—
73	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
74	C₂×C₂₉	C₂×#131	1	2	yes	—
75	C₂×M₁₆	—	29	2	no	—
76	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
78	C₄⋊Q₈	—	29	2	no	—
79	#125	—	29	2	no	—
80	C₂²⋊Q₈	—	29	4	no	—
81	C₄×Q₈	—	29	4	no	—
82	#117	—	29	2	no	—
83	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
84	C₄⋊C₈	—	29	8	no	—
85	C₄⋊C₈	—	29	4	no	—
86	#97	—	29	4	no	—
87	C₄×C₈	—	29	4	no	—
88	C₄×C₈	C₂×D₂₉	1	2	yes	—
89	C₄×C₈	C₂×D₂₉	1	2	yes	—
90	C₂₉	#2960	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	29	4	no	—
92	C₂²×C₄	C₄×D₂₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
94	M₁₆	—	58	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	29	4	no	—
97	C₂×C₈	Dic₅₈	1	4	yes	complemented
98	C₂×C₈	C₂²×D₂₉	1	2	yes	—
99	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
101	C₄⋊C₄	—	29	4	no	—
102	C₄⋊C₄	—	29	8	no	—
103	C₄⋊C₄	—	29	4	no	—
104	C₂²⋊C₄	—	29	4	no	—
105	C₄²	—	29	4	no	—
106	C₄²	C₄×D₂₉	1	1	yes	characteristic
107	C₄²	C₄×D₂₉	1	1	yes	characteristic
108	C₄²	C₂²×D₂₉	1	2	yes	—
109	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
110	Q₈	—	58	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	29	8	no	—
113	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
117	C₂×C₄	#90805	1	2	yes	—
118	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	58	4	no	—
121	C₈	C₂×Dic₅₈	1	2	yes	—
122	C₈	C₂×Dic₅₈	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₅₈	1	2	yes	—
125	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	58	8	no	—
128	C₄	C₄×Dic₅₈	1	2	yes	—
129	C₄	#11745263	1	4	yes	—
130	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
131	C₄	C₂×#90805	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₅₈	1	1	yes	characteristic
134	C₂	C₂×#11745263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic

#14271293

C₂

yes

maximal, complemented

C₈×Dic₅₈

C₂

yes

maximal, complemented

#14271189

C₂

yes

maximal

#14271187

C₂

yes

maximal, characteristic

#14271118

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

#11745286

C₄

yes

—

C₂×#90777

C₂²

yes

—

#11745232

C₄

yes

—

#11745231

C₄

yes

—

C₄×Dic₅₈

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

—

#11745163

C₂²

yes

—

#11745162

C₂²

yes

—

C₈×Dic₂₉

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

—

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

C₂²⋊Dic₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

C₂×C₄

yes

—

C₄×Dic₂₉

C₂³

yes

—

#90777

—

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

C₂²×C₄

yes

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

C₂×Q₈

yes

—

C₈×C₂₉

C₂×Q₈

yes

—

#6597

—

#2960

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄×Q₈

yes

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

#131

yes

—

C₄×C₂₉

C₂²×Q₈

yes

characteristic

C₄×C₂₉

C₂×#41

yes

characteristic

Dic₂₉

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₂₉

yes

characteristic

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₉

C₂×#131

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₂₉

yes

—

C₄×C₈

C₂×D₂₉

yes

—

C₂₉

#2960

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

C₄×D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₈

yes

complemented

C₂×C₈

C₂²×D₂₉

yes

—

C₂×C₈

C₂²×D₂₉

yes

characteristic

100

C₂×C₈

C₂²×D₂₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₂₉

yes

characteristic

107

C₄²

C₄×D₂₉

yes

characteristic

108

C₄²

C₂²×D₂₉

yes

—

109

C₂³

C₂×C₄×D₂₉

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₅₈

yes

—

114

C₂×C₄

C₂×C₄×D₂₉

yes

—

115

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

116

C₂×C₄

C₂×C₄×D₂₉

yes

—

117

C₂×C₄

#90805

yes

—

118

C₂×C₄

C₂³×D₂₉

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

C₂×Dic₅₈

yes

—

122

C₈

C₂×Dic₅₈

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₅₈

yes

—

125

C₂²

C₂²×C₄×D₂₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₄×Dic₅₈

yes

—

129

C₄

#11745263

yes

—

130

C₄

C₂²×Dic₅₈

yes

characteristic

131

C₄

C₂×#90805

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₅₈

yes

characteristic

134

C₂

C₂×#11745263

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₄×C₈×C₂₉	C₂
prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₂₉
hypocenter	C₂₉	#2960
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₂₉

commutator, socle

C₂²×C₂₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂×C₄×C₈×C₂₉

C₂

prime radical

C₂³×C₂₉

C₂²×C₄

cosocle

C₂×C₄×C₂₉

C₂⁴

hypercenter

C₂×C₄×C₈

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14271293	C₂	semidirect
3. C₈×Dic₅₈	C₂	semidirect
27. C₂×C₈×C₂₉	Q₈	semidirect
90. C₂₉	#2960	semidirect
97. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

2. #14271293

C₂

semidirect

3. C₈×Dic₅₈

C₂

semidirect

27. C₂×C₈×C₂₉

Q₈

semidirect

90. C₂₉

#2960

semidirect

97. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
#14271720	2
#14271293	2
C₈×Dic₅₈	2
#14271189	2
#14271187	2
#14271118	2
C₂×C₄×C₈×C₂₉	2
#2960	29

subgroup

index

C₈×Dic₅₈

C₂×C₄×C₈×C₂₉

Subgroups — #16828426

Conjugacy classes of subgroups

Complements

Maximal subgroups