Group #16830015 — subgroups

Subgroups — #16830015

Order 3712.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#14271716	C₂	1	2	yes	maximal
2	#14271571	C₂	1	4	yes	maximal, complemented
3	#14271570	C₂	1	4	yes	maximal, complemented
4	#14271201	C₂	1	1	yes	maximal, characteristic
5	#14271199	C₂	1	2	yes	maximal
6	C₂×#11745163	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂₉×#117	C₂²	1	2	yes	—
10	#11745288	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
12	C₂²×#6597	C₂²	1	1	yes	characteristic
13	#11745252	C₂²	1	4	yes	—
14	#11745250	C₂²	1	4	yes	—
15	#11745240	—	2	4	no	—
16	C₄×Dic₅₈	—	2	4	no	—
17	C₂₉×#107	C₂²	1	2	yes	—
18	C₂₉×#106	C₂²	1	2	yes	—
19	#11745166	C₂²	1	2	yes	—
20	#11745165	C₂²	1	2	yes	—
21	#11745163	C₂²	1	2	yes	—
22	#11745163	C₂²	1	2	yes	—
23	#11745156	C₂²	1	4	yes	—
24	#11745155	C₂²	1	4	yes	—
25	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₉	D₄	1	2	yes	—
27	C₂×Dic₅₈	—	2	4	no	—
28	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
29	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₂₉	—	2	2	no	—
33	C₄²×C₂₉	—	2	2	no	—
34	C₂²⋊Dic₂₉	—	2	2	no	—
35	C₄⋊Dic₂₉	—	2	4	no	—
36	C₄⋊Dic₂₉	C₂³	1	4	yes	—
37	#90779	—	2	4	no	—
38	#90779	—	2	4	no	—
39	C₄×Dic₂₉	—	2	2	no	—
40	C₄×Dic₂₉	D₄	1	2	yes	—
41	C₄×Dic₂₉	C₂³	1	2	yes	—
42	C₂×#6597	Q₈	1	4	yes	complemented
43	C₂×#6597	C₂³	1	2	yes	—
44	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	—	2	4	no	—
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₉	—	2	4	no	—
51	C₂×Dic₂₉	—	4	2	no	—
52	C₂×Dic₂₉	C₂×D₄	1	4	yes	—
53	Dic₅₈	—	4	8	no	—
54	C₈×C₂₉	C₂×Q₈	1	2	yes	—
55	C₈×C₂₉	C₂×Q₈	1	2	yes	—
56	#6597	C₂×Q₈	1	4	yes	—
57	#3162	—	29	1	no	maximal, Sylow
58	C₂²×C₂₉	—	2	2	no	—
59	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₂₉	—	4	4	no	—
62	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
63	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
65	Dic₂₉	—	2	4	no	—
66	Dic₂₉	—	4	4	no	—
67	#533	—	29	2	no	—
68	#501	—	29	4	no	—
69	#501	—	29	4	no	—
70	#427	—	29	1	no	—
71	#427	—	29	2	no	—
72	#427	D₂₉	1	1	yes	characteristic
73	C₂×#97	—	29	1	no	—
74	C₂×C₂₉	—	2	2	no	—
75	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₂₉	#575	1	2	yes	—
77	C₂²×C₈	—	29	1	no	—
78	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
79	C₄⋊Q₈	—	29	4	no	—
80	#125	—	29	4	no	—
81	C₂²⋊Q₈	—	58	4	no	—
82	C₄×Q₈	—	58	4	no	—
83	#117	—	29	2	no	—
84	#117	C₂×D₂₉	1	2	yes	—
85	C₂×C₄²	—	29	1	no	—
86	#107	—	29	2	no	—
87	#107	—	29	4	no	—
88	#107	C₂×D₂₉	1	2	yes	—
89	#106	—	29	2	no	—
90	#106	—	29	4	no	—
91	#106	C₂×D₂₉	1	2	yes	—
92	#97	—	29	2	no	—
93	#97	—	29	2	no	—
94	C₂₉	#3162	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	58	4	no	—
96	C₂²×C₄	—	29	2	no	—
97	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
98	C₂×C₈	—	29	4	no	—
99	C₂×C₈	—	29	2	no	—
100	C₂×C₈	Dic₅₈	1	4	yes	complemented
101	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
103	C₄⋊C₄	—	29	4	no	—
104	C₄⋊C₄	—	58	4	no	—
105	C₄⋊C₄	—	58	4	no	—
106	C₄⋊C₄	—	58	4	no	—
107	C₄⋊C₄	C₂²×D₂₉	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	58	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	29	2	no	—
112	C₄²	—	29	2	no	—
113	C₄²	—	58	2	no	—
114	C₂³	D₄×D₂₉	1	1	yes	characteristic
115	Q₈	—	116	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	29	4	no	—
118	C₂×C₄	—	58	4	no	—
119	C₂×C₄	—	116	2	no	—
120	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
121	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₂₉	1	2	yes	—
123	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
124	C₈	—	29	4	no	—
125	C₈	C₂×Dic₅₈	1	2	yes	—
126	C₈	C₂×Dic₅₈	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#11745250	1	2	yes	—
129	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	58	4	no	—
132	C₄	—	116	4	no	—
133	C₄	#11745250	1	2	yes	—
134	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
135	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#11745250	1	1	yes	characteristic
138	C₂	#14271960	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271716

C₂

yes

maximal

#14271571

C₂

yes

maximal, complemented

#14271570

C₂

yes

maximal, complemented

#14271201

C₂

yes

maximal, characteristic

#14271199

C₂

yes

maximal

C₂×#11745163

C₂

yes

maximal, characteristic

C₂₉×#427

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂₉×#117

C₂²

yes

—

#11745288

C₂²

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

characteristic

C₂²×#6597

C₂²

yes

characteristic

#11745252

C₂²

yes

—

#11745250

C₂²

yes

—

#11745240

—

C₄×Dic₅₈

—

C₂₉×#107

C₂²

yes

—

C₂₉×#106

C₂²

yes

—

#11745166

C₂²

yes

—

#11745165

C₂²

yes

—

#11745163

C₂²

yes

—

#11745163

C₂²

yes

—

#11745156

C₂²

yes

—

#11745155

C₂²

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

D₄

yes

—

C₂×Dic₅₈

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

C₂³

yes

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

D₄

yes

—

C₄×Dic₂₉

C₂³

yes

—

C₂×#6597

Q₈

yes

complemented

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂×D₄

yes

—

Dic₅₈

—

C₈×C₂₉

C₂×Q₈

yes

—

C₈×C₂₉

C₂×Q₈

yes

—

#6597

C₂×Q₈

yes

—

#3162

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄⋊Q₈

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

C₄⋊Q₈

yes

—

C₄×C₂₉

C₂²×Q₈

yes

characteristic

C₄×C₂₉

C₂²×Q₈

yes

characteristic

Dic₂₉

—

Dic₂₉

—

—

—

—

—

—

D₂₉

yes

characteristic

C₂×#97

—

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉

#575

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₉

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₂₉

yes

—

#106

—

#106

—

#106

C₂×D₂₉

yes

—

#97

—

#97

—

C₂₉

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₅₈

yes

complemented

101

C₂×C₈

C₂²×D₂₉

yes

characteristic

102

C₂×C₈

C₂²×D₂₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×D₂₉

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₂₉

yes

characteristic

115

Q₈

—

116

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

116

—

120

C₂×C₄

C₂×Dic₅₈

yes

—

121

C₂×C₄

D₄×D₂₉

yes

characteristic

122

C₂×C₄

Q₈×D₂₉

yes

—

123

C₂×C₄

C₂³×D₂₉

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₅₈

yes

—

126

C₈

C₂×Dic₅₈

yes

—

127

C₂²

—

128

C₂²

#11745250

yes

—

129

C₂²

C₂×D₄×D₂₉

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

116

—

133

C₄

#11745250

yes

—

134

C₄

C₂²×Dic₅₈

yes

characteristic

135

C₄

C₂×Q₈×D₂₉

yes

characteristic

136

C₂

—

137

C₂

C₂×#11745250

yes

characteristic

138

C₂

#14271960

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#427	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	#427	D₂₉
hypocenter	C₂₉	#3162
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₉

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂₉×#427

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂³×C₂₉

C₂×D₄

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #14271571	C₂	semidirect
3. #14271570	C₂	semidirect
28. C₂×C₈×C₂₉	Q₈	semidirect
42. C₂×#6597	Q₈	semidirect
94. C₂₉	#3162	semidirect
100. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

2. #14271571

C₂

semidirect

3. #14271570

C₂

semidirect

28. C₂×C₈×C₂₉

Q₈

semidirect

42. C₂×#6597

Q₈

semidirect

94. C₂₉

#3162

semidirect

100. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
#14271716	2
#14271571	2
#14271570	2
#14271201	2
#14271199	2
C₂×#11745163	2
C₂₉×#427	2
#3162	29

subgroup

index

C₂₉×#427

Subgroups — #16830015

Conjugacy classes of subgroups

Complements

Maximal subgroups