Group #16830002 — subgroups

Subgroups — #16830002

Order 3712.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#14271566	C₂	1	2	yes	maximal, complemented
3	#14271562	C₂	1	4	yes	maximal, complemented
4	#14271559	C₂	1	2	yes	maximal, complemented
5	#14271199	C₂	1	4	yes	maximal
6	C₂×C₄×#6597	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#533	C₂	1	1	yes	maximal, characteristic
8	C₂₉×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂₉×#125	C₂²	1	2	yes	—
10	C₂₉×C₂²⋊Q₈	—	2	2	no	—
11	C₄×Q₈×C₂₉	—	2	2	no	—
12	C₂₉×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
14	#11745288	C₂²	1	2	yes	—
15	#11745286	—	2	2	no	—
16	C₂²×#6597	C₂²	1	2	yes	—
17	#11745232	C₂²	1	2	yes	—
18	#11745231	C₂²	1	2	yes	—
19	C₄×Dic₅₈	—	2	2	no	—
20	#11745156	C₂²	1	8	yes	—
21	#11745155	C₂²	1	8	yes	—
22	C₄×#6597	C₂²	1	1	yes	characteristic
23	C₄×#6597	C₂²	1	2	yes	—
24	C₄×#6597	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₉	—	2	2	no	—
26	C₂²×C₄×C₂₉	D₄	1	2	yes	—
27	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₅₈	—	2	2	no	—
29	C₄⋊C₄×C₂₉	—	2	4	no	—
30	C₄⋊C₄×C₂₉	—	2	2	no	—
31	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₂₉	—	2	2	no	—
33	C₄²×C₂₉	—	2	2	no	—
34	C₄²×C₂₉	D₄	1	2	yes	—
35	C₄²×C₂₉	C₂³	1	1	yes	characteristic
36	C₄²×C₂₉	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₂₉	—	2	2	no	—
38	C₄⋊Dic₂₉	—	2	2	no	—
39	C₄⋊Dic₂₉	C₂³	1	4	yes	—
40	#90779	—	2	4	no	—
41	C₄×Dic₂₉	—	2	2	no	—
42	C₂×#6597	Q₈	1	8	yes	complemented
43	C₂×#6597	C₂³	1	2	yes	—
44	C₂×#6597	C₂³	1	2	yes	—
45	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₉	—	4	4	no	—
47	C₂×C₄×C₂₉	—	2	4	no	—
48	C₂×C₄×C₂₉	—	2	4	no	—
49	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₂₉	—	2	4	no	—
55	Dic₅₈	—	4	4	no	—
56	#6597	C₂×Q₈	1	4	yes	—
57	#6597	C₂×Q₈	1	4	yes	—
58	#3160	—	29	1	no	maximal, Sylow
59	C₂²×C₂₉	—	2	2	no	—
60	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
62	C₄×C₂₉	—	2	2	no	—
63	C₄×C₂₉	—	4	4	no	—
64	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
65	C₄×C₂₉	#135	1	4	yes	—
66	C₄×C₂₉	C₂²×Q₈	1	2	yes	—
67	Dic₂₉	—	4	4	no	—
68	#533	—	29	1	no	—
69	#533	D₂₉	1	1	yes	characteristic
70	#500	—	29	2	no	—
71	#499	—	29	4	no	—
72	#498	—	29	2	no	—
73	#427	—	29	4	no	—
74	C₂×C₄×C₈	—	29	1	no	—
75	C₂×C₂₉	—	2	2	no	—
76	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₂₉	C₂×#135	1	2	yes	—
78	C₂²×C₈	—	29	2	no	—
79	C₄⋊Q₈	—	29	2	no	—
80	C₄⋊Q₈	C₂×D₂₉	1	2	yes	—
81	#125	—	29	2	no	—
82	#125	C₂×D₂₉	1	2	yes	—
83	C₂²⋊Q₈	—	2	2	no	—
84	C₂²⋊Q₈	—	58	2	no	—
85	C₄×Q₈	—	2	2	no	—
86	C₄×Q₈	—	58	2	no	—
87	#117	—	29	2	no	—
88	#117	C₂×D₂₉	1	2	yes	—
89	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
90	#107	—	29	8	no	—
91	#106	—	29	8	no	—
92	C₄×C₈	—	29	1	no	—
93	C₄×C₈	—	29	2	no	—
94	C₄×C₈	—	29	1	no	—
95	C₂₉	#3160	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	58	2	no	—
98	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
100	C₂×C₈	—	29	8	no	—
101	C₂×C₈	—	29	2	no	—
102	C₂×C₈	—	29	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	29	4	no	—
106	C₄⋊C₄	—	58	4	no	—
107	C₄⋊C₄	—	58	2	no	—
108	C₄⋊C₄	C₂²×D₂₉	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	58	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	58	2	no	—
113	C₄²	C₂²⋊D₂₉	1	2	yes	—
114	C₄²	C₂²×D₂₉	1	1	yes	characteristic
115	C₄²	C₂²×D₂₉	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	116	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	58	4	no	—
122	C₂×C₄	Q₈×D₂₉	1	4	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
127	C₈	—	29	4	no	—
128	C₈	—	29	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#11745306	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	116	4	no	—
135	C₄	#11745306	1	2	yes	—
136	C₄	#11745312	1	4	yes	—
137	C₄	C₂×Q₈×D₂₉	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#11745306	1	1	yes	characteristic
140	C₂	C₂×#11745312	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic

#14271566

C₂

yes

maximal, complemented

#14271562

C₂

yes

maximal, complemented

#14271559

C₂

yes

maximal, complemented

#14271199

C₂

yes

maximal

C₂×C₄×#6597

C₂

yes

maximal, characteristic

C₂₉×#533

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₂²

yes

—

C₂₉×#125

C₂²

yes

—

C₂₉×C₂²⋊Q₈

—

C₄×Q₈×C₂₉

—

C₂₉×#117

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

#11745286

—

C₂²×#6597

C₂²

yes

—

#11745232

C₂²

yes

—

#11745231

C₂²

yes

—

C₄×Dic₅₈

—

#11745156

C₂²

yes

—

#11745155

C₂²

yes

—

C₄×#6597

C₂²

yes

characteristic

C₄×#6597

C₂²

yes

—

C₄×#6597

C₂²

yes

characteristic

C₂×Q₈×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂³

yes

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

C₄×Dic₂₉

—

C₂×#6597

Q₈

yes

complemented

C₂×#6597

C₂³

yes

—

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

Dic₅₈

—

#6597

C₂×Q₈

yes

—

#6597

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄⋊Q₈

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄⋊Q₈

yes

—

C₄×C₂₉

#135

yes

—

C₄×C₂₉

C₂²×Q₈

yes

—

Dic₂₉

—

#533

—

#533

D₂₉

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉

C₂×#135

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₉

yes

—

#125

—

#125

C₂×D₂₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₉

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂²×D₂₉

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²⋊D₂₉

yes

—

114

C₄²

C₂²×D₂₉

yes

characteristic

115

C₄²

C₂²×D₂₉

yes

characteristic

116

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

117

Q₈

—

118

Q₈

—

116

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

Q₈×D₂₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

124

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

125

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

126

C₂×C₄

C₂³×D₂₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#11745306

yes

—

131

C₂²

C₂²×C₂²⋊D₂₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

116

—

135

C₄

#11745306

yes

—

136

C₄

#11745312

yes

—

137

C₄

C₂×Q₈×D₂₉

yes

—

138

C₂

—

139

C₂

C₂×#11745306

yes

characteristic

140

C₂

C₂×#11745312

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#533	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	#533	D₂₉
hypocenter	C₂₉	#3160
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₉

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂₉×#533

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 ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14271566	C₂	semidirect
3. #14271562	C₂	semidirect
4. #14271559	C₂	semidirect
42. C₂×#6597	Q₈	semidirect
95. C₂₉	#3160	semidirect

H ≅ G/N

type

2. #14271566

C₂

semidirect

3. #14271562

C₂

semidirect

4. #14271559

C₂

semidirect

42. C₂×#6597

Q₈

semidirect

95. C₂₉

#3160

semidirect

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

subgroup	index
#14271720	2
#14271566	2
#14271562	2
#14271559	2
#14271199	2
C₂×C₄×#6597	2
C₂₉×#533	2
#3160	29

subgroup

index

C₂×C₄×#6597

C₂₉×#533

#3160

Subgroups — #16830002

Conjugacy classes of subgroups

Complements

Maximal subgroups