Group #16835685 — subgroups

Subgroups — #16835685

Order 3712.

260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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	#14272166	C₂	1	1	yes	maximal, characteristic
2	C₄×#90800	C₂	1	1	yes	maximal, characteristic
3	#14272043	C₂	1	4	yes	maximal
4	C₂×#11745250	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₄	1	2	yes	—
6	C₂×#90800	C₂²	1	2	yes	—
7	C₂×#90800	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
10	C₂×#90779	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
12	#11745250	—	2	1	no	—
13	#11745250	—	2	1	no	—
14	#11745250	—	2	1	no	—
15	#11745250	—	2	1	no	—
16	#11745220	—	2	1	no	—
17	#11745218	—	2	1	no	—
18	C₄⋊#6599	—	2	1	no	—
19	C₄×#6599	—	2	1	no	—
20	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	2	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	—	4	4	no	—
28	#90800	—	2	2	no	—
29	#90800	—	2	4	no	—
30	#90800	Q₈	1	4	yes	complemented
31	C₂×#6599	—	2	2	no	—
32	C₂×#6599	—	2	2	no	—
33	C₄⋊C₄×C₂₉	—	2	1	no	—
34	C₄⋊C₄×C₂₉	—	2	1	no	—
35	C₄⋊Dic₂₉	—	2	1	no	—
36	C₄⋊Dic₂₉	—	2	1	no	—
37	#90779	—	2	2	no	—
38	#90779	—	2	2	no	—
39	C₄×Dic₂₉	D₄	1	1	yes	characteristic
40	C₄×Dic₂₉	D₄	1	1	yes	characteristic
41	C₄×Dic₂₉	D₄	1	1	yes	characteristic
42	C₄×Dic₂₉	D₄	1	1	yes	characteristic
43	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	—	2	2	no	—
45	C₂×C₄×C₂₉	—	2	2	no	—
46	C₂×C₄×C₂₉	—	4	2	no	—
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₂₉	—	2	2	no	—
50	C₂×Dic₂₉	—	2	2	no	—
51	C₂×Dic₂₉	—	4	2	no	—
52	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
57	C₂×Dic₂₉	#41	1	2	yes	—
58	Dic₅₈	—	4	4	no	—
59	Dic₅₈	—	4	4	no	—
60	#6599	—	2	4	no	—
61	#6599	—	4	2	no	—
62	#1997	—	29	1	no	maximal, Sylow
63	C₂²×C₂₉	—	2	1	no	—
64	C₂²×C₂₉	—	2	1	no	—
65	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
67	C₄×C₂₉	—	2	1	no	—
68	C₄×C₂₉	—	4	2	no	—
69	C₄×C₂₉	#101	1	2	yes	—
70	Dic₂₉	—	2	1	no	—
71	Dic₂₉	—	4	2	no	—
72	Dic₂₉	#100	1	2	yes	—
73	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
74	Dic₂₉	#124	1	1	yes	characteristic
75	Dic₂₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	—	29	1	no	—
77	#423	—	29	1	no	—
78	C₄×M₁₆	—	29	1	no	—
79	#343	—	29	4	no	—
80	C₂×C₂₉	—	2	2	no	—
81	C₂×C₂₉	#391	1	1	yes	characteristic
82	C₂×C₂₉	C₂×#100	1	1	yes	characteristic
83	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
84	C₂²×Q₈	—	29	2	no	—
85	C₂×M₁₆	—	29	2	no	—
86	C₂×M₁₆	—	29	2	no	—
87	C₄⋊Q₈	—	58	1	no	—
88	C₄⋊Q₈	—	58	1	no	—
89	C₄⋊Q₈	—	58	1	no	—
90	C₄⋊Q₈	—	58	1	no	—
91	C₂×C₄⋊C₄	—	29	2	no	—
92	C₂×C₄⋊C₄	—	29	1	no	—
93	C₂×C₄⋊C₄	#1181	1	1	yes	characteristic
94	C₂×C₄²	—	29	1	no	—
95	C₄⋊C₈	—	58	1	no	—
96	C₄⋊C₈	—	58	1	no	—
97	#97	—	58	1	no	—
98	C₄×C₈	—	58	1	no	—
99	C₂₉	#1997	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	—	58	2	no	—
101	C₂×Q₈	—	58	2	no	—
102	C₂×Q₈	—	116	4	no	—
103	C₂²×C₄	—	29	1	no	—
104	C₂²×C₄	—	29	1	no	—
105	C₂²×C₄	—	29	2	no	—
106	C₂²×C₄	C₂×#1181	1	2	yes	—
107	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
108	M₁₆	—	29	4	no	—
109	M₁₆	—	58	2	no	—
110	M₁₆	—	58	4	no	—
111	C₂×C₈	—	58	2	no	—
112	C₂×C₈	—	58	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	58	2	no	—
116	C₄⋊C₄	—	58	2	no	—
117	C₄⋊C₄	—	58	1	no	—
118	C₄⋊C₄	—	58	1	no	—
119	C₄²	—	29	1	no	—
120	C₄²	—	29	1	no	—
121	C₄²	—	29	1	no	—
122	C₄²	—	29	1	no	—
123	C₂³	C₂²×#1181	1	1	yes	characteristic
124	Q₈	—	116	4	no	—
125	Q₈	—	116	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	29	2	no	—
130	C₂×C₄	—	29	2	no	—
131	C₂×C₄	—	29	1	no	—
132	C₂×C₄	—	29	1	no	—
133	C₂×C₄	—	29	1	no	—
134	C₂×C₄	—	29	1	no	—
135	C₂×C₄	—	58	2	no	—
136	C₂×C₄	—	58	2	no	—
137	C₂×C₄	—	116	2	no	—
138	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
140	C₈	—	58	4	no	—
141	C₈	—	116	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	Q₈×#1181	1	2	yes	—
145	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	29	2	no	—
149	C₄	—	29	1	no	—
150	C₄	—	29	1	no	—
151	C₄	—	29	2	no	—
152	C₄	—	58	1	no	—
153	C₄	—	116	2	no	—
154	C₄	#11745221	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
157	C₂	D₂₉⋊#100	1	1	yes	characteristic
158	C₂	C₂×#11745221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272166

C₂

yes

maximal, characteristic

C₄×#90800

C₂

yes

maximal, characteristic

#14272043

C₂

yes

maximal

C₂×#11745250

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₄

yes

—

C₂×#90800

C₂²

yes

—

C₂×#90800

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

characteristic

C₂×C₄⋊Dic₂₉

C₄

yes

characteristic

C₂×#90779

C₂²

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

—

—

—

—

—

—

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂²×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

#90800

—

#90800

—

#90800

Q₈

yes

complemented

C₂×#6599

—

C₂×#6599

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

#90779

—

#90779

—

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×Q₈

yes

—

C₂×Dic₂₉

#41

yes

—

Dic₅₈

—

Dic₅₈

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#101

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#100

yes

—

Dic₂₉

C₂²⋊Q₈

yes

—

Dic₂₉

#124

yes

characteristic

Dic₂₉

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

—

#343

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#1181

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂₉

#1997

yes

characteristic, complemented, Sylow

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

116

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×#1181

yes

—

107

C₂²×C₄

C₂×#1181

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

C₂²×#1181

yes

characteristic

124

Q₈

—

116

—

125

Q₈

—

116

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

116

—

138

C₂×C₄

C₂²⋊#1181

yes

characteristic

139

C₂×C₄

C₂²⋊#1181

yes

characteristic

140

C₈

—

141

C₈

—

116

—

142

C₂²

—

143

C₂²

—

144

C₂²

Q₈×#1181

yes

—

145

C₂²

C₂×C₂²⋊#1181

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

116

—

154

C₄

#11745221

yes

—

155

C₂

—

156

C₂

(C₂×#1181)⋊Q₈

yes

characteristic

157

C₂

D₂₉⋊#100

yes

characteristic

158

C₂

C₂×#11745221

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator, prime radical	C₂³×C₂₉	C₂²×C₄
Frattini	C₂³	C₂²×#1181
Fitting	C₂×C₄⋊C₄×C₂₉	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₂₉	C₂³
hypercenter	C₂×C₄⋊C₄	#1181
hypocenter	C₂₉	#1997
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator, prime radical

C₂³×C₂₉

C₂²×C₄

Frattini

C₂³

C₂²×#1181

Fitting

C₂×C₄⋊C₄×C₂₉

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₂₉

C₂³

C₂×C₄⋊C₄

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
30. #90800	Q₈	semidirect
99. C₂₉	#1997	semidirect

H ≅ G/N

type

30. #90800

Q₈

semidirect

99. C₂₉

#1997

semidirect

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

subgroup	index
#14272166	2
C₄×#90800	2
#14272043	2
C₂×#11745250	2
#1997	29

subgroup

index

Subgroups — #16835685

Conjugacy classes of subgroups

Complements

Maximal subgroups