Group #16835668 — subgroups

Subgroups — #16835668

Order 3712.

260 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×#90796	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1181	C₂	1	1	yes	maximal, characteristic
3	#14272033	C₂	1	4	yes	maximal
4	#14271718	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#1181	—	2	1	no	—
6	C₂×C₄×#1181	C₂²	1	2	yes	—
7	C₂×#90796	C₂²	1	2	yes	—
8	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#11745308	—	2	1	no	—
10	Q₈×Dic₂₉	—	2	1	no	—
11	#11745259	—	2	1	no	—
12	#11745254	C₂²	1	2	yes	—
13	#11745252	C₄	1	2	yes	complemented
14	#11745249	—	2	1	no	—
15	C₄²×D₂₉	C₂²	1	1	yes	characteristic
16	#11745231	C₄	1	1	yes	characteristic, complemented
17	(C₂×#1181)⋊C₄	—	2	1	no	—
18	#11745218	—	2	1	no	—
19	C₄×#6599	—	2	1	no	—
20	C₂²×#1181	—	2	2	no	—
21	C₂×Q₈×C₂₉	—	2	1	no	—
22	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	1	no	—
26	C₂×#6599	—	2	2	no	—
27	C₄⋊#1181	—	2	4	no	—
28	C₄×#1181	—	2	4	no	—
29	#90796	—	2	2	no	—
30	#90796	Q₈	1	4	yes	complemented
31	C₄⋊C₄×C₂₉	—	2	1	no	—
32	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
33	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
34	#90781	—	2	2	no	—
35	C₄⋊Dic₂₉	—	2	1	no	—
36	C₄⋊Dic₂₉	—	2	2	no	—
37	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
38	#90779	—	2	2	no	—
39	C₄×Dic₂₉	—	2	2	no	—
40	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₂₉	D₄	1	1	yes	characteristic
42	C₄×Dic₂₉	D₄	1	1	yes	characteristic
43	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×#1181	—	2	4	no	—
45	C₂×#1181	—	4	2	no	—
46	C₂×#1181	—	4	2	no	—
47	Q₈×C₂₉	—	4	2	no	—
48	C₂×C₄×C₂₉	—	2	2	no	—
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₂₉	—	2	2	no	—
53	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
56	C₄×D₂₉	—	2	2	no	—
57	C₄×D₂₉	—	4	1	no	—
58	C₄×D₂₉	C₂×Q₈	1	2	yes	—
59	C₄×D₂₉	#41	1	2	yes	—
60	Dic₅₈	—	4	2	no	—
61	#6599	—	2	4	no	—
62	#1986	—	29	1	no	maximal, Sylow
63	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₂₉	—	2	1	no	—
65	C₂×D₂₉	—	2	1	no	—
66	C₂×D₂₉	#124	1	1	yes	characteristic
67	C₂×D₂₉	C₄⋊Q₈	1	1	yes	characteristic
68	#1181	—	4	4	no	—
69	C₄×C₂₉	—	2	1	no	—
70	C₄×C₂₉	—	4	2	no	—
71	C₄×C₂₉	#104	1	2	yes	—
72	C₄×C₂₉	C₄×Q₈	1	2	yes	—
73	Dic₂₉	—	2	1	no	—
74	Dic₂₉	—	2	1	no	—
75	Dic₂₉	—	4	2	no	—
76	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
77	#533	—	29	1	no	—
78	C₄×M₁₆	—	29	1	no	—
79	C₄×C₄⋊C₄	—	29	1	no	—
80	#339	—	29	4	no	—
81	C₂×C₂₉	#391	1	1	yes	characteristic
82	C₂×C₂₉	C₂×#104	1	1	yes	characteristic
83	C₂×C₂₉	#421	1	1	yes	characteristic
84	D₂₉	—	2	1	no	—
85	D₂₉	—	2	1	no	—
86	C₂×M₁₆	—	29	2	no	—
87	C₄⋊Q₈	—	29	1	no	—
88	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
89	#125	—	29	2	no	—
90	C₂²⋊Q₈	—	58	1	no	—
91	C₂²⋊Q₈	—	58	1	no	—
92	C₄×Q₈	—	58	1	no	—
93	C₄×Q₈	—	58	1	no	—
94	#117	—	29	2	no	—
95	C₂×C₄⋊C₄	—	58	1	no	—
96	C₂×C₄²	—	29	1	no	—
97	C₂×C₄²	—	29	2	no	—
98	#97	—	58	1	no	—
99	C₄×C₈	—	58	1	no	—
100	#95	—	58	1	no	—
101	C₂₉	#1986	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	58	1	no	—
104	C₂²×C₄	—	29	1	no	—
105	C₂²×C₄	—	29	1	no	—
106	C₂²×C₄	—	29	1	no	—
107	C₂²×C₄	—	58	2	no	—
108	M₁₆	—	29	4	no	—
109	M₁₆	—	58	2	no	—
110	C₂×C₈	—	58	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	29	2	no	—
113	C₄⋊C₄	—	58	2	no	—
114	C₄⋊C₄	—	58	2	no	—
115	C₄⋊C₄	—	58	4	no	—
116	C₄⋊C₄	—	58	1	no	—
117	C₄⋊C₄	C₂×#1181	1	2	yes	—
118	C₂²⋊C₄	—	58	2	no	—
119	C₄²	—	29	1	no	—
120	C₄²	—	29	1	no	—
121	C₄²	—	29	1	no	—
122	C₄²	—	58	2	no	—
123	C₄²	—	58	4	no	—
124	C₄²	C₂×#1181	1	1	yes	characteristic
125	C₂³	—	29	1	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	116	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	29	1	no	—
130	C₂×C₄	—	29	2	no	—
131	C₂×C₄	—	29	2	no	—
132	C₂×C₄	—	29	1	no	—
133	C₂×C₄	—	29	1	no	—
134	C₂×C₄	—	58	2	no	—
135	C₂×C₄	—	58	4	no	—
136	C₂×C₄	—	58	2	no	—
137	C₂×C₄	—	116	1	no	—
138	C₂×C₄	—	116	2	no	—
139	C₂×C₄	—	116	2	no	—
140	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
142	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
143	C₈	—	58	4	no	—
144	C₂²	—	29	1	no	—
145	C₂²	—	29	1	no	—
146	C₂²	—	58	1	no	—
147	C₂²	—	58	1	no	—
148	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	29	2	no	—
152	C₄	—	58	1	no	—
153	C₄	—	58	1	no	—
154	C₄	—	116	4	no	—
155	C₄	—	116	2	no	—
156	C₄	#11745224	1	2	yes	—
157	C₄	Q₈×#1181	1	2	yes	—
158	C₂	—	58	1	no	—
159	C₂	—	58	1	no	—
160	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
161	C₂	C₂×#11745224	1	1	yes	characteristic
162	C₂	#14272162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#90796

C₂

yes

maximal, characteristic

C₄×C₄⋊#1181

C₂

yes

maximal, characteristic

#14272033

C₂

yes

maximal

#14271718

C₂

yes

maximal, characteristic

C₂×C₄⋊#1181

—

C₂×C₄×#1181

C₂²

yes

—

C₂×#90796

C₂²

yes

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic, complemented

#11745308

—

Q₈×Dic₂₉

—

#11745259

—

#11745254

C₂²

yes

—

#11745252

C₄

yes

complemented

#11745249

—

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic, complemented

(C₂×#1181)⋊C₄

—

#11745218

—

C₄×#6599

—

C₂²×#1181

—

C₂×Q₈×C₂₉

—

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

—

—

—

—

Q₈

yes

complemented

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

#90781

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₂²×D₂₉

C₂×D₄

yes

—

—

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

C₂×Q₈

yes

—

C₄×D₂₉

#41

yes

—

Dic₅₈

—

#6599

—

#1986

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

#124

yes

characteristic

C₂×D₂₉

C₄⋊Q₈

yes

characteristic

#1181

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#104

yes

—

C₄×C₂₉

C₄×Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

C₂²⋊Q₈

yes

—

#533

—

C₄×M₁₆

—

C₄×C₄⋊C₄

—

#339

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

—

D₂₉

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#97

—

C₄×C₈

—

100

#95

—

101

C₂₉

#1986

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂×#1181

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂×#1181

yes

characteristic

125

C₂³

—

126

Q₈

—

127

Q₈

—

116

—

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

—

116

—

139

C₂×C₄

—

116

—

140

C₂×C₄

C₂²⋊#1181

yes

characteristic

141

C₂×C₄

C₂²⋊#1181

yes

characteristic

142

C₂×C₄

C₂²×#1181

yes

characteristic

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊#1181

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

116

—

155

C₄

—

116

—

156

C₄

#11745224

yes

—

157

C₄

Q₈×#1181

yes

—

158

C₂

—

159

C₂

—

160

C₂

(C₂×#1181)⋊Q₈

yes

characteristic

161

C₂

C₂×#11745224

yes

characteristic

162

C₂

#14272162

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉	C₂×D₄
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1986
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂₉×C₄⋊Q₈

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉

C₂×D₄

cosocle

C₂×C₄×D₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
8. C₂₉×C₄⋊Q₈	C₄	semidirect
13. #11745252	C₄	semidirect
16. #11745231	C₄	semidirect
30. #90796	Q₈	semidirect
88. C₄⋊Q₈	#1181	semidirect
101. C₂₉	#1986	semidirect

H ≅ G/N

type

8. C₂₉×C₄⋊Q₈

C₄

semidirect

13. #11745252

C₄

semidirect

16. #11745231

C₄

semidirect

30. #90796

Q₈

semidirect

88. C₄⋊Q₈

#1181

semidirect

101. C₂₉

#1986

semidirect

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

subgroup	index
C₄×#90796	2
C₄×C₄⋊#1181	2
#14272033	2
#14271718	2
#1986	29

subgroup

index

C₄×C₄⋊#1181

Subgroups — #16835668

Conjugacy classes of subgroups

Complements

Maximal subgroups