Group #16835676 — subgroups

Subgroups — #16835676

Order 3712.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14272122	C₂	1	1	yes	maximal, characteristic
2	#14272107	C₂	1	1	yes	maximal, characteristic
3	#14272036	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₂₉	—	2	1	no	—
6	C₂×C₄⋊#1181	C₂²	1	2	yes	—
7	C₂×D₂₉⋊C₈	C₂²	1	2	yes	—
8	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#11745306	—	2	1	no	—
10	C₄⋊C₄×D₂₉	—	2	1	no	—
11	C₄⋊C₄×D₂₉	C₂²	1	2	yes	—
12	#11745250	—	2	1	no	—
13	#11745250	C₄	1	2	yes	complemented
14	C₄²×D₂₉	C₂²	1	1	yes	characteristic
15	#11745231	C₄	1	1	yes	characteristic, complemented
16	(C₂×#1181)⋊C₄	—	2	1	no	—
17	(C₂×#1181)⋊C₄	—	2	1	no	—
18	#11745218	—	2	1	no	—
19	#11745218	—	2	1	no	—
20	C₂²×#1181	—	2	2	no	—
21	C₂×Q₈×C₂₉	—	2	1	no	—
22	Q₈×D₂₉	—	4	4	no	—
23	C₂×C₄×D₂₉	—	2	2	no	—
24	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₅₈	—	2	1	no	—
28	C₂×Dic₅₈	—	2	2	no	—
29	C₂×#6599	—	2	2	no	—
30	C₄⋊#1181	—	2	4	no	—
31	D₂₉⋊C₈	—	2	2	no	—
32	D₂₉⋊C₈	Q₈	1	4	yes	complemented
33	C₄⋊C₄×C₂₉	—	2	1	no	—
34	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
35	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₂₉	—	2	1	no	—
37	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
38	#90779	—	2	2	no	—
39	#90779	—	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	—	4	2	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	—	2	2	no	—
58	C₄×D₂₉	—	4	4	no	—
59	C₄×D₂₉	C₂×Q₈	1	2	yes	—
60	C₄×D₂₉	#41	1	2	yes	—
61	Dic₅₈	—	4	4	no	—
62	Dic₅₈	—	4	2	no	—
63	#6599	—	2	4	no	—
64	#1993	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₂₉	#124	1	1	yes	characteristic
67	C₂×D₂₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×D₂₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×D₂₉	#137	1	2	yes	—
70	#1181	—	8	2	no	—
71	C₄×C₂₉	—	2	1	no	—
72	C₄×C₂₉	—	2	1	no	—
73	C₄×C₂₉	—	4	2	no	—
74	C₄×C₂₉	C₄×Q₈	1	2	yes	—
75	Dic₂₉	—	2	1	no	—
76	Dic₂₉	—	2	1	no	—
77	Dic₂₉	—	4	2	no	—
78	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄⋊Q₈	—	29	1	no	—
80	C₂×#97	—	29	1	no	—
81	#382	—	29	1	no	—
82	#340	—	29	4	no	—
83	C₂×C₂₉	#391	1	1	yes	characteristic
84	C₂×C₂₉	#418	1	1	yes	characteristic
85	C₂×C₂₉	#419	1	1	yes	characteristic
86	D₂₉	#489	1	2	yes	—
87	D₂₉	#501	1	2	yes	—
88	C₂²×Q₈	—	58	1	no	—
89	C₂²×C₈	—	29	2	no	—
90	C₄⋊Q₈	—	29	2	no	—
91	C₄⋊Q₈	—	29	1	no	—
92	C₄⋊Q₈	—	58	1	no	—
93	C₄⋊Q₈	—	58	1	no	—
94	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
95	C₂×C₄⋊C₄	—	29	2	no	—
96	C₂×C₄⋊C₄	—	29	2	no	—
97	C₂×C₄⋊C₄	—	58	1	no	—
98	C₂×C₄²	—	29	1	no	—
99	#97	—	58	1	no	—
100	#97	—	58	1	no	—
101	#95	—	58	1	no	—
102	#95	—	58	1	no	—
103	C₂₉	#1993	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	58	2	no	—
106	C₂×Q₈	—	58	1	no	—
107	C₂×Q₈	—	116	4	no	—
108	C₂²×C₄	—	29	1	no	—
109	C₂²×C₄	—	29	1	no	—
110	C₂²×C₄	—	29	1	no	—
111	C₂²×C₄	—	58	2	no	—
112	C₂²×C₄	—	58	2	no	—
113	C₂×C₈	—	29	4	no	—
114	C₂×C₈	—	58	2	no	—
115	C₂×C₈	—	58	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	29	2	no	—
118	C₄⋊C₄	—	58	2	no	—
119	C₄⋊C₄	—	58	2	no	—
120	C₄⋊C₄	—	58	4	no	—
121	C₄⋊C₄	—	58	1	no	—
122	C₄⋊C₄	C₂×#1181	1	2	yes	—
123	C₄²	—	29	1	no	—
124	C₄²	—	29	1	no	—
125	C₄²	—	29	1	no	—
126	C₄²	C₂×#1181	1	1	yes	characteristic
127	C₂³	—	29	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	116	2	no	—
130	Q₈	—	116	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	29	1	no	—
133	C₂×C₄	—	29	2	no	—
134	C₂×C₄	—	29	2	no	—
135	C₂×C₄	—	29	1	no	—
136	C₂×C₄	—	29	1	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	—	58	2	no	—
139	C₂×C₄	—	58	2	no	—
140	C₂×C₄	—	116	4	no	—
141	C₂×C₄	—	116	2	no	—
142	C₂×C₄	—	116	2	no	—
143	C₂×C₄	—	116	2	no	—
144	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
146	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
147	C₈	—	58	4	no	—
148	C₂²	—	29	1	no	—
149	C₂²	—	29	1	no	—
150	C₂²	—	29	2	no	—
151	C₂²	—	29	2	no	—
152	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	29	2	no	—
157	C₄	—	58	1	no	—
158	C₄	—	58	1	no	—
159	C₄	—	116	2	no	—
160	C₄	—	232	2	no	—
161	C₄	Q₈×#1181	1	2	yes	—
162	C₂	—	29	2	no	—
163	C₂	—	29	2	no	—
164	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
165	C₂	#14272155	1	1	yes	characteristic
166	C₂	#14272156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272122

C₂

yes

maximal, characteristic

#14272107

C₂

yes

maximal, characteristic

#14272036

C₂

yes

maximal

C₄⋊Q₈×D₂₉

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₉

—

C₂×C₄⋊#1181

C₂²

yes

—

C₂×D₂₉⋊C₈

C₂²

yes

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic, complemented

#11745306

—

C₄⋊C₄×D₂₉

—

C₄⋊C₄×D₂₉

C₂²

yes

—

#11745250

—

#11745250

C₄

yes

complemented

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic, complemented

—

—

—

—

C₂²×#1181

—

C₂×Q₈×C₂₉

—

Q₈×D₂₉

—

C₂×C₄×D₂₉

—

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×#6599

—

C₄⋊#1181

—

D₂₉⋊C₈

—

D₂₉⋊C₈

Q₈

yes

complemented

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

#90779

—

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

—

C₄×D₂₉

C₂×Q₈

yes

—

C₄×D₂₉

#41

yes

—

Dic₅₈

—

Dic₅₈

—

#6599

—

#1993

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

#124

yes

characteristic

C₂×D₂₉

C₄⋊Q₈

yes

characteristic

C₂×D₂₉

AGL(1,ℤ/8)

yes

—

C₂×D₂₉

#137

yes

—

#1181

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄×Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

C₂²⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

—

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

yes

—

D₂₉

#501

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#97

—

100

#97

—

101

#95

—

102

#95

—

103

C₂₉

#1993

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

116

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

C₂×#1181

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×#1181

yes

characteristic

127

C₂³

—

128

Q₈

—

129

Q₈

—

116

—

130

Q₈

—

116

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

116

—

141

C₂×C₄

—

116

—

142

C₂×C₄

—

116

—

143

C₂×C₄

—

116

—

144

C₂×C₄

C₂²⋊#1181

yes

characteristic

145

C₂×C₄

C₂²⋊#1181

yes

characteristic

146

C₂×C₄

C₂²×#1181

yes

characteristic

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊#1181

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

116

—

160

C₄

—

232

—

161

C₄

Q₈×#1181

yes

—

162

C₂

—

163

C₂

—

164

C₂

(C₂×#1181)⋊Q₈

yes

characteristic

165

C₂

#14272155

yes

characteristic

166

C₂

#14272156

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	#1993
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. #11745250	C₄	semidirect
15. #11745231	C₄	semidirect
32. D₂₉⋊C₈	Q₈	semidirect
94. C₄⋊Q₈	#1181	semidirect
103. C₂₉	#1993	semidirect

H ≅ G/N

type

8. C₂₉×C₄⋊Q₈

C₄

semidirect

13. #11745250

C₄

semidirect

15. #11745231

C₄

semidirect

32. D₂₉⋊C₈

Q₈

semidirect

94. C₄⋊Q₈

#1181

semidirect

103. C₂₉

#1993

semidirect

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

subgroup	index
#14272122	2
#14272107	2
#14272036	2
C₄⋊Q₈×D₂₉	2
#1993	29

subgroup

index

C₄⋊Q₈×D₂₉

Subgroups — #16835676

Conjugacy classes of subgroups

Complements

Maximal subgroups