Group #16818634 — subgroups

Subgroups — #16818634

Order 3712.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₅₈	C₂	1	1	yes	maximal, characteristic
2	#14271543	C₂	1	1	yes	maximal, characteristic
3	#14271459	C₂	1	1	yes	maximal, characteristic
4	#14271458	C₂	1	1	yes	maximal, characteristic
5	#14270911	C₂	1	1	yes	maximal, characteristic
6	#14270895	C₂	1	1	yes	maximal, characteristic
7	C₂₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₉	—	2	1	no	—
10	C₂×Dic₁₁₆	—	2	2	no	—
11	#11745250	—	2	2	no	—
12	#11745250	—	2	1	no	—
13	#11745231	—	2	1	no	—
14	#11745231	C₂²	1	1	yes	characteristic
15	#11745231	C₂²	1	1	yes	characteristic
16	C₄×Dic₅₈	—	2	1	no	—
17	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
18	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	#11745183	—	2	1	no	—
21	#11745166	—	2	1	no	—
22	#11745164	—	2	1	no	—
23	#11745155	—	2	1	no	—
24	C₄⋊#6597	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
26	C₂×Dic₅₈	—	2	1	no	—
27	C₂×Dic₅₈	—	2	2	no	—
28	C₂×Dic₅₈	D₄	1	1	yes	characteristic
29	C₂×Dic₅₈	D₄	1	2	yes	—
30	C₂×C₈×C₂₉	—	2	1	no	—
31	C₂×C₈×C₂₉	—	2	1	no	—
32	C₄⋊C₄×C₂₉	—	2	1	no	—
33	C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
34	C₄²×C₂₉	—	2	1	no	—
35	C₄²×C₂₉	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₂₉	—	2	2	no	—
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	—	2	1	no	—
39	C₄⋊Dic₂₉	—	2	1	no	—
40	C₄⋊Dic₂₉	—	2	1	no	—
41	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
42	#90779	—	2	2	no	—
43	#90779	—	2	1	no	—
44	C₄×Dic₂₉	—	2	2	no	—
45	C₄×Dic₂₉	—	2	1	no	—
46	C₂×#6597	—	2	1	no	—
47	Dic₁₁₆	—	4	4	no	—
48	Q₈×C₂₉	—	2	1	no	—
49	Q₈×C₂₉	Q₁₆	1	2	yes	complemented
50	C₂×C₄×C₂₉	—	2	1	no	—
51	C₂×C₄×C₂₉	—	2	1	no	—
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₉	—	2	1	no	—
56	C₂×Dic₂₉	—	2	2	no	—
57	C₂×Dic₂₉	—	2	1	no	—
58	C₂×Dic₂₉	—	4	1	no	—
59	Dic₅₈	—	2	1	no	—
60	Dic₅₈	—	4	2	no	—
61	Dic₅₈	—	4	2	no	—
62	Dic₅₈	—	8	2	no	—
63	Dic₅₈	Q₁₆	1	2	yes	complemented
64	C₈×C₂₉	—	2	2	no	—
65	C₈×C₂₉	—	4	1	no	—
66	#6597	—	4	1	no	—
67	#1633	—	29	1	no	maximal, Sylow
68	C₂²×C₂₉	#120	1	1	yes	characteristic
69	C₄×C₂₉	—	2	2	no	—
70	C₄×C₂₉	—	2	1	no	—
71	C₄×C₂₉	—	4	1	no	—
72	C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₂₉	#137	1	1	yes	characteristic
75	C₄×C₂₉	#137	1	1	yes	characteristic
76	Dic₂₉	—	2	2	no	—
77	Dic₂₉	—	4	2	no	—
78	Dic₂₉	—	4	1	no	—
79	Dic₂₉	—	8	1	no	—
80	Q₈²	—	29	1	no	—
81	#494	—	29	1	no	—
82	#477	—	29	1	no	—
83	#477	—	29	1	no	—
84	#333	—	29	1	no	—
85	Q₈⋊C₈	—	29	1	no	—
86	Q₈⋊C₈	D₂₉	1	1	yes	characteristic
87	C₂×C₂₉	#451	1	1	yes	characteristic
88	C₂×C₂₉	#451	1	1	yes	characteristic
89	C₂×C₂₉	#453	1	1	yes	characteristic
90	C₂×Q₁₆	—	58	2	no	—
91	C₄⋊Q₈	—	29	1	no	—
92	C₄⋊Q₈	—	29	1	no	—
93	C₄⋊Q₈	—	58	1	no	—
94	C₄⋊Q₈	—	58	2	no	—
95	C₄⋊Q₈	—	58	1	no	—
96	C₄×Q₈	—	29	1	no	—
97	C₄×Q₈	—	58	1	no	—
98	C₄×Q₈	—	58	1	no	—
99	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
100	#107	—	58	1	no	—
101	#107	—	58	1	no	—
102	C₄⋊C₈	—	29	1	no	—
103	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
104	#103	—	58	1	no	—
105	#103	—	58	1	no	—
106	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
107	C₂₉	#1633	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₈	—	29	1	no	—
109	C₂×Q₈	—	29	2	no	—
110	C₂×Q₈	—	58	2	no	—
111	C₂×Q₈	—	58	1	no	—
112	C₂×Q₈	D₁₁₆	1	1	yes	characteristic
113	Q₁₆	—	116	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	58	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	29	1	no	—
119	C₄⋊C₄	—	58	2	no	—
120	C₄⋊C₄	—	58	2	no	—
121	C₄⋊C₄	—	58	1	no	—
122	C₄⋊C₄	—	58	1	no	—
123	C₄⋊C₄	—	58	1	no	—
124	C₄⋊C₄	—	58	1	no	—
125	C₄⋊C₄	—	58	1	no	—
126	C₄⋊C₄	D₁₁₆	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	58	2	no	—
129	C₄²	—	58	1	no	—
130	C₄²	C₂²×D₂₉	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	29	2	no	—
133	Q₈	—	58	1	no	—
134	Q₈	—	116	2	no	—
135	Q₈	—	116	2	no	—
136	Q₈	—	232	2	no	—
137	Q₈	Dic₁₁₆	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	58	1	no	—
141	C₂×C₄	—	58	1	no	—
142	C₂×C₄	—	58	2	no	—
143	C₂×C₄	—	116	1	no	—
144	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
145	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	4	1	no	—
149	C₈	—	116	1	no	—
150	C₂²	#11745244	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	58	2	no	—
155	C₄	—	116	2	no	—
156	C₄	—	116	1	no	—
157	C₄	—	232	1	no	—
158	C₄	C₂×Dic₁₁₆	1	1	yes	characteristic
159	C₄	#11745271	1	1	yes	characteristic
160	C₄	#11745277	1	1	yes	characteristic
161	C₄	Q₁₆×D₂₉	1	1	yes	characteristic
162	C₂	#14271320	1	1	yes	characteristic
163	C₂	#14271321	1	1	yes	characteristic
164	C₂	#14271332	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₅₈

C₂

yes

maximal, characteristic

#14271543

C₂

yes

maximal, characteristic

#14271459

C₂

yes

maximal, characteristic

#14271458

C₂

yes

maximal, characteristic

#14270911

C₂

yes

maximal, characteristic

#14270895

C₂

yes

maximal, characteristic

C₂₉×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₉

C₂²

yes

characteristic

Q₈×Dic₂₉

—

C₂×Dic₁₁₆

—

—

—

—

C₂²

yes

characteristic

#11745231

C₂²

yes

characteristic

C₄×Dic₅₈

—

C₄×Dic₅₈

C₂²

yes

characteristic

C₂₉×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₂₉

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

D₄

yes

characteristic

C₂×Dic₅₈

D₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

D₄

yes

characteristic

#90779

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₂×#6597

—

Dic₁₁₆

—

Q₈×C₂₉

—

Q₈×C₂₉

Q₁₆

yes

complemented

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

Q₁₆

yes

complemented

C₈×C₂₉

—

C₈×C₂₉

—

—

—

C₂²×C₂₉

yes

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×Q₁₆

yes

characteristic

C₄×C₂₉

C₂×Q₁₆

yes

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₉

yes

characteristic

100

#107

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₂₉

yes

characteristic

104

#103

—

105

#103

—

106

C₄×C₈

C₂×D₂₉

yes

characteristic

107

C₂₉

#1633

yes

characteristic, complemented, Sylow

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₁₁₆

yes

characteristic

113

Q₁₆

—

116

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

D₁₁₆

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²×D₂₉

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

116

—

135

Q₈

—

116

—

136

Q₈

—

232

—

137

Q₈

Dic₁₁₆

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

116

—

144

C₂×C₄

C₂×D₁₁₆

yes

characteristic

145

C₂×C₄

D₄×D₂₉

yes

characteristic

146

C₂×C₄

D₄×D₂₉

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

116

—

150

C₂²

#11745244

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

116

—

156

C₄

—

116

—

157

C₄

—

232

—

158

C₄

C₂×Dic₁₁₆

yes

characteristic

159

C₄

#11745271

yes

characteristic

160

C₄

#11745277

yes

characteristic

161

C₄

Q₁₆×D₂₉

yes

characteristic

162

C₂

#14271320

yes

characteristic

163

C₂

#14271321

yes

characteristic

164

C₂

#14271332

yes

characteristic

	subgroup	G/H
center	C₂²	#11745244
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×Q₈⋊C₈	C₂
socle, prime radical	C₂²×C₂₉	#120
hypercenter	Q₈⋊C₈	D₂₉
hypocenter	C₂₉	#1633
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#11745244

commutator, cosocle

C₄²×C₂₉

C₂³

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×Q₈⋊C₈

C₂

socle, prime radical

C₂²×C₂₉

#120

hypercenter

Q₈⋊C₈

D₂₉

hypocenter

C₂₉

#1633

LCS

G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
49. Q₈×C₂₉	Q₁₆	semidirect
63. Dic₅₈	Q₁₆	semidirect
107. C₂₉	#1633	semidirect
137. Q₈	Dic₁₁₆	semidirect

H ≅ G/N

type

49. Q₈×C₂₉

Q₁₆

semidirect

63. Dic₅₈

Q₁₆

semidirect

107. C₂₉

#1633

semidirect

137. Q₈

Dic₁₁₆

semidirect

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

subgroup	index
Q₈×Dic₅₈	2
#14271543	2
#14271459	2
#14271458	2
#14270911	2
#14270895	2
C₂₉×Q₈⋊C₈	2
#1633	29

subgroup

index

Q₈×Dic₅₈

C₂₉×Q₈⋊C₈

Subgroups — #16818634

Conjugacy classes of subgroups

Complements

Maximal subgroups