Group #16834834 — subgroups

Subgroups — #16834834

Order 3712.

252 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 72 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	#14272184	C₂	1	1	yes	maximal, characteristic
2	C₄×#90800	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6599	C₂	1	4	yes	maximal, complemented
4	C₂×Q₈×Dic₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₉	C₄	1	1	yes	characteristic
6	C₂×#90800	—	2	1	no	—
7	Q₈×Dic₂₉	—	2	1	no	—
8	Q₈×Dic₂₉	—	2	1	no	—
9	Q₈×Dic₂₉	C₄	1	2	yes	—
10	Q₈×Dic₂₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₉	—	2	1	no	—
12	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₂₉	—	2	1	no	—
14	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
15	C₂²⋊#6599	—	2	1	no	—
16	#11745218	—	2	1	no	—
17	#11745218	—	2	1	no	—
18	C₄⋊#6599	C₂²	1	2	yes	—
19	C₄×#6599	C₂²	1	2	yes	—
20	C₂×Q₈×C₂₉	—	2	4	no	—
21	C₂×Q₈×C₂₉	—	2	1	no	—
22	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₉	—	2	1	no	—
24	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₉	—	2	1	no	—
26	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
28	#90800	—	2	4	no	—
29	C₂×#6599	—	2	2	no	—
30	C₂×#6599	—	2	2	no	—
31	C₄⋊Dic₂₉	—	2	1	no	—
32	C₄⋊Dic₂₉	—	2	2	no	—
33	C₄⋊Dic₂₉	—	2	2	no	—
34	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₉	—	2	2	no	—
36	C₄×Dic₂₉	—	2	2	no	—
37	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₂₉	D₄	1	1	yes	characteristic
39	C₄×Dic₂₉	D₄	1	1	yes	characteristic
40	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
41	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₂₉	—	2	4	no	—
43	Q₈×C₂₉	—	2	2	no	—
44	Q₈×C₂₉	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₂₉	—	2	4	no	—
46	C₂×C₄×C₂₉	—	2	2	no	—
47	C₂×C₄×C₂₉	—	2	2	no	—
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₂₉	—	2	2	no	—
51	C₂×Dic₂₉	—	2	2	no	—
52	C₂×Dic₂₉	—	2	2	no	—
53	C₂×Dic₂₉	—	4	2	no	—
54	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
58	#6599	—	2	4	no	—
59	#6599	—	4	2	no	—
60	#1491	—	29	1	no	maximal, Sylow
61	C₂²×C₂₉	—	2	1	no	—
62	C₂²×C₂₉	—	2	1	no	—
63	C₂²×C₂₉	#104	1	2	yes	—
64	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₉	—	2	4	no	—
66	C₄×C₂₉	—	2	1	no	—
67	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
68	Dic₂₉	—	2	1	no	—
69	Dic₂₉	—	2	1	no	—
70	Dic₂₉	—	2	1	no	—
71	Dic₂₉	—	4	2	no	—
72	Dic₂₉	#137	1	2	yes	—
73	C₂×C₄×Q₈	—	29	1	no	—
74	#432	—	29	1	no	—
75	C₄×M₁₆	—	29	1	no	—
76	Q₈⋊C₈	—	29	4	no	—
77	C₂×C₂₉	—	2	2	no	—
78	C₂×C₂₉	#407	1	1	yes	characteristic
79	C₂×C₂₉	#419	1	1	yes	characteristic
80	C₂×C₂₉	C₂×#104	1	1	yes	characteristic
81	C₂²×Q₈	#1181	1	1	yes	characteristic
82	C₂×M₁₆	—	58	1	no	—
83	C₄×Q₈	—	29	2	no	—
84	C₄×Q₈	—	29	2	no	—
85	C₄×Q₈	—	58	1	no	—
86	C₄×Q₈	—	58	1	no	—
87	C₂×C₄⋊C₄	—	29	1	no	—
88	C₂×C₄⋊C₄	—	58	1	no	—
89	C₂×C₄²	—	29	1	no	—
90	C₂×C₄²	—	58	1	no	—
91	C₄⋊C₈	—	29	2	no	—
92	C₂²⋊C₈	—	58	1	no	—
93	#97	—	58	1	no	—
94	#97	—	58	1	no	—
95	C₄×C₈	—	29	2	no	—
96	C₂₉	#1491	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	C₂×#1181	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	29	1	no	—
102	C₂²×C₄	—	29	1	no	—
103	C₂²×C₄	—	58	1	no	—
104	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
105	M₁₆	—	58	4	no	—
106	C₂×C₈	—	58	2	no	—
107	C₂×C₈	—	58	2	no	—
108	C₄⋊C₄	—	29	2	no	—
109	C₄⋊C₄	—	58	2	no	—
110	C₄⋊C₄	—	58	2	no	—
111	C₄⋊C₄	—	58	1	no	—
112	C₄²	—	29	1	no	—
113	C₄²	—	29	1	no	—
114	C₄²	—	29	1	no	—
115	C₄²	—	29	1	no	—
116	C₄²	—	58	2	no	—
117	C₄²	—	58	2	no	—
118	C₂³	C₂²⋊#1181	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	2	no	—
121	Q₈	#90800	1	4	yes	complemented
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	29	1	no	—
126	C₂×C₄	—	29	1	no	—
127	C₂×C₄	—	29	1	no	—
128	C₂×C₄	—	29	1	no	—
129	C₂×C₄	—	58	2	no	—
130	C₂×C₄	—	58	2	no	—
131	C₂×C₄	—	58	2	no	—
132	C₂×C₄	—	116	2	no	—
133	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
134	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
135	C₈	—	58	4	no	—
136	C₈	—	116	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#11745226	1	2	yes	—
140	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	29	2	no	—
144	C₄	—	58	1	no	—
145	C₄	—	58	1	no	—
146	C₄	—	58	1	no	—
147	C₄	—	116	2	no	—
148	C₄	C₂×#90800	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#14272133	1	1	yes	characteristic
151	C₂	#14272156	1	1	yes	characteristic
152	C₂	C₂×#11745226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272184

C₂

yes

maximal, characteristic

C₄×#90800

C₂

yes

maximal, characteristic

Q₈⋊#6599

C₂

yes

maximal, complemented

C₂×Q₈×Dic₂₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₉

C₄

yes

characteristic

C₂×#90800

—

Q₈×Dic₂₉

—

Q₈×Dic₂₉

—

Q₈×Dic₂₉

C₄

yes

—

Q₈×Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₄

yes

characteristic

C₂×C₄×Dic₂₉

—

C₂×C₄×Dic₂₉

C₂²

yes

characteristic

C₂²⋊#6599

—

#11745218

—

#11745218

—

C₄⋊#6599

C₂²

yes

—

C₄×#6599

C₂²

yes

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

D₄

yes

characteristic

C₂²×Dic₂₉

D₄

yes

—

—

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

C₂³

yes

characteristic

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

M₁₆

yes

complemented

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

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#104

yes

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×M₁₆

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#137

yes

—

C₂×C₄×Q₈

—

#432

—

C₄×M₁₆

—

Q₈⋊C₈

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

yes

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂₉

#1491

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#1181

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×#1181

yes

characteristic

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₂³

C₂²⋊#1181

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

#90800

yes

complemented

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

116

—

133

C₂×C₄

C₂²⋊#1181

yes

characteristic

134

C₂×C₄

C₂²×#1181

yes

characteristic

135

C₈

—

136

C₈

—

116

—

137

C₂²

—

138

C₂²

—

139

C₂²

#11745226

yes

—

140

C₂²

C₂×C₂²⋊#1181

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

116

—

148

C₄

C₂×#90800

yes

—

149

C₂

—

150

C₂

#14272133

yes

characteristic

151

C₂

#14272156

yes

characteristic

152

C₂

C₂×#11745226

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂²×Q₈×C₂₉	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₄×Dic₂₉	C₂³
hypercenter	C₂²×Q₈	#1181
hypocenter	C₂₉	#1491
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂²×Q₈×C₂₉

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₄×Dic₂₉

C₂³

C₂²×Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
3. Q₈⋊#6599	C₂	semidirect
44. Q₈×C₂₉	M₁₆	semidirect
96. C₂₉	#1491	semidirect
121. Q₈	#90800	semidirect

H ≅ G/N

type

3. Q₈⋊#6599

C₂

semidirect

44. Q₈×C₂₉

M₁₆

semidirect

96. C₂₉

#1491

semidirect

121. Q₈

#90800

semidirect

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

subgroup	index
#14272184	2
C₄×#90800	2
Q₈⋊#6599	2
C₂×Q₈×Dic₂₉	2
#1491	29

subgroup

index

C₂×Q₈×Dic₂₉

Subgroups — #16834834

Conjugacy classes of subgroups

Complements

Maximal subgroups