Group #16835008 — subgroups

Subgroups — #16835008

Order 3712.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14272181	C₂	1	1	yes	maximal, characteristic
2	#14272168	C₂	1	1	yes	maximal, characteristic
3	#14272022	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₂₉	—	2	1	no	—
6	C₂×#90796	—	2	1	no	—
7	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	#11745306	—	2	1	no	—
9	C₄⋊C₄×D₂₉	—	2	1	no	—
10	C₄⋊C₄×D₂₉	C₄	1	2	yes	—
11	#11745250	—	2	1	no	—
12	#11745250	C₂²	1	2	yes	—
13	C₄²×D₂₉	C₂²	1	1	yes	characteristic
14	#11745231	C₄	1	1	yes	characteristic
15	#11745220	—	2	1	no	—
16	#11745220	C₂²	1	2	yes	—
17	#11745220	C₂²	1	2	yes	—
18	#11745219	—	2	1	no	—
19	C₄×#6599	—	2	1	no	—
20	C₂×Q₈×C₂₉	—	2	1	no	—
21	Q₈×D₂₉	—	4	4	no	—
22	C₂×C₄×D₂₉	—	2	2	no	—
23	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
26	C₂×Dic₅₈	—	2	1	no	—
27	C₂×Dic₅₈	—	2	2	no	—
28	C₂×#6599	—	2	2	no	—
29	C₂×#6599	—	2	2	no	—
30	#90796	—	4	2	no	—
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	C₄⋊Dic₂₉	—	2	1	no	—
35	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
36	#90779	—	2	2	no	—
37	#90779	—	2	2	no	—
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	C₂²×D₂₉	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₂₉	—	4	2	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₂₉	—	2	2	no	—
48	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
51	C₄×D₂₉	—	2	2	no	—
52	C₄×D₂₉	—	2	2	no	—
53	C₄×D₂₉	—	4	4	no	—
54	C₄×D₂₉	SD₁₆	1	2	yes	—
55	C₄×D₂₉	Q₁₆	1	2	yes	—
56	Dic₅₈	—	4	2	no	—
57	Dic₅₈	—	4	4	no	—
58	#6599	—	4	2	no	—
59	#6599	—	4	2	no	—
60	#1549	—	29	1	no	maximal, Sylow
61	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×D₂₉	—	2	1	no	—
63	C₂×D₂₉	—	2	1	no	—
64	C₂×D₂₉	#103	1	2	yes	—
65	C₄×C₂₉	—	2	1	no	—
66	C₄×C₂₉	—	2	1	no	—
67	C₄×C₂₉	—	4	2	no	—
68	C₄×C₂₉	#103	1	2	yes	—
69	Dic₂₉	—	4	2	no	—
70	Dic₂₉	#101	1	2	yes	—
71	Dic₂₉	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₂₉	C₂×Q₁₆	1	1	yes	characteristic
73	Dic₂₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄⋊Q₈	—	29	1	no	—
75	#431	—	29	1	no	—
76	#423	—	29	1	no	—
77	#332	—	29	4	no	—
78	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
79	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
80	C₂×C₂₉	#418	1	1	yes	characteristic
81	D₂₉	—	2	2	no	—
82	C₂²×Q₈	—	58	1	no	—
83	C₂×M₁₆	—	58	1	no	—
84	C₄⋊Q₈	—	29	2	no	—
85	C₄⋊Q₈	—	29	1	no	—
86	C₄⋊Q₈	—	58	1	no	—
87	C₄⋊Q₈	—	58	1	no	—
88	C₄⋊Q₈	#1181	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	29	2	no	—
90	C₂×C₄⋊C₄	—	58	1	no	—
91	C₂×C₄²	—	29	1	no	—
92	C₄⋊C₈	—	29	2	no	—
93	C₄⋊C₈	—	29	2	no	—
94	C₄⋊C₈	—	58	1	no	—
95	C₂²⋊C₈	—	58	1	no	—
96	C₄×C₈	—	58	1	no	—
97	C₂₉	#1549	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	58	2	no	—
100	C₂×Q₈	—	58	1	no	—
101	C₂×Q₈	—	116	4	no	—
102	C₂²×C₄	—	29	1	no	—
103	C₂²×C₄	—	29	1	no	—
104	C₂²×C₄	—	29	1	no	—
105	C₂²×C₄	—	58	2	no	—
106	M₁₆	—	116	2	no	—
107	C₂×C₈	—	58	2	no	—
108	C₂×C₈	—	58	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	29	2	no	—
111	C₄⋊C₄	—	58	2	no	—
112	C₄⋊C₄	—	58	2	no	—
113	C₄⋊C₄	—	58	1	no	—
114	C₄⋊C₄	C₂×#1181	1	2	yes	—
115	C₄²	—	29	1	no	—
116	C₄²	—	29	1	no	—
117	C₄²	—	29	1	no	—
118	C₄²	C₂×#1181	1	1	yes	characteristic
119	C₂³	—	29	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	116	2	no	—
122	Q₈	—	116	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	29	1	no	—
125	C₂×C₄	—	29	1	no	—
126	C₂×C₄	—	29	1	no	—
127	C₂×C₄	—	29	2	no	—
128	C₂×C₄	—	29	2	no	—
129	C₂×C₄	—	58	2	no	—
130	C₂×C₄	—	58	2	no	—
131	C₂×C₄	—	58	2	no	—
132	C₂×C₄	—	116	4	no	—
133	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
135	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
136	C₈	—	116	2	no	—
137	C₈	—	116	2	no	—
138	C₂²	—	29	2	no	—
139	C₂²	—	58	1	no	—
140	C₂²	—	58	1	no	—
141	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	29	2	no	—
146	C₄	—	29	1	no	—
147	C₄	—	29	1	no	—
148	C₄	—	29	2	no	—
149	C₄	—	116	2	no	—
150	C₄	#11745225	1	2	yes	—
151	C₂	—	58	2	no	—
152	C₂	D₂₉⋊#101	1	1	yes	characteristic
153	C₂	C₂×#11745225	1	1	yes	characteristic
154	C₂	#14272155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272181

C₂

yes

maximal, characteristic

#14272168

C₂

yes

maximal, characteristic

#14272022

C₂

yes

maximal, complemented

C₄⋊Q₈×D₂₉

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₉

—

C₂×#90796

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic

#11745306

—

C₄⋊C₄×D₂₉

—

C₄⋊C₄×D₂₉

C₄

yes

—

#11745250

—

#11745250

C₂²

yes

—

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic

#11745220

—

#11745220

C₂²

yes

—

#11745220

C₂²

yes

—

#11745219

—

C₄×#6599

—

C₂×Q₈×C₂₉

—

Q₈×D₂₉

—

C₂×C₄×D₂₉

—

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×#6599

—

C₂×#6599

—

#90796

—

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

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

C₂³

yes

characteristic

C₂²×D₂₉

C₂²⋊C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

SD₁₆

yes

—

C₄×D₂₉

Q₁₆

yes

—

Dic₅₈

—

Dic₅₈

—

—

—

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

#103

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#103

yes

—

Dic₂₉

—

Dic₂₉

#101

yes

—

Dic₂₉

C₂×SD₁₆

yes

characteristic

Dic₂₉

C₂×Q₁₆

yes

characteristic

Dic₂₉

AGL(1,ℤ/8)

yes

—

C₂×C₄⋊Q₈

—

#431

—

#423

—

#332

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₉

#1549

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

116

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

116

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×#1181

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#1181

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

116

—

122

Q₈

—

116

—

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

C₂×C₄

C₂²×#1181

yes

characteristic

136

C₈

—

116

—

137

C₈

—

116

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#1181

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

116

—

150

C₄

#11745225

yes

—

151

C₂

—

152

C₂

D₂₉⋊#101

yes

characteristic

153

C₂

C₂×#11745225

yes

characteristic

154

C₂

#14272155

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	C₂²⋊C₄
cosocle	C₄×Dic₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1549
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×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

C₂²⋊C₄

cosocle

C₄×Dic₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #14272022	C₂	semidirect
97. C₂₉	#1549	semidirect

H ≅ G/N

type

3. #14272022

C₂

semidirect

97. C₂₉

#1549

semidirect

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

subgroup	index
#14272181	2
#14272168	2
#14272022	2
C₄⋊Q₈×D₂₉	2
#1549	29

subgroup

index

C₄⋊Q₈×D₂₉

Subgroups — #16835008

Conjugacy classes of subgroups

Complements

Maximal subgroups