Group #16834924 — subgroups

Subgroups — #16834924

Order 3712.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14272107	C₂	1	1	yes	maximal, characteristic
3	#14272016	C₂	1	4	yes	maximal
4	#14271718	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#1181	—	2	1	no	—
6	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	#11745308	—	2	1	no	—
8	Q₈×Dic₂₉	—	2	1	no	—
9	#11745259	C₂²	1	2	yes	—
10	#11745254	—	2	1	no	—
11	#11745252	—	2	1	no	—
12	#11745249	C₄	1	2	yes	complemented
13	C₄²×D₂₉	C₂²	1	1	yes	characteristic
14	#11745231	C₄	1	1	yes	characteristic, complemented
15	(C₂×#1181)⋊C₄	—	2	1	no	—
16	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
17	#11745220	—	2	1	no	—
18	#11745219	C₂²	1	2	yes	—
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	C₂×C₄	1	2	yes	—
26	C₂×#6599	—	2	2	no	—
27	C₄⋊#1181	—	4	2	no	—
28	C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
30	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
31	#90781	—	2	2	no	—
32	C₄⋊Dic₂₉	—	2	1	no	—
33	C₄⋊Dic₂₉	—	2	1	no	—
34	C₄⋊Dic₂₉	—	2	2	no	—
35	#90779	—	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	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
41	C₂×#1181	—	4	2	no	—
42	C₂×#1181	—	4	2	no	—
43	C₂×#1181	—	4	2	no	—
44	Q₈×C₂₉	—	4	2	no	—
45	C₂×C₄×C₂₉	—	2	2	no	—
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₂₉	—	2	2	no	—
50	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
53	C₄×D₂₉	—	4	1	no	—
54	C₄×D₂₉	—	4	1	no	—
55	C₄×D₂₉	SD₁₆	1	2	yes	—
56	C₄×D₂₉	Q₁₆	1	2	yes	—
57	Dic₅₈	—	4	2	no	—
58	#6599	—	4	2	no	—
59	#1518	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₂₉	—	2	1	no	—
62	C₂×D₂₉	—	2	1	no	—
63	C₂×D₂₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×D₂₉	C₂×Q₁₆	1	1	yes	characteristic
65	#1181	—	8	2	no	—
66	C₄×C₂₉	—	2	1	no	—
67	C₄×C₂₉	—	2	1	no	—
68	C₄×C₂₉	—	4	2	no	—
69	C₄×C₂₉	#103	1	2	yes	—
70	Dic₂₉	—	2	1	no	—
71	Dic₂₉	—	2	1	no	—
72	Dic₂₉	—	4	2	no	—
73	Dic₂₉	#103	1	2	yes	—
74	#533	—	29	1	no	—
75	#431	—	29	1	no	—
76	#382	—	29	1	no	—
77	#328	—	29	4	no	—
78	C₂×C₂₉	#410	1	1	yes	characteristic
79	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
80	C₂×C₂₉	#421	1	1	yes	characteristic
81	D₂₉	—	2	1	no	—
82	D₂₉	—	2	1	no	—
83	C₄⋊Q₈	—	29	1	no	—
84	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
85	#125	—	58	1	no	—
86	C₂²⋊Q₈	—	29	2	no	—
87	C₂²⋊Q₈	—	58	1	no	—
88	C₄×Q₈	—	29	2	no	—
89	C₄×Q₈	—	58	1	no	—
90	#117	—	58	1	no	—
91	C₂×C₄⋊C₄	—	58	1	no	—
92	C₂×C₄²	—	29	1	no	—
93	C₄⋊C₈	—	58	1	no	—
94	C₂²⋊C₈	—	29	2	no	—
95	C₄×C₈	—	58	1	no	—
96	#95	—	29	2	no	—
97	#95	—	58	1	no	—
98	C₂₉	#1518	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	29	2	no	—
101	C₂²×C₄	—	29	1	no	—
102	C₂²×C₄	—	29	1	no	—
103	C₂²×C₄	—	29	1	no	—
104	C₂²×C₄	—	58	2	no	—
105	C₂×C₈	—	58	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	58	2	no	—
108	C₄⋊C₄	—	58	2	no	—
109	C₄⋊C₄	—	58	1	no	—
110	C₄⋊C₄	—	58	1	no	—
111	C₄⋊C₄	—	116	2	no	—
112	C₄⋊C₄	C₂×#1181	1	2	yes	—
113	C₂²⋊C₄	—	58	2	no	—
114	C₄²	—	29	1	no	—
115	C₄²	—	29	1	no	—
116	C₄²	—	29	1	no	—
117	C₄²	—	58	2	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	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	29	1	no	—
124	C₂×C₄	—	29	2	no	—
125	C₂×C₄	—	29	2	no	—
126	C₂×C₄	—	29	1	no	—
127	C₂×C₄	—	29	1	no	—
128	C₂×C₄	—	58	2	no	—
129	C₂×C₄	—	116	1	no	—
130	C₂×C₄	—	116	1	no	—
131	C₂×C₄	—	116	2	no	—
132	C₂×C₄	—	116	2	no	—
133	C₂×C₄	—	116	2	no	—
134	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
136	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
137	C₈	—	116	2	no	—
138	C₂²	—	29	1	no	—
139	C₂²	—	29	1	no	—
140	C₂²	—	58	1	no	—
141	C₂²	—	58	1	no	—
142	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	29	2	no	—
147	C₄	—	58	1	no	—
148	C₄	—	58	1	no	—
149	C₄	—	116	2	no	—
150	C₄	—	232	2	no	—
151	C₄	#11745225	1	2	yes	—
152	C₂	—	58	1	no	—
153	C₂	—	58	1	no	—
154	C₂	#14272144	1	1	yes	characteristic
155	C₂	C₂×#11745225	1	1	yes	characteristic
156	C₂	#14272161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272181

C₂

yes

maximal, characteristic

#14272107

C₂

yes

maximal, characteristic

#14272016

C₂

yes

maximal

#14271718

C₂

yes

maximal, characteristic

C₂×C₄⋊#1181

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic, complemented

#11745308

—

Q₈×Dic₂₉

—

#11745259

C₂²

yes

—

#11745254

—

#11745252

—

#11745249

C₄

yes

complemented

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic, complemented

(C₂×#1181)⋊C₄

—

(C₂×#1181)⋊C₄

C₂²

yes

—

#11745220

—

#11745219

C₂²

yes

—

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

C₂×C₄

yes

—

C₂×#6599

—

C₄⋊#1181

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

#90781

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

#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

SD₁₆

yes

—

C₄×D₂₉

Q₁₆

yes

—

Dic₅₈

—

#6599

—

#1518

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

C₂×SD₁₆

yes

characteristic

C₂×D₂₉

C₂×Q₁₆

yes

characteristic

#1181

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#103

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#103

yes

—

—

—

—

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

—

D₂₉

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

#95

—

C₂₉

#1518

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

116

—

112

C₄⋊C₄

C₂×#1181

yes

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#1181

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

116

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

116

—

130

C₂×C₄

—

116

—

131

C₂×C₄

—

116

—

132

C₂×C₄

—

116

—

133

C₂×C₄

—

116

—

134

C₂×C₄

C₂²⋊#1181

yes

characteristic

135

C₂×C₄

C₂²⋊#1181

yes

characteristic

136

C₂×C₄

C₂²×#1181

yes

characteristic

137

C₈

—

116

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×C₂²⋊#1181

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

116

—

150

C₄

—

232

—

151

C₄

#11745225

yes

—

152

C₂

—

153

C₂

—

154

C₂

#14272144

yes

characteristic

155

C₂

C₂×#11745225

yes

characteristic

156

C₂

#14272161

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	#1518
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

C₂×D₄

cosocle

C₂×C₄×D₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
6. C₂₉×C₄⋊Q₈	C₄	semidirect
12. #11745249	C₄	semidirect
14. #11745231	C₄	semidirect
84. C₄⋊Q₈	#1181	semidirect
98. C₂₉	#1518	semidirect

H ≅ G/N

type

6. C₂₉×C₄⋊Q₈

C₄

semidirect

12. #11745249

C₄

semidirect

14. #11745231

C₄

semidirect

84. C₄⋊Q₈

#1181

semidirect

98. C₂₉

#1518

semidirect

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

subgroup	index
#14272181	2
#14272107	2
#14272016	2
#14271718	2
#1518	29

subgroup

index

Subgroups — #16834924

Conjugacy classes of subgroups

Complements

Maximal subgroups