Group #16834855 — subgroups

Subgroups — #16834855

Order 3712.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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	#14272185	C₂	1	1	yes	maximal, characteristic
2	#14272181	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6599	C₂	1	4	yes	maximal, complemented
4	#14271718	C₂	1	1	yes	maximal, characteristic
5	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	#11745308	C₄	1	2	yes	—
7	Q₈×Dic₂₉	C₂²	1	2	yes	—
8	#11745259	—	2	1	no	—
9	#11745254	—	2	1	no	—
10	#11745252	—	2	1	no	—
11	#11745249	—	2	1	no	—
12	C₄²×D₂₉	C₂²	1	1	yes	characteristic
13	#11745231	C₄	1	1	yes	characteristic
14	#11745220	—	2	1	no	—
15	#11745219	—	2	1	no	—
16	#11745219	—	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	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
24	C₂×Dic₅₈	—	2	1	no	—
25	C₂×#6599	—	2	2	no	—
26	C₂×#6599	—	2	2	no	—
27	C₄⋊C₄×C₂₉	—	2	1	no	—
28	C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
30	#90781	—	2	2	no	—
31	C₄⋊Dic₂₉	—	2	1	no	—
32	C₄⋊Dic₂₉	—	2	2	no	—
33	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
34	#90779	—	2	2	no	—
35	C₄×Dic₂₉	—	2	2	no	—
36	C₄×Dic₂₉	D₄	1	1	yes	characteristic
37	C₄×Dic₂₉	D₄	1	1	yes	characteristic
38	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
39	C₂²×D₂₉	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₂₉	—	2	4	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₂₉	—	2	2	no	—
46	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
49	C₄×D₂₉	—	2	2	no	—
50	C₄×D₂₉	—	2	2	no	—
51	C₄×D₂₉	—	4	1	no	—
52	Dic₅₈	—	4	2	no	—
53	#6599	—	4	2	no	—
54	#6599	—	4	2	no	—
55	#1497	—	29	1	no	maximal, Sylow
56	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₂₉	—	2	1	no	—
58	C₂×D₂₉	—	2	1	no	—
59	C₂×D₂₉	—	2	1	no	—
60	C₄×C₂₉	—	2	1	no	—
61	C₄×C₂₉	—	2	1	no	—
62	C₄×C₂₉	—	4	2	no	—
63	C₄×C₂₉	#131	1	2	yes	—
64	Dic₂₉	—	2	1	no	—
65	Dic₂₉	—	2	1	no	—
66	Dic₂₉	—	4	2	no	—
67	Dic₂₉	#137	1	2	yes	—
68	#533	—	29	1	no	—
69	#432	—	29	1	no	—
70	#431	—	29	1	no	—
71	Q₈⋊C₈	—	29	4	no	—
72	C₂×C₂₉	#408	1	1	yes	characteristic
73	C₂×C₂₉	#419	1	1	yes	characteristic
74	C₂×C₂₉	#421	1	1	yes	characteristic
75	D₂₉	—	4	1	no	—
76	C₄⋊Q₈	—	29	1	no	—
77	C₄⋊Q₈	#1181	1	1	yes	characteristic
78	#125	—	58	1	no	—
79	C₂²⋊Q₈	—	29	2	no	—
80	C₂²⋊Q₈	—	58	1	no	—
81	C₄×Q₈	—	29	2	no	—
82	C₄×Q₈	—	58	1	no	—
83	#117	—	58	1	no	—
84	C₂×C₄²	—	29	1	no	—
85	C₄⋊C₈	—	29	2	no	—
86	C₄⋊C₈	—	58	1	no	—
87	C₂²⋊C₈	—	58	1	no	—
88	C₂²⋊C₈	—	58	1	no	—
89	#97	—	58	1	no	—
90	C₄×C₈	—	29	2	no	—
91	C₂₉	#1497	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	58	1	no	—
93	C₂×Q₈	C₂×#1181	1	2	yes	—
94	C₂²×C₄	—	29	1	no	—
95	C₂²×C₄	—	29	1	no	—
96	C₂²×C₄	—	29	1	no	—
97	C₂×C₈	—	58	2	no	—
98	C₂×C₈	—	58	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	29	2	no	—
102	C₄⋊C₄	—	58	2	no	—
103	C₄⋊C₄	—	58	2	no	—
104	C₄⋊C₄	—	58	1	no	—
105	C₂²⋊C₄	—	58	2	no	—
106	C₄²	—	29	1	no	—
107	C₄²	—	29	1	no	—
108	C₄²	—	29	1	no	—
109	C₄²	—	58	2	no	—
110	C₄²	C₂×#1181	1	1	yes	characteristic
111	C₂³	—	29	1	no	—
112	Q₈	—	2	4	no	—
113	Q₈	—	116	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	29	1	no	—
116	C₂×C₄	—	29	1	no	—
117	C₂×C₄	—	29	1	no	—
118	C₂×C₄	—	58	2	no	—
119	C₂×C₄	—	58	2	no	—
120	C₂×C₄	—	58	2	no	—
121	C₂×C₄	—	116	1	no	—
122	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
124	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
125	C₈	—	116	2	no	—
126	C₈	—	116	2	no	—
127	C₂²	—	58	1	no	—
128	C₂²	—	58	1	no	—
129	C₂²	—	58	1	no	—
130	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	29	2	no	—
135	C₄	—	58	1	no	—
136	C₄	—	58	1	no	—
137	C₄	—	116	2	no	—
138	C₄	#11745346	1	2	yes	—
139	C₂	—	116	1	no	—
140	C₂	#14272137	1	1	yes	characteristic
141	C₂	#14272156	1	1	yes	characteristic
142	C₂	#14272162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272185

C₂

yes

maximal, characteristic

#14272181

C₂

yes

maximal, characteristic

Q₈⋊#6599

C₂

yes

maximal, complemented

#14271718

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₄

yes

characteristic

#11745308

C₄

yes

—

Q₈×Dic₂₉

C₂²

yes

—

—

—

—

—

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

—

—

—

—

C₂²

yes

—

C₄×#6599

C₂²

yes

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×Dic₅₈

—

C₂×#6599

—

C₂×#6599

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

#90781

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

C₄×Dic₂₉

—

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

—

Dic₅₈

—

—

—

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#131

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#137

yes

—

#533

—

#432

—

#431

—

Q₈⋊C₈

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂₉

#1497

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×#1181

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×#1181

yes

characteristic

111

C₂³

—

112

Q₈

—

113

Q₈

—

116

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

116

—

122

C₂×C₄

C₂²⋊#1181

yes

characteristic

123

C₂×C₄

C₂²⋊#1181

yes

characteristic

124

C₂×C₄

C₂²×#1181

yes

characteristic

125

C₈

—

116

—

126

C₈

—

116

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊#1181

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

116

—

138

C₄

#11745346

yes

—

139

C₂

—

116

—

140

C₂

#14272137

yes

characteristic

141

C₂

#14272156

yes

characteristic

142

C₂

#14272162

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	#1497
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

C₂²⋊C₄

cosocle

C₄×Dic₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. Q₈⋊#6599	C₂	semidirect
91. C₂₉	#1497	semidirect

H ≅ G/N

type

3. Q₈⋊#6599

C₂

semidirect

91. C₂₉

#1497

semidirect

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

subgroup	index
#14272185	2
#14272181	2
Q₈⋊#6599	2
#14271718	2
#1497	29

subgroup

index

Subgroups — #16834855

Conjugacy classes of subgroups

Complements

Maximal subgroups