Group #16820482 — subgroups

Subgroups — #16820482

Order 3712.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14271187	C₂	1	1	yes	maximal, characteristic
2	C₄×#90777	C₂	1	1	yes	maximal, characteristic
3	#14270932	C₂	1	4	yes	maximal
4	C₂×C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₉	C₄	1	2	yes	—
6	C₂₉×C₄⋊Q₈	—	2	1	no	—
7	C₂₉×C₄⋊Q₈	—	2	1	no	—
8	C₂₉×C₄⋊Q₈	—	2	1	no	—
9	C₂₉×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₄⋊C₄×C₂₉	C₄	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90777	C₂²	1	2	yes	—
14	C₂×#90777	C₂²	1	2	yes	—
15	C₄⋊#6597	—	2	1	no	—
16	C₄⋊#6597	—	2	1	no	—
17	#11745151	—	2	1	no	—
18	C₄×#6597	—	2	1	no	—
19	C₂×Q₈×C₂₉	—	2	2	no	—
20	C₂×Q₈×C₂₉	—	2	2	no	—
21	C₂×Q₈×C₂₉	—	4	4	no	—
22	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₉	—	2	2	no	—
27	C₄⋊C₄×C₂₉	—	2	2	no	—
28	C₄⋊C₄×C₂₉	—	2	2	no	—
29	C₄⋊C₄×C₂₉	—	2	2	no	—
30	C₄²×C₂₉	D₄	1	1	yes	characteristic
31	C₄²×C₂₉	D₄	1	1	yes	characteristic
32	C₄²×C₂₉	D₄	1	1	yes	characteristic
33	C₄²×C₂₉	D₄	1	1	yes	characteristic
34	#90777	—	2	2	no	—
35	#90777	—	2	4	no	—
36	#90777	Q₈	1	4	yes	complemented
37	C₂×#6597	—	2	2	no	—
38	C₂×#6597	—	2	2	no	—
39	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₂₉	—	4	4	no	—
41	Q₈×C₂₉	—	4	4	no	—
42	C₂×C₄×C₂₉	—	2	2	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	—	2	4	no	—
45	C₂×C₄×C₂₉	—	4	4	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×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₉	#41	1	2	yes	—
54	#6597	—	2	4	no	—
55	#6597	—	4	2	no	—
56	#1997	—	29	1	no	maximal, Sylow
57	C₂²×C₂₉	—	2	1	no	—
58	C₂²×C₂₉	—	2	1	no	—
59	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
61	C₄×C₂₉	—	2	1	no	—
62	C₄×C₂₉	—	2	1	no	—
63	C₄×C₂₉	—	4	4	no	—
64	C₄×C₂₉	#100	1	2	yes	—
65	C₄×C₂₉	#101	1	2	yes	—
66	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₂₉	#124	1	1	yes	characteristic
68	C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	D₂₉	1	1	yes	characteristic
70	#423	—	29	1	no	—
71	C₄×M₁₆	—	29	1	no	—
72	#343	—	29	4	no	—
73	C₂×C₂₉	—	2	2	no	—
74	C₂×C₂₉	#391	1	1	yes	characteristic
75	C₂×C₂₉	C₂×#100	1	1	yes	characteristic
76	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
77	C₂²×Q₈	Dic₂₉	1	2	yes	—
78	C₂×M₁₆	—	29	2	no	—
79	C₂×M₁₆	—	29	2	no	—
80	C₄⋊Q₈	—	2	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₂×C₄⋊C₄	Dic₂₉	1	2	yes	—
85	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
86	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
87	C₄⋊C₈	—	58	1	no	—
88	C₄⋊C₈	—	58	1	no	—
89	#97	—	58	1	no	—
90	C₄×C₈	—	58	1	no	—
91	C₂₉	#1997	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
98	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
99	M₁₆	—	29	4	no	—
100	M₁₆	—	58	2	no	—
101	M₁₆	—	58	4	no	—
102	C₂×C₈	—	58	2	no	—
103	C₂×C₈	—	58	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
109	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
110	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
112	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
123	C₂×C₄	Q₈×D₂₉	1	2	yes	—
124	C₂×C₄	#90809	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
127	C₈	—	58	4	no	—
128	C₈	—	116	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	Q₈×Dic₂₉	1	2	yes	—
132	C₂²	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#11745181	1	2	yes	—
137	C₄	#11745184	1	2	yes	—
138	C₄	#11745306	1	1	yes	characteristic
139	C₄	#11745308	1	2	yes	—
140	C₄	#11745309	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#14271070	1	1	yes	characteristic
143	C₂	C₂×#11745181	1	1	yes	characteristic
144	C₂	C₂×#11745184	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271187

C₂

yes

maximal, characteristic

C₄×#90777

C₂

yes

maximal, characteristic

#14270932

C₂

yes

maximal

C₂×C₂₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₉

C₄

yes

—

C₂₉×C₄⋊Q₈

—

C₂₉×C₄⋊Q₈

—

C₂₉×C₄⋊Q₈

—

C₂₉×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×#90777

C₂²

yes

—

C₂×#90777

C₂²

yes

—

—

—

—

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

—

—

Q₈

yes

complemented

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×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×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

#41

yes

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#100

yes

—

C₄×C₂₉

#101

yes

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

#124

yes

characteristic

C₄×C₂₉

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊Q₈

D₂₉

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

Dic₂₉

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₂₉

yes

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂₉

#1997

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²⋊D₂₉

yes

characteristic

109

C₄²

C₂²⋊D₂₉

yes

characteristic

110

C₄²

C₂²⋊D₂₉

yes

characteristic

111

C₄²

C₂²⋊D₂₉

yes

characteristic

112

C₂³

C₂²×Dic₂₉

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

123

C₂×C₄

Q₈×D₂₉

yes

—

124

C₂×C₄

#90809

yes

—

125

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

127

C₈

—

128

C₈

—

116

—

129

C₂²

—

130

C₂²

—

131

C₂²

Q₈×Dic₂₉

yes

—

132

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#11745181

yes

—

137

C₄

#11745184

yes

—

138

C₄

#11745306

yes

characteristic

139

C₄

#11745308

yes

—

140

C₄

#11745309

yes

characteristic

141

C₂

—

142

C₂

#14271070

yes

characteristic

143

C₂

C₂×#11745181

yes

characteristic

144

C₂

C₂×#11745184

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₉
commutator, prime radical	C₂³×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×C₄⋊Q₈	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₉
hypocenter	C₂₉	#1997
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₉

commutator, prime radical

C₂³×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×C₄⋊Q₈

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₉

hypocenter

C₂₉

#1997

LCS

G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
36. #90777	Q₈	semidirect
91. C₂₉	#1997	semidirect

H ≅ G/N

type

36. #90777

Q₈

semidirect

91. C₂₉

#1997

semidirect

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

subgroup	index
#14271187	2
C₄×#90777	2
#14270932	2
C₂×C₂₉×C₄⋊Q₈	2
#1997	29

subgroup

index

C₂×C₂₉×C₄⋊Q₈

Subgroups — #16820482

Conjugacy classes of subgroups

Complements

Maximal subgroups