Group #16818172 — subgroups

Subgroups — #16818172

Order 3712.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#11745231	C₂	1	1	yes	maximal, characteristic
2	#14271187	C₂	1	1	yes	maximal, characteristic
3	#14270911	C₂	1	4	yes	maximal, complemented
4	C₂₉×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₄	1	2	yes	—
6	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₉	—	2	1	no	—
8	C₂×C₄⋊Dic₂₉	—	2	1	no	—
9	C₂×#90777	—	2	1	no	—
10	#11745231	—	2	1	no	—
11	#11745231	—	2	1	no	—
12	#11745231	C₄	1	2	yes	—
13	#11745231	C₂²	1	2	yes	—
14	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂₉×C₂²⋊C₈	—	2	1	no	—
16	C₄×C₈×C₂₉	—	2	1	no	—
17	C₄⋊#6597	—	2	1	no	—
18	C₄⋊#6597	C₂²	1	2	yes	—
19	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
22	C₂²×Dic₂₉	—	2	2	no	—
23	C₂×Dic₅₈	—	2	2	no	—
24	C₂×Dic₅₈	—	4	4	no	—
25	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₂₉	—	2	2	no	—
27	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₉	D₄	1	1	yes	characteristic
29	C₄²×C₂₉	D₄	1	1	yes	characteristic
30	C₄²×C₂₉	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₂₉	—	2	2	no	—
32	C₄⋊Dic₂₉	—	2	2	no	—
33	C₄⋊Dic₂₉	—	2	2	no	—
34	C₄⋊Dic₂₉	—	2	2	no	—
35	#90777	—	4	2	no	—
36	C₂×#6597	—	2	2	no	—
37	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	—	2	2	no	—
39	C₂×C₄×C₂₉	—	2	2	no	—
40	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
45	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₂₉	—	2	4	no	—
49	C₂×Dic₂₉	—	4	4	no	—
50	Dic₅₈	—	4	4	no	—
51	Dic₅₈	—	4	4	no	—
52	C₈×C₂₉	—	4	2	no	—
53	#6597	—	4	2	no	—
54	#1548	—	29	1	no	maximal, Sylow
55	C₂²×C₂₉	—	2	1	no	—
56	C₂²×C₂₉	—	2	1	no	—
57	C₂²×C₂₉	#103	1	2	yes	—
58	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₂₉	—	2	1	no	—
60	C₄×C₂₉	—	2	1	no	—
61	C₄×C₂₉	#100	1	2	yes	—
62	C₄×C₂₉	#103	1	2	yes	—
63	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₂₉	#137	1	2	yes	—
66	Dic₂₉	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	29	1	no	—
68	#431	D₂₉	1	1	yes	characteristic
69	#423	—	29	1	no	—
70	#333	—	29	4	no	—
71	C₂×C₂₉	—	2	2	no	—
72	C₂×C₂₉	C₂×#100	1	1	yes	characteristic
73	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
74	C₂×C₂₉	#419	1	1	yes	characteristic
75	C₂²×Q₈	—	29	2	no	—
76	C₂×M₁₆	—	58	1	no	—
77	C₄⋊Q₈	—	29	2	no	—
78	C₄⋊Q₈	—	29	2	no	—
79	C₄⋊Q₈	—	58	1	no	—
80	C₄⋊Q₈	—	58	1	no	—
81	C₂×C₄⋊C₄	—	58	1	no	—
82	C₂×C₄⋊C₄	—	58	1	no	—
83	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
84	C₄⋊C₈	—	29	2	no	—
85	C₄⋊C₈	—	58	1	no	—
86	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
87	C₂²⋊C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₂₉	#1548	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	29	4	no	—
91	C₂×Q₈	—	58	2	no	—
92	C₂×Q₈	—	116	4	no	—
93	C₂²×C₄	—	58	2	no	—
94	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
95	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
97	M₁₆	—	116	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	58	2	no	—
100	C₄⋊C₄	—	58	2	no	—
101	C₄⋊C₄	—	58	2	no	—
102	C₄⋊C₄	—	58	2	no	—
103	C₄⋊C₄	—	58	2	no	—
104	C₄²	C₄×D₂₉	1	1	yes	characteristic
105	C₄²	D₁₁₆	1	1	yes	characteristic
106	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
107	C₄²	C₂²×D₂₉	1	1	yes	characteristic
108	C₂³	#90781	1	1	yes	characteristic
109	Q₈	—	116	4	no	—
110	Q₈	—	116	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	58	4	no	—
114	C₂×C₄	—	116	4	no	—
115	C₂×C₄	#90773	1	2	yes	—
116	C₂×C₄	Dic₁₁₆	1	2	yes	—
117	C₂×C₄	#90781	1	1	yes	characteristic
118	C₂×C₄	#90781	1	1	yes	characteristic
119	C₂×C₄	#90781	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
121	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	116	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#11745164	1	2	yes	—
128	C₂²	C₂×#90781	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	116	4	no	—
132	C₄	#11745164	1	2	yes	—
133	C₄	#11745171	1	2	yes	—
134	C₄	C₂×#90773	1	1	yes	characteristic
135	C₄	C₂×Dic₁₁₆	1	1	yes	characteristic
136	C₄	#11745313	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#11745171	1	1	yes	characteristic
139	C₂	C₂×#11745164	1	1	yes	characteristic
140	C₂	#14271171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

#14271187

C₂

yes

maximal, characteristic

#14270911

C₂

yes

maximal, complemented

C₂₉×#431

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₄

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

—

—

—

C₄

yes

—

#11745231

C₂²

yes

—

C₂₉×C₄⋊C₈

C₂²

yes

—

C₂₉×C₂²⋊C₈

—

C₄×C₈×C₂₉

—

C₄⋊#6597

—

C₄⋊#6597

C₂²

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

#90777

—

C₂×#6597

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

#6597

—

#1548

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#103

yes

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#100

yes

—

C₄×C₂₉

#103

yes

—

C₄×C₂₉

C₂×SD₁₆

yes

characteristic

C₄×C₂₉

C₂×Q₁₆

yes

characteristic

C₄×C₂₉

#137

yes

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

#431

D₂₉

yes

characteristic

#423

—

#333

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₉

#1548

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

M₁₆

—

116

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₂₉

yes

characteristic

105

C₄²

D₁₁₆

yes

characteristic

106

C₄²

C₂²⋊D₂₉

yes

characteristic

107

C₄²

C₂²×D₂₉

yes

characteristic

108

C₂³

#90781

yes

characteristic

109

Q₈

—

116

—

110

Q₈

—

116

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

116

—

115

C₂×C₄

#90773

yes

—

116

C₂×C₄

Dic₁₁₆

yes

—

117

C₂×C₄

#90781

yes

characteristic

118

C₂×C₄

#90781

yes

characteristic

119

C₂×C₄

#90781

yes

characteristic

120

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

121

C₂×C₄

C₂×D₁₁₆

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

123

C₈

—

124

C₈

—

116

—

125

C₂²

—

126

C₂²

—

127

C₂²

#11745164

yes

—

128

C₂²

C₂×#90781

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

116

—

132

C₄

#11745164

yes

—

133

C₄

#11745171

yes

—

134

C₄

C₂×#90773

yes

characteristic

135

C₄

C₂×Dic₁₁₆

yes

characteristic

136

C₄

#11745313

yes

—

137

C₂

—

138

C₂

C₂×#11745171

yes

characteristic

139

C₂

C₂×#11745164

yes

characteristic

140

C₂

#14271171

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#90781
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#431	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₄²×C₂₉	C₂³
hypercenter	#431	D₂₉
hypocenter	C₂₉	#1548
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#90781

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#431

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₄²×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14270911	C₂	semidirect
89. C₂₉	#1548	semidirect

H ≅ G/N

type

3. #14270911

C₂

semidirect

89. C₂₉

#1548

semidirect

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

subgroup	index
C₂×#11745231	2
#14271187	2
#14270911	2
C₂₉×#431	2
#1548	29

subgroup

index

C₂₉×#431

Subgroups — #16818172

Conjugacy classes of subgroups

Complements

Maximal subgroups