Group #16819231 — subgroups

Subgroups — #16819231

Order 3712.

388 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×#90777	C₂	1	2	yes	maximal
2	#14270932	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₉×#117	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
5	C₂₉×#117	—	2	2	no	—
6	C₂₉×#117	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
8	C₂×C₂²⋊C₄×C₂₉	C₄	1	2	yes	—
9	C₂×C₄²×C₂₉	C₄	1	2	yes	—
10	C₂×#90777	—	2	4	no	—
11	C₂×#90777	—	2	2	no	—
12	C₂×#90777	C₄	1	8	yes	complemented
13	C₂×#90777	C₂²	1	4	yes	—
14	C₂²×#6597	—	2	2	no	—
15	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	D₄	1	2	yes	—
20	C₂²×C₄×C₂₉	D₄	1	2	yes	—
21	C₂²×C₄×C₂₉	D₄	1	2	yes	—
22	C₂²×C₄×C₂₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₉	—	2	4	no	—
25	C₂²⋊C₄×C₂₉	—	2	4	no	—
26	C₄²×C₂₉	—	2	2	no	—
27	C₄²×C₂₉	—	2	2	no	—
28	#90777	—	2	8	no	—
29	#90777	—	2	4	no	—
30	#90777	C₂×C₄	1	8	yes	—
31	C₂×#6597	—	2	8	no	—
32	C₂×#6597	—	2	4	no	—
33	C₂³×C₂₉	—	4	2	no	—
34	C₂³×C₂₉	C₄²	1	2	yes	—
35	C₂³×C₂₉	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	—	2	4	no	—
39	C₂×C₄×C₂₉	—	4	4	no	—
40	C₂×C₄×C₂₉	C₄²	1	2	yes	—
41	C₂×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
46	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
52	#6597	—	2	8	no	—
53	#1742	—	29	1	no	maximal, Sylow
54	C₂²×C₂₉	—	2	2	no	—
55	C₂²×C₂₉	—	2	4	no	—
56	C₂²×C₂₉	—	4	2	no	—
57	C₂²×C₂₉	—	4	2	no	—
58	C₂²×C₂₉	#95	1	4	yes	—
59	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
60	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₂₉	—	4	4	no	—
63	C₄×C₂₉	#95	1	2	yes	—
64	C₄×C₂₉	#95	1	2	yes	—
65	C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
66	C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₂²×M₁₆	—	29	2	no	—
68	C₂×#117	D₂₉	1	1	yes	characteristic
69	#343	—	29	4	no	—
70	C₂×C₂₉	—	2	4	no	—
71	C₂×C₂₉	—	2	1	no	—
72	C₂×C₂₉	—	2	1	no	—
73	C₂×C₂₉	C₂×#95	1	1	yes	characteristic
74	C₂×C₂₉	#413	1	2	yes	—
75	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
76	C₂×M₁₆	—	29	8	no	—
77	C₂×M₁₆	—	29	4	no	—
78	C₂×M₁₆	—	58	4	no	—
79	C₂×M₁₆	—	58	2	no	—
80	C₂²×C₈	—	58	2	no	—
81	#117	—	2	2	no	—
82	#117	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
84	C₂×C₂²⋊C₄	Dic₂₉	1	2	yes	—
85	C₂×C₄²	Dic₂₉	1	2	yes	—
86	C₂₉	#1742	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
88	C₂²×C₄	Dic₅₈	1	2	yes	—
89	C₂²×C₄	C₄×D₂₉	1	4	yes	—
90	C₂²×C₄	D₁₁₆	1	2	yes	—
91	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
92	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
96	M₁₆	—	29	8	no	—
97	M₁₆	—	58	8	no	—
98	M₁₆	—	58	4	no	—
99	C₂×C₈	—	58	8	no	—
100	C₂×C₈	—	58	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₄×Dic₂₉	1	2	yes	—
107	C₂³	C₄⋊Dic₂₉	1	2	yes	—
108	C₂³	C₂²⋊Dic₂₉	1	2	yes	—
109	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₄×Dic₂₉	1	2	yes	—
113	C₂×C₄	#90779	1	4	yes	—
114	C₂×C₄	#90779	1	4	yes	—
115	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
116	C₂×C₄	#90781	1	4	yes	—
117	C₂×C₄	#90781	1	4	yes	—
118	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
122	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
124	C₈	—	58	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	#11745179	1	4	yes	—
130	C₂²	C₂×C₄×Dic₂₉	1	1	yes	characteristic
131	C₂²	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
132	C₂²	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
133	C₄	—	4	4	no	—
134	C₄	#11745179	1	2	yes	—
135	C₄	#11745179	1	2	yes	—
136	C₄	C₂×#90779	1	2	yes	—
137	C₄	C₂×#90781	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	4	no	—
141	C₂	C₂×#11745179	1	1	yes	characteristic
142	C₂	#14271147	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#90777

C₂

yes

maximal

#14270932

C₂

yes

maximal, complemented

C₂×C₂₉×#117

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₉

C₂²

yes

characteristic

C₂₉×#117

—

C₂₉×#117

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₉

C₄

yes

—

C₂×C₄²×C₂₉

C₄

yes

—

C₂×#90777

—

C₂×#90777

—

C₂×#90777

C₄

yes

complemented

C₂×#90777

C₂²

yes

—

C₂²×#6597

—

C₂⁴×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

Q₈

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

—

#90777

—

#90777

—

#90777

C₂×C₄

yes

—

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

—

C₂³×C₂₉

C₄²

yes

—

C₂³×C₂₉

C₂²⋊C₄

yes

—

C₂³×C₂₉

C₄⋊C₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄²

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#95

yes

—

C₂²×C₂₉

C₂×C₄²

yes

characteristic

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

#95

yes

—

C₄×C₂₉

#95

yes

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₉

C₂×C₄⋊C₄

yes

—

C₂²×M₁₆

—

C₂×#117

D₂₉

yes

characteristic

#343

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#95

yes

characteristic

C₂×C₂₉

#413

yes

—

C₂³×C₄

C₂×D₂₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₂²⋊C₄

Dic₂₉

yes

—

C₂×C₄²

Dic₂₉

yes

—

C₂₉

#1742

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

Dic₅₈

yes

—

C₂²×C₄

C₄×D₂₉

yes

—

C₂²×C₄

D₁₁₆

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₂³

—

106

C₂³

C₄×Dic₂₉

yes

—

107

C₂³

C₄⋊Dic₂₉

yes

—

108

C₂³

C₂²⋊Dic₂₉

yes

—

109

C₂³

C₂²×Dic₂₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₄×Dic₂₉

yes

—

113

C₂×C₄

#90779

yes

—

114

C₂×C₄

#90779

yes

—

115

C₂×C₄

C₄⋊Dic₂₉

yes

—

116

C₂×C₄

#90781

yes

—

117

C₂×C₄

#90781

yes

—

118

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

120

C₂×C₄

C₂×Dic₅₈

yes

characteristic

121

C₂×C₄

C₂×C₄×D₂₉

yes

—

122

C₂×C₄

C₂×D₁₁₆

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11745179

yes

—

130

C₂²

C₂×C₄×Dic₂₉

yes

characteristic

131

C₂²

C₂×C₄⋊Dic₂₉

yes

characteristic

132

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

133

C₄

—

134

C₄

#11745179

yes

—

135

C₄

#11745179

yes

—

136

C₄

C₂×#90779

yes

—

137

C₄

C₂×#90781

yes

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂×#11745179

yes

characteristic

142

C₂

#14271147

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₉
commutator	C₂²×C₂₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×#117	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×#117	D₂₉
hypocenter	C₂₉	#1742
LCS	G ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₉

commutator

C₂²×C₂₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×#117

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₉

C₂×C₄

cosocle

C₂²×C₄×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #14270932	C₂	semidirect
12. C₂×#90777	C₄	semidirect
86. C₂₉	#1742	semidirect

H ≅ G/N

type

2. #14270932

C₂

semidirect

12. C₂×#90777

C₄

semidirect

86. C₂₉

#1742

semidirect

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

subgroup	index
C₂²×#90777	2
#14270932	2
C₂×C₂₉×#117	2
#1742	29

subgroup

index

C₂²×#90777

#14270932

C₂×C₂₉×#117

#1742

Subgroups — #16819231

Conjugacy classes of subgroups

Complements

Maximal subgroups