Group #16822684 — subgroups

Subgroups — #16822684

Order 3712.

456 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 213 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	#14271062	C₂	1	3	yes	maximal, complemented
2	#14271056	C₂	1	3	yes	maximal
3	#14271049	C₂	1	3	yes	maximal, complemented
4	#14271038	C₂	1	3	yes	maximal
5	#14271033	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271031	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₉×#126	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₂₉	C₂²	1	1	yes	characteristic
9	C₂₉×#126	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₂₉	C₂²	1	3	yes	—
11	C₂×C₂²⋊C₄×C₂₉	C₂²	1	3	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₂₉	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₂₉	C₂²	1	3	yes	—
15	C₂×C₄⋊Dic₂₉	C₂²	1	3	yes	—
16	C₂×C₄⋊Dic₂₉	C₂²	1	3	yes	—
17	C₂×C₄×Dic₂₉	C₂²	1	3	yes	—
18	#11745179	C₂²	1	3	yes	—
19	#11745179	C₂²	1	3	yes	—
20	#11745179	C₂²	1	3	yes	—
21	#11745179	C₂²	1	3	yes	—
22	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
25	C₂²×Dic₂₉	—	4	3	no	—
26	C₂²×Dic₂₉	C₂³	1	3	yes	—
27	C₂²×Dic₂₉	C₂³	1	3	yes	—
28	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₉	C₂×C₄	1	12	yes	complemented
31	C₂²⋊C₄×C₂₉	C₂×C₄	1	12	yes	—
32	C₄²×C₂₉	C₂×C₄	1	4	yes	complemented
33	C₂²⋊Dic₂₉	—	4	3	no	—
34	C₄⋊Dic₂₉	—	4	3	no	—
35	C₄⋊Dic₂₉	—	4	3	no	—
36	C₄⋊Dic₂₉	—	4	3	no	—
37	C₄×Dic₂₉	—	4	3	no	—
38	C₂³×C₂₉	—	2	3	no	—
39	C₂³×C₂₉	—	2	3	no	—
40	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
41	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	2	3	no	—
43	C₂×C₄×C₂₉	—	2	3	no	—
44	C₂×C₄×C₂₉	—	2	3	no	—
45	C₂×C₄×C₂₉	—	2	3	no	—
46	C₂×C₄×C₂₉	C₂²×C₄	1	6	yes	—
47	C₂×C₄×C₂₉	C₂²×C₄	1	6	yes	—
48	C₂×Dic₂₉	—	2	3	no	—
49	C₂×Dic₂₉	—	2	3	no	—
50	C₂×Dic₂₉	—	2	3	no	—
51	C₂×Dic₂₉	—	2	3	no	—
52	C₂×Dic₂₉	—	2	3	no	—
53	C₂×Dic₂₉	—	2	3	no	—
54	C₂×Dic₂₉	—	2	6	no	—
55	C₂×Dic₂₉	—	4	4	no	—
56	#2383	—	29	1	no	maximal, Sylow
57	C₂²×C₂₉	—	2	6	no	—
58	C₂²×C₂₉	—	4	1	no	—
59	C₂²×C₂₉	—	4	3	no	—
60	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₂₉	H(5,𝔽₂)	1	3	yes	—
62	C₂²×C₂₉	2₋¹⁺⁴	1	3	yes	—
63	C₄×C₂₉	—	2	6	no	—
64	C₄×C₂₉	—	2	6	no	—
65	Dic₂₉	—	4	3	no	—
66	Dic₂₉	—	4	1	no	—
67	Dic₂₉	—	4	1	no	—
68	Dic₂₉	—	4	3	no	—
69	C₂×#126	D₂₉	1	1	yes	characteristic
70	#389	—	29	3	no	—
71	#388	—	29	3	no	—
72	#387	—	29	3	no	—
73	#385	—	29	3	no	—
74	#384	—	29	1	no	—
75	#383	—	29	1	no	—
76	C₂×C₂₉	—	4	2	no	—
77	C₂×C₂₉	#520	1	3	yes	—
78	C₂×C₂₉	#560	1	1	yes	characteristic
79	C₂×C₂₉	#563	1	3	yes	—
80	C₂³×C₄	—	29	1	no	—
81	#126	Dic₂₉	1	8	yes	complemented
82	C₂×C₄⋊C₄	—	29	3	no	—
83	C₂×C₄⋊C₄	—	29	3	no	—
84	C₂×C₄⋊C₄	—	29	3	no	—
85	C₂×C₄⋊C₄	C₂×D₂₉	1	3	yes	—
86	C₂×C₂²⋊C₄	—	29	3	no	—
87	C₂×C₂²⋊C₄	C₂×D₂₉	1	3	yes	—
88	C₂×C₄²	—	29	3	no	—
89	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
90	#95	—	29	3	no	—
91	#95	—	29	3	no	—
92	#95	—	29	3	no	—
93	#95	—	29	3	no	—
94	C₂₉	#2383	1	1	yes	characteristic, complemented, Sylow
95	C₂⁴	C₂²×D₂₉	1	1	yes	characteristic
96	C₂²×C₄	—	29	3	no	—
97	C₂²×C₄	—	29	1	no	—
98	C₂²×C₄	—	29	1	no	—
99	C₂²×C₄	—	29	3	no	—
100	C₂²×C₄	—	116	3	no	—
101	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
102	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
103	C₄⋊C₄	—	116	3	no	—
104	C₄⋊C₄	—	116	3	no	—
105	C₄⋊C₄	—	116	3	no	—
106	C₄⋊C₄	C₂×Dic₂₉	1	12	yes	complemented
107	C₂²⋊C₄	—	116	3	no	—
108	C₂²⋊C₄	C₂×Dic₂₉	1	12	yes	—
109	C₄²	—	116	3	no	—
110	C₄²	C₂×Dic₂₉	1	4	yes	complemented
111	C₂³	—	2	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	C₂²×Dic₂₉	1	2	yes	—
114	C₂³	C₂³×D₂₉	1	1	yes	characteristic
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	58	3	no	—
120	C₂×C₄	—	58	3	no	—
121	C₂×C₄	—	58	3	no	—
122	C₂×C₄	—	58	3	no	—
123	C₂×C₄	—	58	3	no	—
124	C₂×C₄	—	58	3	no	—
125	C₂×C₄	—	58	6	no	—
126	C₂×C₄	—	116	4	no	—
127	C₂×C₄	C₂²×Dic₂₉	1	6	yes	—
128	C₂×C₄	C₂²×Dic₂₉	1	6	yes	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	3	no	—
132	C₂²	#11745364	1	3	yes	—
133	C₂²	#11745365	1	3	yes	—
134	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
135	C₄	—	2	6	no	—
136	C₄	—	2	6	no	—
137	C₄	—	116	1	no	—
138	C₄	—	116	1	no	—
139	C₄	—	116	3	no	—
140	C₄	—	116	3	no	—
141	C₂	—	4	2	no	—
142	C₂	#14271646	1	3	yes	—
143	C₂	#14271904	1	1	yes	characteristic
144	C₂	#14271915	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271062

C₂

yes

maximal, complemented

#14271056

C₂

yes

maximal

#14271049

C₂

yes

maximal, complemented

#14271038

C₂

yes

maximal

#14271033

C₂

yes

maximal, characteristic, complemented

#14271031

C₂

yes

maximal, characteristic, complemented

C₂×C₂₉×#126

C₂

yes

maximal, characteristic

C₂³×Dic₂₉

C₂²

yes

characteristic

C₂₉×#126

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

C₂⁴×C₂₉

C₂³

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×Dic₂₉

—

C₂²×Dic₂₉

C₂³

yes

—

C₂²×Dic₂₉

C₂³

yes

—

C₂²×Dic₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

C₂×C₄

yes

complemented

C₂²⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

complemented

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄×Dic₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

C₂²×C₄

yes

—

C₂³×C₂₉

C₂⁴

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

#2383

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₂²×C₂₉

H(5,𝔽₂)

yes

—

C₂²×C₂₉

2₋¹⁺⁴

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

C₂×#126

D₂₉

yes

—

—

—

—

—

—

C₂×C₂₉

—

C₂×C₂₉

#520

yes

—

C₂×C₂₉

#560

yes

characteristic

C₂×C₂₉

#563

yes

—

C₂³×C₄

—

#126

Dic₂₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

—

—

—

—

C₂₉

#2383

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₂₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

116

—

101

C₂²×C₄

C₂²×D₂₉

yes

—

102

C₂²×C₄

C₂²×D₂₉

yes

—

103

C₄⋊C₄

—

116

—

104

C₄⋊C₄

—

116

—

105

C₄⋊C₄

—

116

—

106

C₄⋊C₄

C₂×Dic₂₉

yes

complemented

107

C₂²⋊C₄

—

116

—

108

C₂²⋊C₄

C₂×Dic₂₉

yes

—

109

C₄²

—

116

—

110

C₄²

C₂×Dic₂₉

yes

complemented

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²×Dic₂₉

yes

—

114

C₂³

C₂³×D₂₉

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

116

—

127

C₂×C₄

C₂²×Dic₂₉

yes

—

128

C₂×C₄

C₂²×Dic₂₉

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#11745364

yes

—

133

C₂²

#11745365

yes

—

134

C₂²

C₂³×Dic₂₉

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

116

—

138

C₄

—

116

—

139

C₄

—

116

—

140

C₄

—

116

—

141

C₂

—

142

C₂

#14271646

yes

—

143

C₂

#14271904

yes

characteristic

144

C₂

#14271915

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×#126	C₂
socle, cosocle	C₂³×C₂₉	C₂⁴
prime radical	C₂⁴×C₂₉	C₂³
hypercenter	C₂×#126	D₂₉
hypocenter	C₂₉	#2383
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₉

commutator

C₂²×C₂₉

C₂³×C₄

Fitting

C₂×C₂₉×#126

C₂

socle, cosocle

C₂³×C₂₉

C₂⁴

prime radical

C₂⁴×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. #14271062	C₂	semidirect
3. #14271049	C₂	semidirect
5. #14271033	C₂	semidirect
6. #14271031	C₂	semidirect
9. C₂₉×#126	C₄	semidirect
30. C₄⋊C₄×C₂₉	C₂×C₄	semidirect
32. C₄²×C₂₉	C₂×C₄	semidirect
81. #126	Dic₂₉	semidirect
94. C₂₉	#2383	semidirect
106. C₄⋊C₄	C₂×Dic₂₉	semidirect
110. C₄²	C₂×Dic₂₉	semidirect

H ≅ G/N

type

1. #14271062

C₂

semidirect

3. #14271049

C₂

semidirect

5. #14271033

C₂

semidirect

6. #14271031

C₂

semidirect

9. C₂₉×#126

C₄

semidirect

30. C₄⋊C₄×C₂₉

C₂×C₄

semidirect

32. C₄²×C₂₉

C₂×C₄

semidirect

81. #126

Dic₂₉

semidirect

94. C₂₉

#2383

semidirect

106. C₄⋊C₄

C₂×Dic₂₉

semidirect

110. C₄²

C₂×Dic₂₉

semidirect

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

subgroup	index
#14271062	2
#14271056	2
#14271049	2
#14271038	2
#14271033	2
#14271031	2
C₂×C₂₉×#126	2
#2383	29

subgroup

index

C₂×C₂₉×#126

Subgroups — #16822684

Conjugacy classes of subgroups

Complements

Maximal subgroups