Group #16829866 — subgroups

Subgroups — #16829866

Order 3712.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14271720	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271527	C₂	1	4	yes	maximal, complemented
3	#14271509	C₂	1	4	yes	maximal, complemented
4	#14271168	C₂	1	4	yes	maximal, complemented
5	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₉×#127	C₂	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₂₉	—	2	2	no	—
8	C₂²×D₄×C₂₉	C₂²	1	2	yes	—
9	C₂₉×#127	—	2	2	no	—
10	C₂₉×#127	C₂²	1	2	yes	—
11	C₂₉×#127	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	#11745288	C₂²	1	2	yes	—
14	#11745286	—	2	2	no	—
15	C₂×#90777	C₂²	1	2	yes	—
16	#11745232	C₂²	1	2	yes	complemented
17	#11745231	C₂²	1	2	yes	complemented
18	C₄×Dic₅₈	—	2	2	no	—
19	#11745180	C₂²	1	16	yes	complemented
20	#11745151	C₂²	1	2	yes	complemented
21	C₄×#6597	C₂²	1	2	yes	complemented
22	C₂⁴×C₂₉	—	2	2	no	—
23	C₂×D₄×C₂₉	—	2	2	no	—
24	C₂×D₄×C₂₉	—	2	4	no	—
25	C₂×D₄×C₂₉	—	2	4	no	—
26	C₂×D₄×C₂₉	—	2	8	no	—
27	C₂×D₄×C₂₉	—	4	8	no	—
28	C₂×D₄×C₂₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₉	D₄	1	2	yes	—
30	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₅₈	—	2	2	no	—
32	C₄²×C₂₉	D₄	1	2	yes	—
33	C₄²×C₂₉	C₂³	1	1	yes	characteristic
34	C₄²×C₂₉	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₂₉	—	2	2	no	—
36	C₄⋊Dic₂₉	—	2	2	no	—
37	C₄⋊Dic₂₉	C₂³	1	4	yes	—
38	#90779	—	2	4	no	—
39	C₄×Dic₂₉	—	2	2	no	—
40	#90777	—	2	4	no	—
41	C₂×#6597	C₂³	1	4	yes	—
42	C₂³×C₂₉	—	2	4	no	—
43	C₂³×C₂₉	—	4	8	no	—
44	C₂³×C₂₉	—	4	4	no	—
45	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
46	D₄×C₂₉	—	2	8	no	—
47	D₄×C₂₉	—	4	8	no	—
48	D₄×C₂₉	—	4	4	no	—
49	D₄×C₂₉	—	4	8	no	—
50	C₂×C₄×C₂₉	—	2	4	no	—
51	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₉	#41	1	4	yes	—
55	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₂₉	—	2	4	no	—
57	Dic₅₈	—	4	4	no	—
58	#6597	—	2	4	no	—
59	#3138	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	—	2	2	no	—
61	C₂²×C₂₉	—	4	4	no	—
62	C₂²×C₂₉	—	4	8	no	—
63	C₂²×C₂₉	—	8	8	no	—
64	C₂²×C₂₉	#124	1	2	yes	—
65	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₂₉	—	2	2	no	—
67	C₄×C₂₉	#124	1	2	yes	—
68	C₄×C₂₉	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₂₉	C₂×#41	1	2	yes	—
70	Dic₂₉	—	4	4	no	—
71	#533	—	29	1	no	—
72	C₂×#127	D₂₉	1	1	yes	characteristic
73	#490	—	29	4	no	—
74	#486	—	29	4	no	—
75	#418	—	29	4	no	—
76	C₄×M₁₆	—	29	1	no	—
77	C₂×C₂₉	—	2	2	no	—
78	C₂×C₂₉	—	8	4	no	—
79	C₂×C₂₉	C₂×#124	1	1	yes	characteristic
80	C₂×C₂₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
81	C₂²×D₄	—	2	2	no	—
82	C₂²×D₄	C₂×D₂₉	1	2	yes	—
83	C₂×M₁₆	—	29	2	no	—
84	C₄⋊Q₈	—	29	2	no	—
85	#127	—	2	2	no	—
86	#127	C₂×D₂₉	1	2	yes	—
87	#127	C₂×D₂₉	1	2	yes	—
88	#125	—	29	2	no	—
89	C₂²⋊Q₈	—	58	2	no	—
90	C₄×Q₈	—	58	2	no	—
91	#117	—	29	2	no	—
92	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
93	#102	—	29	16	no	—
94	#97	—	29	2	no	—
95	C₄×C₈	—	29	2	no	—
96	C₂₉	#3138	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	2	2	no	—
98	C₂×Q₈	—	58	2	no	—
99	C₂×D₄	—	2	8	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	4	no	—
103	C₂×D₄	—	4	8	no	—
104	C₂×D₄	C₂²×D₂₉	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
107	M₁₆	—	58	4	no	—
108	C₂×C₈	—	29	4	no	—
109	C₄⋊C₄	—	29	4	no	—
110	C₄⋊C₄	—	58	4	no	—
111	C₄⋊C₄	—	58	2	no	—
112	C₂²⋊C₄	—	58	2	no	—
113	C₄²	—	58	2	no	—
114	C₄²	C₂²⋊D₂₉	1	2	yes	—
115	C₄²	C₂²×D₂₉	1	1	yes	characteristic
116	C₄²	C₂²×D₂₉	1	1	yes	characteristic
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
121	Q₈	—	116	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	8	no	—
125	D₄	—	4	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	58	4	no	—
128	C₂×C₄	#90807	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
133	C₈	—	58	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	4	no	—
136	C₂²	—	4	8	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#11745298	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	116	4	no	—
142	C₄	#11745294	1	4	yes	—
143	C₄	#11745298	1	2	yes	—
144	C₄	C₂×#90807	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#11745298	1	1	yes	characteristic
148	C₂	C₂×#11745294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic, complemented

#14271527

C₂

yes

maximal, complemented

#14271509

C₂

yes

maximal, complemented

#14271168

C₂

yes

maximal, complemented

C₄×#90777

C₂

yes

maximal, characteristic, complemented

C₂×C₂₉×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₉

—

C₂²×D₄×C₂₉

C₂²

yes

—

C₂₉×#127

—

C₂₉×#127

C₂²

yes

—

C₂₉×#127

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

#11745286

—

C₂×#90777

C₂²

yes

—

#11745232

C₂²

yes

complemented

#11745231

C₂²

yes

complemented

C₄×Dic₅₈

—

#11745180

C₂²

yes

complemented

#11745151

C₂²

yes

complemented

C₄×#6597

C₂²

yes

complemented

C₂⁴×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

C₄×Dic₂₉

—

#90777

—

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

Dic₅₈

—

#6597

—

#3138

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#124

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

#124

yes

—

C₄×C₂₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₉

C₂×#41

yes

—

Dic₂₉

—

#533

—

C₂×#127

D₂₉

yes

—

—

—

C₄×M₁₆

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#124

yes

characteristic

C₂×C₂₉

C₂×AGL(1,ℤ/8)

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₂₉

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₂₉

yes

—

#127

C₂×D₂₉

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂₉

#3138

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂²×D₂₉

yes

—

105

C₂²×C₄

C₂²⋊D₂₉

yes

—

106

C₂²×C₄

C₂²×D₂₉

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₂₉

yes

—

115

C₄²

C₂²×D₂₉

yes

characteristic

116

C₄²

C₂²×D₂₉

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

121

Q₈

—

116

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#90807

yes

—

129

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

132

C₂×C₄

C₂³×D₂₉

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#11745298

yes

—

139

C₂²

C₂²×C₂²⋊D₂₉

yes

characteristic

140

C₄

—

141

C₄

—

116

—

142

C₄

#11745294

yes

—

143

C₄

#11745298

yes

—

144

C₄

C₂×#90807

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#11745298

yes

characteristic

148

C₂

C₂×#11745294

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting, prime radical	C₂×C₂₉×#127	C₂
socle	C₂²×C₂₉	C₂²×D₄
hypercenter	C₂×#127	D₂₉
hypocenter	C₂₉	#3138
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₉

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting, prime radical

C₂×C₂₉×#127

C₂

socle

C₂²×C₂₉

C₂²×D₄

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #14271720	C₂	semidirect
2. #14271527	C₂	semidirect
3. #14271509	C₂	semidirect
4. #14271168	C₂	semidirect
5. C₄×#90777	C₂	semidirect
16. #11745232	C₂²	semidirect
17. #11745231	C₂²	semidirect
19. #11745180	C₂²	semidirect
20. #11745151	C₂²	semidirect
21. C₄×#6597	C₂²	semidirect
96. C₂₉	#3138	semidirect

H ≅ G/N

type

1. #14271720

C₂

semidirect

2. #14271527

C₂

semidirect

3. #14271509

C₂

semidirect

4. #14271168

C₂

semidirect

5. C₄×#90777

C₂

semidirect

16. #11745232

C₂²

semidirect

17. #11745231

C₂²

semidirect

19. #11745180

C₂²

semidirect

20. #11745151

C₂²

semidirect

21. C₄×#6597

C₂²

semidirect

96. C₂₉

#3138

semidirect

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

subgroup	index
#14271720	2
#14271527	2
#14271509	2
#14271168	2
C₄×#90777	2
C₂×C₂₉×#127	2
#3138	29

subgroup

index

C₂×C₂₉×#127

Subgroups — #16829866

Conjugacy classes of subgroups

Complements

Maximal subgroups