Group #16836426 — subgroups

Subgroups — #16836426

Order 3712.

394 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#6599	C₂	1	2	yes	maximal, complemented
2	#14272197	C₂	1	6	yes	maximal, complemented
3	#14272182	C₂	1	3	yes	maximal, complemented
4	#14272167	C₂	1	3	yes	maximal, complemented
5	C₄×#90809	C₂	1	1	yes	maximal, characteristic
6	C₂×#90809	C₄	1	1	yes	characteristic
7	C₂×#90796	C₂²	1	6	yes	—
8	C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
9	Q₈×Dic₂₉	C₂²	1	1	yes	characteristic
10	#11745255	C₂²	1	3	yes	—
11	#11745254	C₄	1	3	yes	—
12	C₄×D₁₁₆	C₄	1	3	yes	—
13	C₄²×D₂₉	C₂²	1	3	yes	—
14	#11745219	C₂²	1	6	yes	—
15	C₄⋊#6599	C₂²	1	3	yes	—
16	C₄⋊#6599	C₂²	1	6	yes	—
17	C₄⋊#6599	C₂²	1	1	yes	characteristic
18	C₄×#6599	C₂²	1	6	yes	—
19	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
20	#90809	—	2	4	no	—
21	C₂×D₁₁₆	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₂₉	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₂₉	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₂₉	C₂³	1	3	yes	—
25	C₂×#6599	C₂³	1	6	yes	—
26	C₂×#6599	C₂³	1	2	yes	—
27	#90796	—	4	6	no	—
28	C₄⋊C₄×C₂₉	C₂×C₄	1	3	yes	—
29	C₄²×C₂₉	C₂×C₄	1	3	yes	—
30	#90781	C₂×C₄	1	6	yes	—
31	C₄⋊Dic₂₉	C₂×C₄	1	3	yes	—
32	C₄×Dic₂₉	C₂×C₄	1	3	yes	—
33	C₄×Dic₂₉	C₂³	1	3	yes	—
34	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
35	C₂²×D₂₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₂₉	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₂₉	C₂²×C₄	1	3	yes	—
42	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
43	D₁₁₆	—	2	6	no	—
44	C₄×D₂₉	—	2	6	no	—
45	C₄×D₂₉	—	2	6	no	—
46	C₄×D₂₉	—	2	3	no	—
47	C₄×D₂₉	—	2	3	no	—
48	#6599	—	2	6	no	—
49	#6599	—	2	2	no	—
50	#2991	—	29	1	no	maximal, Sylow
51	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₂₉	—	2	3	no	—
53	C₂×D₂₉	—	2	3	no	—
54	C₂×D₂₉	—	2	3	no	—
55	C₄×C₂₉	—	2	3	no	—
56	C₄×C₂₉	C₂×M₁₆	1	6	yes	—
57	C₄×C₂₉	#131	1	2	yes	—
58	Dic₂₉	—	2	1	no	—
59	Dic₂₉	—	2	3	no	—
60	Dic₂₉	H(5,𝔽₂)	1	1	yes	characteristic
61	Dic₂₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×#41	—	29	1	no	—
63	C₈×Q₈	—	29	2	no	—
64	#436	—	29	6	no	—
65	#431	—	29	3	no	—
66	#423	—	29	3	no	—
67	C₂×C₂₉	#520	1	1	yes	characteristic
68	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
69	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
70	D₂₉	—	4	3	no	—
71	C₂×#41	—	29	1	no	—
72	C₂×M₁₆	—	29	6	no	—
73	C₄×Q₈	—	29	1	no	—
74	C₄×Q₈	#1181	1	1	yes	characteristic
75	C₄×D₄	—	29	3	no	—
76	C₄×D₄	—	29	3	no	—
77	#117	—	29	3	no	—
78	C₂×C₄²	—	29	3	no	—
79	C₄⋊C₈	—	29	3	no	—
80	C₄⋊C₈	—	29	6	no	—
81	C₄⋊C₈	—	29	1	no	—
82	C₂²⋊C₈	—	29	6	no	—
83	C₄×C₈	—	29	6	no	—
84	C₂₉	#2991	1	1	yes	characteristic, complemented, Sylow
85	#41	—	58	4	no	—
86	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
87	C₂×D₄	—	29	3	no	—
88	C₂²×C₄	—	29	3	no	—
89	C₂²×C₄	—	29	3	no	—
90	C₂²×C₄	—	29	3	no	—
91	M₁₆	—	116	6	no	—
92	C₂×C₈	—	29	2	no	—
93	C₂×C₈	—	29	6	no	—
94	C₄⋊C₄	—	29	3	no	—
95	C₄⋊C₄	C₂×#1181	1	3	yes	—
96	C₂²⋊C₄	—	29	6	no	—
97	C₄²	—	29	3	no	—
98	C₄²	—	29	1	no	—
99	C₄²	—	29	3	no	—
100	C₄²	C₂×#1181	1	3	yes	—
101	C₂³	—	29	3	no	—
102	Q₈	#90796	1	4	yes	complemented
103	D₄	—	58	6	no	—
104	C₂×C₄	—	29	1	no	—
105	C₂×C₄	—	29	1	no	—
106	C₂×C₄	—	29	3	no	—
107	C₂×C₄	—	58	3	no	—
108	C₂×C₄	—	58	3	no	—
109	C₂×C₄	—	58	6	no	—
110	C₂×C₄	—	58	6	no	—
111	C₂×C₄	C₂²×#1181	1	3	yes	—
112	C₂×C₄	C₂²×#1181	1	3	yes	—
113	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
114	C₈	—	58	6	no	—
115	C₈	—	58	2	no	—
116	C₂²	—	58	3	no	—
117	C₂²	—	58	3	no	—
118	C₂²	—	58	3	no	—
119	C₂²	C₂³×#1181	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	29	1	no	—
122	C₄	—	29	1	no	—
123	C₄	—	58	1	no	—
124	C₄	—	58	3	no	—
125	C₄	C₂×#90796	1	6	yes	—
126	C₄	#11745348	1	2	yes	—
127	C₂	—	116	3	no	—
128	C₂	#14272232	1	1	yes	characteristic
129	C₂	C₂²×#90796	1	1	yes	characteristic
130	C₂	C₂×#11745348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#6599

C₂

yes

maximal, complemented

#14272197

C₂

yes

maximal, complemented

#14272182

C₂

yes

maximal, complemented

#14272167

C₂

yes

maximal, complemented

C₄×#90809

C₂

yes

maximal, characteristic

C₂×#90809

C₄

yes

characteristic

C₂×#90796

C₂²

yes

—

C₄×Q₈×C₂₉

C₄

yes

characteristic

Q₈×Dic₂₉

C₂²

yes

characteristic

#11745255

C₂²

yes

—

#11745254

C₄

yes

—

C₄×D₁₁₆

C₄

yes

—

C₄²×D₂₉

C₂²

yes

—

#11745219

C₂²

yes

—

C₄⋊#6599

C₂²

yes

—

C₄⋊#6599

C₂²

yes

—

C₄⋊#6599

C₂²

yes

characteristic

C₄×#6599

C₂²

yes

—

C₂×Q₈×C₂₉

C₂×C₄

yes

characteristic

#90809

—

C₂×D₁₁₆

C₂×C₄

yes

—

C₂×C₄×D₂₉

C₂×C₄

yes

—

C₂×C₄×D₂₉

C₂×C₄

yes

—

C₂×C₄×D₂₉

C₂³

yes

—

C₂×#6599

C₂³

yes

—

C₂×#6599

C₂³

yes

—

#90796

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

—

#90781

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₄×Dic₂₉

C₂×C₄

yes

—

C₄×Dic₂₉

C₂³

yes

—

C₄×Dic₂₉

C₂³

yes

characteristic

C₂²×D₂₉

C₂²×C₄

yes

—

Q₈×C₂₉

M₁₆

yes

complemented

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

C₂²×C₄

yes

—

C₂×Dic₂₉

C₂⁴

yes

characteristic

D₁₁₆

—

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

—

—

—

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×M₁₆

yes

—

C₄×C₂₉

#131

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

H(5,𝔽₂)

yes

characteristic

Dic₂₉

2₋¹⁺⁴

yes

characteristic

C₄×#41

—

C₈×Q₈

—

#436

—

#431

—

#423

—

C₂×C₂₉

#520

yes

characteristic

C₂×C₂₉

C₂²×M₁₆

yes

characteristic

C₂×C₂₉

C₂×#131

yes

characteristic

D₂₉

—

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

#1181

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₉

#2991

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×#1181

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

116

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#1181

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#1181

yes

—

101

C₂³

—

102

Q₈

#90796

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×#1181

yes

—

112

C₂×C₄

C₂²×#1181

yes

—

113

C₂×C₄

C₂²×#1181

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×#1181

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×#90796

yes

—

126

C₄

#11745348

yes

—

127

C₂

—

116

—

128

C₂

#14272232

yes

characteristic

129

C₂

C₂²×#90796

yes

characteristic

130

C₂

C₂×#11745348

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₄×Q₈×C₂₉	C₄
prime radical	C₂×#90809	C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	C₄×Q₈	#1181
hypocenter	C₂₉	#2991
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#1181

commutator, socle

C₂²×C₂₉

C₂³×C₄

Fitting

C₄×Q₈×C₂₉

C₄

prime radical

C₂×#90809

C₄

cosocle

C₂×Dic₂₉

C₂⁴

C₄×Q₈

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#6599	C₂	semidirect
2. #14272197	C₂	semidirect
3. #14272182	C₂	semidirect
4. #14272167	C₂	semidirect
36. Q₈×C₂₉	M₁₆	semidirect
84. C₂₉	#2991	semidirect
102. Q₈	#90796	semidirect

H ≅ G/N

type

1. Q₈×#6599

C₂

semidirect

2. #14272197

C₂

semidirect

3. #14272182

C₂

semidirect

4. #14272167

C₂

semidirect

36. Q₈×C₂₉

M₁₆

semidirect

84. C₂₉

#2991

semidirect

102. Q₈

#90796

semidirect

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

subgroup	index
Q₈×#6599	2
#14272197	2
#14272182	2
#14272167	2
C₄×#90809	2
#2991	29

subgroup

index

Subgroups — #16836426

Conjugacy classes of subgroups

Complements

Maximal subgroups