Group #21462753 — subgroups

Subgroups — #21462753

Order 7808.

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×#92420	C₂	1	2	yes	maximal, complemented
2	#16840796	C₂	1	6	yes	maximal, complemented
3	#16840781	C₂	1	3	yes	maximal, complemented
4	#16840766	C₂	1	3	yes	maximal, complemented
5	C₄×#11758226	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758213	C₂²	1	6	yes	—
7	C₄⋊#92420	C₂²	1	3	yes	—
8	C₄⋊#92420	C₂²	1	6	yes	—
9	C₄⋊#92420	C₂²	1	1	yes	characteristic
10	#14519834	C₂²	1	6	yes	—
11	C₄×#92420	C₂²	1	6	yes	—
12	C₂×#11758226	C₄	1	1	yes	characteristic
13	Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄×D₂₄₄	C₄	1	3	yes	—
15	#14519739	C₂²	1	3	yes	—
16	#14519735	C₄	1	3	yes	—
17	C₄²×D₆₁	C₂²	1	3	yes	—
18	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	#11758226	—	2	4	no	—
21	C₂×D₂₄₄	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₆₁	C₂³	1	3	yes	—
25	C₂×#92420	C₂³	1	2	yes	—
26	C₂×#92420	C₂³	1	6	yes	—
27	#11758213	—	4	6	no	—
28	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
29	C₄²×C₆₁	C₂×C₄	1	3	yes	—
30	#11758198	C₂×C₄	1	6	yes	—
31	C₄⋊Dic₆₁	C₂×C₄	1	3	yes	—
32	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
33	C₄×Dic₆₁	C₂³	1	3	yes	—
34	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
35	C₂²×D₆₁	C₂²×C₄	1	3	yes	—
36	Q₈×C₆₁	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
42	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
43	D₂₄₄	—	2	6	no	—
44	C₄×D₆₁	—	2	6	no	—
45	C₄×D₆₁	—	2	6	no	—
46	C₄×D₆₁	—	2	3	no	—
47	C₄×D₆₁	—	2	3	no	—
48	#92420	—	2	6	no	—
49	#92420	—	2	2	no	—
50	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₆₁	—	2	3	no	—
52	C₂×D₆₁	—	2	3	no	—
53	C₂×D₆₁	—	2	3	no	—
54	C₄×C₆₁	—	2	3	no	—
55	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
56	C₄×C₆₁	#131	1	2	yes	—
57	Dic₆₁	—	2	1	no	—
58	Dic₆₁	—	2	3	no	—
59	Dic₆₁	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₆₁	2₋¹⁺⁴	1	1	yes	characteristic
61	#2991	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	#520	1	1	yes	characteristic
63	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
65	D₆₁	—	4	3	no	—
66	C₄×#41	—	61	1	no	—
67	C₈×Q₈	—	61	2	no	—
68	#436	—	61	6	no	—
69	#431	—	61	3	no	—
70	#423	—	61	3	no	—
71	C₆₁	#2991	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	61	1	no	—
73	C₂×M₁₆	—	61	6	no	—
74	C₄×Q₈	—	61	1	no	—
75	C₄×Q₈	#6923	1	1	yes	characteristic
76	C₄×D₄	—	61	3	no	—
77	C₄×D₄	—	61	3	no	—
78	#117	—	61	3	no	—
79	C₂×C₄²	—	61	3	no	—
80	C₄⋊C₈	—	61	3	no	—
81	C₄⋊C₈	—	61	6	no	—
82	C₄⋊C₈	—	61	1	no	—
83	C₂²⋊C₈	—	61	6	no	—
84	C₄×C₈	—	61	6	no	—
85	#41	—	122	4	no	—
86	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
87	C₂×D₄	—	61	3	no	—
88	C₂²×C₄	—	61	3	no	—
89	C₂²×C₄	—	61	3	no	—
90	C₂²×C₄	—	61	3	no	—
91	M₁₆	—	244	6	no	—
92	C₂×C₈	—	61	2	no	—
93	C₂×C₈	—	61	6	no	—
94	C₄⋊C₄	—	61	3	no	—
95	C₄⋊C₄	C₂×#6923	1	3	yes	—
96	C₂²⋊C₄	—	61	6	no	—
97	C₄²	—	61	3	no	—
98	C₄²	—	61	1	no	—
99	C₄²	—	61	3	no	—
100	C₄²	C₂×#6923	1	3	yes	—
101	C₂³	—	61	3	no	—
102	Q₈	#11758213	1	4	yes	complemented
103	D₄	—	122	6	no	—
104	C₂×C₄	—	61	1	no	—
105	C₂×C₄	—	61	1	no	—
106	C₂×C₄	—	61	3	no	—
107	C₂×C₄	—	122	3	no	—
108	C₂×C₄	—	122	3	no	—
109	C₂×C₄	—	122	6	no	—
110	C₂×C₄	—	122	6	no	—
111	C₂×C₄	C₂²×#6923	1	3	yes	—
112	C₂×C₄	C₂²×#6923	1	3	yes	—
113	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
114	C₈	—	122	2	no	—
115	C₈	—	122	6	no	—
116	C₂²	—	122	3	no	—
117	C₂²	—	122	3	no	—
118	C₂²	—	122	3	no	—
119	C₂²	C₂³×#6923	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	61	1	no	—
122	C₄	—	61	1	no	—
123	C₄	—	122	1	no	—
124	C₄	—	122	3	no	—
125	C₄	C₂×#11758213	1	6	yes	—
126	C₄	#14519865	1	2	yes	—
127	C₂	—	244	3	no	—
128	C₂	#16840831	1	1	yes	characteristic
129	C₂	C₂²×#11758213	1	1	yes	characteristic
130	C₂	C₂×#14519865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#92420

C₂

yes

maximal, complemented

#16840796

C₂

yes

maximal, complemented

#16840781

C₂

yes

maximal, complemented

#16840766

C₂

yes

maximal, complemented

C₄×#11758226

C₂

yes

maximal, characteristic

C₂×#11758213

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

C₄⋊#92420

C₂²

yes

characteristic

#14519834

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₂×#11758226

C₄

yes

characteristic

Q₈×Dic₆₁

C₂²

yes

characteristic

C₄×D₂₄₄

C₄

yes

—

#14519739

C₂²

yes

—

#14519735

C₄

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

#11758226

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

#11758213

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

#11758198

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²×C₄

yes

—

Q₈×C₆₁

M₁₆

yes

complemented

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

#92420

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

#131

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

2₋¹⁺⁴

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

D₆₁

—

C₄×#41

—

C₈×Q₈

—

#436

—

#431

—

#423

—

C₆₁

#2991

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

#6923

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

122

—

C₂×Q₈

C₂×#6923

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6923

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#6923

yes

—

101

C₂³

—

102

Q₈

#11758213

yes

complemented

103

D₄

—

122

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

122

—

111

C₂×C₄

C₂²×#6923

yes

—

112

C₂×C₄

C₂²×#6923

yes

—

113

C₂×C₄

C₂²×#6923

yes

characteristic

114

C₈

—

122

—

115

C₈

—

122

—

116

C₂²

—

122

—

117

C₂²

—

122

—

118

C₂²

—

122

—

119

C₂²

C₂³×#6923

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

122

—

124

C₄

—

122

—

125

C₄

C₂×#11758213

yes

—

126

C₄

#14519865

yes

—

127

C₂

—

244

—

128

C₂

#16840831

yes

characteristic

129

C₂

C₂²×#11758213

yes

characteristic

130

C₂

C₂×#14519865

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×Q₈×C₆₁	C₄
hypercenter	C₄×Q₈	#6923
hypocenter	C₆₁	#2991
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₄×Q₈×C₆₁

C₄

C₄×Q₈

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#92420	C₂	semidirect
2. #16840796	C₂	semidirect
3. #16840781	C₂	semidirect
4. #16840766	C₂	semidirect
36. Q₈×C₆₁	M₁₆	semidirect
71. C₆₁	#2991	semidirect
102. Q₈	#11758213	semidirect

H ≅ G/N

type

1. Q₈×#92420

C₂

semidirect

2. #16840796

C₂

semidirect

3. #16840781

C₂

semidirect

4. #16840766

C₂

semidirect

36. Q₈×C₆₁

M₁₆

semidirect

71. C₆₁

#2991

semidirect

102. Q₈

#11758213

semidirect

Subgroups — #21462753

Conjugacy classes of subgroups

Complements