Group #21454753 — subgroups

Subgroups — #21454753

Order 7808.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₁₂₂	C₂	1	4	yes	maximal, complemented
4	#16839788	C₂	1	2	yes	maximal
5	#16839786	C₂	1	1	yes	maximal, characteristic
6	#16839717	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758194	C₂²	1	2	yes	—
9	#14519777	C₄	1	2	yes	—
10	#14519769	C₄	1	2	yes	—
11	#14519756	C₄	1	4	yes	—
12	C₄×Dic₁₂₂	C₂²	1	4	yes	—
13	#14519736	C₂²	1	2	yes	—
14	C₄⋊#92418	C₂²	1	4	yes	—
15	#14519706	C₂²	1	8	yes	—
16	#14519687	C₂²	1	4	yes	—
17	C₈×Dic₆₁	C₂²	1	4	yes	—
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₆₁	C₂²	1	2	yes	—
22	C₄×C₈×C₆₁	C₂²	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
28	C₂×C₈×C₆₁	C₂³	1	2	yes	—
29	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂³	1	2	yes	—
34	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
35	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
37	#11758196	C₂×C₄	1	8	yes	—
38	C₄×Dic₆₁	C₂³	1	4	yes	—
39	#11758194	—	2	4	no	—
40	C₂×#92418	C₂³	1	4	yes	—
41	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	4	no	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₆₁	#41	1	2	yes	—
48	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂²×C₄	1	8	yes	—
50	Dic₁₂₂	—	2	8	no	—
51	C₈×C₆₁	—	2	2	no	—
52	C₈×C₆₁	C₂×Q₈	1	2	yes	—
53	C₈×C₆₁	C₂×Q₈	1	2	yes	—
54	#92418	—	2	4	no	—
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
57	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	C₄×Q₈	1	2	yes	—
60	C₄×C₆₁	#131	1	4	yes	—
61	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
63	Dic₆₁	—	2	8	no	—
64	#2960	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₂×C₆₁	C₂×#131	1	2	yes	—
68	#533	—	61	1	no	—
69	#446	—	61	4	no	—
70	C₈×Q₈	—	61	4	no	—
71	#424	—	61	2	no	—
72	#423	—	61	1	no	—
73	#405	—	61	2	no	—
74	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
75	C₆₁	#2960	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	61	2	no	—
77	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
79	C₄⋊Q₈	—	61	2	no	—
80	#125	—	61	2	no	—
81	C₂²⋊Q₈	—	61	4	no	—
82	C₄×Q₈	—	61	4	no	—
83	#117	—	61	2	no	—
84	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
85	C₄⋊C₈	—	61	8	no	—
86	C₄⋊C₈	—	61	4	no	—
87	#97	—	61	4	no	—
88	C₄×C₈	—	61	4	no	—
89	C₄×C₈	C₂×D₆₁	1	2	yes	—
90	C₄×C₈	C₂×D₆₁	1	2	yes	—
91	C₂×Q₈	—	61	4	no	—
92	C₂²×C₄	C₄×D₆₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
94	M₁₆	—	122	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	61	4	no	—
97	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
98	C₂×C₈	C₂²×D₆₁	1	2	yes	—
99	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
101	C₄⋊C₄	—	61	8	no	—
102	C₄⋊C₄	—	61	4	no	—
103	C₄⋊C₄	—	61	4	no	—
104	C₂²⋊C₄	—	61	4	no	—
105	C₄²	—	61	4	no	—
106	C₄²	C₄×D₆₁	1	1	yes	characteristic
107	C₄²	C₄×D₆₁	1	1	yes	characteristic
108	C₄²	C₂²×D₆₁	1	2	yes	—
109	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
110	Q₈	—	122	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	61	8	no	—
113	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
117	C₂×C₄	#11758222	1	2	yes	—
118	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	122	4	no	—
121	C₈	C₂×Dic₁₂₂	1	2	yes	—
122	C₈	C₂×Dic₁₂₂	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₁₂₂	1	2	yes	—
125	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	122	8	no	—
128	C₄	C₄×Dic₁₂₂	1	2	yes	—
129	C₄	#14519783	1	4	yes	—
130	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
131	C₄	C₂×#11758222	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
134	C₂	C₂×#14519783	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, complemented

C₈×Dic₁₂₂

C₂

yes

maximal, complemented

#16839788

C₂

yes

maximal

#16839786

C₂

yes

maximal, characteristic

#16839717

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

#14519777

C₄

yes

—

#14519769

C₄

yes

—

#14519756

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

#2960

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂×#131

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₆₁

yes

characteristic

C₆₁

#2960

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

complemented

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

100

C₂×C₈

C₂²×D₆₁

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₆₁

yes

characteristic

107

C₄²

C₄×D₆₁

yes

characteristic

108

C₄²

C₂²×D₆₁

yes

—

109

C₂³

C₂×C₄×D₆₁

yes

characteristic

110

Q₈

—

122

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₁₂₂

yes

—

114

C₂×C₄

C₂×C₄×D₆₁

yes

—

115

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

116

C₂×C₄

C₂×C₄×D₆₁

yes

—

117

C₂×C₄

#11758222

yes

—

118

C₂×C₄

C₂³×D₆₁

yes

characteristic

119

C₈

—

120

C₈

—

122

—

121

C₈

C₂×Dic₁₂₂

yes

—

122

C₈

C₂×Dic₁₂₂

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₁₂₂

yes

—

125

C₂²

C₂²×C₄×D₆₁

yes

characteristic

126

C₄

—

127

C₄

—

122

—

128

C₄

C₄×Dic₁₂₂

yes

—

129

C₄

#14519783

yes

—

130

C₄

C₂²×Dic₁₂₂

yes

characteristic

131

C₄

C₂×#11758222

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₁₂₂

yes

characteristic

134

C₂

C₂×#14519783

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	#2960
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

hypercenter

C₂×C₄×C₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. C₈×Dic₁₂₂	C₂	semidirect
27. C₂×C₈×C₆₁	Q₈	semidirect
75. C₆₁	#2960	semidirect
97. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16839892

C₂

semidirect

3. C₈×Dic₁₂₂

C₂

semidirect

27. C₂×C₈×C₆₁

Q₈

semidirect

75. C₆₁

#2960

semidirect

97. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21454753

Conjugacy classes of subgroups

Complements