Group #21454756 — subgroups

Subgroups — #21454756

Order 7808.

348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 181 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	#16839891	C₂	1	8	yes	maximal, complemented
3	#16839790	C₂	1	2	yes	maximal
4	#16839720	C₂	1	2	yes	maximal
5	C₂×#14519688	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×#92418	C₂²	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
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	#14519706	C₂²	1	8	yes	—
15	#14519688	C₂²	1	4	yes	—
16	#14519687	C₂²	1	4	yes	—
17	C₈×Dic₆₁	C₂²	1	4	yes	—
18	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
24	M₁₆×C₆₁	—	2	4	no	—
25	C₂×C₈×C₆₁	C₂³	1	4	yes	—
26	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₆₁	C₂³	1	2	yes	—
29	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
32	#11758196	C₂×C₄	1	8	yes	—
33	C₄×Dic₆₁	C₂³	1	4	yes	—
34	C₂×#92418	—	2	2	no	—
35	C₂×#92418	Q₈	1	4	yes	complemented
36	C₂×#92418	C₂³	1	2	yes	—
37	C₂×#92418	C₂³	1	2	yes	—
38	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	4	2	no	—
40	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₁	#41	1	2	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂²×C₄	1	8	yes	—
47	Dic₁₂₂	—	4	4	no	—
48	C₈×C₆₁	—	2	4	no	—
49	#92418	—	2	2	no	—
50	#92418	C₂×Q₈	1	4	yes	—
51	C₂²×C₆₁	—	2	2	no	—
52	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
53	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	C₄×Q₈	1	2	yes	—
57	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
59	Dic₆₁	—	2	8	no	—
60	#2961	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	—	2	2	no	—
62	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₆₁	#568	1	2	yes	—
64	#533	—	61	1	no	—
65	#446	—	61	8	no	—
66	#424	—	61	2	no	—
67	#423	D₆₁	1	1	yes	characteristic
68	#405	—	61	2	no	—
69	C₂×#97	—	61	1	no	—
70	C₆₁	#2961	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
72	C₂²×C₈	—	61	1	no	—
73	C₂²×C₈	—	61	1	no	—
74	C₄⋊Q₈	—	61	2	no	—
75	#125	—	61	2	no	—
76	C₂²⋊Q₈	—	61	4	no	—
77	C₄×Q₈	—	61	4	no	—
78	#117	—	61	2	no	—
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊C₈	—	61	8	no	—
81	C₄⋊C₈	C₂×D₆₁	1	4	yes	—
82	#97	—	61	4	no	—
83	#97	—	61	4	no	—
84	C₄×C₈	—	61	4	no	—
85	C₂×Q₈	—	61	4	no	—
86	C₂²×C₄	C₄×D₆₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	61	4	no	—
90	C₂×C₈	—	61	2	no	—
91	C₂×C₈	—	61	2	no	—
92	C₂×C₈	—	122	2	no	—
93	C₂×C₈	C₂²×D₆₁	1	4	yes	—
94	C₄⋊C₄	—	61	8	no	—
95	C₄⋊C₄	—	61	4	no	—
96	C₄⋊C₄	—	61	4	no	—
97	C₂²⋊C₄	—	61	4	no	—
98	C₄²	—	61	4	no	—
99	C₄²	C₄×D₆₁	1	1	yes	characteristic
100	C₄²	C₄×D₆₁	1	1	yes	characteristic
101	C₄²	C₂²×D₆₁	1	2	yes	—
102	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
103	Q₈	—	244	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	61	8	no	—
106	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
109	C₂×C₄	#11758224	1	2	yes	—
110	C₂×C₄	Q₈×D₆₁	1	2	yes	—
111	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	61	4	no	—
114	C₈	—	122	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#14519742	1	2	yes	—
117	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	122	8	no	—
121	C₄	#14519742	1	2	yes	—
122	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
123	C₄	C₂×#11758224	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×#14519742	1	1	yes	characteristic
126	C₂	#16840528	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839891

C₂

yes

maximal, complemented

#16839790

C₂

yes

maximal

#16839720

C₂

yes

maximal

C₂×#14519688

C₂

yes

maximal, characteristic

C₆₁×#423

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

#14519777

C₄

yes

—

#14519769

C₄

yes

—

#14519756

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

C₂²

yes

—

#14519706

C₂²

yes

—

#14519688

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

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

—

C₂×#92418

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

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

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

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

—

#92418

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

#2961

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

#568

yes

—

—

—

—

D₆₁

yes

characteristic

#405

—

C₂×#97

—

C₆₁

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₆₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₆₁

yes

characteristic

100

C₄²

C₄×D₆₁

yes

characteristic

101

C₄²

C₂²×D₆₁

yes

—

102

C₂³

C₂×C₄×D₆₁

yes

characteristic

103

Q₈

—

244

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₆₁

yes

—

107

C₂×C₄

C₂×C₄×D₆₁

yes

—

108

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

109

C₂×C₄

#11758224

yes

—

110

C₂×C₄

Q₈×D₆₁

yes

—

111

C₂×C₄

C₂³×D₆₁

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

122

—

115

C₂²

—

116

C₂²

#14519742

yes

—

117

C₂²

C₂²×C₄×D₆₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

122

—

121

C₄

#14519742

yes

—

122

C₄

C₂×Q₈×D₆₁

yes

characteristic

123

C₄

C₂×#11758224

yes

characteristic

124

C₂

—

125

C₂

C₂×#14519742

yes

characteristic

126

C₂

#16840528

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#423	C₂
hypercenter	#423	D₆₁
hypocenter	C₆₁	#2961
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#423

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #16839891	C₂	semidirect
35. C₂×#92418	Q₈	semidirect
70. C₆₁	#2961	semidirect

H ≅ G/N

type

2. #16839891

C₂

semidirect

35. C₂×#92418

Q₈

semidirect

70. C₆₁

#2961

semidirect

Subgroups — #21454756

Conjugacy classes of subgroups

Complements