Group #21461027 — subgroups

Subgroups — #21461027

Order 7808.

208 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 90 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	#16840780	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#92420⋊C₈	C₂	1	2	yes	maximal, complemented
4	#92420⋊C₈	C₂	1	2	yes	maximal, complemented
5	#16839714	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₈	C₄	1	2	yes	—
7	#14519844	C₄	1	2	yes	—
8	#14519834	C₄	1	2	yes	—
9	C₄×#92420	C₄	1	2	yes	—
10	C₄×#92420	C₂²	1	1	yes	characteristic
11	C₄×#92420	C₂²	1	1	yes	characteristic
12	C₄×#92420	C₂²	1	2	yes	—
13	C₂×#11758189	C₄	1	2	yes	—
14	C₄²×D₆₁	C₂²	1	1	yes	characteristic
15	#14519688	C₄	1	1	yes	characteristic
16	C₈×Dic₆₁	C₂²	1	2	yes	—
17	#97×C₆₁	C₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×#92420	C₂×C₄	1	2	yes	—
22	C₂×#92420	C₂×C₄	1	4	yes	—
23	C₂×#92420	C₂×C₄	1	2	yes	—
24	D₆₁⋊C₈	—	2	4	no	—
25	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
30	C₂×#92418	C₂×C₄	1	2	yes	—
31	#11758189	—	2	4	no	—
32	C₂²×D₆₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₆₁	C₄²	1	1	yes	characteristic
37	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄×D₆₁	—	2	2	no	—
40	C₄×D₆₁	—	2	2	no	—
41	C₄×D₆₁	M₁₆	1	4	yes	—
42	#92420	—	2	2	no	—
43	#92420	—	2	4	no	—
44	#92420	M₁₆	1	4	yes	complemented
45	C₈×C₆₁	—	2	2	no	—
46	#92418	—	2	2	no	—
47	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×D₆₁	—	2	1	no	—
49	C₂×D₆₁	—	2	1	no	—
50	C₂×D₆₁	#97	1	2	yes	—
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
54	Dic₆₁	#97	1	2	yes	—
55	Dic₆₁	C₂×M₁₆	1	2	yes	—
56	Dic₆₁	#131	1	2	yes	—
57	#1451	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	C₂×#97	1	1	yes	characteristic
59	C₂×C₆₁	C₄×M₁₆	1	1	yes	characteristic
60	C₂×C₆₁	#405	1	1	yes	characteristic
61	D₆₁	—	2	2	no	—
62	#431	—	61	1	no	—
63	C₄×M₁₆	—	61	1	no	—
64	C₂×C₄×C₈	—	61	1	no	—
65	#322	—	61	2	no	—
66	#322	—	61	2	no	—
67	C₆₁	#1451	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	61	2	no	—
69	C₂²×C₈	—	61	2	no	—
70	C₂×C₄²	—	61	1	no	—
71	C₄⋊C₈	—	61	2	no	—
72	C₂²⋊C₈	—	61	2	no	—
73	#97	—	61	1	no	—
74	#97	#6923	1	1	yes	characteristic
75	C₄×C₈	—	61	2	no	—
76	C₄×C₈	—	61	1	no	—
77	C₄×C₈	—	61	1	no	—
78	C₄×C₈	—	61	2	no	—
79	C₄×C₈	—	61	2	no	—
80	C₂²×C₄	—	61	1	no	—
81	C₂²×C₄	—	61	1	no	—
82	C₂²×C₄	—	61	1	no	—
83	M₁₆	—	122	4	no	—
84	C₂×C₈	—	61	2	no	—
85	C₂×C₈	—	61	2	no	—
86	C₂×C₈	—	61	4	no	—
87	C₂×C₈	—	61	2	no	—
88	C₂×C₈	—	122	4	no	—
89	C₂×C₈	C₂×#6923	1	2	yes	—
90	C₄²	—	61	1	no	—
91	C₄²	—	61	1	no	—
92	C₄²	—	61	1	no	—
93	C₄²	C₂×#6923	1	1	yes	characteristic
94	C₂³	—	61	1	no	—
95	C₂×C₄	—	61	1	no	—
96	C₂×C₄	—	61	1	no	—
97	C₂×C₄	—	61	1	no	—
98	C₂×C₄	—	61	4	no	—
99	C₂×C₄	—	122	2	no	—
100	C₂×C₄	—	122	2	no	—
101	C₂×C₄	C₄×#6923	1	1	yes	characteristic
102	C₂×C₄	C₄×#6923	1	1	yes	characteristic
103	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	61	4	no	—
106	C₈	—	122	4	no	—
107	C₈	—	122	2	no	—
108	C₈	—	122	2	no	—
109	C₂²	—	61	2	no	—
110	C₂²	—	122	1	no	—
111	C₂²	—	122	1	no	—
112	C₂²	C₂×C₄×#6923	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	61	2	no	—
116	C₄	—	61	2	no	—
117	C₄	—	61	2	no	—
118	C₄	D₆₁⋊M₁₆	1	2	yes	—
119	C₂	—	122	2	no	—
120	C₂	#16840721	1	1	yes	characteristic
121	C₂	M₁₆×#6923	1	1	yes	characteristic
122	C₂	#16840728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840780

C₂

yes

maximal, characteristic

C₄×D₆₁⋊C₈

C₂

yes

maximal, characteristic

#92420⋊C₈

C₂

yes

maximal, complemented

#92420⋊C₈

C₂

yes

maximal, complemented

#16839714

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

C₄

yes

—

#14519844

C₄

yes

—

#14519834

C₄

yes

—

C₄×#92420

C₄

yes

—

C₄×#92420

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

—

C₂×#11758189

C₄

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519688

C₄

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

#97×C₆₁

C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×#92420

C₂×C₄

yes

—

C₂×#92420

C₂×C₄

yes

—

C₂×#92420

C₂×C₄

yes

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

—

#11758189

—

C₂²×D₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₄²

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

M₁₆

yes

—

#92420

—

#92420

—

#92420

M₁₆

yes

complemented

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#97

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

Dic₆₁

#97

yes

—

Dic₆₁

C₂×M₁₆

yes

—

Dic₆₁

#131

yes

—

#1451

—

maximal, Sylow

C₂×C₆₁

C₂×#97

yes

characteristic

C₂×C₆₁

C₄×M₁₆

yes

characteristic

C₂×C₆₁

#405

yes

characteristic

D₆₁

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₆₁

#1451

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂×#6923

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

100

C₂×C₄

—

122

—

101

C₂×C₄

C₄×#6923

yes

characteristic

102

C₂×C₄

C₄×#6923

yes

characteristic

103

C₂×C₄

C₂²×#6923

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

122

—

107

C₈

—

122

—

108

C₈

—

122

—

109

C₂²

—

110

C₂²

—

122

—

111

C₂²

—

122

—

112

C₂²

C₂×C₄×#6923

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

D₆₁⋊M₁₆

yes

—

119

C₂

—

122

—

120

C₂

#16840721

yes

characteristic

121

C₂

M₁₆×#6923

yes

characteristic

122

C₂

#16840728

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂×C₄²
Fitting	#97×C₆₁	C₄
hypercenter	#97	#6923
hypocenter	C₆₁	#1451
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6923

commutator, socle

C₂²×C₆₁

C₂×C₄²

Fitting

#97×C₆₁

C₄

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #92420⋊C₈	C₂	semidirect
4. #92420⋊C₈	C₂	semidirect
44. #92420	M₁₆	semidirect
67. C₆₁	#1451	semidirect

H ≅ G/N

type

3. #92420⋊C₈

C₂

semidirect

4. #92420⋊C₈

C₂

semidirect

44. #92420

M₁₆

semidirect

67. C₆₁

#1451

semidirect

Subgroups — #21461027

Conjugacy classes of subgroups

Complements