Group #21445977 — subgroups

Subgroups — #21445977

Order 7808.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#14519711	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519710	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839631	C₂	1	2	yes	maximal
5	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×#11758196	—	2	4	no	—
9	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	#14519711	—	2	2	no	—
11	#14519710	—	2	4	no	—
12	C₈×Dic₆₁	C₄	1	4	yes	complemented
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	#107×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	—	2	2	no	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₂²×Dic₆₁	D₄	1	2	yes	—
20	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₆₁	D₄	1	2	yes	—
22	C₂×C₈×C₆₁	Q₈	1	2	yes	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	#11758196	—	4	8	no	—
28	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
29	C₄×Dic₆₁	Q₈	1	2	yes	—
30	C₂×#92418	C₂×C₄	1	2	yes	—
31	C₂×#92418	D₄	1	2	yes	—
32	C₂×#92418	Q₈	1	2	yes	—
33	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	4	2	no	—
35	C₂×C₄×C₆₁	—	4	4	no	—
36	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₆₁	—	4	2	no	—
43	C₂×Dic₆₁	—	4	4	no	—
44	C₂×Dic₆₁	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₆₁	D₈	1	4	yes	—
46	C₂×Dic₆₁	Q₁₆	1	4	yes	—
47	C₈×C₆₁	C₄⋊C₄	1	4	yes	complemented
48	#92418	C₄⋊C₄	1	4	yes	complemented
49	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₆₁	#127	1	1	yes	characteristic
51	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₆₁	C₂×D₈	1	2	yes	—
53	C₂²×C₆₁	C₂×Q₁₆	1	2	yes	—
54	C₄×C₆₁	—	8	2	no	—
55	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₆₁	—	8	2	no	—
60	Dic₆₁	#107	1	8	yes	complemented
61	#1845	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	#384	1	1	yes	characteristic
63	C₂×C₆₁	C₂×#107	1	2	yes	—
64	C₂×C₆₁	#493	1	1	yes	characteristic
65	C₂×C₆₁	#494	1	1	yes	characteristic
66	C₂×C₆₁	#500	1	2	yes	—
67	C₂×#107	—	61	1	no	—
68	C₂×#107	—	61	2	no	—
69	C₂×#107	D₆₁	1	1	yes	characteristic
70	C₂×C₄×C₈	—	61	1	no	—
71	#384	—	61	2	no	—
72	C₆₁	#1845	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	61	1	no	—
74	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	61	2	no	—
76	C₂×C₄⋊C₄	—	122	4	no	—
77	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
78	C₂×C₄²	—	61	1	no	—
79	#107	—	2	2	no	—
80	#107	—	122	4	no	—
81	#107	—	122	2	no	—
82	C₄×C₈	—	61	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	61	2	no	—
85	C₂²×C₄	—	122	2	no	—
86	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
87	C₂×C₈	—	61	2	no	—
88	C₂×C₈	—	61	2	no	—
89	C₂×C₈	—	61	2	no	—
90	C₂×C₈	Dic₁₂₂	1	2	yes	—
91	C₂×C₈	C₄×D₆₁	1	2	yes	—
92	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	122	2	no	—
96	C₄⋊C₄	—	122	2	no	—
97	C₄⋊C₄	—	244	8	no	—
98	C₄²	—	61	2	no	—
99	C₄²	—	61	2	no	—
100	C₂³	D₄×D₆₁	1	1	yes	characteristic
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	61	4	no	—
104	C₂×C₄	—	61	4	no	—
105	C₂×C₄	—	61	4	no	—
106	C₂×C₄	—	244	2	no	—
107	C₂×C₄	—	244	4	no	—
108	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
110	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₈	—	61	4	no	—
115	C₈	#11758196	1	4	yes	complemented
116	C₂²	C₄⋊C₄×D₆₁	1	1	yes	characteristic
117	C₂²	#14519773	1	1	yes	characteristic
118	C₂²	#14519776	1	1	yes	characteristic
119	C₂²	D₈×D₆₁	1	2	yes	—
120	C₂²	Q₁₆×D₆₁	1	2	yes	—
121	C₄	—	8	2	no	—
122	C₄	—	61	8	no	—
123	C₄	—	488	2	no	—
124	C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
125	C₄	C₂×#11758196	1	1	yes	characteristic
126	C₄	#14519775	1	1	yes	characteristic
127	C₄	#14519776	1	1	yes	characteristic
128	C₂	#16839631	1	1	yes	characteristic
129	C₂	#107×D₆₁	1	2	yes	—
130	C₂	#16840137	1	1	yes	characteristic
131	C₂	#16840140	1	1	yes	characteristic
132	C₂	#16840166	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519711

C₂

yes

maximal, characteristic

C₂×#14519710

C₂

yes

maximal

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#16839631

C₂

yes

maximal

C₂×#107×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519711

—

#14519710

—

C₈×Dic₆₁

C₄

yes

complemented

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#107×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

D₄

yes

—

C₂×#92418

Q₈

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

D₈

yes

—

C₂×Dic₆₁

Q₁₆

yes

—

C₈×C₆₁

C₄⋊C₄

yes

complemented

#92418

C₄⋊C₄

yes

complemented

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#127

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×#107

—

C₂×#107

—

C₂×#107

D₆₁

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₆₁

#1845

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#107

—

122

—

#107

—

122

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

C₄×D₆₁

yes

—

C₂×C₈

C₂²⋊D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

—

C₄²

—

100

C₂³

D₄×D₆₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

244

—

107

C₂×C₄

—

244

—

108

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

109

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

110

C₂×C₄

D₄×D₆₁

yes

characteristic

111

C₂×C₄

Q₈×D₆₁

yes

characteristic

112

C₂×C₄

Q₈×D₆₁

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₈

—

115

C₈

#11758196

yes

complemented

116

C₂²

C₄⋊C₄×D₆₁

yes

characteristic

117

C₂²

#14519773

yes

characteristic

118

C₂²

#14519776

yes

characteristic

119

C₂²

D₈×D₆₁

yes

—

120

C₂²

Q₁₆×D₆₁

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

488

—

124

C₄

C₄⋊C₄×D₆₁

yes

characteristic

125

C₄

C₂×#11758196

yes

characteristic

126

C₄

#14519775

yes

characteristic

127

C₄

#14519776

yes

characteristic

128

C₂

#16839631

yes

characteristic

129

C₂

#107×D₆₁

yes

—

130

C₂

#16840137

yes

characteristic

131

C₂

#16840140

yes

characteristic

132

C₂

#16840166

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	#1845
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
12. C₈×Dic₆₁	C₄	semidirect
47. C₈×C₆₁	C₄⋊C₄	semidirect
48. #92418	C₄⋊C₄	semidirect
60. Dic₆₁	#107	semidirect
72. C₆₁	#1845	semidirect
115. C₈	#11758196	semidirect

H ≅ G/N

type

12. C₈×Dic₆₁

C₄

semidirect

47. C₈×C₆₁

C₄⋊C₄

semidirect

48. #92418

C₄⋊C₄

semidirect

60. Dic₆₁

#107

semidirect

72. C₆₁

#1845

semidirect

115. C₈

#11758196

semidirect

Subgroups — #21445977

Conjugacy classes of subgroups

Complements