Group #21445815 — subgroups

Subgroups — #21445815

Order 7808.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16839786	C₂	1	2	yes	maximal
2	C₂×#14519696	C₂	1	2	yes	maximal
3	#16839531	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	C₂²	1	4	yes	—
6	C₄⋊#92418	—	2	2	no	—
7	C₄⋊#92418	—	2	2	no	—
8	#14519696	—	2	4	no	—
9	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₆₁	—	2	2	no	—
11	C₄×Q₈×C₆₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	C₄	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₆₁	—	2	2	no	—
17	C₂×Q₈×C₆₁	—	2	2	no	—
18	C₂×Q₈×C₆₁	D₄	1	2	yes	—
19	C₂×Q₈×C₆₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₁	—	2	4	no	—
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄²×C₆₁	—	2	2	no	—
28	C₄²×C₆₁	—	2	2	no	—
29	C₄²×C₆₁	D₄	1	2	yes	—
30	C₄²×C₆₁	D₄	1	2	yes	—
31	#11758194	—	2	4	no	—
32	#11758194	—	2	4	no	—
33	C₂×#92418	—	2	4	no	—
34	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₆₁	—	2	4	no	—
36	Q₈×C₆₁	—	2	4	no	—
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#41	1	2	yes	—
51	#92418	—	4	4	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	1	no	—
54	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₆₁	#117	1	1	yes	characteristic
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	—	4	2	no	—
61	C₄×C₆₁	#101	1	2	yes	complemented
62	C₄×C₆₁	#121	1	2	yes	—
63	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
64	#1801	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	1	no	—
66	C₂×C₆₁	—	2	1	no	—
67	C₂×C₆₁	#380	1	1	yes	characteristic
68	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
69	C₂×C₆₁	#413	1	1	yes	characteristic
70	C₂×C₄×Q₈	D₆₁	1	1	yes	characteristic
71	#423	—	61	2	no	—
72	C₂×#101	—	61	2	no	—
73	#343	—	61	2	no	—
74	C₆₁	#1801	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
76	C₂×M₁₆	—	61	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₆₁	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
81	C₂×C₄²	Dic₆₁	1	2	yes	—
82	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
83	C₄⋊C₈	—	122	2	no	—
84	C₄⋊C₈	—	122	2	no	—
85	#101	—	122	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₁₂₂	1	2	yes	—
89	C₂×Q₈	D₂₄₄	1	2	yes	—
90	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
91	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
95	M₁₆	—	122	4	no	—
96	M₁₆	—	122	4	no	—
97	C₂×C₈	—	122	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₆₁	1	2	yes	—
103	C₄²	C₂²⋊D₆₁	1	2	yes	—
104	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄⋊Dic₆₁	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
119	C₂×C₄	#11758222	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
121	C₈	—	244	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
126	C₂²	#14519736	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	#14519696	1	2	yes	complemented
132	C₄	#14519751	1	2	yes	—
133	C₄	#14519756	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#16839622	1	1	yes	characteristic
137	C₂	C₂×#14519696	1	1	yes	characteristic
138	C₂	#16839746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal

C₂×#14519696

C₂

yes

maximal

#16839531

C₂

yes

maximal

C₂×C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₄⋊#92418

—

C₄⋊#92418

—

#14519696

—

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

D₄

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#101

yes

complemented

C₄×C₆₁

#121

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

#1801

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

D₆₁

yes

—

—

—

C₆₁

#1801

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄²

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

#101

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₂₂

yes

—

C₂×Q₈

D₂₄₄

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

M₁₆

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₆₁

yes

—

103

C₄²

C₂²⋊D₆₁

yes

—

104

C₂³

C₂²×Dic₆₁

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₆₁

yes

—

114

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

115

C₂×C₄

C₂²⋊Dic₆₁

yes

—

116

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

117

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

118

C₂×C₄

C₂×D₂₄₄

yes

characteristic

119

C₂×C₄

#11758222

yes

—

120

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

121

C₈

—

244

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

126

C₂²

#14519736

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#14519696

yes

complemented

132

C₄

#14519751

yes

—

133

C₄

#14519756

yes

—

134

C₂

—

135

C₂

—

136

C₂

#16839622

yes

characteristic

137

C₂

C₂×#14519696

yes

characteristic

138

C₂

#16839746

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₆₁
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₆₁
hypocenter	C₆₁	#1801
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₆₁

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₆₁

hypocenter

C₆₁

#1801

LCS

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₆₁	#101	semidirect
74. C₆₁	#1801	semidirect
131. C₄	#14519696	semidirect

H ≅ G/N

type

61. C₄×C₆₁

#101

semidirect

74. C₆₁

#1801

semidirect

131. C₄

#14519696

semidirect

Subgroups — #21445815

Conjugacy classes of subgroups

Complements