Group #21446809 — subgroups

Subgroups — #21446809

Order 7808.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	1	yes	maximal, characteristic
2	C₄×#11758194	C₂	1	1	yes	maximal, characteristic
3	#16839531	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	C₂²	1	2	yes	—
6	C₂×#11758194	C₂²	1	2	yes	—
7	C₄⋊#92418	—	2	1	no	—
8	C₄⋊#92418	—	2	1	no	—
9	#14519688	—	2	1	no	—
10	C₄×#92418	—	2	1	no	—
11	C₂²×Q₈×C₆₁	C₄	1	2	yes	—
12	C₄⋊Q₈×C₆₁	—	2	1	no	—
13	C₄⋊Q₈×C₆₁	—	2	1	no	—
14	C₄⋊Q₈×C₆₁	—	2	1	no	—
15	C₄⋊Q₈×C₆₁	—	2	1	no	—
16	C₂×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	—	2	2	no	—
20	C₂×Q₈×C₆₁	—	2	2	no	—
21	C₂×Q₈×C₆₁	—	4	4	no	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄²×C₆₁	D₄	1	1	yes	characteristic
31	C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	D₄	1	1	yes	characteristic
34	#11758194	—	2	2	no	—
35	#11758194	—	2	4	no	—
36	#11758194	Q₈	1	4	yes	complemented
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₆₁	—	4	4	no	—
41	Q₈×C₆₁	—	4	4	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	4	4	no	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₆₁	#41	1	2	yes	—
54	#92418	—	2	4	no	—
55	#92418	—	4	2	no	—
56	C₂²×C₆₁	—	2	1	no	—
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	—	2	1	no	—
62	C₄×C₆₁	—	4	4	no	—
63	C₄×C₆₁	#100	1	2	yes	—
64	C₄×C₆₁	#101	1	2	yes	—
65	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₆₁	#124	1	1	yes	characteristic
67	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
68	#1997	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	—	2	2	no	—
70	C₂×C₆₁	#391	1	1	yes	characteristic
71	C₂×C₆₁	C₂×#100	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
73	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
74	#423	—	61	1	no	—
75	C₄×M₁₆	—	61	1	no	—
76	#343	—	61	4	no	—
77	C₆₁	#1997	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	Dic₆₁	1	2	yes	—
79	C₂×M₁₆	—	61	2	no	—
80	C₂×M₁₆	—	61	2	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₂×C₄⋊C₄	Dic₆₁	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
87	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
88	C₄⋊C₈	—	122	1	no	—
89	C₄⋊C₈	—	122	1	no	—
90	#97	—	122	1	no	—
91	C₄×C₈	—	122	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₆₁	1	4	yes	—
98	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
99	M₁₆	—	61	4	no	—
100	M₁₆	—	122	2	no	—
101	M₁₆	—	122	4	no	—
102	C₂×C₈	—	122	2	no	—
103	C₂×C₈	—	122	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
109	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
111	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
123	C₂×C₄	Q₈×D₆₁	1	2	yes	—
124	C₂×C₄	#11758226	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₈	—	122	4	no	—
128	C₈	—	244	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
132	C₂²	Q₈×Dic₆₁	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#14519694	1	2	yes	—
137	C₄	#14519696	1	2	yes	—
138	C₄	#14519753	1	2	yes	—
139	C₄	#14519766	1	1	yes	characteristic
140	C₄	#14519775	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#16839669	1	1	yes	characteristic
143	C₂	C₂×#14519694	1	1	yes	characteristic
144	C₂	C₂×#14519696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal, characteristic

C₄×#11758194

C₂

yes

maximal, characteristic

#16839531

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂×#11758194

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₆₁

C₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

—

Q₈

yes

complemented

C₂×#92418

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₆₁

#41

yes

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#100

yes

—

C₄×C₆₁

#101

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

#124

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

#1997

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₆₁

#1997

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₆₁

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

100

M₁₆

—

122

—

101

M₁₆

—

122

—

102

C₂×C₈

—

122

—

103

C₂×C₈

—

122

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²⋊D₆₁

yes

characteristic

109

C₄²

C₂²⋊D₆₁

yes

characteristic

110

C₄²

C₂²⋊D₆₁

yes

characteristic

111

C₄²

C₂²⋊D₆₁

yes

characteristic

112

C₂³

C₂²×Dic₆₁

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

123

C₂×C₄

Q₈×D₆₁

yes

—

124

C₂×C₄

#11758226

yes

—

125

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

127

C₈

—

122

—

128

C₈

—

244

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

132

C₂²

Q₈×Dic₆₁

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#14519694

yes

—

137

C₄

#14519696

yes

—

138

C₄

#14519753

yes

—

139

C₄

#14519766

yes

characteristic

140

C₄

#14519775

yes

characteristic

141

C₂

—

142

C₂

#16839669

yes

characteristic

143

C₂

C₂×#14519694

yes

characteristic

144

C₂

C₂×#14519696

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	#1997
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

#1997

LCS

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
36. #11758194	Q₈	semidirect
77. C₆₁	#1997	semidirect

H ≅ G/N

type

36. #11758194

Q₈

semidirect

77. C₆₁

#1997

semidirect

Subgroups — #21446809

Conjugacy classes of subgroups

Complements