Group #21446768 — subgroups

Subgroups — #21446768

Order 7808.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839524	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
7	C₂×C₄×Dic₆₁	—	2	1	no	—
8	C₄⋊#92418	—	2	2	no	—
9	#14519684	—	2	1	no	—
10	C₂²×Q₈×C₆₁	—	2	1	no	—
11	C₄⋊Q₈×C₆₁	—	2	2	no	—
12	C₄⋊Q₈×C₆₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₆₁	—	2	4	no	—
17	C₂×Q₈×C₆₁	—	4	4	no	—
18	C₂²×C₄×C₆₁	—	2	2	no	—
19	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₄⋊C₄×C₆₁	—	2	4	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
26	C₄²×C₆₁	C₂×C₄	1	2	yes	—
27	C₄²×C₆₁	D₄	1	2	yes	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	Q₈	1	4	yes	complemented
30	C₄×Dic₆₁	—	4	2	no	—
31	C₂×#92418	—	2	2	no	—
32	C₂×#92418	—	2	4	no	—
33	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₆₁	—	4	8	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	2	4	no	—
37	C₂×C₄×C₆₁	—	4	4	no	—
38	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₆₁	#41	1	2	yes	—
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	—	4	2	no	—
49	#92418	—	4	2	no	—
50	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	4	4	no	—
57	C₄×C₆₁	#103	1	4	yes	—
58	C₄×C₆₁	C₄×Q₈	1	2	yes	—
59	C₄×C₆₁	#124	1	1	yes	characteristic
60	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₆₁	—	4	4	no	—
62	#1987	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	#391	1	1	yes	characteristic
64	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
65	C₂×C₆₁	#418	1	1	yes	characteristic
66	C₂×C₆₁	#476	1	2	yes	—
67	C₂×C₆₁	#477	1	2	yes	—
68	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	61	1	no	—
70	C₄×C₄⋊C₄	—	61	1	no	—
71	#340	—	61	4	no	—
72	C₆₁	#1987	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	61	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₆₁	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	61	2	no	—
79	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
80	C₂×C₄²	—	122	1	no	—
81	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
82	C₄⋊C₈	—	122	2	no	—
83	#95	—	122	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	122	2	no	—
88	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
91	C₂×C₈	—	122	4	no	—
92	C₂×C₈	—	122	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	61	4	no	—
96	C₄⋊C₄	—	122	2	no	—
97	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
98	C₄²	—	244	2	no	—
99	C₄²	C₂×Dic₆₁	1	2	yes	—
100	C₄²	C₂²⋊D₆₁	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	—	244	2	no	—
108	C₂×C₄	—	244	2	no	—
109	C₂×C₄	#11758201	1	2	yes	—
110	C₂×C₄	#11758202	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
112	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
113	C₂×C₄	Q₈×D₆₁	1	2	yes	—
114	C₂×C₄	#11758226	1	2	yes	—
115	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
117	C₈	—	244	2	no	—
118	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
119	C₂²	#14519753	1	2	yes	—
120	C₂²	C₂×#11758201	1	1	yes	characteristic
121	C₂²	C₂×#11758202	1	1	yes	characteristic
122	C₂²	#14519803	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	244	4	no	—
126	C₄	#14519701	1	4	yes	—
127	C₄	Q₈×Dic₆₁	1	2	yes	—
128	C₄	#14519766	1	1	yes	characteristic
129	C₄	#14519775	1	1	yes	characteristic
130	C₂	#16839669	1	1	yes	characteristic
131	C₂	C₂×#14519701	1	1	yes	characteristic
132	C₂	#16839767	1	1	yes	characteristic
133	C₂	#16840049	1	2	yes	—
134	C₂	#16840054	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₆₁

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

—

C₄⋊#92418

—

#14519684

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

Q₈

yes

complemented

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

characteristic

C₂²×C₆₁

AGL(1,ℤ/8)

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

#124

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#477

yes

—

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₆₁

#1987

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₆₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄²

—

244

—

C₄²

C₂×Dic₆₁

yes

—

100

C₄²

C₂²⋊D₆₁

yes

—

101

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

244

—

108

C₂×C₄

—

244

—

109

C₂×C₄

#11758201

yes

—

110

C₂×C₄

#11758202

yes

—

111

C₂×C₄

C₂²⋊Dic₆₁

yes

—

112

C₂×C₄

C₂²⋊Dic₆₁

yes

—

113

C₂×C₄

Q₈×D₆₁

yes

—

114

C₂×C₄

#11758226

yes

—

115

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

117

C₈

—

244

—

118

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

119

C₂²

#14519753

yes

—

120

C₂²

C₂×#11758201

yes

characteristic

121

C₂²

C₂×#11758202

yes

characteristic

122

C₂²

#14519803

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

244

—

126

C₄

#14519701

yes

—

127

C₄

Q₈×Dic₆₁

yes

—

128

C₄

#14519766

yes

characteristic

129

C₄

#14519775

yes

characteristic

130

C₂

#16839669

yes

characteristic

131

C₂

C₂×#14519701

yes

characteristic

132

C₂

#16839767

yes

characteristic

133

C₂

#16840049

yes

—

134

C₂

#16840054

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#1987
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

#1987

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₆₁	C₄	semidirect
29. C₄⋊Dic₆₁	Q₈	semidirect
72. C₆₁	#1987	semidirect
76. C₄⋊Q₈	Dic₆₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₆₁

C₄

semidirect

29. C₄⋊Dic₆₁

Q₈

semidirect

72. C₆₁

#1987

semidirect

76. C₄⋊Q₈

Dic₆₁

semidirect

Subgroups — #21446768

Conjugacy classes of subgroups

Complements