Group #21446758 — subgroups

Subgroups — #21446758

Order 7808.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519688	C₂	1	1	yes	maximal, characteristic
3	#16839516	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
6	C₂²×#92418	C₂²	1	2	yes	—
7	C₄×Dic₁₂₂	—	2	2	no	—
8	C₄×Dic₁₂₂	—	2	2	no	—
9	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×#11758196	C₄	1	2	yes	—
11	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
12	#14519688	—	2	2	no	—
13	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₂₂	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	4	no	—
23	C₂×C₈×C₆₁	—	2	2	no	—
24	C₂×C₈×C₆₁	—	2	4	no	—
25	C₄²×C₆₁	D₄	1	2	yes	—
26	C₄²×C₆₁	D₄	1	2	yes	—
27	C₄⋊Dic₆₁	—	2	2	no	—
28	#11758196	—	2	4	no	—
29	C₄×Dic₆₁	—	2	2	no	—
30	C₄×Dic₆₁	—	2	2	no	—
31	C₂×#92418	—	2	2	no	—
32	C₂×#92418	—	2	2	no	—
33	C₂×#92418	Q₈	1	4	yes	complemented
34	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₆₁	#41	1	2	yes	—
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	—	2	2	no	—
45	C₂×Dic₆₁	—	2	2	no	—
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	M₁₆	1	4	yes	—
48	Dic₁₂₂	—	2	8	no	—
49	C₈×C₆₁	—	4	2	no	—
50	#92418	—	2	4	no	—
51	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
53	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
54	C₂²×C₆₁	#131	1	2	yes	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₆₁	#124	1	1	yes	characteristic
59	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₆₁	—	2	4	no	—
61	Dic₆₁	—	4	2	no	—
62	#1985	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	#391	1	1	yes	characteristic
64	C₂×C₆₁	#407	1	1	yes	characteristic
65	C₂×C₆₁	#408	1	1	yes	characteristic
66	C₂×C₆₁	#446	1	4	yes	—
67	C₂×C₄×Q₈	—	61	1	no	—
68	C₂×C₄⋊C₈	D₆₁	1	1	yes	characteristic
69	C₂×#97	—	61	1	no	—
70	#336	—	61	4	no	—
71	C₆₁	#1985	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	61	1	no	—
73	C₂²×C₈	—	61	2	no	—
74	C₂²×C₈	C₂×D₆₁	1	2	yes	—
75	C₄×Q₈	—	122	2	no	—
76	C₄×Q₈	—	122	2	no	—
77	C₂×C₄⋊C₄	—	61	2	no	—
78	C₂×C₄⋊C₄	—	61	1	no	—
79	C₂×C₄²	—	61	2	no	—
80	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	122	2	no	—
83	C₂×Q₈	—	122	4	no	—
84	C₂×Q₈	—	122	2	no	—
85	C₂²×C₄	—	61	2	no	—
86	C₂²×C₄	—	61	2	no	—
87	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	61	4	no	—
93	C₂×C₈	—	122	2	no	—
94	C₂×C₈	—	122	2	no	—
95	C₄⋊C₄	—	122	4	no	—
96	C₄⋊C₄	—	122	2	no	—
97	C₄²	—	122	2	no	—
98	C₄²	—	122	2	no	—
99	C₄²	D₂₄₄	1	2	yes	—
100	C₄²	C₂²⋊D₆₁	1	2	yes	—
101	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
102	Q₈	—	122	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	61	4	no	—
106	C₂×C₄	—	122	2	no	—
107	C₂×C₄	—	122	2	no	—
108	C₂×C₄	—	122	2	no	—
109	C₂×C₄	—	122	4	no	—
110	C₂×C₄	#11758198	1	2	yes	—
111	C₂×C₄	#11758198	1	2	yes	—
112	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
113	C₂×C₄	#11758224	1	2	yes	—
114	C₂×C₄	Q₈×D₆₁	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
116	C₈	—	4	2	no	—
117	C₈	—	122	4	no	—
118	C₂²	C₂×#11758198	1	1	yes	characteristic
119	C₂²	#14519742	1	2	yes	—
120	C₂²	M₁₆×D₆₁	1	2	yes	—
121	C₂²	#14519783	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	122	4	no	—
125	C₄	—	244	2	no	—
126	C₄	#14519757	1	2	yes	—
127	C₄	#14519764	1	1	yes	characteristic
128	C₄	#14519775	1	1	yes	characteristic
129	C₂	#16839667	1	1	yes	characteristic
130	C₂	#16839725	1	1	yes	characteristic
131	C₂	#16839728	1	1	yes	characteristic
132	C₂	#16839891	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×#14519688

C₂

yes

maximal, characteristic

#16839516

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₄

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519688

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

Q₈

yes

complemented

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

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

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

M₁₆

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

#131

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

#124

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₆₁

yes

characteristic

C₂×#97

—

#336

—

C₆₁

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

#97

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

D₂₄₄

yes

—

100

C₄²

C₂²⋊D₆₁

yes

—

101

C₂³

C₂×C₄×D₆₁

yes

characteristic

102

Q₈

—

122

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

#11758198

yes

—

111

C₂×C₄

#11758198

yes

—

112

C₂×C₄

C₂×D₂₄₄

yes

characteristic

113

C₂×C₄

#11758224

yes

—

114

C₂×C₄

Q₈×D₆₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

116

C₈

—

117

C₈

—

122

—

118

C₂²

C₂×#11758198

yes

characteristic

119

C₂²

#14519742

yes

—

120

C₂²

M₁₆×D₆₁

yes

—

121

C₂²

#14519783

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

122

—

125

C₄

—

244

—

126

C₄

#14519757

yes

—

127

C₄

#14519764

yes

characteristic

128

C₄

#14519775

yes

characteristic

129

C₂

#16839667

yes

characteristic

130

C₂

#16839725

yes

characteristic

131

C₂

#16839728

yes

characteristic

132

C₂

#16839891

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₁
commutator, socle	C₂³×C₆₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₆₁	C₂
hypercenter	C₂×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#1985
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₁

commutator, socle

C₂³×C₆₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₆₁

C₂

hypercenter

C₂×C₄⋊C₈

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
33. C₂×#92418	Q₈	semidirect
71. C₆₁	#1985	semidirect

H ≅ G/N

type

33. C₂×#92418

Q₈

semidirect

71. C₆₁

#1985

semidirect

Subgroups — #21446758

Conjugacy classes of subgroups

Complements