Group #21445970 — subgroups

Subgroups — #21445970

Order 7808.

300 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂×#14519706	C₂	1	4	yes	maximal
3	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	C₂²	1	2	yes	—
6	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
10	C₄⋊#92418	—	2	2	no	—
11	#14519706	—	2	8	no	—
12	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	—	2	2	no	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
19	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₆₁	—	2	2	no	—
21	C₂×C₈×C₆₁	—	2	4	no	—
22	C₄²×C₆₁	D₄	1	2	yes	—
23	C₄²×C₆₁	Q₈	1	2	yes	—
24	C₄×Dic₆₁	—	2	2	no	—
25	C₄×Dic₆₁	—	2	4	no	—
26	C₄×Dic₆₁	C₈	1	4	yes	complemented
27	C₄×Dic₆₁	D₄	1	4	yes	—
28	C₄×Dic₆₁	Q₈	1	4	yes	—
29	C₂×#92418	—	2	4	no	—
30	C₂×#92418	—	2	2	no	—
31	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
37	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
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	2	yes	—
41	C₂×Dic₆₁	—	2	2	no	—
42	C₂×Dic₆₁	—	2	2	no	—
43	C₂×Dic₆₁	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₆₁	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₆₁	C₂×C₈	1	4	yes	—
46	C₂×Dic₆₁	M₁₆	1	4	yes	—
47	C₈×C₆₁	—	4	2	no	—
48	#92418	—	4	2	no	—
49	C₂²×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
50	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
53	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	C₄⋊C₈	1	4	yes	complemented
56	C₄×C₆₁	#127	1	1	yes	characteristic
57	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
59	Dic₆₁	—	2	4	no	—
60	Dic₆₁	C₄⋊C₈	1	8	yes	complemented
61	#1842	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	#384	1	1	yes	characteristic
63	C₂×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
64	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₆₁	#423	1	2	yes	—
66	C₂×C₆₁	#423	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	61	4	no	—
68	C₂×C₄⋊C₈	—	61	1	no	—
69	C₂×C₄⋊C₈	D₆₁	1	1	yes	characteristic
70	C₄³	—	61	1	no	—
71	C₆₁	#1842	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₈	—	61	2	no	—
73	C₂²×C₈	C₂×D₆₁	1	2	yes	—
74	C₂×C₄²	—	61	1	no	—
75	C₂×C₄²	—	61	2	no	—
76	C₂×C₄²	—	61	2	no	—
77	C₂×C₄²	—	61	1	no	—
78	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	122	8	no	—
81	C₄⋊C₈	—	122	2	no	—
82	C₂²×C₄	—	61	2	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	122	4	no	—
90	C₂×C₈	—	122	2	no	—
91	C₄²	—	61	4	no	—
92	C₄²	—	61	4	no	—
93	C₄²	—	61	4	no	—
94	C₄²	—	122	4	no	—
95	C₄²	—	122	2	no	—
96	C₄²	Dic₁₂₂	1	2	yes	—
97	C₄²	C₂²⋊D₆₁	1	2	yes	—
98	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	61	4	no	—
101	C₂×C₄	—	61	4	no	—
102	C₂×C₄	—	61	4	no	—
103	C₂×C₄	—	61	4	no	—
104	C₂×C₄	—	122	2	no	—
105	C₂×C₄	—	122	2	no	—
106	C₂×C₄	C₈×D₆₁	1	2	yes	—
107	C₂×C₄	#11758189	1	2	yes	—
108	C₂×C₄	#11758196	1	2	yes	—
109	C₂×C₄	#11758196	1	2	yes	—
110	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
111	C₂×C₄	D₄×D₆₁	1	2	yes	—
112	C₂×C₄	Q₈×D₆₁	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	244	2	no	—
116	C₂²	C₄⋊C₄×D₆₁	1	2	yes	—
117	C₂²	C₂×#11758196	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₆₁	1	1	yes	characteristic
119	C₂²	M₁₆×D₆₁	1	2	yes	—
120	C₂²	C₂×#11758189	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	61	8	no	—
123	C₄	—	122	4	no	—
124	C₄	#14519706	1	4	yes	complemented
125	C₄	#14519773	1	1	yes	characteristic
126	C₄	#14519775	1	1	yes	characteristic
127	C₄	#14519776	1	2	yes	—
128	C₂	#16839631	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₆₁	1	2	yes	—
130	C₂	C₂×#14519706	1	1	yes	characteristic
131	C₂	#16839784	1	2	yes	—
132	C₂	#16839785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

C₂×#14519706

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₄

yes

—

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄⋊#92418

—

#14519706

—

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×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₈

yes

complemented

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂×C₈

yes

—

C₂×Dic₆₁

M₁₆

yes

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

C₂²×C₈

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

complemented

C₄×C₆₁

#127

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

C₄⋊C₈

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊C₈

yes

—

C₂×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₆₁

#423

yes

—

C₂×C₆₁

#423

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₆₁

yes

characteristic

C₄³

—

C₆₁

#1842

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

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

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

Dic₁₂₂

yes

—

C₄²

C₂²⋊D₆₁

yes

—

C₂³

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

122

—

106

C₂×C₄

C₈×D₆₁

yes

—

107

C₂×C₄

#11758189

yes

—

108

C₂×C₄

#11758196

yes

—

109

C₂×C₄

#11758196

yes

—

110

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

111

C₂×C₄

D₄×D₆₁

yes

—

112

C₂×C₄

Q₈×D₆₁

yes

—

113

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₈

—

115

C₈

—

244

—

116

C₂²

C₄⋊C₄×D₆₁

yes

—

117

C₂²

C₂×#11758196

yes

characteristic

118

C₂²

C₂×C₈×D₆₁

yes

characteristic

119

C₂²

M₁₆×D₆₁

yes

—

120

C₂²

C₂×#11758189

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

122

—

124

C₄

#14519706

yes

complemented

125

C₄

#14519773

yes

characteristic

126

C₄

#14519775

yes

characteristic

127

C₄

#14519776

yes

—

128

C₂

#16839631

yes

characteristic

129

C₂

C₄⋊C₈×D₆₁

yes

—

130

C₂

C₂×#14519706

yes

characteristic

131

C₂

#16839784

yes

—

132

C₂

#16839785

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#1842
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
26. C₄×Dic₆₁	C₈	semidirect
55. C₄×C₆₁	C₄⋊C₈	semidirect
60. Dic₆₁	C₄⋊C₈	semidirect
71. C₆₁	#1842	semidirect
124. C₄	#14519706	semidirect

H ≅ G/N

type

26. C₄×Dic₆₁

C₈

semidirect

55. C₄×C₆₁

C₄⋊C₈

semidirect

60. Dic₆₁

C₄⋊C₈

semidirect

71. C₆₁

#1842

semidirect

124. C₄

#14519706

semidirect

Subgroups — #21445970

Conjugacy classes of subgroups

Complements