Group #70101 — subgroups

Subgroups — #70101

Order 384.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#5812	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#101	C₂	1	1	yes	maximal, characteristic
3	C₂×#836	C₂	1	1	yes	maximal, characteristic
4	#4559	C₂	1	2	yes	maximal
5	#4556	C₂	1	2	yes	maximal
6	#1830	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	Q₈×Dic₃	—	2	2	no	—
10	Q₈×Dic₃	—	2	2	no	—
11	#926	C₄	1	4	yes	—
12	#926	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	2	yes	—
15	C₃×#101	—	2	2	no	—
16	#836	—	2	2	no	—
17	#519	—	3	1	no	—
18	C₂×#101	—	3	1	no	—
19	C₂×#101	D₃	1	1	yes	characteristic
20	#344	—	3	2	no	—
21	#344	—	3	2	no	—
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	D₄	1	2	yes	—
25	C₂×C₃×Q₈	D₄	1	2	yes	—
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₃×M₁₆	—	2	2	no	—
29	C₃×M₁₆	—	2	2	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	#217	C₂×C₄	1	2	yes	—
32	#217	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₃	—	2	4	no	—
34	C₄⋊Dic₃	—	2	1	no	—
35	C₄⋊Dic₃	—	2	1	no	—
36	C₄×Dic₃	—	2	4	no	—
37	C₄×Dic₃	—	2	1	no	—
38	C₄×Dic₃	—	2	1	no	—
39	#208	—	2	2	no	—
40	#208	—	2	2	no	—
41	C₂×C₃⋊C₈	—	2	2	no	—
42	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
43	C₂×M₁₆	—	3	2	no	—
44	C₂×M₁₆	C₂×D₃	1	2	yes	—
45	C₄×Q₈	—	6	2	no	—
46	C₄×Q₈	—	6	2	no	—
47	#117	—	3	4	no	—
48	#117	—	3	1	no	—
49	#117	—	3	1	no	—
50	#101	—	2	2	no	—
51	#101	—	6	2	no	—
52	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
53	C₃×Q₈	—	2	4	no	—
54	C₃×Q₈	—	2	4	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	#41	1	2	yes	—
61	C₂×C₃×C₄	#41	1	2	yes	—
62	C₂×Dic₃	—	2	2	no	—
63	C₂×Dic₃	—	2	2	no	—
64	C₃×C₈	—	4	2	no	—
65	C₃⋊C₈	—	4	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	D₁₂	1	2	yes	—
69	C₂×Q₈	#67	1	2	yes	—
70	C₂²×C₄	C₄×D₃	1	2	yes	—
71	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
72	M₁₆	—	2	2	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	—	6	2	no	—
75	M₁₆	—	6	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	6	2	no	—
78	C₄⋊C₄	—	6	4	no	—
79	C₄⋊C₄	—	6	1	no	—
80	C₄⋊C₄	—	6	1	no	—
81	C₂²⋊C₄	—	3	2	no	—
82	C₂²⋊C₄	—	3	2	no	—
83	C₄²	—	6	4	no	—
84	C₄²	—	6	1	no	—
85	C₄²	—	6	1	no	—
86	C₂²×C₃	—	4	1	no	—
87	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₃	#117	1	1	yes	characteristic
89	C₂²×C₃	#117	1	1	yes	characteristic
90	C₃×C₄	—	2	4	no	—
91	C₃×C₄	#123	1	2	yes	—
92	C₃×C₄	#123	1	2	yes	—
93	Dic₃	—	4	2	no	—
94	Dic₃	—	4	2	no	—
95	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
96	Q₈	—	2	4	no	—
97	Q₈	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	6	2	no	—
101	C₂×C₄	—	6	2	no	—
102	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
103	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
104	C₂×C₄	#237	1	2	yes	—
105	C₂×C₄	#237	1	2	yes	—
106	C₂×C₄	C₂×#67	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	12	2	no	—
109	C₂×C₃	—	2	1	no	—
110	C₂×C₃	—	2	1	no	—
111	C₂×C₃	—	2	1	no	—
112	C₂×C₃	#381	1	1	yes	characteristic
113	C₂²	—	4	1	no	—
114	C₂²	#877	1	1	yes	characteristic
115	C₂²	#877	1	1	yes	characteristic
116	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
117	C₄	—	2	4	no	—
118	C₄	—	12	2	no	—
119	C₄	—	12	2	no	—
120	C₄	#886	1	2	yes	—
121	C₄	#935	1	2	yes	—
122	C₃	#1830	1	1	yes	characteristic, complemented, Sylow
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#4943	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5812

C₂

yes

maximal, characteristic

C₂×C₃×#101

C₂

yes

maximal, characteristic

C₂×#836

C₂

yes

maximal, characteristic

#4559

C₂

yes

maximal

#4556

C₂

yes

maximal

#1830

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

Q₈×Dic₃

—

Q₈×Dic₃

—

#926

C₄

yes

—

#926

C₂²

yes

characteristic

#926

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

#344

—

#344

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂×C₃×Q₈

D₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#217

C₂×C₄

yes

—

#217

C₂×C₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

—

—

—

—

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂

yes

—

C₂×Q₈

#67

yes

—

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

—

C₃×C₄

#123

yes

—

C₃×C₄

#123

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂×C₄×D₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

103

C₂×C₄

C₂×D₁₂

yes

characteristic

104

C₂×C₄

#237

yes

—

105

C₂×C₄

#237

yes

—

106

C₂×C₄

C₂×#67

yes

characteristic

107

C₈

—

108

C₈

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

#381

yes

characteristic

113

C₂²

—

114

C₂²

#877

yes

characteristic

115

C₂²

#877

yes

characteristic

116

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#886

yes

—

121

C₄

#935

yes

—

122

C₃

#1830

yes

characteristic, complemented, Sylow

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#4943

yes

characteristic

	subgroup	G/H
center	C₂	#4943
commutator, prime radical	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#101	C₂
socle	C₂×C₃	#381
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#101	D₃
hypocenter	C₃	#1830
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#4943

commutator, prime radical

C₂³×C₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#101

C₂

socle

C₂×C₃

#381

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
122. C₃	#1830	semidirect

H ≅ G/N

type

122. C₃

#1830

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#5812	2
C₂×C₃×#101	2
C₂×#836	2
#4559	2
#4556	2
#1830	3

subgroup

index

C₂×C₃×#101

Subgroups — #70101

Conjugacy classes of subgroups

Complements

Maximal subgroups