Group #70102 — subgroups

Subgroups — #70102

Order 384.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#5229	C₂	1	1	yes	maximal, characteristic
4	#5121	C₂	1	2	yes	maximal
5	#4555	C₂	1	2	yes	maximal
6	#1857	—	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	—	2	2	no	—
12	#926	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×#101	C₄	1	4	yes	complemented
16	C₃⋊#95	—	2	2	no	—
17	#818	—	2	2	no	—
18	#817	—	2	2	no	—
19	#519	—	3	1	no	—
20	#428	—	3	2	no	—
21	C₂×#101	D₃	1	1	yes	characteristic
22	#391	—	3	1	no	—
23	#339	—	3	2	no	—
24	C₂×C₃×Q₈	—	2	2	no	—
25	C₂×C₃×Q₈	—	2	2	no	—
26	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×Q₈	Q₈	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂²×Dic₃	—	2	1	no	—
31	C₂²×Dic₃	—	2	1	no	—
32	C₃×M₁₆	—	2	2	no	—
33	C₃×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₈	—	2	2	no	—
35	#217	—	2	2	no	—
36	C₄⋊Dic₃	—	2	2	no	—
37	C₄⋊Dic₃	—	2	2	no	—
38	C₄⋊Dic₃	—	2	2	no	—
39	C₄⋊Dic₃	D₄	1	2	yes	—
40	C₄⋊Dic₃	D₄	1	2	yes	—
41	C₄×Dic₃	—	2	4	no	—
42	C₄×Dic₃	—	2	2	no	—
43	C₄×Dic₃	—	2	2	no	—
44	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
45	C₂×M₁₆	C₂×D₃	1	2	yes	—
46	C₄×Q₈	—	6	2	no	—
47	C₄×Q₈	—	6	2	no	—
48	#117	—	3	2	no	—
49	#117	—	6	2	no	—
50	C₂×C₄⋊C₄	—	3	1	no	—
51	C₂×C₄²	—	3	1	no	—
52	#107	—	6	2	no	—
53	#106	—	6	2	no	—
54	#101	Dic₃	1	4	yes	complemented
55	#95	—	6	2	no	—
56	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
57	C₃×Q₈	—	2	4	no	—
58	C₃×Q₈	—	2	4	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
62	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₄	#41	1	2	yes	—
66	C₂×Dic₃	—	2	2	no	—
67	C₂×Dic₃	—	2	2	no	—
68	C₂×Dic₃	—	2	2	no	—
69	C₂×Dic₃	—	4	1	no	—
70	C₂×Dic₃	—	4	1	no	—
71	C₂×Dic₃	—	4	2	no	—
72	C₂×Dic₃	—	4	1	no	—
73	C₃×C₈	—	2	4	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	Dic₆	1	2	yes	—
77	C₂×Q₈	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄	—	6	1	no	—
79	C₂²×C₄	—	6	1	no	—
80	C₂²×C₄	D₁₂	1	2	yes	—
81	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	C₂×Dic₃	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₄⋊C₄	—	3	2	no	—
86	C₄⋊C₄	—	3	2	no	—
87	C₄⋊C₄	—	6	2	no	—
88	C₄⋊C₄	—	6	2	no	—
89	C₄⋊C₄	—	6	2	no	—
90	C₂²⋊C₄	—	6	2	no	—
91	C₄²	—	6	4	no	—
92	C₄²	—	6	2	no	—
93	C₄²	—	6	2	no	—
94	C₂²×C₃	—	2	1	no	—
95	C₂²×C₃	—	2	1	no	—
96	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₃	#120	1	1	yes	characteristic
98	C₂²×C₃	#123	1	1	yes	characteristic
99	C₃×C₄	—	2	4	no	—
100	C₃×C₄	C₄×D₄	1	2	yes	—
101	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
102	Dic₃	—	4	2	no	—
103	Dic₃	—	4	2	no	—
104	Dic₃	—	4	2	no	—
105	Dic₃	—	8	1	no	—
106	C₂³	C₂×D₁₂	1	1	yes	characteristic
107	Q₈	—	2	4	no	—
108	Q₈	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	6	2	no	—
112	C₂×C₄	—	6	2	no	—
113	C₂×C₄	—	6	2	no	—
114	C₂×C₄	—	12	1	no	—
115	C₂×C₄	—	12	1	no	—
116	C₂×C₄	—	12	2	no	—
117	C₂×C₄	—	12	1	no	—
118	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
119	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
120	C₂×C₄	D₃×D₄	1	2	yes	—
121	C₂×C₄	#237	1	2	yes	—
122	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₂×C₃	—	2	1	no	—
125	C₂×C₃	—	2	1	no	—
126	C₂×C₃	#385	1	1	yes	characteristic
127	C₂×C₃	#454	1	1	yes	characteristic
128	C₂×C₃	#456	1	1	yes	characteristic
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#882	1	1	yes	characteristic
132	C₂²	#886	1	1	yes	characteristic
133	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	12	2	no	—
136	C₄	—	12	2	no	—
137	C₄	—	12	2	no	—
138	C₄	—	24	1	no	—
139	C₄	#878	1	2	yes	—
140	C₄	D₄×Dic₃	1	2	yes	—
141	C₃	#1857	1	1	yes	characteristic, complemented, Sylow
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#4752	1	1	yes	characteristic
145	C₂	#4753	1	1	yes	characteristic
146	C₂	#4950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5812

C₂

yes

maximal, characteristic

C₂×C₃×#101

C₂

yes

maximal, characteristic

#5229

C₂

yes

maximal, characteristic

#5121

C₂

yes

maximal

#4555

C₂

yes

maximal

#1857

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

Q₈×Dic₃

—

Q₈×Dic₃

—

#926

—

#926

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₃×#101

C₄

yes

—

—

—

—

—

D₃

yes

characteristic

#391

—

#339

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂×C₃×Q₈

Q₈

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

C₂×C₄

yes

—

C₂×C₃×C₈

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

D₄

yes

—

C₄⋊Dic₃

D₄

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂×M₁₆

C₂×D₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#106

—

#101

Dic₃

yes

complemented

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₆

yes

—

C₂×Q₈

C₂×Dic₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₃

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

—

100

C₃×C₄

C₄×D₄

yes

—

101

C₃×C₄

C₂²⋊Q₈

yes

—

102

Dic₃

—

103

Dic₃

—

104

Dic₃

—

105

Dic₃

—

106

C₂³

C₂×D₁₂

yes

characteristic

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄⋊Dic₃

yes

—

119

C₂×C₄

C₂×Dic₆

yes

characteristic

120

C₂×C₄

D₃×D₄

yes

—

121

C₂×C₄

#237

yes

—

122

C₂×C₄

C₂²×Dic₃

yes

characteristic

123

C₈

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

#385

yes

characteristic

127

C₂×C₃

#454

yes

characteristic

128

C₂×C₃

#456

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

#882

yes

characteristic

132

C₂²

#886

yes

characteristic

133

C₂²

C₂×C₄⋊Dic₃

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#878

yes

—

140

C₄

D₄×Dic₃

yes

—

141

C₃

#1857

yes

characteristic, complemented, Sylow

142

C₂

—

143

C₂

—

144

C₂

#4752

yes

characteristic

145

C₂

#4753

yes

characteristic

146

C₂

#4950

yes

characteristic

	subgroup	G/H
center	C₂²	#882
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#101	C₂
socle	C₂²×C₃	#120
prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#101	D₃
hypocenter	C₃	#1857
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#882

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#101

C₂

socle

C₂²×C₃

#120

prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
15. C₃×#101	C₄	semidirect
54. #101	Dic₃	semidirect
141. C₃	#1857	semidirect

H ≅ G/N

type

15. C₃×#101

C₄

semidirect

54. #101

Dic₃

semidirect

141. C₃

#1857

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
#5229	2
#5121	2
#4555	2
#1857	3

subgroup

index

C₂×C₃×#101

Subgroups — #70102

Conjugacy classes of subgroups

Complements

Maximal subgroups