Group #70134 — subgroups

Subgroups — #70134

Order 384.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#948	C₂	1	1	yes	maximal, characteristic
2	C₃×#413	C₂	1	1	yes	maximal, characteristic
3	#5237	C₂	1	1	yes	maximal, characteristic
4	C₂×#822	C₂	1	2	yes	maximal, complemented
5	C₃⋊#338	C₂	1	2	yes	maximal, complemented
6	#1813	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	#948	—	2	4	no	—
10	C₂×#208	C₄	1	2	yes	—
11	C₂×#208	C₂²	1	2	yes	—
12	C₂×#208	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₈	C₄	1	2	yes	—
14	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#101	—	2	1	no	—
16	C₃×#100	—	2	1	no	—
17	#836	—	2	1	no	—
18	#833	—	2	1	no	—
19	#822	—	2	4	no	—
20	C₂×#131	—	3	1	no	—
21	C₂×#108	—	3	2	no	—
22	#413	—	3	1	no	—
23	#413	D₃	1	1	yes	characteristic
24	#338	—	3	2	no	—
25	C₃×#41	—	2	2	no	—
26	C₃×#41	—	2	2	no	—
27	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
28	C₂×C₃×D₄	D₄	1	1	yes	characteristic
29	C₂×C₃×D₄	Q₈	1	2	yes	—
30	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₃×M₁₆	—	2	2	no	—
33	C₃×M₁₆	—	2	2	no	—
34	C₂×C₃×C₈	—	2	2	no	—
35	#208	—	2	2	no	—
36	#208	—	2	4	no	—
37	#208	—	2	2	no	—
38	#208	—	2	2	no	—
39	C₂×C₃⋊C₈	—	2	4	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊C₈	D₄	1	4	yes	—
45	C₂×#41	C₂×D₃	1	1	yes	characteristic
46	#131	—	6	4	no	—
47	C₂×M₁₆	—	3	2	no	—
48	C₂×M₁₆	—	3	1	no	—
49	C₂×M₁₆	—	3	2	no	—
50	C₂×M₁₆	C₂×D₃	1	2	yes	—
51	C₂²×C₈	—	3	1	no	—
52	C₂²×C₈	—	3	2	no	—
53	#108	—	6	4	no	—
54	#101	—	2	1	no	—
55	#101	—	6	1	no	—
56	#100	—	2	1	no	—
57	#100	—	6	1	no	—
58	C₂³×C₃	C₄⋊C₄	1	2	yes	—
59	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
60	C₃×Q₈	—	2	2	no	—
61	C₃×D₄	—	2	4	no	—
62	C₃×D₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	#41	1	2	yes	—
70	C₃×C₈	—	4	2	no	—
71	C₃⋊C₈	—	2	2	no	—
72	C₃⋊C₈	—	4	2	no	—
73	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
74	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	#67	1	1	yes	characteristic
78	C₂×D₄	Dic₆	1	2	yes	—
79	C₂×D₄	#67	1	1	yes	characteristic
80	C₂²×C₄	C₄×D₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	6	4	no	—
85	M₁₆	—	6	2	no	—
86	M₁₆	—	6	2	no	—
87	M₁₆	—	6	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	3	1	no	—
90	C₂×C₈	—	3	1	no	—
91	C₂×C₈	—	3	4	no	—
92	C₂×C₈	—	3	2	no	—
93	C₂×C₈	—	6	4	no	—
94	C₂×C₈	—	6	2	no	—
95	C₂²×C₃	—	2	2	no	—
96	C₂²×C₃	—	4	1	no	—
97	C₂²×C₃	—	4	2	no	—
98	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
100	C₃×C₄	—	2	2	no	—
101	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₃×C₄	#117	1	1	yes	characteristic
103	C₃×C₄	#121	1	2	yes	—
104	C₂³	Dic₃⋊C₄	1	2	yes	—
105	C₂³	C₂×Dic₆	1	1	yes	characteristic
106	Q₈	—	2	2	no	—
107	D₄	—	2	4	no	—
108	D₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
113	C₂×C₄	D₃×D₄	1	2	yes	—
114	C₂×C₄	#237	1	2	yes	—
115	C₂×C₄	C₂×#67	1	1	yes	characteristic
116	C₈	—	3	2	no	—
117	C₈	—	3	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	6	2	no	—
120	C₈	—	12	2	no	—
121	C₂×C₃	—	2	1	no	—
122	C₂×C₃	—	2	2	no	—
123	C₂×C₃	—	4	2	no	—
124	C₂×C₃	#380	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	4	2	no	—
128	C₂²	#878	1	2	yes	—
129	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	#877	1	1	yes	characteristic
132	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
133	C₄	#939	1	2	yes	—
134	C₃	#1813	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#4942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#948

C₂

yes

maximal, characteristic

C₃×#413

C₂

yes

maximal, characteristic

#5237

C₂

yes

maximal, characteristic

C₂×#822

C₂

yes

maximal, complemented

C₃⋊#338

C₂

yes

maximal, complemented

#1813

—

maximal, Sylow

C₂×C₃×#41

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

#948

—

C₂×#208

C₄

yes

—

C₂×#208

C₂²

yes

—

C₂×#208

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₄

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

—

—

—

—

—

—

—

D₃

yes

characteristic

#338

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

D₄

yes

characteristic

C₂×C₃×D₄

D₄

yes

characteristic

C₂×C₃×D₄

Q₈

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

—

—

—

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

D₄

yes

—

C₂×#41

C₂×D₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

—

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₂×Q₈

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

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×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

C₂²⋊C₄

yes

—

C₃⋊C₈

C₂²⋊C₄

yes

—

#41

—

#41

—

C₂×Q₈

#67

yes

characteristic

C₂×D₄

Dic₆

yes

—

C₂×D₄

#67

yes

characteristic

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

100

C₃×C₄

—

101

C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

102

C₃×C₄

#117

yes

characteristic

103

C₃×C₄

#121

yes

—

104

C₂³

Dic₃⋊C₄

yes

—

105

C₂³

C₂×Dic₆

yes

characteristic

106

Q₈

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

Dic₃⋊C₄

yes

—

112

C₂×C₄

C₂×C₄×D₃

yes

characteristic

113

C₂×C₄

D₃×D₄

yes

—

114

C₂×C₄

#237

yes

—

115

C₂×C₄

C₂×#67

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂×C₃

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

#380

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#878

yes

—

129

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

130

C₄

—

131

C₄

#877

yes

characteristic

132

C₄

D₃×C₂²⋊C₄

yes

characteristic

133

C₄

#939

yes

—

134

C₃

#1813

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#4942

yes

characteristic

	subgroup	G/H
center	C₄	#877
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#413	C₂
socle	C₂×C₃	#380
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#413	D₃
hypocenter	C₃	#1813
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#877

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂×C₃×#41

C₂²

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
4. C₂×#822	C₂	semidirect
5. C₃⋊#338	C₂	semidirect
134. C₃	#1813	semidirect

H ≅ G/N

type

4. C₂×#822

C₂

semidirect

5. C₃⋊#338

C₂

semidirect

134. C₃

#1813

semidirect

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

subgroup	index
C₂×#948	2
C₃×#413	2
#5237	2
C₂×#822	2
C₃⋊#338	2
#1813	3

subgroup

index

Subgroups — #70134

Conjugacy classes of subgroups

Complements

Maximal subgroups