Group #70041 — subgroups

Subgroups — #70041

Order 384.

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

normal subgroups only

This group has 260 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×#410	C₂	1	1	yes	maximal, characteristic
3	#5240	C₂	1	1	yes	maximal, characteristic
4	#4998	C₂	1	2	yes	maximal, complemented
5	#4531	C₂	1	2	yes	maximal, complemented
6	#1812	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₃×#117	C₂²	1	2	yes	—
9	#948	—	2	2	no	—
10	#948	—	2	2	no	—
11	#926	C₂²	1	2	yes	—
12	C₂×#208	C₄	1	2	yes	complemented
13	C₂×#208	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₄	1	2	yes	complemented
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#99	—	2	2	no	—
17	#834	—	2	2	no	—
18	#808	—	2	2	no	—
19	#807	—	2	2	no	—
20	C₂×#131	—	3	1	no	—
21	#427	—	3	2	no	—
22	#410	—	3	1	no	—
23	#410	D₃	1	1	yes	characteristic
24	H(3,ℤ/4)	—	3	2	no	—
25	C₃×#41	—	2	2	no	—
26	C₃×#41	—	2	2	no	—
27	C₂×C₃×Q₈	Q₈	1	1	yes	characteristic
28	C₂×C₃×D₄	D₄	1	2	yes	—
29	C₂×C₃×D₄	Q₈	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₃×C₄⋊C₄	—	2	2	no	—
33	C₃×C₂²⋊C₄	—	2	2	no	—
34	C₃×C₄²	—	2	2	no	—
35	#217	—	2	2	no	—
36	C₄⋊Dic₃	—	2	2	no	—
37	C₄×Dic₃	—	2	2	no	—
38	#208	—	2	2	no	—
39	#208	—	2	4	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃⋊C₈	—	2	2	no	—
42	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃⋊C₈	D₄	1	2	yes	—
46	C₂×C₃⋊C₈	D₄	1	2	yes	—
47	C₂×#41	C₂×D₃	1	1	yes	characteristic
48	#131	—	6	2	no	—
49	#131	—	6	2	no	—
50	C₂×M₁₆	—	3	2	no	—
51	C₂×M₁₆	—	3	1	no	—
52	C₂²×C₈	—	3	1	no	—
53	C₂²×C₈	—	3	2	no	—
54	#117	—	3	2	no	—
55	#117	C₂×D₃	1	2	yes	—
56	#107	—	6	2	no	—
57	#106	—	6	2	no	—
58	#99	—	2	2	no	—
59	#99	—	6	2	no	—
60	C₂³×C₃	C₄⋊C₄	1	2	yes	—
61	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
62	C₃×Q₈	—	2	2	no	—
63	C₃×D₄	—	2	4	no	—
64	C₃×D₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	4	2	no	—
68	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₄	#41	1	2	yes	—
73	C₂×Dic₃	—	4	2	no	—
74	C₃⋊C₈	—	2	2	no	—
75	C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Dic₆	1	1	yes	characteristic
81	C₂×D₄	Dic₆	1	1	yes	characteristic
82	C₂×D₄	#67	1	2	yes	—
83	C₂²×C₄	C₄×D₃	1	2	yes	—
84	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
85	M₁₆	—	6	4	no	—
86	M₁₆	—	6	2	no	—
87	C₂×C₈	—	3	1	no	—
88	C₂×C₈	—	3	1	no	—
89	C₂×C₈	—	3	2	no	—
90	C₂×C₈	—	3	2	no	—
91	C₂×C₈	—	3	2	no	—
92	C₂×C₈	—	6	2	no	—
93	C₂×C₈	—	6	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	6	2	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₂²⋊C₄	—	6	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	6	2	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	—	4	1	no	—
102	C₂²×C₃	—	4	2	no	—
103	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃	#121	1	2	yes	—
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	—	8	2	no	—
107	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₃×C₄	#117	1	1	yes	characteristic
109	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
110	Dic₃	—	8	2	no	—
111	C₂³	Dic₃⋊C₄	1	2	yes	—
112	C₂³	C₂×#67	1	1	yes	characteristic
113	Q₈	—	2	2	no	—
114	D₄	—	2	4	no	—
115	D₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	12	2	no	—
120	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
121	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₄	D₃×D₄	1	2	yes	—
124	C₂×C₄	#237	1	2	yes	—
125	C₈	—	3	1	no	—
126	C₈	—	3	1	no	—
127	C₈	—	3	2	no	—
128	C₈	—	6	2	no	—
129	C₂×C₃	—	2	2	no	—
130	C₂×C₃	—	2	1	no	—
131	C₂×C₃	—	4	2	no	—
132	C₂×C₃	#380	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
137	C₂²	#939	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	8	2	no	—
140	C₄	—	24	2	no	—
141	C₄	#877	1	1	yes	characteristic
142	C₄	#878	1	2	yes	—
143	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
144	C₃	#1812	1	1	yes	characteristic, complemented, Sylow
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#4942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#948

C₂

yes

maximal, characteristic

C₃×#410

C₂

yes

maximal, characteristic

#5240

C₂

yes

maximal, characteristic

#4998

C₂

yes

maximal, complemented

#4531

C₂

yes

maximal, complemented

#1812

—

maximal, Sylow

C₂×C₃×#41

C₂²

yes

characteristic

C₃×#117

C₂²

yes

—

#948

—

#948

—

#926

C₂²

yes

—

C₂×#208

C₄

yes

complemented

C₂×#208

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₄

yes

complemented

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₃×#99

—

—

—

—

—

—

—

D₃

yes

characteristic

H(3,ℤ/4)

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

Q₈

yes

characteristic

C₂×C₃×D₄

D₄

yes

—

C₂×C₃×D₄

Q₈

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

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×C₃⋊C₈

D₄

yes

—

C₂×#41

C₂×D₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₃

yes

—

—

—

—

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

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

—

C₃⋊C₈

—

C₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₃⋊C₈

C₂²⋊C₄

yes

—

#41

—

#41

—

C₂×Q₈

Dic₆

yes

characteristic

C₂×D₄

Dic₆

yes

characteristic

C₂×D₄

#67

yes

—

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₂²×C₃

—

101

C₂²×C₃

—

102

C₂²×C₃

—

103

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

104

C₂²×C₃

#121

yes

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

108

C₃×C₄

#117

yes

characteristic

109

C₃×C₄

C₂²⋊Q₈

yes

—

110

Dic₃

—

111

C₂³

Dic₃⋊C₄

yes

—

112

C₂³

C₂×#67

yes

characteristic

113

Q₈

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

Dic₃⋊C₄

yes

—

121

C₂×C₄

C₂×Dic₆

yes

characteristic

122

C₂×C₄

C₂×C₄×D₃

yes

characteristic

123

C₂×C₄

D₃×D₄

yes

—

124

C₂×C₄

#237

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

#380

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

137

C₂²

#939

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#877

yes

characteristic

142

C₄

#878

yes

—

143

C₄

D₃×C₂²⋊C₄

yes

characteristic

144

C₃

#1812

yes

characteristic, complemented, Sylow

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#4942

yes

characteristic

	subgroup	G/H
center	C₄	#877
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#410	C₂
socle	C₂×C₃	#380
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#410	D₃
hypocenter	C₃	#1812
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

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 ◁ #410

N	H ≅ G/N	type
4. #4998	C₂	semidirect
5. #4531	C₂	semidirect
12. C₂×#208	C₄	semidirect
14. C₂²×C₃⋊C₈	C₄	semidirect
144. C₃	#1812	semidirect

H ≅ G/N

type

4. #4998

C₂

semidirect

5. #4531

C₂

semidirect

12. C₂×#208

C₄

semidirect

14. C₂²×C₃⋊C₈

C₄

semidirect

144. C₃

#1812

semidirect

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

subgroup	index
C₂×#948	2
C₃×#410	2
#5240	2
#4998	2
#4531	2
#1812	3

subgroup

index

Subgroups — #70041

Conjugacy classes of subgroups

Complements

Maximal subgroups