Group #71260 — subgroups

Subgroups — #71260

Order 384.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	#5093	C₂	1	1	yes	maximal, characteristic, complemented
3	#5088	C₂	1	1	yes	maximal, characteristic, complemented
4	#5073	C₂	1	1	yes	maximal, characteristic
5	#5070	C₂	1	1	yes	maximal, characteristic, complemented
6	#4694	C₂	1	1	yes	maximal, characteristic, complemented
7	#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	#1717	—	3	1	no	maximal, Sylow
9	C₂×C₃×SD₁₆	—	2	2	no	—
10	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×#127	C₂²	1	1	yes	characteristic
12	C₂×#213	—	2	2	no	—
13	#874	C₂²	1	1	yes	characteristic
14	#869	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#832	—	2	2	no	—
17	#821	—	2	2	no	—
18	#810	—	2	2	no	—
19	#809	—	2	2	no	—
20	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#493	—	3	1	no	—
23	#492	D₃	1	1	yes	characteristic, complemented
24	#487	—	3	1	no	—
25	#486	—	3	1	no	—
26	#486	—	3	1	no	—
27	#486	—	3	1	no	—
28	#322	—	3	1	no	—
29	C₂×C₃×Q₈	—	2	1	no	—
30	C₂×C₃×D₄	—	2	1	no	—
31	C₂×C₃×D₄	—	2	1	no	—
32	C₂×C₃×D₄	—	2	1	no	—
33	C₂×D₁₂	—	2	1	no	—
34	C₂×D₁₂	—	2	1	no	—
35	C₂×D₁₂	—	2	1	no	—
36	C₂×Dic₆	—	2	1	no	—
37	C₃×SD₁₆	—	4	4	no	—
38	C₂×C₃×C₈	D₄	1	2	yes	—
39	C₃×C₄⋊C₄	—	2	1	no	—
40	C₃×C₄⋊C₄	—	2	1	no	—
41	C₃×C₄²	C₂³	1	1	yes	characteristic
42	#213	—	4	4	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	C₂×C₃⋊C₈	D₄	1	2	yes	—
46	C₂×C₃⋊C₈	D₄	1	2	yes	—
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×D₈	—	6	2	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
51	#127	—	3	1	no	—
52	#127	C₂×D₃	1	1	yes	characteristic
53	#103	—	6	2	no	—
54	#102	—	6	2	no	—
55	#102	—	6	2	no	—
56	#102	—	6	2	no	—
57	C₄×C₈	—	3	1	no	—
58	C₄×C₈	—	3	1	no	—
59	C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂³×C₃	—	4	1	no	—
61	C₂²×D₃	—	4	1	no	—
62	C₃×Q₈	—	4	2	no	—
63	C₃×D₄	—	4	2	no	—
64	C₃×D₄	—	4	2	no	—
65	C₃×D₄	—	4	2	no	—
66	C₂×C₃×C₄	—	4	1	no	—
67	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₃	—	4	1	no	—
71	D₁₂	—	4	2	no	—
72	D₁₂	—	4	2	no	—
73	D₁₂	—	4	2	no	—
74	Dic₆	—	4	2	no	—
75	C₃×C₈	—	2	2	no	—
76	C₃⋊C₈	—	2	2	no	—
77	C₃⋊C₈	—	2	2	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	6	1	no	—
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	6	1	no	—
84	C₂×D₄	—	6	1	no	—
85	C₂×D₄	—	6	1	no	—
86	SD₁₆	—	4	4	no	—
87	D₈	—	12	4	no	—
88	C₂×C₈	—	3	2	no	—
89	C₂×C₈	—	3	2	no	—
90	C₂×C₈	#67	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	6	1	no	—
94	C₄⋊C₄	—	6	1	no	—
95	C₄²	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃	—	8	1	no	—
97	C₂²×C₃	—	8	1	no	—
98	C₂²×C₃	—	8	1	no	—
99	C₂²×C₃	#127	1	1	yes	characteristic
100	C₂×D₃	—	8	1	no	—
101	C₂×D₃	—	8	1	no	—
102	C₂×D₃	—	8	1	no	—
103	C₃×C₄	—	8	1	no	—
104	C₃×C₄	#135	1	2	yes	—
105	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
106	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
107	Dic₃	—	8	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	12	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	12	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	12	2	no	—
116	D₄	—	12	2	no	—
117	D₄	—	12	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	12	1	no	—
120	C₂×C₄	D₃×D₄	1	1	yes	characteristic
121	C₂×C₄	D₃×D₄	1	1	yes	characteristic
122	C₂×C₄	C₂×#67	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	6	2	no	—
125	C₈	—	6	2	no	—
126	C₂×C₃	—	16	1	no	—
127	C₂×C₃	#495	1	1	yes	characteristic
128	C₂×C₃	#496	1	1	yes	characteristic
129	C₂×C₃	#496	1	1	yes	characteristic
130	D₃	—	16	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	24	1	no	—
135	C₂²	—	24	1	no	—
136	C₂²	—	24	1	no	—
137	C₂²	#940	1	1	yes	characteristic
138	C₄	—	8	1	no	—
139	C₄	—	24	1	no	—
140	C₄	#914	1	2	yes	—
141	C₄	#916	1	2	yes	—
142	C₄	#932	1	2	yes	—
143	C₃	#1717	1	1	yes	characteristic, complemented, Sylow
144	C₂	—	16	1	no	—
145	C₂	—	48	1	no	—
146	C₂	#5074	1	1	yes	characteristic
147	C₂	#5168	1	1	yes	characteristic
148	C₂	#5176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#492

C₂

yes

maximal, characteristic, complemented

#5093

C₂

yes

maximal, characteristic, complemented

#5088

C₂

yes

maximal, characteristic, complemented

#5073

C₂

yes

maximal, characteristic

#5070

C₂

yes

maximal, characteristic, complemented

#4694

C₂

yes

maximal, characteristic, complemented

#4454

C₂

yes

maximal, characteristic, complemented

#1717

—

maximal, Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×#127

C₂²

yes

characteristic

C₂×#213

—

#874

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

—

—

—

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

characteristic

#493

—

#492

D₃

yes

characteristic, complemented

—

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#213

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

—

#127

C₂×D₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂³×C₃

—

C₂²×D₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#127

yes

characteristic

100

C₂×D₃

—

101

C₂×D₃

—

102

C₂×D₃

—

103

C₃×C₄

—

104

C₃×C₄

#135

yes

—

105

C₃×C₄

AGL(1,ℤ/8)

yes

—

106

C₃×C₄

AGL(1,ℤ/8)

yes

—

107

Dic₃

—

108

C₂³

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₃×D₄

yes

characteristic

121

C₂×C₄

D₃×D₄

yes

characteristic

122

C₂×C₄

C₂×#67

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂×C₃

—

127

C₂×C₃

#495

yes

characteristic

128

C₂×C₃

#496

yes

characteristic

129

C₂×C₃

#496

yes

characteristic

130

D₃

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#940

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

#914

yes

—

141

C₄

#916

yes

—

142

C₄

#932

yes

—

143

C₃

#1717

yes

characteristic, complemented, Sylow

144

C₂

—

145

C₂

—

146

C₂

#5074

yes

characteristic

147

C₂

#5168

yes

characteristic

148

C₂

#5176

yes

characteristic

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#492	C₂
socle	C₂²×C₃	#127
prime radical	#5073	C₂
hypercenter	#492	D₃
hypocenter	C₃	#1717
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#940

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₃×#492	C₂	semidirect
2. #5093	C₂	semidirect
3. #5088	C₂	semidirect
5. #5070	C₂	semidirect
6. #4694	C₂	semidirect
7. #4454	C₂	semidirect
23. #492	D₃	semidirect
143. C₃	#1717	semidirect

H ≅ G/N

type

1. C₃×#492

C₂

semidirect

2. #5093

C₂

semidirect

3. #5088

C₂

semidirect

5. #5070

C₂

semidirect

6. #4694

C₂

semidirect

7. #4454

C₂

semidirect

23. #492

D₃

semidirect

143. C₃

#1717

semidirect

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

subgroup	index
C₃×#492	2
#5093	2
#5088	2
#5073	2
#5070	2
#4694	2
#4454	2
#1717	3

subgroup

index

Subgroups — #71260

Conjugacy classes of subgroups

Complements

Maximal subgroups