Group #71257 — subgroups

Subgroups — #71257

Order 384.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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
2	#5080	C₂	1	1	yes	maximal, characteristic, complemented
3	#5078	C₂	1	1	yes	maximal, characteristic, complemented
4	#5076	C₂	1	1	yes	maximal, characteristic
5	#5070	C₂	1	1	yes	maximal, characteristic
6	#4681	C₂	1	1	yes	maximal, characteristic, complemented
7	#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	#1714	—	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×#214	—	2	2	no	—
13	#869	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	#816	—	2	2	no	—
18	#808	—	2	2	no	—
19	#807	—	2	2	no	—
20	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#500	—	3	1	no	—
23	#498	—	3	1	no	—
24	#492	—	3	1	no	—
25	#492	D₃	1	1	yes	characteristic
26	#487	—	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×Dic₆	—	2	1	no	—
34	C₂×Dic₆	—	2	1	no	—
35	C₃×SD₁₆	—	4	4	no	—
36	C₂×C₃×C₈	D₄	1	2	yes	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄⋊C₄	—	2	1	no	—
39	C₃×C₄²	C₂³	1	1	yes	characteristic
40	#214	—	4	4	no	—
41	C₄⋊Dic₃	—	2	1	no	—
42	C₄⋊Dic₃	—	2	1	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×SD₁₆	—	6	2	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	—	3	1	no	—
51	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
52	#127	C₂×D₃	1	1	yes	characteristic
53	#107	—	6	2	no	—
54	#106	—	6	2	no	—
55	#103	—	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	C₃×Q₈	—	4	2	no	—
62	C₃×D₄	—	4	2	no	—
63	C₃×D₄	—	4	2	no	—
64	C₃×D₄	—	4	2	no	—
65	C₂×C₃×C₄	—	4	1	no	—
66	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×Dic₃	—	4	1	no	—
70	C₂×Dic₃	—	4	1	no	—
71	Dic₆	—	4	2	no	—
72	Dic₆	—	4	2	no	—
73	C₃×C₈	—	2	2	no	—
74	C₃⋊C₈	—	2	2	no	—
75	C₃⋊C₈	SD₁₆	1	4	yes	complemented
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	6	1	no	—
78	C₂×Q₈	—	6	1	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	—	2	1	no	—
82	SD₁₆	—	4	4	no	—
83	SD₁₆	—	12	4	no	—
84	C₂×C₈	—	3	2	no	—
85	C₂×C₈	—	3	2	no	—
86	C₂×C₈	#67	1	2	yes	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	6	1	no	—
90	C₄⋊C₄	—	6	1	no	—
91	C₄⋊C₄	—	6	1	no	—
92	C₄⋊C₄	—	6	1	no	—
93	C₄²	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃	—	8	1	no	—
95	C₂²×C₃	—	8	1	no	—
96	C₂²×C₃	—	8	1	no	—
97	C₂²×C₃	#127	1	1	yes	characteristic
98	C₃×C₄	—	8	1	no	—
99	C₃×C₄	C₂×SD₁₆	1	2	yes	—
100	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
101	C₃×C₄	#137	1	2	yes	—
102	Dic₃	—	8	1	no	—
103	Dic₃	—	8	1	no	—
104	C₂³	—	4	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	12	2	no	—
107	Q₈	—	12	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	12	1	no	—
113	C₂×C₄	—	12	1	no	—
114	C₂×C₄	D₃×D₄	1	1	yes	characteristic
115	C₂×C₄	D₃×D₄	1	1	yes	characteristic
116	C₂×C₄	C₂×#67	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	3	4	no	—
119	C₈	—	6	2	no	—
120	C₂×C₃	—	16	1	no	—
121	C₂×C₃	#492	1	1	yes	characteristic
122	C₂×C₃	#496	1	1	yes	characteristic
123	C₂×C₃	#497	1	1	yes	characteristic
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	#940	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	24	1	no	—
130	C₄	—	24	1	no	—
131	C₄	D₃×SD₁₆	1	2	yes	—
132	C₄	#915	1	2	yes	—
133	C₄	#932	1	2	yes	—
134	C₃	#1714	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	16	1	no	—
136	C₂	#5074	1	1	yes	characteristic
137	C₂	#5167	1	1	yes	characteristic
138	C₂	#5175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#492

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, characteristic, complemented

#5078

C₂

yes

maximal, characteristic, complemented

#5076

C₂

yes

maximal, characteristic

#5070

C₂

yes

maximal, characteristic

#4681

C₂

yes

maximal, characteristic, complemented

#4454

C₂

yes

maximal, characteristic, complemented

#1714

—

maximal, Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×#127

C₂²

yes

characteristic

C₂×#214

—

#869

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

—

—

—

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

—

—

—

D₃

yes

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#214

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

C₂×D₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂³×C₃

—

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

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

SD₁₆

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#127

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂×SD₁₆

yes

—

100

C₃×C₄

AGL(1,ℤ/8)

yes

—

101

C₃×C₄

#137

yes

—

102

Dic₃

—

103

Dic₃

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₃×D₄

yes

characteristic

115

C₂×C₄

D₃×D₄

yes

characteristic

116

C₂×C₄

C₂×#67

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂×C₃

—

121

C₂×C₃

#492

yes

characteristic

122

C₂×C₃

#496

yes

characteristic

123

C₂×C₃

#497

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#940

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₃×SD₁₆

yes

—

132

C₄

#915

yes

—

133

C₄

#932

yes

—

134

C₃

#1714

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

#5074

yes

characteristic

137

C₂

#5167

yes

characteristic

138

C₂

#5175

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	C₃×#127	C₂²
hypercenter	#492	D₃
hypocenter	C₃	#1714
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

C₂²

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #5080	C₂	semidirect
3. #5078	C₂	semidirect
6. #4681	C₂	semidirect
7. #4454	C₂	semidirect
75. C₃⋊C₈	SD₁₆	semidirect
134. C₃	#1714	semidirect

H ≅ G/N

type

2. #5080

C₂

semidirect

3. #5078

C₂

semidirect

6. #4681

C₂

semidirect

7. #4454

C₂

semidirect

75. C₃⋊C₈

SD₁₆

semidirect

134. C₃

#1714

semidirect

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

subgroup	index
C₃×#492	2
#5080	2
#5078	2
#5076	2
#5070	2
#4681	2
#4454	2
#1714	3

subgroup

index

Subgroups — #71257

Conjugacy classes of subgroups

Complements

Maximal subgroups