Group #71363 — subgroups

Subgroups — #71363

Order 384.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	#5088	C₂	1	1	yes	maximal, characteristic
3	#5085	C₂	1	1	yes	maximal, characteristic
4	#5080	C₂	1	1	yes	maximal, characteristic, complemented
5	#5079	C₂	1	1	yes	maximal, characteristic, complemented
6	#4693	C₂	1	1	yes	maximal, characteristic
7	#4455	C₂	1	1	yes	maximal, characteristic, complemented
8	#1714	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×#215	—	2	2	no	—
12	C₂×#204	—	2	2	no	—
13	#874	C₂²	1	1	yes	characteristic
14	#869	C₂²	1	1	yes	characteristic
15	C₃×#107	—	2	2	no	—
16	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	#835	—	2	2	no	—
18	#809	—	2	2	no	—
19	#808	—	2	2	no	—
20	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#500	D₃	1	1	yes	characteristic, complemented
23	#498	—	3	1	no	—
24	#492	—	3	1	no	—
25	#492	—	3	1	no	—
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×Q₈	—	2	1	no	—
31	C₂×D₁₂	—	2	1	no	—
32	C₂×D₁₂	—	2	1	no	—
33	C₂×D₁₂	—	2	1	no	—
34	C₂×Dic₆	—	2	1	no	—
35	C₂×C₃×C₈	D₄	1	2	yes	—
36	C₃×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄⋊C₄	—	2	1	no	—
39	C₃×C₄⋊C₄	—	2	1	no	—
40	C₃×C₄²	C₂³	1	1	yes	characteristic
41	#215	—	4	4	no	—
42	C₄⋊Dic₃	—	2	1	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₂×C₃⋊C₈	D₄	1	2	yes	—
45	C₂×C₃⋊C₈	D₄	1	2	yes	—
46	#204	—	4	4	no	—
47	C₂×SD₁₆	—	6	2	no	—
48	C₂×SD₁₆	—	6	2	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
51	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
52	#127	—	3	1	no	—
53	#107	—	2	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	C₂²×D₃	—	4	1	no	—
61	C₃×Q₈	—	4	2	no	—
62	C₃×Q₈	—	4	2	no	—
63	C₂×C₃×C₄	—	4	1	no	—
64	C₂×C₃×C₄	—	4	1	no	—
65	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₃	—	4	1	no	—
69	D₁₂	—	4	2	no	—
70	D₁₂	—	4	2	no	—
71	D₁₂	—	4	2	no	—
72	Dic₆	—	4	2	no	—
73	C₃×C₈	SD₁₆	1	4	yes	complemented
74	C₃⋊C₈	—	2	2	no	—
75	C₃⋊C₈	—	2	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	6	1	no	—
79	C₂×D₄	—	6	1	no	—
80	C₂×D₄	—	6	1	no	—
81	C₂×D₄	—	6	1	no	—
82	SD₁₆	—	12	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	—	2	1	no	—
90	C₄⋊C₄	—	2	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	#127	1	1	yes	characteristic
95	C₂×D₃	—	8	1	no	—
96	C₂×D₃	—	8	1	no	—
97	C₂×D₃	—	8	1	no	—
98	C₃×C₄	—	8	1	no	—
99	C₃×C₄	—	8	1	no	—
100	C₃×C₄	C₂×SD₁₆	1	2	yes	—
101	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
102	C₃×C₄	#137	1	2	yes	—
103	Dic₃	—	8	1	no	—
104	C₂³	—	12	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	12	2	no	—
108	D₄	—	12	2	no	—
109	D₄	—	12	2	no	—
110	D₄	—	12	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	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	—	6	2	no	—
118	C₈	—	6	2	no	—
119	C₈	#215	1	4	yes	complemented
120	C₂×C₃	#492	1	1	yes	characteristic
121	C₂×C₃	#496	1	1	yes	characteristic
122	C₂×C₃	#497	1	1	yes	characteristic
123	D₃	—	16	1	no	—
124	C₂²	—	24	1	no	—
125	C₂²	—	24	1	no	—
126	C₂²	—	24	1	no	—
127	C₂²	#940	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	24	1	no	—
131	C₄	#911	1	2	yes	—
132	C₄	#918	1	2	yes	—
133	C₄	C₂×#215	1	2	yes	—
134	C₃	#1714	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	48	1	no	—
136	C₂	#5085	1	1	yes	characteristic
137	C₂	#5154	1	1	yes	characteristic
138	C₂	#5190	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#500

C₂

yes

maximal, characteristic, complemented

#5088

C₂

yes

maximal, characteristic

#5085

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, characteristic, complemented

#5079

C₂

yes

maximal, characteristic, complemented

#4693

C₂

yes

maximal, characteristic

#4455

C₂

yes

maximal, characteristic, complemented

#1714

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×C₄⋊Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×#107

—

C₃×C₄×C₈

C₂²

yes

—

—

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

characteristic

#500

D₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#215

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

#204

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

C₄⋊Q₈

C₂×D₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

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

SD₁₆

yes

complemented

C₃⋊C₈

—

C₃⋊C₈

—

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

#127

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

C₂×SD₁₆

yes

—

101

C₃×C₄

AGL(1,ℤ/8)

yes

—

102

C₃×C₄

#137

yes

—

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

#215

yes

complemented

120

C₂×C₃

#492

yes

characteristic

121

C₂×C₃

#496

yes

characteristic

122

C₂×C₃

#497

yes

characteristic

123

D₃

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#940

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#911

yes

—

132

C₄

#918

yes

—

133

C₄

C₂×#215

yes

—

134

C₃

#1714

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

#5085

yes

characteristic

137

C₂

#5154

yes

characteristic

138

C₂

#5190

yes

characteristic

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#500	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	#500	D₃
hypocenter	C₃	#1714
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

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 ◁ #500

N	H ≅ G/N	type
1. C₃×#500	C₂	semidirect
4. #5080	C₂	semidirect
5. #5079	C₂	semidirect
7. #4455	C₂	semidirect
22. #500	D₃	semidirect
73. C₃×C₈	SD₁₆	semidirect
119. C₈	#215	semidirect
134. C₃	#1714	semidirect

H ≅ G/N

type

1. C₃×#500

C₂

semidirect

4. #5080

C₂

semidirect

5. #5079

C₂

semidirect

7. #4455

C₂

semidirect

22. #500

D₃

semidirect

73. C₃×C₈

SD₁₆

semidirect

119. C₈

#215

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×#500	2
#5088	2
#5085	2
#5080	2
#5079	2
#4693	2
#4455	2
#1714	3

subgroup

index

Subgroups — #71363

Conjugacy classes of subgroups

Complements

Maximal subgroups