Group #71213 — subgroups

Subgroups — #71213

Order 384.

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

normal subgroups only

This group has 190 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×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	#5093	C₂	1	1	yes	maximal, characteristic, complemented
3	#5088	C₂	1	1	yes	maximal, characteristic
4	#5085	C₂	1	1	yes	maximal, characteristic
5	#5080	C₂	1	1	yes	maximal, characteristic, complemented
6	#4694	C₂	1	1	yes	maximal, characteristic
7	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#1708	—	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	#874	C₂²	1	1	yes	characteristic
13	#869	C₂²	1	1	yes	characteristic
14	C₃×#103	—	2	2	no	—
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#821	—	2	2	no	—
17	#810	—	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	#498	—	3	1	no	—
23	#492	—	3	1	no	—
24	#487	—	3	1	no	—
25	#487	D₃	1	1	yes	characteristic, complemented
26	#486	—	3	1	no	—
27	#486	—	3	1	no	—
28	C₈²	—	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	C₂×SD₁₆	—	6	2	no	—
47	C₄⋊Q₈	—	3	1	no	—
48	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
49	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
50	#127	—	3	1	no	—
51	#106	—	6	2	no	—
52	#103	—	2	2	no	—
53	#103	—	6	2	no	—
54	#102	—	6	2	no	—
55	#102	—	6	2	no	—
56	C₄×C₈	—	3	1	no	—
57	C₄×C₈	—	3	1	no	—
58	C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂²×D₃	—	4	1	no	—
60	C₃×Q₈	—	4	2	no	—
61	C₃×Q₈	—	4	2	no	—
62	C₂×C₃×C₄	—	4	1	no	—
63	C₂×C₃×C₄	—	4	1	no	—
64	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₃	—	4	1	no	—
68	D₁₂	—	4	2	no	—
69	D₁₂	—	4	2	no	—
70	D₁₂	—	4	2	no	—
71	Dic₆	—	4	2	no	—
72	C₃×C₈	—	2	2	no	—
73	C₃⋊C₈	—	2	2	no	—
74	C₃⋊C₈	SD₁₆	1	4	yes	—
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	6	1	no	—
78	C₂×D₄	—	6	1	no	—
79	C₂×D₄	—	6	1	no	—
80	C₂×D₄	—	6	1	no	—
81	SD₁₆	—	12	4	no	—
82	C₂×C₈	—	3	2	no	—
83	C₂×C₈	—	3	2	no	—
84	C₂×C₈	#67	1	2	yes	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
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	C₄²	C₂²×D₃	1	1	yes	characteristic
92	C₂²×C₃	#127	1	1	yes	characteristic
93	C₂×D₃	—	8	1	no	—
94	C₂×D₃	—	8	1	no	—
95	C₂×D₃	—	8	1	no	—
96	C₃×C₄	—	8	1	no	—
97	C₃×C₄	—	8	1	no	—
98	C₃×C₄	C₂×SD₁₆	1	2	yes	—
99	C₃×C₄	#135	1	2	yes	—
100	C₃×C₄	#135	1	2	yes	—
101	Dic₃	—	8	1	no	—
102	C₂³	—	12	1	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	12	2	no	—
106	D₄	—	12	2	no	—
107	D₄	—	12	2	no	—
108	D₄	—	12	2	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	12	1	no	—
112	C₂×C₄	D₃×D₄	1	1	yes	characteristic
113	C₂×C₄	D₃×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂×#67	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	—	3	4	no	—
117	C₈	—	6	2	no	—
118	C₂×C₃	#492	1	1	yes	characteristic
119	C₂×C₃	#495	1	1	yes	characteristic
120	C₂×C₃	#495	1	1	yes	characteristic
121	D₃	—	16	1	no	—
122	C₂²	—	24	1	no	—
123	C₂²	—	24	1	no	—
124	C₂²	—	24	1	no	—
125	C₂²	#940	1	1	yes	characteristic
126	C₄	—	8	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	24	1	no	—
129	C₄	D₃×SD₁₆	1	2	yes	—
130	C₄	#919	1	2	yes	—
131	C₄	#950	1	2	yes	—
132	C₃	#1708	1	1	yes	characteristic, complemented, Sylow
133	C₂	—	48	1	no	—
134	C₂	#5059	1	1	yes	characteristic
135	C₂	#5175	1	1	yes	characteristic
136	C₂	#5191	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#487

C₂

yes

maximal, characteristic, complemented

#5093

C₂

yes

maximal, characteristic, complemented

#5088

C₂

yes

maximal, characteristic

#5085

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, characteristic, complemented

#4694

C₂

yes

maximal, characteristic

C₈×C₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#1708

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#215

—

#874

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×#103

—

C₃×C₄×C₈

C₂²

yes

—

—

—

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

—

—

—

D₃

yes

characteristic, complemented

#486

—

#486

—

C₈²

—

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

—

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

—

C₃⋊C₈

—

C₃⋊C₈

SD₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

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

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

#135

yes

—

100

C₃×C₄

#135

yes

—

101

Dic₃

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

D₃×D₄

yes

characteristic

113

C₂×C₄

D₃×D₄

yes

characteristic

114

C₂×C₄

C₂×#67

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂×C₃

#492

yes

characteristic

119

C₂×C₃

#495

yes

characteristic

120

C₂×C₃

#495

yes

characteristic

121

D₃

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#940

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

D₃×SD₁₆

yes

—

130

C₄

#919

yes

—

131

C₄

#950

yes

—

132

C₃

#1708

yes

characteristic, complemented, Sylow

133

C₂

—

134

C₂

#5059

yes

characteristic

135

C₂

#5175

yes

characteristic

136

C₂

#5191

yes

characteristic

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#487	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	#487	D₃
hypocenter	C₃	#1708
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

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 ◁ #487

N	H ≅ G/N	type
1. C₃×#487	C₂	semidirect
2. #5093	C₂	semidirect
5. #5080	C₂	semidirect
7. C₈×C₃⋊C₈	C₂	semidirect
25. #487	D₃	semidirect
132. C₃	#1708	semidirect

H ≅ G/N

type

1. C₃×#487

C₂

semidirect

2. #5093

C₂

semidirect

5. #5080

C₂

semidirect

7. C₈×C₃⋊C₈

C₂

semidirect

25. #487

D₃

semidirect

132. C₃

#1708

semidirect

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

subgroup	index
C₃×#487	2
#5093	2
#5088	2
#5085	2
#5080	2
#4694	2
C₈×C₃⋊C₈	2
#1708	3

subgroup

index

C₈×C₃⋊C₈

Subgroups — #71213

Conjugacy classes of subgroups

Complements

Maximal subgroups