Group #71259 — subgroups

Subgroups — #71259

Order 384.

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

normal subgroups only

This group has 220 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, complemented
2	#5088	C₂	1	1	yes	maximal, characteristic, complemented
3	#5080	C₂	1	1	yes	maximal, characteristic, complemented
4	#5076	C₂	1	1	yes	maximal, characteristic, complemented
5	#5073	C₂	1	1	yes	maximal, characteristic
6	#4694	C₂	1	1	yes	maximal, characteristic, complemented
7	#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	#1715	—	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	C₂×#213	—	2	2	no	—
14	#874	C₂²	1	1	yes	characteristic
15	#869	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	#821	—	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	#493	—	3	1	no	—
24	#492	—	3	1	no	—
25	#492	D₃	1	1	yes	characteristic, complemented
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	#214	—	4	4	no	—
43	#213	—	4	4	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	C₄⋊Dic₃	—	2	1	no	—
46	C₂×C₃⋊C₈	D₄	1	2	yes	—
47	C₂×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	6	2	no	—
50	C₂×D₈	—	6	2	no	—
51	C₄⋊Q₈	—	3	1	no	—
52	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
53	#127	—	3	1	no	—
54	#127	C₂×D₃	1	1	yes	characteristic
55	#106	—	6	2	no	—
56	#102	—	6	2	no	—
57	#102	—	6	2	no	—
58	C₄×C₈	—	3	1	no	—
59	C₄×C₈	—	3	1	no	—
60	C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂³×C₃	—	4	1	no	—
62	C₂²×D₃	—	4	1	no	—
63	C₃×Q₈	—	4	2	no	—
64	C₃×D₄	—	4	2	no	—
65	C₃×D₄	—	4	2	no	—
66	C₃×D₄	—	4	2	no	—
67	C₂×C₃×C₄	—	4	1	no	—
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×Dic₃	—	4	1	no	—
72	D₁₂	—	4	2	no	—
73	D₁₂	—	4	2	no	—
74	D₁₂	—	4	2	no	—
75	Dic₆	—	4	2	no	—
76	C₃×C₈	—	2	2	no	—
77	C₃⋊C₈	—	2	2	no	—
78	C₃⋊C₈	SD₁₆	1	4	yes	complemented
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	6	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	6	1	no	—
85	C₂×D₄	—	6	1	no	—
86	C₂×D₄	—	6	1	no	—
87	SD₁₆	—	4	4	no	—
88	SD₁₆	—	12	4	no	—
89	D₈	—	12	4	no	—
90	C₂×C₈	—	3	2	no	—
91	C₂×C₈	—	3	2	no	—
92	C₂×C₈	#67	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	6	1	no	—
96	C₄⋊C₄	—	6	1	no	—
97	C₄²	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃	—	8	1	no	—
99	C₂²×C₃	—	8	1	no	—
100	C₂²×C₃	—	8	1	no	—
101	C₂²×C₃	#127	1	1	yes	characteristic
102	C₂×D₃	—	8	1	no	—
103	C₂×D₃	—	8	1	no	—
104	C₂×D₃	—	8	1	no	—
105	C₃×C₄	—	8	1	no	—
106	C₃×C₄	C₂×SD₁₆	1	2	yes	—
107	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
108	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
109	Dic₃	—	8	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	12	1	no	—
112	Q₈	—	4	2	no	—
113	Q₈	—	12	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	12	2	no	—
118	D₄	—	12	2	no	—
119	D₄	—	12	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	12	1	no	—
122	C₂×C₄	D₃×D₄	1	1	yes	characteristic
123	C₂×C₄	D₃×D₄	1	1	yes	characteristic
124	C₂×C₄	C₂×#67	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	3	4	no	—
127	C₈	—	6	2	no	—
128	C₂×C₃	—	16	1	no	—
129	C₂×C₃	#492	1	1	yes	characteristic
130	C₂×C₃	#496	1	1	yes	characteristic
131	C₂×C₃	#496	1	1	yes	characteristic
132	D₃	—	16	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	24	1	no	—
137	C₂²	—	24	1	no	—
138	C₂²	—	24	1	no	—
139	C₂²	#940	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	24	1	no	—
142	C₄	D₃×SD₁₆	1	2	yes	—
143	C₄	#914	1	2	yes	—
144	C₄	#932	1	2	yes	—
145	C₃	#1715	1	1	yes	characteristic, complemented, Sylow
146	C₂	—	16	1	no	—
147	C₂	—	48	1	no	—
148	C₂	#5074	1	1	yes	characteristic
149	C₂	#5175	1	1	yes	characteristic
150	C₂	#5176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#492

C₂

yes

maximal, characteristic, complemented

#5088

C₂

yes

maximal, characteristic, complemented

#5080

C₂

yes

maximal, characteristic, complemented

#5076

C₂

yes

maximal, characteristic, complemented

#5073

C₂

yes

maximal, characteristic

#4694

C₂

yes

maximal, characteristic, complemented

#4454

C₂

yes

maximal, characteristic, complemented

#1715

—

maximal, Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×#127

C₂²

yes

—

—

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

characteristic, complemented

#486

—

#486

—

#322

—

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

#214

—

#213

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×SD₁₆

—

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

SD₁₆

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

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

—

100

C₂²×C₃

—

101

C₂²×C₃

#127

yes

characteristic

102

C₂×D₃

—

103

C₂×D₃

—

104

C₂×D₃

—

105

C₃×C₄

—

106

C₃×C₄

C₂×SD₁₆

yes

—

107

C₃×C₄

AGL(1,ℤ/8)

yes

—

108

C₃×C₄

AGL(1,ℤ/8)

yes

—

109

Dic₃

—

110

C₂³

—

111

C₂³

—

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₃×D₄

yes

characteristic

123

C₂×C₄

D₃×D₄

yes

characteristic

124

C₂×C₄

C₂×#67

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂×C₃

—

129

C₂×C₃

#492

yes

characteristic

130

C₂×C₃

#496

yes

characteristic

131

C₂×C₃

#496

yes

characteristic

132

D₃

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

#940

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

D₃×SD₁₆

yes

—

143

C₄

#914

yes

—

144

C₄

#932

yes

—

145

C₃

#1715

yes

characteristic, complemented, Sylow

146

C₂

—

147

C₂

—

148

C₂

#5074

yes

characteristic

149

C₂

#5175

yes

characteristic

150

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	#1715
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. #5088	C₂	semidirect
3. #5080	C₂	semidirect
4. #5076	C₂	semidirect
6. #4694	C₂	semidirect
7. #4454	C₂	semidirect
25. #492	D₃	semidirect
78. C₃⋊C₈	SD₁₆	semidirect
145. C₃	#1715	semidirect

H ≅ G/N

type

1. C₃×#492

C₂

semidirect

2. #5088

C₂

semidirect

3. #5080

C₂

semidirect

4. #5076

C₂

semidirect

6. #4694

C₂

semidirect

7. #4454

C₂

semidirect

25. #492

D₃

semidirect

78. C₃⋊C₈

SD₁₆

semidirect

145. C₃

#1715

semidirect

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

subgroup	index
C₃×#492	2
#5088	2
#5080	2
#5076	2
#5073	2
#4694	2
#4454	2
#1715	3

subgroup

index

Subgroups — #71259

Conjugacy classes of subgroups

Complements

Maximal subgroups