Group #70771 — subgroups

Subgroups — #70771

Order 384.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	#5470	C₂	1	1	yes	maximal, characteristic
2	C₃×#431	C₂	1	1	yes	maximal, characteristic
3	#4926	C₂	1	1	yes	maximal, characteristic
4	#4456	C₂	1	4	yes	maximal, complemented
5	#1497	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	#926	—	2	1	no	—
8	#924	—	2	1	no	—
9	#924	C₄	1	2	yes	—
10	#870	—	2	1	no	—
11	#869	C₄	1	1	yes	characteristic
12	#869	C₄	1	1	yes	characteristic
13	C₄×Dic₆	—	2	1	no	—
14	C₄×Dic₆	C₂²	1	2	yes	—
15	C₃×C₄⋊C₈	—	2	1	no	—
16	C₃×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₈	C₂²	1	2	yes	—
18	#830	—	2	1	no	—
19	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	#803	—	2	1	no	—
21	#533	—	3	1	no	—
22	#432	—	3	1	no	—
23	#431	D₃	1	1	yes	characteristic
24	Q₈⋊C₈	—	3	4	no	—
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂×Dic₆	—	2	1	no	—
29	C₂×Dic₆	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²	D₄	1	1	yes	characteristic
33	C₃×C₄²	D₄	1	1	yes	characteristic
34	C₃×C₄²	C₂³	1	1	yes	characteristic
35	#217	—	2	2	no	—
36	C₄⋊Dic₃	—	2	1	no	—
37	C₄⋊Dic₃	—	2	1	no	—
38	C₄⋊Dic₃	—	2	1	no	—
39	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
40	Dic₃⋊C₄	—	2	2	no	—
41	Dic₃⋊C₄	—	2	2	no	—
42	C₄×Dic₃	—	2	2	no	—
43	C₂×C₃⋊C₈	—	2	2	no	—
44	C₄⋊Q₈	—	3	1	no	—
45	C₄⋊Q₈	—	3	1	no	—
46	#125	—	6	1	no	—
47	C₂²⋊Q₈	—	3	2	no	—
48	C₂²⋊Q₈	—	6	1	no	—
49	C₄×Q₈	—	3	2	no	—
50	C₄×Q₈	—	6	1	no	—
51	#117	—	6	1	no	—
52	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	3	2	no	—
55	C₂²⋊C₈	—	2	1	no	—
56	C₂²⋊C₈	—	6	1	no	—
57	#97	—	6	1	no	—
58	C₄×C₈	C₂×D₃	1	2	yes	—
59	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	—	2	2	no	—
62	C₂×C₃×C₄	—	4	1	no	—
63	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂²×C₄	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	—	2	2	no	—
70	C₂×Dic₃	—	2	2	no	—
71	Dic₆	—	2	4	no	—
72	Dic₆	—	4	2	no	—
73	C₃×C₈	—	4	2	no	—
74	C₃⋊C₈	—	4	2	no	—
75	C₂×Q₈	—	3	2	no	—
76	C₂×Q₈	—	6	1	no	—
77	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
78	C₂²×C₄	D₁₂	1	1	yes	characteristic
79	C₂²×C₄	#67	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	6	2	no	—
82	C₄⋊C₄	—	3	2	no	—
83	C₄⋊C₄	—	6	1	no	—
84	C₄⋊C₄	—	6	1	no	—
85	C₄⋊C₄	—	6	2	no	—
86	C₄⋊C₄	—	6	1	no	—
87	C₄⋊C₄	—	6	2	no	—
88	C₂²⋊C₄	—	6	2	no	—
89	C₄²	—	6	2	no	—
90	C₄²	C₄×D₃	1	1	yes	characteristic
91	C₄²	D₁₂	1	1	yes	characteristic
92	C₄²	#67	1	1	yes	characteristic
93	C₄²	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃	—	2	1	no	—
95	C₂²×C₃	—	2	1	no	—
96	C₂²×C₃	—	2	1	no	—
97	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄	—	2	1	no	—
99	C₃×C₄	—	2	1	no	—
100	C₃×C₄	—	2	1	no	—
101	C₃×C₄	—	2	1	no	—
102	C₃×C₄	#131	1	2	yes	—
103	C₃×C₄	#137	1	2	yes	—
104	Dic₃	—	4	2	no	—
105	Dic₃	—	4	2	no	—
106	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
107	Q₈	—	6	4	no	—
108	Q₈	—	12	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	6	2	no	—
113	C₂×C₄	—	6	2	no	—
114	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
116	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
119	C₂×C₄	C₂×#67	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	12	2	no	—
122	C₂×C₃	—	4	1	no	—
123	C₂×C₃	#408	1	1	yes	characteristic
124	C₂×C₃	#419	1	1	yes	characteristic
125	C₂×C₃	#421	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	12	2	no	—
135	C₄	—	12	2	no	—
136	C₄	#901	1	2	yes	—
137	C₄	#909	1	2	yes	—
138	C₃	#1497	1	1	yes	characteristic, complemented, Sylow
139	C₂	—	4	1	no	—
140	C₂	#5005	1	1	yes	characteristic
141	C₂	#5110	1	1	yes	characteristic
142	C₂	#5120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5470

C₂

yes

maximal, characteristic

C₃×#431

C₂

yes

maximal, characteristic

#4926

C₂

yes

maximal, characteristic

#4456

C₂

yes

maximal, complemented

#1497

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

—

—

C₄

yes

—

#870

—

#869

C₄

yes

characteristic

#869

C₄

yes

characteristic

C₄×Dic₆

—

C₄×Dic₆

C₂²

yes

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

—

C₃×C₄×C₈

C₂²

yes

—

#830

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

D₃

yes

characteristic

Q₈⋊C₈

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×D₃

yes

—

C₂³×C₃

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

#67

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₃

yes

characteristic

C₄²

D₁₂

yes

characteristic

C₄²

#67

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

#131

yes

—

103

C₃×C₄

#137

yes

—

104

Dic₃

—

105

Dic₃

—

106

C₂³

(C₂×D₃)⋊C₄

yes

characteristic

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

115

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

116

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₃

yes

characteristic

118

C₂×C₄

C₂×D₁₂

yes

characteristic

119

C₂×C₄

C₂×#67

yes

characteristic

120

C₈

—

121

C₈

—

122

C₂×C₃

—

123

C₂×C₃

#408

yes

characteristic

124

C₂×C₃

#419

yes

characteristic

125

C₂×C₃

#421

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#901

yes

—

137

C₄

#909

yes

—

138

C₃

#1497

yes

characteristic, complemented, Sylow

139

C₂

—

140

C₂

#5005

yes

characteristic

141

C₂

#5110

yes

characteristic

142

C₂

#5120

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#431	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#431	D₃
hypocenter	C₃	#1497
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting

C₃×#431

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₃

C₂²⋊C₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
4. #4456	C₂	semidirect
138. C₃	#1497	semidirect

H ≅ G/N

type

4. #4456

C₂

semidirect

138. C₃

#1497

semidirect

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

subgroup	index
#5470	2
C₃×#431	2
#4926	2
#4456	2
#1497	3

subgroup

index

Subgroups — #70771

Conjugacy classes of subgroups

Complements

Maximal subgroups