Group #69641 — subgroups

Subgroups — #69641

Order 384.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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×#390	C₂	1	1	yes	maximal, characteristic
2	C₂×#832	C₂	1	2	yes	maximal
3	#4936	C₂	1	1	yes	maximal, characteristic, complemented
4	#4529	C₂	1	2	yes	maximal, complemented
5	#4524	C₂	1	1	yes	maximal, characteristic, complemented
6	#2073	—	3	1	no	maximal, Sylow
7	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
13	#832	—	2	4	no	—
14	C₃⋊#95	—	2	2	no	—
15	C₂×#102	—	3	2	no	—
16	#390	D₃	1	1	yes	characteristic
17	#382	—	3	1	no	—
18	#340	—	3	2	no	—
19	#336	—	3	1	no	—
20	C₂⁴×C₃	—	2	1	no	—
21	C₂×C₃×D₄	—	2	2	no	—
22	C₂×C₃×D₄	—	2	2	no	—
23	C₂×C₃×D₄	—	2	1	no	—
24	C₂×C₃×D₄	D₄	1	2	yes	—
25	C₂²×C₃×C₄	—	2	1	no	—
26	C₂²×C₃×C₄	D₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃	—	2	2	no	—
29	C₃×C₄⋊C₄	—	2	2	no	—
30	C₃×C₂²⋊C₄	—	4	4	no	—
31	C₃×C₄²	—	2	2	no	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂²×D₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₈	—	3	2	no	—
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	#102	—	6	4	no	—
44	#95	—	6	2	no	—
45	C₂³×C₃	—	2	2	no	—
46	C₂³×C₃	—	4	1	no	—
47	C₂³×C₃	—	4	1	no	—
48	C₂³×C₃	—	4	2	no	—
49	C₂³×C₃	—	4	2	no	—
50	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
51	C₃×D₄	—	2	4	no	—
52	C₃×D₄	—	4	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	4	1	no	—
57	C₂×C₃×C₄	—	4	2	no	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	#41	1	1	yes	characteristic
60	C₂×C₃×C₄	#41	1	2	yes	—
61	C₂×C₃×C₄	#41	1	2	yes	—
62	C₂×Dic₃	—	4	2	no	—
63	C₂×Dic₃	—	4	2	no	—
64	C₂×Dic₃	—	4	2	no	—
65	C₃⋊C₈	—	4	2	no	—
66	C₂⁴	—	2	1	no	—
67	C₂×D₄	—	2	1	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	D₁₂	1	2	yes	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	—	6	2	no	—
73	C₂²×C₄	#67	1	2	yes	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₈	—	6	2	no	—
76	C₂×C₈	—	6	2	no	—
77	C₂×C₈	—	6	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	6	2	no	—
81	C₂²⋊C₄	—	4	4	no	—
82	C₄²	—	2	2	no	—
83	C₂²×C₃	—	4	2	no	—
84	C₂²×C₃	—	4	4	no	—
85	C₂²×C₃	—	8	2	no	—
86	C₂²×C₃	—	8	1	no	—
87	C₂²×C₃	—	8	1	no	—
88	C₂²×C₃	#121	1	1	yes	characteristic
89	C₂²×C₃	#123	1	1	yes	characteristic
90	C₂²×C₃	#124	1	1	yes	characteristic
91	C₂²×C₃	#135	1	2	yes	—
92	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₃×C₄	—	4	2	no	—
94	C₃×C₄	—	8	1	no	—
95	C₃×C₄	#123	1	2	yes	—
96	C₃×C₄	#126	1	2	yes	—
97	Dic₃	—	8	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	1	no	—
101	C₂³	—	4	1	no	—
102	C₂³	—	4	2	no	—
103	C₂³	C₂×#67	1	1	yes	characteristic
104	D₄	—	2	4	no	—
105	D₄	—	4	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	12	2	no	—
112	C₂×C₄	—	12	2	no	—
113	C₂×C₄	—	12	2	no	—
114	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄	#235	1	1	yes	characteristic
116	C₂×C₄	#237	1	2	yes	—
117	C₂×C₄	#237	1	2	yes	—
118	C₈	—	12	2	no	—
119	C₂×C₃	—	8	2	no	—
120	C₂×C₃	#397	1	1	yes	characteristic
121	C₂×C₃	#463	1	2	yes	—
122	C₂×C₃	#482	1	2	yes	—
123	C₂×C₃	#488	1	1	yes	characteristic
124	C₂×C₃	#490	1	1	yes	characteristic
125	C₂²	—	4	2	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	1	no	—
129	C₂²	—	8	1	no	—
130	C₂²	#930	1	1	yes	characteristic
131	C₂²	#932	1	2	yes	—
132	C₂²	#935	1	1	yes	characteristic
133	C₂²	#936	1	1	yes	characteristic
134	C₂²	#950	1	2	yes	—
135	C₄	—	4	2	no	—
136	C₄	—	8	1	no	—
137	C₄	—	24	2	no	—
138	C₄	#879	1	2	yes	—
139	C₄	#886	1	2	yes	—
140	C₃	#2073	1	1	yes	characteristic, complemented, Sylow
141	C₂	—	8	2	no	—
142	C₂	#4953	1	1	yes	characteristic
143	C₂	#5016	1	2	yes	—
144	C₂	#5035	1	2	yes	—
145	C₂	#5056	1	1	yes	characteristic
146	C₂	#5071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#390

C₂

yes

maximal, characteristic

C₂×#832

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic, complemented

#4529

C₂

yes

maximal, complemented

#4524

C₂

yes

maximal, characteristic, complemented

#2073

—

maximal, Sylow

C₂²×C₃×D₄

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₃×C₂²⋊C₄

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₂⁴×C₃

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×D₄

C₂×D₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#102

—

#95

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₁₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#123

yes

—

C₃×C₄

#126

yes

—

Dic₃

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

C₂×#67

yes

characteristic

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₁₂

yes

characteristic

115

C₂×C₄

#235

yes

characteristic

116

C₂×C₄

#237

yes

—

117

C₂×C₄

#237

yes

—

118

C₈

—

119

C₂×C₃

—

120

C₂×C₃

#397

yes

characteristic

121

C₂×C₃

#463

yes

—

122

C₂×C₃

#482

yes

—

123

C₂×C₃

#488

yes

characteristic

124

C₂×C₃

#490

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#930

yes

characteristic

131

C₂²

#932

yes

—

132

C₂²

#935

yes

characteristic

133

C₂²

#936

yes

characteristic

134

C₂²

#950

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#879

yes

—

139

C₄

#886

yes

—

140

C₃

#2073

yes

characteristic, complemented, Sylow

141

C₂

—

142

C₂

#4953

yes

characteristic

143

C₂

#5016

yes

—

144

C₂

#5035

yes

—

145

C₂

#5056

yes

characteristic

146

C₂

#5071

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#390	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×C₃×D₄	C₂²
hypercenter	#390	D₃
hypocenter	C₃	#2073
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×#67

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#390

C₂

socle

C₂³×C₃

C₂×D₄

prime radical

C₂²×C₃×D₄

C₂²

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
3. #4936	C₂	semidirect
4. #4529	C₂	semidirect
5. #4524	C₂	semidirect
140. C₃	#2073	semidirect

H ≅ G/N

type

3. #4936

C₂

semidirect

4. #4529

C₂

semidirect

5. #4524

C₂

semidirect

140. C₃

#2073

semidirect

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

subgroup	index
C₃×#390	2
C₂×#832	2
#4936	2
#4529	2
#4524	2
#2073	3

subgroup

index

Subgroups — #69641

Conjugacy classes of subgroups

Complements

Maximal subgroups