Group #69656 — subgroups

Subgroups — #69656

Order 384.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#808	C₂	1	2	yes	maximal, complemented
4	#4935	C₂	1	1	yes	maximal, characteristic, complemented
5	#4524	C₂	1	1	yes	maximal, characteristic, complemented
6	#2094	—	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	—	2	2	no	—
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
14	#832	—	2	4	no	—
15	#808	—	2	4	no	—
16	C₂×#106	—	3	2	no	—
17	C₂×#102	—	3	2	no	—
18	#390	D₃	1	1	yes	characteristic
19	#384	—	3	1	no	—
20	#336	—	3	1	no	—
21	C₂⁴×C₃	—	2	1	no	—
22	C₂×C₃×D₄	—	2	2	no	—
23	C₂×C₃×D₄	—	2	2	no	—
24	C₂×C₃×D₄	—	2	1	no	—
25	C₂×C₃×D₄	Q₈	1	2	yes	—
26	C₂²×C₃×C₄	—	2	1	no	—
27	C₂²×C₃×C₄	D₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃	—	2	2	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₂²⋊C₄	—	4	4	no	—
32	C₃×C₄²	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	4	4	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	2	no	—
38	C₂×C₃⋊C₈	D₄	1	4	yes	—
39	C₂²×D₄	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₈	—	3	2	no	—
41	C₂×C₄⋊C₄	—	3	2	no	—
42	C₂×C₄⋊C₄	—	6	2	no	—
43	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
46	#106	—	6	4	no	—
47	#102	—	6	4	no	—
48	C₂³×C₃	—	2	2	no	—
49	C₂³×C₃	—	4	1	no	—
50	C₂³×C₃	—	4	1	no	—
51	C₂³×C₃	—	4	2	no	—
52	C₂³×C₃	—	4	2	no	—
53	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
54	C₃×D₄	—	2	4	no	—
55	C₃×D₄	—	4	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	2	no	—
58	C₂×C₃×C₄	—	4	1	no	—
59	C₂×C₃×C₄	—	4	2	no	—
60	C₂×C₃×C₄	SD₁₆	1	4	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄	#41	1	1	yes	characteristic
64	C₂×C₃×C₄	#41	1	2	yes	—
65	C₂×Dic₃	—	4	2	no	—
66	C₂×Dic₃	—	4	4	no	—
67	C₃⋊C₈	—	2	4	no	—
68	C₂⁴	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	Dic₆	1	2	yes	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	6	2	no	—
75	C₂²×C₄	#67	1	2	yes	—
76	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₈	—	3	4	no	—
78	C₂×C₈	—	6	2	no	—
79	C₂×C₈	—	6	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	6	2	no	—
82	C₄⋊C₄	—	6	2	no	—
83	C₄⋊C₄	—	12	4	no	—
84	C₂²⋊C₄	—	4	4	no	—
85	C₄²	—	2	2	no	—
86	C₂²×C₃	—	4	2	no	—
87	C₂²×C₃	—	4	4	no	—
88	C₂²×C₃	—	8	2	no	—
89	C₂²×C₃	—	8	1	no	—
90	C₂²×C₃	—	8	1	no	—
91	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂²×C₃	#123	1	1	yes	characteristic
93	C₂²×C₃	#127	1	1	yes	characteristic
94	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
95	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₃×C₄	—	2	4	no	—
97	C₃×C₄	—	8	1	no	—
98	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
99	C₃×C₄	#123	1	2	yes	—
100	Dic₃	—	8	2	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	4	1	no	—
104	C₂³	—	4	1	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂×#67	1	1	yes	characteristic
107	D₄	—	2	4	no	—
108	D₄	—	4	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	12	2	no	—
114	C₂×C₄	—	12	4	no	—
115	C₂×C₄	#214	1	4	yes	—
116	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
117	C₂×C₄	#235	1	1	yes	characteristic
118	C₂×C₄	D₃×D₄	1	2	yes	—
119	C₂×C₄	#237	1	2	yes	—
120	C₈	—	6	4	no	—
121	C₂×C₃	—	8	2	no	—
122	C₂×C₃	#399	1	1	yes	characteristic
123	C₂×C₃	#476	1	2	yes	—
124	C₂×C₃	#481	1	2	yes	—
125	C₂×C₃	#492	1	1	yes	characteristic
126	C₂×C₃	#496	1	1	yes	characteristic
127	C₂²	—	4	2	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	8	2	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	#924	1	1	yes	characteristic
133	C₂²	C₂×#214	1	2	yes	—
134	C₂²	#935	1	1	yes	characteristic
135	C₂²	#940	1	1	yes	characteristic
136	C₂²	#949	1	2	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	8	1	no	—
139	C₄	—	24	2	no	—
140	C₄	#878	1	2	yes	—
141	C₄	#883	1	2	yes	—
142	C₃	#2094	1	1	yes	characteristic, complemented, Sylow
143	C₂	—	8	2	no	—
144	C₂	#4949	1	1	yes	characteristic
145	C₂	#5008	1	2	yes	—
146	C₂	#5034	1	2	yes	—
147	C₂	#5058	1	1	yes	characteristic
148	C₂	#5076	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#390

C₂

yes

maximal, characteristic

C₂×#832

C₂

yes

maximal

C₂×#808

C₂

yes

maximal, complemented

#4935

C₂

yes

maximal, characteristic, complemented

#4524

C₂

yes

maximal, characteristic, complemented

#2094

—

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

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

#384

—

#336

—

C₂⁴×C₃

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

Q₈

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

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

—

C₂²×D₄

C₂×D₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#106

—

#102

—

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

SD₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₆

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

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²⋊Q₈

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×SD₁₆

yes

—

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#123

yes

—

100

Dic₃

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂×#67

yes

characteristic

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#214

yes

—

116

C₂×C₄

C₂×Dic₆

yes

characteristic

117

C₂×C₄

#235

yes

characteristic

118

C₂×C₄

D₃×D₄

yes

—

119

C₂×C₄

#237

yes

—

120

C₈

—

121

C₂×C₃

—

122

C₂×C₃

#399

yes

characteristic

123

C₂×C₃

#476

yes

—

124

C₂×C₃

#481

yes

—

125

C₂×C₃

#492

yes

characteristic

126

C₂×C₃

#496

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#924

yes

characteristic

133

C₂²

C₂×#214

yes

—

134

C₂²

#935

yes

characteristic

135

C₂²

#940

yes

characteristic

136

C₂²

#949

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#878

yes

—

141

C₄

#883

yes

—

142

C₃

#2094

yes

characteristic, complemented, Sylow

143

C₂

—

144

C₂

#4949

yes

characteristic

145

C₂

#5008

yes

—

146

C₂

#5034

yes

—

147

C₂

#5058

yes

characteristic

148

C₂

#5076

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	#2094
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. C₂×#808	C₂	semidirect
4. #4935	C₂	semidirect
5. #4524	C₂	semidirect
142. C₃	#2094	semidirect

H ≅ G/N

type

3. C₂×#808

C₂

semidirect

4. #4935

C₂

semidirect

5. #4524

C₂

semidirect

142. C₃

#2094

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
C₂×#808	2
#4935	2
#4524	2
#2094	3

subgroup

index

Subgroups — #69656

Conjugacy classes of subgroups

Complements

Maximal subgroups