Group #68637 — subgroups

Subgroups — #68637

Order 384.

258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#821	C₂	1	2	yes	maximal
2	#4939	C₂	1	1	yes	maximal, characteristic
3	#4936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	#4551	C₂	1	2	yes	maximal, complemented
6	#2073	—	3	1	no	maximal, Sylow
7	C₂²×D₁₂	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₃⋊#95	—	2	2	no	—
14	#821	—	2	4	no	—
15	C₂×#102	—	3	2	no	—
16	#390	—	3	1	no	—
17	#382	—	3	1	no	—
18	#340	—	3	2	no	—
19	#336	D₃	1	1	yes	characteristic, complemented
20	C₂³×D₃	—	2	1	no	—
21	C₂²×C₃×C₄	D₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	2	no	—
24	C₂²×Dic₃	—	2	1	no	—
25	C₂×D₁₂	—	2	2	no	—
26	C₂×D₁₂	—	2	2	no	—
27	C₂×D₁₂	—	2	1	no	—
28	C₂×D₁₂	D₄	1	2	yes	—
29	C₂×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₃×C₄²	—	2	2	no	—
33	(C₂×D₃)⋊C₄	—	4	4	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	2	2	no	—
36	C₄⋊Dic₃	—	2	2	no	—
37	C₂²×D₄	—	3	1	no	—
38	C₂²×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₄⋊C₄	—	3	1	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₂²⋊C₄	—	6	2	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	#102	—	6	4	no	—
44	#95	—	6	2	no	—
45	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₃	—	2	2	no	—
47	C₂²×D₃	—	4	1	no	—
48	C₂²×D₃	—	4	1	no	—
49	C₂²×D₃	—	4	2	no	—
50	C₂²×D₃	—	4	2	no	—
51	C₂×C₃×C₄	—	2	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄	#41	1	2	yes	—
56	C₂×C₃×C₄	#41	1	2	yes	—
57	C₂×C₃×C₄	#41	1	1	yes	characteristic
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	1	no	—
60	C₂×Dic₃	—	4	2	no	—
61	C₂×Dic₃	—	4	2	no	—
62	C₂×Dic₃	—	4	2	no	—
63	D₁₂	—	2	4	no	—
64	D₁₂	—	4	2	no	—
65	C₃×C₈	—	4	2	no	—
66	C₂⁴	—	6	1	no	—
67	C₂×D₄	—	3	2	no	—
68	C₂×D₄	—	6	2	no	—
69	C₂×D₄	—	6	2	no	—
70	C₂×D₄	—	6	1	no	—
71	C₂²×C₄	—	6	2	no	—
72	C₂²×C₄	—	6	1	no	—
73	C₂²×C₄	D₁₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₄⋊C₄	—	6	2	no	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	6	2	no	—
81	C₂²⋊C₄	—	12	4	no	—
82	C₄²	—	2	2	no	—
83	C₂²×C₃	#121	1	1	yes	characteristic
84	C₂²×C₃	#123	1	1	yes	characteristic
85	C₂²×C₃	#124	1	1	yes	characteristic
86	C₂²×C₃	#135	1	2	yes	—
87	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×D₃	—	4	2	no	—
89	C₂×D₃	—	4	4	no	—
90	C₂×D₃	—	8	2	no	—
91	C₂×D₃	—	8	1	no	—
92	C₂×D₃	—	8	1	no	—
93	C₃×C₄	—	4	2	no	—
94	C₃×C₄	#123	1	2	yes	—
95	C₃×C₄	#126	1	2	yes	—
96	Dic₃	—	8	2	no	—
97	Dic₃	—	8	1	no	—
98	C₂³	—	6	2	no	—
99	C₂³	—	12	1	no	—
100	C₂³	—	12	1	no	—
101	C₂³	—	12	2	no	—
102	C₂³	—	12	2	no	—
103	C₂³	C₂×D₁₂	1	1	yes	characteristic
104	D₄	—	6	4	no	—
105	D₄	—	12	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	12	1	no	—
110	C₂×C₄	—	12	2	no	—
111	C₂×C₄	—	12	2	no	—
112	C₂×C₄	—	12	2	no	—
113	C₂×C₄	—	12	2	no	—
114	C₂×C₄	#235	1	2	yes	—
115	C₂×C₄	D₃×D₄	1	1	yes	characteristic
116	C₂×C₄	#237	1	2	yes	—
117	C₂×C₄	#239	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₂×C₃	#397	1	1	yes	characteristic
120	C₂×C₃	#463	1	2	yes	—
121	C₂×C₃	#482	1	2	yes	—
122	C₂×C₃	#488	1	1	yes	characteristic
123	C₂×C₃	#490	1	1	yes	characteristic
124	D₃	—	8	2	no	—
125	C₂²	—	12	2	no	—
126	C₂²	—	12	4	no	—
127	C₂²	—	24	1	no	—
128	C₂²	—	24	1	no	—
129	C₂²	—	24	2	no	—
130	C₂²	#875	1	1	yes	characteristic
131	C₂²	#886	1	1	yes	characteristic
132	C₂²	#895	1	1	yes	characteristic
133	C₂²	#904	1	2	yes	—
134	C₂²	#908	1	2	yes	—
135	C₄	—	4	2	no	—
136	C₄	—	24	1	no	—
137	C₄	—	24	2	no	—
138	C₄	#883	1	2	yes	—
139	C₄	#898	1	2	yes	—
140	C₃	#2073	1	1	yes	characteristic, complemented, Sylow
141	C₂	—	24	2	no	—
142	C₂	#4674	1	1	yes	characteristic
143	C₂	#4697	1	1	yes	characteristic
144	C₂	#4713	1	1	yes	characteristic
145	C₂	#4737	1	2	yes	—
146	C₂	#4844	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#821

C₂

yes

maximal

#4939

C₂

yes

maximal, characteristic

#4936

C₂

yes

maximal, characteristic, complemented

C₃×#336

C₂

yes

maximal, characteristic, complemented

#4551

C₂

yes

maximal, complemented

#2073

—

maximal, Sylow

C₂²×D₁₂

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×(C₂×D₃)⋊C₄

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂³×D₃

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

D₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×D₄

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#102

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

#123

yes

—

C₃×C₄

#126

yes

—

Dic₃

—

Dic₃

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

C₂×D₁₂

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

#235

yes

—

115

C₂×C₄

D₃×D₄

yes

characteristic

116

C₂×C₄

#237

yes

—

117

C₂×C₄

#239

yes

characteristic

118

C₈

—

119

C₂×C₃

#397

yes

characteristic

120

C₂×C₃

#463

yes

—

121

C₂×C₃

#482

yes

—

122

C₂×C₃

#488

yes

characteristic

123

C₂×C₃

#490

yes

characteristic

124

D₃

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#875

yes

characteristic

131

C₂²

#886

yes

characteristic

132

C₂²

#895

yes

characteristic

133

C₂²

#904

yes

—

134

C₂²

#908

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#883

yes

—

139

C₄

#898

yes

—

140

C₃

#2073

yes

characteristic, complemented, Sylow

141

C₂

—

142

C₂

#4674

yes

characteristic

143

C₂

#4697

yes

characteristic

144

C₂

#4713

yes

characteristic

145

C₂

#4737

yes

—

146

C₂

#4844

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#336	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	#336	D₃
hypocenter	C₃	#2073
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₂

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#336

C₂

socle

C₂³×C₃

C₂×D₄

prime radical

C₂²×D₁₂

C₂²

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
3. #4936	C₂	semidirect
4. C₃×#336	C₂	semidirect
5. #4551	C₂	semidirect
19. #336	D₃	semidirect
140. C₃	#2073	semidirect

H ≅ G/N

type

3. #4936

C₂

semidirect

4. C₃×#336

C₂

semidirect

5. #4551

C₂

semidirect

19. #336

D₃

semidirect

140. C₃

#2073

semidirect

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

subgroup	index
C₂×#821	2
#4939	2
#4936	2
C₃×#336	2
#4551	2
#2073	3

subgroup

index

Subgroups — #68637

Conjugacy classes of subgroups

Complements

Maximal subgroups