Group #83339 — subgroups

Subgroups — #83339

Order 384.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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×#533	C₂	1	1	yes	maximal, characteristic
2	#5470	C₂	1	1	yes	maximal, characteristic
3	#5080	C₂	1	2	yes	maximal, complemented
4	#5078	C₂	1	2	yes	maximal, complemented
5	#5065	C₂	1	4	yes	maximal, complemented
6	#4998	C₂	1	4	yes	maximal
7	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#3160	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×#125	C₂²	1	2	yes	—
11	C₃×C₂²⋊Q₈	—	2	2	no	—
12	C₃×C₄×Q₈	—	2	2	no	—
13	C₃×#117	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	#926	C₂²	1	2	yes	—
16	#924	—	2	2	no	—
17	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
18	#870	C₂²	1	2	yes	—
19	#869	C₂²	1	2	yes	—
20	C₄×Dic₆	—	2	2	no	—
21	#808	C₂²	1	8	yes	—
22	#807	C₂²	1	8	yes	—
23	C₄×C₃⋊C₈	C₂²	1	2	yes	—
24	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#533	—	3	1	no	—
27	#533	D₃	1	1	yes	characteristic
28	#500	—	3	2	no	—
29	#499	—	3	4	no	—
30	#498	—	3	2	no	—
31	#427	—	3	4	no	—
32	C₂×C₄×C₈	—	3	1	no	—
33	C₂×C₃×Q₈	—	2	2	no	—
34	C₂²×C₃×C₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂×Dic₆	—	2	2	no	—
37	C₃×C₄⋊C₄	—	2	4	no	—
38	C₃×C₄⋊C₄	—	2	2	no	—
39	C₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₃×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²	—	2	2	no	—
42	C₃×C₄²	D₄	1	2	yes	—
43	C₃×C₄²	C₂³	1	1	yes	characteristic
44	C₃×C₄²	C₂³	1	1	yes	characteristic
45	#217	—	2	2	no	—
46	C₄⋊Dic₃	—	2	2	no	—
47	C₄⋊Dic₃	C₂³	1	4	yes	—
48	Dic₃⋊C₄	—	2	4	no	—
49	C₄×Dic₃	—	2	2	no	—
50	C₂×C₃⋊C₈	Q₈	1	8	yes	complemented
51	C₂×C₃⋊C₈	C₂³	1	2	yes	—
52	C₂×C₃⋊C₈	C₂³	1	2	yes	—
53	C₂²×C₈	—	3	2	no	—
54	C₄⋊Q₈	—	3	2	no	—
55	C₄⋊Q₈	C₂×D₃	1	2	yes	—
56	#125	—	3	2	no	—
57	#125	C₂×D₃	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₂²⋊Q₈	—	6	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	—	6	2	no	—
62	#117	—	3	2	no	—
63	#117	C₂×D₃	1	2	yes	—
64	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
65	#107	—	3	8	no	—
66	#106	—	3	8	no	—
67	C₄×C₈	—	3	1	no	—
68	C₄×C₈	—	3	2	no	—
69	C₄×C₈	—	3	1	no	—
70	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
71	C₃×Q₈	—	4	4	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
78	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₃	—	2	4	no	—
80	Dic₆	—	4	4	no	—
81	C₃⋊C₈	C₂×Q₈	1	4	yes	—
82	C₃⋊C₈	C₂×Q₈	1	4	yes	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	6	2	no	—
85	C₂²×C₄	#67	1	2	yes	—
86	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₈	—	3	8	no	—
88	C₂×C₈	—	3	2	no	—
89	C₂×C₈	—	3	2	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	3	4	no	—
93	C₄⋊C₄	—	6	4	no	—
94	C₄⋊C₄	—	6	2	no	—
95	C₄⋊C₄	C₂²×D₃	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₂²⋊C₄	—	6	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	6	2	no	—
100	C₄²	#67	1	2	yes	—
101	C₄²	C₂²×D₃	1	1	yes	characteristic
102	C₄²	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₃	—	2	2	no	—
104	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
105	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
106	C₃×C₄	—	2	2	no	—
107	C₃×C₄	—	4	4	no	—
108	C₃×C₄	C₄⋊Q₈	1	2	yes	—
109	C₃×C₄	#135	1	4	yes	—
110	C₃×C₄	C₂²×Q₈	1	2	yes	—
111	Dic₃	—	4	4	no	—
112	C₂³	C₂×#67	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	12	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	6	4	no	—
118	C₂×C₄	D₃×Q₈	1	4	yes	—
119	C₂×C₄	C₂×#67	1	2	yes	—
120	C₂×C₄	C₂×#67	1	2	yes	—
121	C₂×C₄	C₂×#67	1	1	yes	characteristic
122	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
123	C₈	—	3	4	no	—
124	C₈	—	3	4	no	—
125	C₂×C₃	—	2	2	no	—
126	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₃	C₂×#135	1	2	yes	—
128	C₂²	—	2	2	no	—
129	C₂²	#944	1	2	yes	—
130	C₂²	C₂²×#67	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	12	4	no	—
134	C₄	#944	1	2	yes	—
135	C₄	#950	1	4	yes	—
136	C₄	C₂×D₃×Q₈	1	2	yes	—
137	C₃	#3160	1	1	yes	characteristic, complemented, Sylow
138	C₂	—	2	2	no	—
139	C₂	C₂×#944	1	1	yes	characteristic
140	C₂	C₂×#950	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

#5470

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, complemented

#5078

C₂

yes

maximal, complemented

#5065

C₂

yes

maximal, complemented

#4998

C₂

yes

maximal

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#3160

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#125

C₂²

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

#924

—

C₂²×C₃⋊C₈

C₂²

yes

—

#870

C₂²

yes

—

#869

C₂²

yes

—

C₄×Dic₆

—

#808

C₂²

yes

—

#807

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

characteristic

#533

—

#533

D₃

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂³

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

Q₈

yes

complemented

C₂×C₃⋊C₈

C₂³

yes

—

C₂×C₃⋊C₈

C₂³

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

—

#125

—

#125

C₂×D₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

C₂×Q₈

yes

—

C₃⋊C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

#67

yes

—

101

C₄²

C₂²×D₃

yes

characteristic

102

C₄²

C₂²×D₃

yes

characteristic

103

C₂²×C₃

—

104

C₂²×C₃

C₄⋊Q₈

yes

—

105

C₂²×C₃

C₂²×D₄

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

C₄⋊Q₈

yes

—

109

C₃×C₄

#135

yes

—

110

C₃×C₄

C₂²×Q₈

yes

—

111

Dic₃

—

112

C₂³

C₂×#67

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₃×Q₈

yes

—

119

C₂×C₄

C₂×#67

yes

—

120

C₂×C₄

C₂×#67

yes

—

121

C₂×C₄

C₂×#67

yes

characteristic

122

C₂×C₄

C₂³×D₃

yes

characteristic

123

C₈

—

124

C₈

—

125

C₂×C₃

—

126

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

127

C₂×C₃

C₂×#135

yes

—

128

C₂²

—

129

C₂²

#944

yes

—

130

C₂²

C₂²×#67

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#944

yes

—

135

C₄

#950

yes

—

136

C₄

C₂×D₃×Q₈

yes

—

137

C₃

#3160

yes

characteristic, complemented, Sylow

138

C₂

—

139

C₂

C₂×#944

yes

characteristic

140

C₂

C₂×#950

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	#533	D₃
hypocenter	C₃	#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×#67

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₃×#533

C₂

socle

C₂²×C₃

C₂²×D₄

prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #5080	C₂	semidirect
4. #5078	C₂	semidirect
5. #5065	C₂	semidirect
50. C₂×C₃⋊C₈	Q₈	semidirect
137. C₃	#3160	semidirect

H ≅ G/N

type

3. #5080

C₂

semidirect

4. #5078

C₂

semidirect

5. #5065

C₂

semidirect

50. C₂×C₃⋊C₈

Q₈

semidirect

137. C₃

#3160

semidirect

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

subgroup	index
C₃×#533	2
#5470	2
#5080	2
#5078	2
#5065	2
#4998	2
C₂×C₄×C₃⋊C₈	2
#3160	3

subgroup

index

C₂×C₄×C₃⋊C₈

Subgroups — #83339

Conjugacy classes of subgroups

Complements

Maximal subgroups