Group #83012 — subgroups

Subgroups — #83012

Order 384.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#5477	C₂	1	1	yes	maximal, characteristic
3	#5069	C₂	1	2	yes	maximal, complemented
4	#5068	C₂	1	4	yes	maximal, complemented
5	#5067	C₂	1	2	yes	maximal, complemented
6	#4966	C₂	1	4	yes	maximal, complemented
7	C₄×#208	C₂	1	1	yes	maximal, characteristic, complemented
8	#3142	—	3	1	no	maximal, Sylow
9	C₂×#235	C₂²	1	2	yes	—
10	C₃×#125	—	2	2	no	—
11	C₃×#125	C₂²	1	2	yes	—
12	C₃×#125	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	#930	—	2	2	no	—
17	C₂×#208	C₂²	1	2	yes	—
18	#875	C₂²	1	2	yes	—
19	#874	C₂²	1	1	yes	characteristic
20	C₄×D₁₂	—	2	2	no	—
21	#869	C₂²	1	1	yes	characteristic
22	#810	C₂²	1	8	yes	—
23	#809	C₂²	1	8	yes	—
24	#803	C₂²	1	2	yes	—
25	C₄×C₃⋊C₈	C₂²	1	2	yes	—
26	#532	—	3	1	no	—
27	C₂×#125	D₃	1	1	yes	characteristic, complemented
28	#491	—	3	2	no	—
29	#490	—	3	2	no	—
30	#488	—	3	4	no	—
31	#417	—	3	4	no	—
32	C₄×M₁₆	—	3	1	no	—
33	C₂²×C₃×C₄	—	2	2	no	—
34	C₂²×C₃×C₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂×#67	—	2	2	no	—
37	#235	—	2	8	no	—
38	C₂×D₁₂	—	2	2	no	—
39	C₂×D₁₂	C₂³	1	2	yes	—
40	C₂×C₄×D₃	—	2	2	no	—
41	C₂×Dic₆	C₂³	1	2	yes	—
42	C₃×C₄⋊C₄	—	2	4	no	—
43	C₃×C₄⋊C₄	—	2	4	no	—
44	C₃×C₄⋊C₄	—	2	2	no	—
45	C₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₃×C₄²	D₄	1	2	yes	—
47	C₃×C₄²	C₂³	1	1	yes	characteristic
48	C₃×C₄²	C₂³	1	1	yes	characteristic
49	(C₂×D₃)⋊C₄	—	2	4	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	#208	—	2	4	no	—
52	C₂×C₃⋊C₈	C₂³	1	4	yes	—
53	C₂×#41	—	3	2	no	—
54	C₂×M₁₆	—	3	2	no	—
55	C₄⋊Q₈	—	3	1	no	—
56	#127	—	3	1	no	—
57	#125	—	2	2	no	—
58	#125	C₂×D₃	1	2	yes	—
59	#125	C₂×D₃	1	2	yes	—
60	#124	—	3	2	no	—
61	#121	—	6	2	no	—
62	C₄×D₄	—	6	2	no	—
63	C₂×C₄⋊C₄	—	2	2	no	—
64	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
65	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
66	#103	—	3	8	no	—
67	#102	—	3	8	no	—
68	#97	—	3	2	no	—
69	C₄×C₈	—	3	2	no	—
70	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₃	—	2	2	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	4	2	no	—
74	C₂×C₃×C₄	—	4	4	no	—
75	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	#41	1	4	yes	—
79	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
80	#67	—	4	4	no	—
81	C₂×Dic₃	—	2	2	no	—
82	D₁₂	—	2	4	no	—
83	D₁₂	—	4	4	no	—
84	C₄×D₃	—	4	4	no	—
85	Dic₆	—	2	4	no	—
86	C₃⋊C₈	—	2	4	no	—
87	#41	—	6	8	no	—
88	C₂×Q₈	—	3	2	no	—
89	C₂×D₄	—	3	2	no	—
90	C₂×D₄	—	6	2	no	—
91	C₂×D₄	—	6	2	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	6	2	no	—
94	C₂²×C₄	#67	1	2	yes	—
95	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
96	M₁₆	—	6	4	no	—
97	C₂×C₈	—	3	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	6	2	no	—
102	C₄⋊C₄	C₂²×D₃	1	4	yes	—
103	C₂²⋊C₄	—	6	4	no	—
104	C₄²	#67	1	2	yes	—
105	C₄²	C₂²×D₃	1	1	yes	characteristic
106	C₄²	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	#124	1	2	yes	—
109	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×D₃	—	4	2	no	—
111	C₂×D₃	—	4	4	no	—
112	C₃×C₄	—	2	2	no	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	—	4	4	no	—
115	C₃×C₄	#124	1	2	yes	—
116	C₃×C₄	C₂×#41	1	2	yes	—
117	Dic₃	—	4	2	no	—
118	C₂³	—	6	2	no	—
119	C₂³	C₂×#67	1	1	yes	characteristic
120	Q₈	—	6	4	no	—
121	D₄	—	6	4	no	—
122	D₄	—	12	4	no	—
123	D₄	—	12	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	6	2	no	—
128	C₂×C₄	—	12	4	no	—
129	C₂×C₄	#239	1	4	yes	—
130	C₂×C₄	C₂×#67	1	2	yes	—
131	C₂×C₄	C₂×#67	1	2	yes	—
132	C₂×C₄	C₂×#67	1	1	yes	characteristic
133	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
134	C₈	—	6	4	no	—
135	C₂×C₃	—	2	2	no	—
136	C₂×C₃	C₂×#124	1	1	yes	characteristic
137	C₂×C₃	#575	1	2	yes	—
138	D₃	—	8	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	12	2	no	—
141	C₂²	—	12	4	no	—
142	C₂²	#947	1	2	yes	—
143	C₂²	C₂²×#67	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	4	4	no	—
147	C₄	—	12	2	no	—
148	C₄	#947	1	2	yes	—
149	C₄	C₂×#239	1	2	yes	—
150	C₃	#3142	1	1	yes	characteristic, complemented, Sylow
151	C₂	—	2	2	no	—
152	C₂	—	24	2	no	—
153	C₂	C₂×#947	1	1	yes	characteristic
154	C₂	#5822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic, complemented

#5477

C₂

yes

maximal, characteristic

#5069

C₂

yes

maximal, complemented

#5068

C₂

yes

maximal, complemented

#5067

C₂

yes

maximal, complemented

#4966

C₂

yes

maximal, complemented

C₄×#208

C₂

yes

maximal, characteristic, complemented

#3142

—

maximal, Sylow

C₂×#235

C₂²

yes

—

C₃×#125

—

C₃×#125

C₂²

yes

—

C₃×#125

C₂²

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#930

—

C₂×#208

C₂²

yes

—

#875

C₂²

yes

—

#874

C₂²

yes

characteristic

C₄×D₁₂

—

#869

C₂²

yes

characteristic

#810

C₂²

yes

—

#809

C₂²

yes

—

#803

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

#532

—

C₂×#125

D₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×#67

—

#235

—

C₂×D₁₂

—

C₂×D₁₂

C₂³

yes

—

C₂×C₄×D₃

—

C₂×Dic₆

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

#208

—

C₂×C₃⋊C₈

C₂³

yes

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₃

yes

—

#125

C₂×D₃

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

—

—

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

—

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

#41

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

#67

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₄×D₃

—

Dic₆

—

C₃⋊C₈

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂²×D₃

yes

—

103

C₂²⋊C₄

—

104

C₄²

#67

yes

—

105

C₄²

C₂²×D₃

yes

characteristic

106

C₄²

C₂²×D₃

yes

characteristic

107

C₂²×C₃

—

108

C₂²×C₃

#124

yes

—

109

C₂²×C₃

C₂²×D₄

yes

characteristic

110

C₂×D₃

—

111

C₂×D₃

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

#124

yes

—

116

C₃×C₄

C₂×#41

yes

—

117

Dic₃

—

118

C₂³

—

119

C₂³

C₂×#67

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#239

yes

—

130

C₂×C₄

C₂×#67

yes

—

131

C₂×C₄

C₂×#67

yes

—

132

C₂×C₄

C₂×#67

yes

characteristic

133

C₂×C₄

C₂³×D₃

yes

characteristic

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

C₂×#124

yes

characteristic

137

C₂×C₃

#575

yes

—

138

D₃

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#947

yes

—

143

C₂²

C₂²×#67

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#947

yes

—

149

C₄

C₂×#239

yes

—

150

C₃

#3142

yes

characteristic, complemented, Sylow

151

C₂

—

152

C₂

—

153

C₂

C₂×#947

yes

characteristic

154

C₂

#5822

yes

—

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#125	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5477	C₂
hypercenter	C₂×#125	D₃
hypocenter	C₃	#3142
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×#67

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₂×C₃×#125

C₂

socle

C₂²×C₃

C₂²×D₄

C₂

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
1. C₂×C₃×#125	C₂	semidirect
3. #5069	C₂	semidirect
4. #5068	C₂	semidirect
5. #5067	C₂	semidirect
6. #4966	C₂	semidirect
7. C₄×#208	C₂	semidirect
27. C₂×#125	D₃	semidirect
150. C₃	#3142	semidirect

H ≅ G/N

type

1. C₂×C₃×#125

C₂

semidirect

3. #5069

C₂

semidirect

4. #5068

C₂

semidirect

5. #5067

C₂

semidirect

6. #4966

C₂

semidirect

7. C₄×#208

C₂

semidirect

27. C₂×#125

D₃

semidirect

150. C₃

#3142

semidirect

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

subgroup	index
C₂×C₃×#125	2
#5477	2
#5069	2
#5068	2
#5067	2
#4966	2
C₄×#208	2
#3142	3

subgroup

index

C₂×C₃×#125

Subgroups — #83012

Conjugacy classes of subgroups

Complements

Maximal subgroups