Group #74222 — subgroups

Subgroups — #74222

Order 384.

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

normal subgroups only

This group has 400 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×#382	C₂	1	1	yes	maximal, characteristic
2	#4984	C₂	1	2	yes	maximal
3	#4972	C₂	1	2	yes	maximal
4	#4936	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
6	#4655	C₂	1	4	yes	maximal
7	#4654	C₂	1	4	yes	maximal
8	#2864	—	3	1	no	maximal, Sylow
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×#95	C₂²	1	2	yes	—
19	C₃×#95	C₂²	1	2	yes	—
20	C₃⋊#95	C₂²	1	2	yes	—
21	C₃⋊#95	C₂²	1	4	yes	—
22	C₃⋊#95	C₂²	1	4	yes	—
23	C₃⋊#95	C₂²	1	2	yes	—
24	#389	—	3	2	no	—
25	#389	—	3	2	no	—
26	#387	—	3	4	no	—
27	#387	—	3	4	no	—
28	#382	—	3	1	no	—
29	#382	D₃	1	1	yes	characteristic
30	C₄³	—	3	1	no	—
31	C₂²×C₃×C₄	C₂³	1	4	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	C₂³	1	4	yes	—
36	C₂²×Dic₃	C₂³	1	2	yes	—
37	C₂²×Dic₃	C₂³	1	2	yes	—
38	C₃×C₄⋊C₄	—	2	4	no	—
39	C₃×C₄²	Q₈	1	4	yes	—
40	C₄⋊Dic₃	—	2	4	no	—
41	C₄⋊Dic₃	—	2	8	no	—
42	Dic₃⋊C₄	—	2	8	no	—
43	C₄×Dic₃	—	2	2	no	—
44	C₄×Dic₃	—	2	2	no	—
45	C₄×Dic₃	—	2	4	no	—
46	C₄×Dic₃	—	2	4	no	—
47	C₂×C₄⋊C₄	—	3	2	no	—
48	C₂×C₄⋊C₄	—	3	4	no	—
49	C₂×C₄⋊C₄	—	3	4	no	—
50	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×C₄²	—	3	2	no	—
52	C₂×C₄²	—	3	2	no	—
53	C₂×C₄²	—	3	1	no	—
54	C₂×C₄²	—	3	1	no	—
55	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
56	#95	—	3	2	no	—
57	#95	—	3	2	no	—
58	#95	—	3	4	no	—
59	#95	—	3	4	no	—
60	#95	C₂×D₃	1	2	yes	—
61	#95	C₂×D₃	1	2	yes	—
62	C₂³×C₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	—	2	4	no	—
67	C₂×C₃×C₄	—	2	4	no	—
68	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃×C₄	#41	1	2	yes	—
72	C₂×C₃×C₄	#41	1	2	yes	—
73	C₂×Dic₃	—	2	2	no	—
74	C₂×Dic₃	—	2	2	no	—
75	C₂×Dic₃	—	2	2	no	—
76	C₂×Dic₃	—	2	2	no	—
77	C₂×Dic₃	—	2	4	no	—
78	C₂×Dic₃	—	2	4	no	—
79	C₂×Dic₃	—	2	4	no	—
80	C₂×Dic₃	#41	1	4	yes	—
81	C₂×Dic₃	#41	1	4	yes	—
82	C₂²×C₄	—	3	2	no	—
83	C₂²×C₄	—	3	2	no	—
84	C₂²×C₄	—	3	4	no	—
85	C₂²×C₄	C₂²×D₃	1	4	yes	—
86	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	6	8	no	—
91	C₄⋊C₄	—	6	8	no	—
92	C₄⋊C₄	—	6	4	no	—
93	C₄²	—	6	4	no	—
94	C₄²	—	6	4	no	—
95	C₄²	—	6	2	no	—
96	C₄²	—	6	2	no	—
97	C₄²	Dic₆	1	4	yes	—
98	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₃	C₂×#41	1	1	yes	characteristic
100	C₂²×C₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃	C₂×#41	1	2	yes	—
102	C₂²×C₃	C₂×#41	1	2	yes	—
103	C₃×C₄	—	2	2	no	—
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	4	4	no	—
106	C₃×C₄	#125	1	4	yes	—
107	Dic₃	—	2	4	no	—
108	Dic₃	—	2	4	no	—
109	Dic₃	—	4	4	no	—
110	C₂³	C₂³×D₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	3	4	no	—
117	C₂×C₄	—	3	4	no	—
118	C₂×C₄	—	6	2	no	—
119	C₂×C₄	—	6	2	no	—
120	C₂×C₄	—	6	2	no	—
121	C₂×C₄	—	6	2	no	—
122	C₂×C₄	—	6	4	no	—
123	C₂×C₄	—	6	4	no	—
124	C₂×C₄	—	6	4	no	—
125	C₂×C₄	C₂×Dic₆	1	2	yes	—
126	C₂×C₄	C₂×Dic₆	1	2	yes	—
127	C₂×C₄	C₂×Dic₆	1	2	yes	—
128	C₂×C₄	#237	1	2	yes	—
129	C₂×C₄	#239	1	2	yes	—
130	C₂×C₃	C₂×#125	1	1	yes	characteristic
131	C₂×C₃	#529	1	2	yes	—
132	C₂×C₃	#529	1	2	yes	—
133	C₂×C₃	#533	1	1	yes	characteristic
134	C₂×C₃	#533	1	1	yes	characteristic
135	C₂²	C₂²×Dic₆	1	1	yes	characteristic
136	C₂²	C₂×#237	1	1	yes	characteristic
137	C₂²	C₂×#239	1	1	yes	characteristic
138	C₂²	D₃×#41	1	2	yes	—
139	C₂²	D₃×#41	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	—	6	4	no	—
144	C₄	—	6	4	no	—
145	C₄	—	12	4	no	—
146	C₄	#890	1	4	yes	—
147	C₃	#2864	1	1	yes	characteristic, complemented, Sylow
148	C₂	C₂×#890	1	1	yes	characteristic
149	C₂	#5517	1	1	yes	characteristic
150	C₂	#5517	1	1	yes	characteristic
151	C₂	#5680	1	2	yes	—
152	C₂	#5691	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#382

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

#4972

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic

C₄²×Dic₃

C₂

yes

maximal, characteristic

#4655

C₂

yes

maximal

#4654

C₂

yes

maximal

#2864

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₄³

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂×D₃

yes

—

#95

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

#41

yes

—

C₂×Dic₃

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

characteristic

100

C₂²×C₃

C₂×#41

yes

characteristic

101

C₂²×C₃

C₂×#41

yes

—

102

C₂²×C₃

C₂×#41

yes

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

#125

yes

—

107

Dic₃

—

108

Dic₃

—

109

Dic₃

—

110

C₂³

C₂³×D₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₆

yes

—

126

C₂×C₄

C₂×Dic₆

yes

—

127

C₂×C₄

C₂×Dic₆

yes

—

128

C₂×C₄

#237

yes

—

129

C₂×C₄

#239

yes

—

130

C₂×C₃

C₂×#125

yes

characteristic

131

C₂×C₃

#529

yes

—

132

C₂×C₃

#529

yes

—

133

C₂×C₃

#533

yes

characteristic

134

C₂×C₃

#533

yes

characteristic

135

C₂²

C₂²×Dic₆

yes

characteristic

136

C₂²

C₂×#237

yes

characteristic

137

C₂²

C₂×#239

yes

characteristic

138

C₂²

D₃×#41

yes

—

139

C₂²

D₃×#41

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#890

yes

—

147

C₃

#2864

yes

characteristic, complemented, Sylow

148

C₂

C₂×#890

yes

characteristic

149

C₂

#5517

yes

characteristic

150

C₂

#5517

yes

characteristic

151

C₂

#5680

yes

—

152

C₂

#5691

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#382	C₂
hypercenter	#382	D₃
hypocenter	C₃	#2864
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
147. C₃	#2864	semidirect

H ≅ G/N

type

147. C₃

#2864

semidirect

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

subgroup	index
C₃×#382	2
#4984	2
#4972	2
#4936	2
C₄²×Dic₃	2
#4655	2
#4654	2
#2864	3

subgroup

index

C₄²×Dic₃

Subgroups — #74222

Conjugacy classes of subgroups

Complements

Maximal subgroups