Group #74195 — subgroups

Subgroups — #74195

Order 384.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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	#4935	C₂	1	1	yes	maximal, characteristic
5	#4932	C₂	1	1	yes	maximal, characteristic
6	#4661	C₂	1	4	yes	maximal
7	#4660	C₂	1	4	yes	maximal
8	#2647	—	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	2	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
14	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×#95	C₂²	1	4	yes	—
18	C₃⋊#95	C₂²	1	2	yes	—
19	C₃⋊#95	C₂²	1	4	yes	—
20	C₃⋊#95	C₂²	1	4	yes	—
21	C₃⋊#95	C₂²	1	2	yes	—
22	#400	—	3	4	no	—
23	#398	—	3	4	no	—
24	#389	—	3	2	no	—
25	#389	—	3	2	no	—
26	#384	—	3	1	no	—
27	#382	D₃	1	1	yes	characteristic
28	#376	—	3	1	no	—
29	C₂²×C₃×C₄	C₂³	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	C₂³	1	2	yes	—
34	C₂²×Dic₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	C₂³	1	4	yes	—
36	C₃×C₄⋊C₄	—	2	4	no	—
37	C₃×C₄²	Q₈	1	4	yes	—
38	C₄⋊Dic₃	—	2	4	no	—
39	C₄⋊Dic₃	—	4	4	no	—
40	C₄⋊Dic₃	—	4	4	no	—
41	Dic₃⋊C₄	—	4	4	no	—
42	C₄×Dic₃	—	2	2	no	—
43	C₄×Dic₃	—	2	2	no	—
44	C₂×C₄⋊C₄	—	3	2	no	—
45	C₂×C₄⋊C₄	—	3	4	no	—
46	C₂×C₄⋊C₄	—	3	4	no	—
47	C₂×C₄⋊C₄	—	3	4	no	—
48	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂×C₄²	—	3	1	no	—
50	C₂×C₄²	—	3	1	no	—
51	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
52	#95	—	3	2	no	—
53	#95	—	3	2	no	—
54	#95	—	3	4	no	—
55	#95	—	3	4	no	—
56	#95	C₂×D₃	1	4	yes	—
57	C₂³×C₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	4	no	—
63	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
64	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
65	C₂×C₃×C₄	#41	1	2	yes	—
66	C₂×C₃×C₄	#41	1	2	yes	—
67	C₂×Dic₃	—	2	2	no	—
68	C₂×Dic₃	—	2	1	no	—
69	C₂×Dic₃	—	2	1	no	—
70	C₂×Dic₃	—	2	4	no	—
71	C₂×Dic₃	—	2	2	no	—
72	C₂×Dic₃	—	2	4	no	—
73	C₂×Dic₃	—	2	2	no	—
74	C₂×Dic₃	—	2	4	no	—
75	C₂×Dic₃	—	2	2	no	—
76	C₂×Dic₃	—	2	2	no	—
77	C₂²×C₄	—	3	1	no	—
78	C₂²×C₄	—	3	2	no	—
79	C₂²×C₄	—	3	1	no	—
80	C₂²×C₄	—	3	4	no	—
81	C₂²×C₄	C₂²×D₃	1	4	yes	—
82	C₂²×C₄	C₂²×D₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₄⋊C₄	—	2	4	no	—
85	C₄⋊C₄	—	6	4	no	—
86	C₄⋊C₄	—	12	4	no	—
87	C₄⋊C₄	—	12	4	no	—
88	C₄⋊C₄	—	12	4	no	—
89	C₄²	—	6	2	no	—
90	C₄²	—	6	2	no	—
91	C₄²	Dic₆	1	4	yes	—
92	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₃	C₂×#41	1	1	yes	characteristic
94	C₂²×C₃	C₂×#41	1	1	yes	characteristic
95	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
96	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
97	C₃×C₄	—	2	4	no	—
98	C₃×C₄	—	4	4	no	—
99	C₃×C₄	#125	1	4	yes	—
100	Dic₃	—	4	1	no	—
101	Dic₃	—	4	4	no	—
102	Dic₃	—	4	1	no	—
103	Dic₃	—	4	2	no	—
104	C₂³	C₂³×D₃	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	6	2	no	—
111	C₂×C₄	—	6	1	no	—
112	C₂×C₄	—	6	1	no	—
113	C₂×C₄	—	6	4	no	—
114	C₂×C₄	—	6	2	no	—
115	C₂×C₄	—	6	4	no	—
116	C₂×C₄	—	6	2	no	—
117	C₂×C₄	—	6	4	no	—
118	C₂×C₄	—	6	2	no	—
119	C₂×C₄	—	6	2	no	—
120	C₂×C₄	C₂×Dic₆	1	4	yes	—
121	C₂×C₄	C₂×Dic₆	1	2	yes	—
122	C₂×C₄	#237	1	2	yes	—
123	C₂×C₄	#239	1	2	yes	—
124	C₂×C₃	C₂×#125	1	1	yes	characteristic
125	C₂×C₃	#540	1	1	yes	characteristic
126	C₂×C₃	#541	1	1	yes	characteristic
127	C₂×C₃	#542	1	2	yes	—
128	C₂×C₃	#544	1	2	yes	—
129	C₂²	C₂²×Dic₆	1	1	yes	characteristic
130	C₂²	C₂×#237	1	1	yes	characteristic
131	C₂²	C₂×#239	1	1	yes	characteristic
132	C₂²	#1009	1	2	yes	—
133	C₂²	#1010	1	2	yes	—
134	C₄	—	2	4	no	—
135	C₄	—	4	4	no	—
136	C₄	—	12	2	no	—
137	C₄	—	12	1	no	—
138	C₄	—	12	4	no	—
139	C₄	—	12	1	no	—
140	C₄	#890	1	4	yes	—
141	C₃	#2647	1	1	yes	characteristic, complemented, Sylow
142	C₂	C₂×#890	1	1	yes	characteristic
143	C₂	#5519	1	2	yes	—
144	C₂	#5682	1	2	yes	—
145	C₂	#5697	1	1	yes	characteristic
146	C₂	#5698	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#382

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

#4972

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

#4932

C₂

yes

maximal, characteristic

#4661

C₂

yes

maximal

#4660

C₂

yes

maximal

#2647

—

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×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

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

—

—

—

—

—

—

D₃

yes

characteristic

#376

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

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

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

characteristic

C₂²×C₃

C₂×#41

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#125

yes

—

100

Dic₃

—

101

Dic₃

—

102

Dic₃

—

103

Dic₃

—

104

C₂³

C₂³×D₃

yes

characteristic

105

C₂×C₄

—

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

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×Dic₆

yes

—

121

C₂×C₄

C₂×Dic₆

yes

—

122

C₂×C₄

#237

yes

—

123

C₂×C₄

#239

yes

—

124

C₂×C₃

C₂×#125

yes

characteristic

125

C₂×C₃

#540

yes

characteristic

126

C₂×C₃

#541

yes

characteristic

127

C₂×C₃

#542

yes

—

128

C₂×C₃

#544

yes

—

129

C₂²

C₂²×Dic₆

yes

characteristic

130

C₂²

C₂×#237

yes

characteristic

131

C₂²

C₂×#239

yes

characteristic

132

C₂²

#1009

yes

—

133

C₂²

#1010

yes

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#890

yes

—

141

C₃

#2647

yes

characteristic, complemented, Sylow

142

C₂

C₂×#890

yes

characteristic

143

C₂

#5519

yes

—

144

C₂

#5682

yes

—

145

C₂

#5697

yes

characteristic

146

C₂

#5698

yes

characteristic

	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	#2647
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
141. C₃	#2647	semidirect

H ≅ G/N

type

141. C₃

#2647

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
#4935	2
#4932	2
#4661	2
#4660	2
#2647	3

subgroup

index

Subgroups — #74195

Conjugacy classes of subgroups

Complements

Maximal subgroups