Group #69491 — subgroups

Subgroups — #69491

Order 384.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	2	yes	maximal
5	#4529	C₂	1	2	yes	maximal
6	#1989	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	—	2	1	no	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃⋊C₄	—	2	1	no	—
13	C₂×Dic₃⋊C₄	—	2	1	no	—
14	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
15	#870	—	2	2	no	—
16	#870	C₄	1	4	yes	complemented
17	C₃×#95	—	2	1	no	—
18	C₄⋊(C₃⋊C₈)	—	2	2	no	—
19	C₂×#125	—	3	1	no	—
20	C₂×C₄⋊C₈	—	3	1	no	—
21	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
22	#340	—	3	2	no	—
23	#340	—	3	2	no	—
24	C₂²×C₃×C₄	—	2	2	no	—
25	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃	—	2	1	no	—
29	C₂²×Dic₃	—	2	1	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄⋊C₄	Q₈	1	4	yes	—
32	C₃×C₄²	—	4	2	no	—
33	C₃×C₄²	C₂×C₄	1	2	yes	—
34	C₃×C₄²	D₄	1	2	yes	—
35	C₄⋊Dic₃	—	2	1	no	—
36	C₄⋊Dic₃	—	2	1	no	—
37	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
39	Dic₃⋊C₄	—	2	4	no	—
40	Dic₃⋊C₄	—	2	4	no	—
41	C₂×C₃⋊C₈	—	2	2	no	—
42	C₂×C₃⋊C₈	—	2	4	no	—
43	C₂²×C₈	—	3	2	no	—
44	#125	—	3	4	no	—
45	#125	—	6	2	no	—
46	C₂×C₄⋊C₄	—	3	1	no	—
47	C₂×C₄⋊C₄	—	3	1	no	—
48	C₂×C₄⋊C₄	—	6	1	no	—
49	C₂×C₄⋊C₄	—	6	1	no	—
50	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₄⋊C₈	—	6	2	no	—
54	#95	—	2	1	no	—
55	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	1	no	—
58	C₂×C₃×C₄	—	2	1	no	—
59	C₂×C₃×C₄	—	2	4	no	—
60	C₂×C₃×C₄	—	4	2	no	—
61	C₂×C₃×C₄	—	4	2	no	—
62	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
63	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	#41	1	2	yes	—
68	C₂×Dic₃	—	2	2	no	—
69	C₂×Dic₃	—	2	2	no	—
70	C₂×Dic₃	—	4	2	no	—
71	C₂×Dic₃	—	4	1	no	—
72	C₂×Dic₃	—	4	1	no	—
73	C₃⋊C₈	—	4	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	6	1	no	—
76	C₂²×C₄	—	6	1	no	—
77	C₂²×C₄	#67	1	1	yes	characteristic
78	C₂²×C₄	#67	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₈	—	6	2	no	—
81	C₂×C₈	—	6	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	3	2	no	—
84	C₄⋊C₄	—	3	2	no	—
85	C₄⋊C₄	—	6	4	no	—
86	C₄⋊C₄	—	6	4	no	—
87	C₄⋊C₄	—	6	1	no	—
88	C₄⋊C₄	—	6	1	no	—
89	C₄⋊C₄	Dic₆	1	4	yes	—
90	C₄²	—	4	2	no	—
91	C₄²	C₄×D₃	1	2	yes	—
92	C₄²	D₁₂	1	2	yes	—
93	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
95	C₂²×C₃	#135	1	2	yes	—
96	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×C₃	#137	1	1	yes	characteristic
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	—	4	4	no	—
101	C₃×C₄	C₄×Q₈	1	2	yes	—
102	C₃×C₄	#124	1	1	yes	characteristic
103	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
104	Dic₃	—	4	2	no	—
105	Dic₃	—	4	2	no	—
106	C₂³	C₂×#67	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	6	2	no	—
114	C₂×C₄	—	6	2	no	—
115	C₂×C₄	—	12	1	no	—
116	C₂×C₄	—	12	1	no	—
117	C₂×C₄	—	12	2	no	—
118	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
119	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
120	C₂×C₄	C₂×Dic₆	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
123	C₂×C₄	#235	1	2	yes	—
124	C₈	—	12	2	no	—
125	C₂×C₃	#391	1	1	yes	characteristic
126	C₂×C₃	#416	1	1	yes	characteristic
127	C₂×C₃	#417	1	1	yes	characteristic
128	C₂×C₃	#478	1	2	yes	—
129	C₂×C₃	#479	1	2	yes	—
130	C₂²	#924	1	2	yes	—
131	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₂²	#932	1	1	yes	characteristic
133	C₂²	#942	1	1	yes	characteristic
134	C₂²	#950	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	4	no	—
138	C₄	—	12	2	no	—
139	C₄	—	12	2	no	—
140	C₄	C₄×Dic₆	1	2	yes	—
141	C₄	#869	1	1	yes	characteristic
142	C₄	#875	1	1	yes	characteristic
143	C₃	#1989	1	1	yes	characteristic, complemented, Sylow
144	C₂	#4929	1	1	yes	characteristic
145	C₂	#4966	1	1	yes	characteristic
146	C₂	#5002	1	1	yes	characteristic
147	C₂	#5009	1	2	yes	—
148	C₂	#5022	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#870

C₂

yes

maximal, characteristic

C₃×C₄×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#4529

C₂

yes

maximal

#1989

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×Dic₃⋊C₄

—

C₂×Dic₃⋊C₄

—

C₂²×C₃⋊C₈

C₂²

yes

—

#870

—

#870

C₄

yes

complemented

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₃

yes

characteristic

#340

—

#340

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

Q₈

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₆

yes

—

C₄²

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₂²×C₃

#135

yes

—

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

C₄×Q₈

yes

—

102

C₃×C₄

#124

yes

characteristic

103

C₃×C₄

C₄⋊Q₈

yes

characteristic

104

Dic₃

—

105

Dic₃

—

106

C₂³

C₂×#67

yes

characteristic

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

(C₂×D₃)⋊C₄

yes

—

119

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

120

C₂×C₄

C₂×Dic₆

yes

—

121

C₂×C₄

C₂×C₄×D₃

yes

characteristic

122

C₂×C₄

C₂×D₁₂

yes

characteristic

123

C₂×C₄

#235

yes

—

124

C₈

—

125

C₂×C₃

#391

yes

characteristic

126

C₂×C₃

#416

yes

characteristic

127

C₂×C₃

#417

yes

characteristic

128

C₂×C₃

#478

yes

—

129

C₂×C₃

#479

yes

—

130

C₂²

#924

yes

—

131

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

132

C₂²

#932

yes

characteristic

133

C₂²

#942

yes

characteristic

134

C₂²

#950

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₄×Dic₆

yes

—

141

C₄

#869

yes

characteristic

142

C₄

#875

yes

characteristic

143

C₃

#1989

yes

characteristic, complemented, Sylow

144

C₂

#4929

yes

characteristic

145

C₂

#4966

yes

characteristic

146

C₂

#5002

yes

characteristic

147

C₂

#5009

yes

—

148

C₂

#5022

yes

—

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#1989
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×C₄×C₄⋊C₄

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₄×C₄⋊C₄

D₃

hypocenter

C₃

#1989

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
16. #870	C₄	semidirect
143. C₃	#1989	semidirect

H ≅ G/N

type

16. #870

C₄

semidirect

143. C₃

#1989

semidirect

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

subgroup	index
C₂×#870	2
C₃×C₄×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊C₈)	2
#4529	2
#1989	3

subgroup

index

C₂×#870

C₃×C₄×C₄⋊C₄

C₂×C₄⋊(C₃⋊C₈)

#4529

#1989

Subgroups — #69491

Conjugacy classes of subgroups

Complements

Maximal subgroups