Group #70524 — subgroups

Subgroups — #70524

Order 384.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#4554	C₂	1	4	yes	maximal
5	#1997	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₄	1	2	yes	—
7	C₂×C₃×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	2	yes	—
12	#869	—	2	1	no	—
13	#869	—	2	1	no	—
14	#869	—	2	1	no	—
15	#869	—	2	1	no	—
16	C₃×C₄⋊C₈	—	2	1	no	—
17	C₃×C₄⋊C₈	—	2	1	no	—
18	#803	—	2	1	no	—
19	C₄×C₃⋊C₈	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	3	1	no	—
21	#423	D₃	1	1	yes	characteristic
22	C₄×M₁₆	—	3	1	no	—
23	#343	—	3	4	no	—
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃	C₂×C₄	1	4	yes	—
28	C₂×Dic₆	—	2	2	no	—
29	C₂×Dic₆	—	2	2	no	—
30	C₂×Dic₆	—	4	4	no	—
31	C₃×M₁₆	—	2	4	no	—
32	C₂×C₃×C₈	—	2	2	no	—
33	C₃×C₄²	D₄	1	1	yes	characteristic
34	C₃×C₄²	D₄	1	1	yes	characteristic
35	C₃×C₄²	D₄	1	1	yes	characteristic
36	C₃×C₄²	D₄	1	1	yes	characteristic
37	C₄⋊Dic₃	—	2	2	no	—
38	C₄⋊Dic₃	—	2	2	no	—
39	C₄⋊Dic₃	—	2	2	no	—
40	C₄⋊Dic₃	—	2	2	no	—
41	#208	—	2	2	no	—
42	#208	Q₈	1	4	yes	complemented
43	C₂×C₃⋊C₈	—	2	2	no	—
44	C₂²×Q₈	—	3	2	no	—
45	C₂×M₁₆	—	3	2	no	—
46	C₂×M₁₆	C₂×D₃	1	2	yes	—
47	C₄⋊Q₈	—	6	1	no	—
48	C₄⋊Q₈	—	6	1	no	—
49	C₄⋊Q₈	—	6	1	no	—
50	C₄⋊Q₈	—	6	1	no	—
51	C₂×C₄⋊C₄	—	3	2	no	—
52	C₂×C₄⋊C₄	—	3	2	no	—
53	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	#97	—	6	1	no	—
57	C₄×C₈	—	6	1	no	—
58	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
68	C₂×C₃×C₄	#41	1	2	yes	—
69	C₂×Dic₃	—	2	4	no	—
70	C₂×Dic₃	—	4	4	no	—
71	Dic₆	—	4	4	no	—
72	Dic₆	—	4	4	no	—
73	C₃×C₈	—	4	2	no	—
74	C₃⋊C₈	—	2	4	no	—
75	C₂×Q₈	—	6	2	no	—
76	C₂×Q₈	—	6	2	no	—
77	C₂×Q₈	—	12	4	no	—
78	C₂²×C₄	—	3	4	no	—
79	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	3	4	no	—
84	M₁₆	—	6	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	6	2	no	—
87	C₄⋊C₄	—	6	2	no	—
88	C₄⋊C₄	—	6	2	no	—
89	C₄⋊C₄	—	6	2	no	—
90	C₄⋊C₄	—	6	2	no	—
91	C₄²	D₁₂	1	1	yes	characteristic
92	C₄²	D₁₂	1	1	yes	characteristic
93	C₄²	#67	1	1	yes	characteristic
94	C₄²	#67	1	1	yes	characteristic
95	C₂²×C₃	—	2	1	no	—
96	C₂²×C₃	—	2	1	no	—
97	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₃	C₄×Q₈	1	2	yes	—
99	C₃×C₄	—	2	1	no	—
100	C₃×C₄	—	2	1	no	—
101	C₃×C₄	#100	1	2	yes	—
102	C₃×C₄	#101	1	2	yes	—
103	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
104	C₃×C₄	#124	1	1	yes	characteristic
105	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
106	Dic₃	—	4	4	no	—
107	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
108	Q₈	—	12	4	no	—
109	Q₈	—	12	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	6	4	no	—
113	C₂×C₄	—	12	4	no	—
114	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
116	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
119	C₂×C₄	#237	1	2	yes	—
120	C₂×C₄	D₃×Q₈	1	2	yes	—
121	C₂×C₄	C₂×#67	1	1	yes	characteristic
122	C₈	—	4	2	no	—
123	C₈	—	6	4	no	—
124	C₂×C₃	—	2	2	no	—
125	C₂×C₃	#391	1	1	yes	characteristic
126	C₂×C₃	C₂×#100	1	1	yes	characteristic
127	C₂×C₃	C₂×#101	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#887	1	2	yes	—
131	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	12	4	no	—
135	C₄	#823	1	2	yes	—
136	C₄	#824	1	2	yes	—
137	C₄	#897	1	2	yes	—
138	C₄	#936	1	1	yes	characteristic
139	C₄	#944	1	1	yes	characteristic
140	C₃	#1997	1	1	yes	characteristic, complemented, Sylow
141	C₂	—	2	2	no	—
142	C₂	#4973	1	1	yes	characteristic
143	C₂	C₂×#823	1	1	yes	characteristic
144	C₂	C₂×#824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×#423

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#4554

C₂

yes

maximal

#1997

—

maximal, Sylow

C₂²×Dic₆

C₄

yes

—

C₂×C₃×M₁₆

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₄

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×#208

C₂²

yes

—

—

—

—

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

#803

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

—

#423

D₃

yes

characteristic

C₄×M₁₆

—

#343

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

#208

Q₈

yes

complemented

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₂

yes

characteristic

C₄²

D₁₂

yes

C₄²

yes

C₄²

yes

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₄×Q₈

yes

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

#100

yes

—

102

C₃×C₄

#101

yes

—

103

C₃×C₄

C₂²⋊Q₈

yes

—

104

C₃×C₄

#124

yes

characteristic

105

C₃×C₄

C₄⋊Q₈

yes

characteristic

106

Dic₃

—

107

C₂³

C₂×C₄×D₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

115

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

116

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

117

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

118

C₂×C₄

C₂×D₁₂

yes

characteristic

119

C₂×C₄

#237

yes

—

120

C₂×C₄

D₃×Q₈

yes

—

121

C₂×C₄

C₂×#67

yes

characteristic

122

C₈

—

123

C₈

—

124

C₂×C₃

—

125

C₂×C₃

#391

yes

characteristic

126

C₂×C₃

C₂×#100

yes

characteristic

127

C₂×C₃

C₂×#101

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

#887

yes

—

131

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#823

yes

—

136

C₄

#824

yes

—

137

C₄

#897

yes

—

138

C₄

#936

yes

characteristic

139

C₄

#944

yes

characteristic

140

C₃

#1997

yes

characteristic, complemented, Sylow

141

C₂

—

142

C₂

#4973

yes

characteristic

143

C₂

C₂×#823

yes

characteristic

144

C₂

C₂×#824

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊C₄
commutator, prime radical	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#423	D₃
hypocenter	C₃	#1997
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₃)⋊C₄

commutator, prime radical

C₂³×C₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#423

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
42. #208	Q₈	semidirect
140. C₃	#1997	semidirect

H ≅ G/N

type

42. #208

Q₈

semidirect

140. C₃

#1997

semidirect

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

subgroup	index
C₂×#869	2
C₃×#423	2
C₄×#208	2
#4554	2
#1997	3

subgroup

index

Subgroups — #70524

Conjugacy classes of subgroups

Complements

Maximal subgroups