Group #31884528 — subgroups

Subgroups — #31884528

Order 18176.

270 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22588378	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556064	—	2	2	no	—
4	#17985963	—	2	2	no	—
5	#17985764	C₄	1	4	yes	—
6	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
8	C₂²⋊C₁₆×C₇₁	—	2	2	no	—
9	C₂×#11765849	—	2	4	no	—
10	#14556064	—	4	4	no	—
11	C₂²⋊(C₇₁⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₇₁⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	—	4	2	no	—
15	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₁	D₄	1	2	yes	—
18	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	—	4	2	no	—
21	#11765849	—	4	4	no	—
22	#11765849	—	4	4	no	—
23	C₂×C₇₁⋊C₈	—	4	4	no	—
24	C₂⁴×C₇₁	Q₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	—	4	1	no	—
26	C₂²×C₄×C₇₁	—	4	1	no	—
27	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	4	no	—
33	C₂×C₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	—	4	1	no	—
35	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₇₁	—	4	2	no	—
39	C₂³×C₇₁	—	2	2	no	—
40	C₂³×C₇₁	—	4	1	no	—
41	C₂³×C₇₁	—	4	1	no	—
42	C₂³×C₇₁	#103	1	2	yes	—
43	C₂³×C₇₁	#107	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	—	4	1	no	—
49	C₂×C₄×C₇₁	—	4	2	no	—
50	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#102	1	2	yes	—
52	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
53	C₈×C₇₁	—	2	1	no	—
54	C₈×C₇₁	—	2	2	no	—
55	C₈×C₇₁	#99	1	2	yes	—
56	C₇₁⋊C₈	—	8	4	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	2	2	no	—
60	C₂²×C₇₁	—	2	2	no	—
61	C₂²×C₇₁	—	2	2	no	—
62	C₂²×C₇₁	—	4	4	no	—
63	C₂²×C₇₁	#340	1	1	yes	characteristic
64	C₂²×C₇₁	#368	1	2	yes	—
65	C₄×C₇₁	—	2	2	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
68	C₄×C₇₁	#342	1	1	yes	characteristic
69	#7231	—	71	1	no	maximal, Sylow
70	C₂×C₇₁	—	2	2	no	—
71	C₂×C₇₁	—	4	2	no	—
72	C₂×C₇₁	#1289	1	1	yes	characteristic
73	C₂×C₇₁	#1381	1	1	yes	characteristic
74	C₂×C₇₁	#1385	1	1	yes	characteristic
75	C₂×C₂²⋊C₁₆	D₇₁	1	1	yes	characteristic
76	#1810	—	71	2	no	—
77	C₇₁	#7231	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
79	C₂²×C₁₆	Dic₇₁	1	2	yes	—
80	C₂×#108	—	142	2	no	—
81	#407	—	142	2	no	—
82	C₂²⋊C₁₆	—	2	2	no	—
83	#341	—	71	4	no	—
84	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
85	C₂×M₁₆	—	142	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
88	C₂²×C₈	C₄×D₇₁	1	2	yes	—
89	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	4	2	no	—
93	#108	—	284	4	no	—
94	C₂²⋊C₈	—	284	2	no	—
95	C₂²⋊C₈	—	284	2	no	—
96	C₂⁴	Dic₂₈₄	1	1	yes	characteristic
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
100	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
101	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
102	C₂²×C₄	#11765854	1	2	yes	—
103	M₁₆	—	284	4	no	—
104	M₁₆	—	284	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	284	4	no	—
110	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
112	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
113	C₁₆	—	4	2	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	#14556053	1	2	yes	—
118	C₂³	#14556062	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
126	C₂×C₄	#14556050	1	2	yes	—
127	C₂×C₄	#14556060	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	2	no	—
130	C₈	—	568	4	no	—
131	C₈	#14556043	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#17985762	1	1	yes	characteristic
139	C₂²	#17985829	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	#17985756	1	1	yes	characteristic
143	C₄	#17985766	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	—	4	2	no	—
146	C₂	#22586208	1	1	yes	characteristic
147	C₂	#22586434	1	1	yes	characteristic
148	C₂	#22586444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588378

C₂

yes

maximal

C₂×C₂²⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17985963

—

#17985764

C₄

yes

—

C₂³×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₄

yes

—

C₂²⋊C₁₆×C₇₁

—

C₂×#11765849

—

#14556064

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₂³×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂⁴×C₇₁

Q₁₆

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

#103

yes

—

C₂³×C₇₁

#107

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#106

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#99

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#340

yes

characteristic

C₂²×C₇₁

#368

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₂²⋊C₁₆

D₇₁

yes

characteristic

#1810

—

C₇₁

#7231

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₁₆

Dic₇₁

yes

—

C₂×#108

—

142

—

#407

—

142

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₄

D₂₈₄

yes

characteristic

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

Dic₁₄₂

yes

characteristic

C₂²×C₈

C₄×D₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

284

—

C₂²⋊C₈

—

284

—

C₂²⋊C₈

—

284

—

C₂⁴

Dic₂₈₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

—

100

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

101

C₂²×C₄

D₅₆₈

yes

characteristic

102

C₂²×C₄

#11765854

yes

—

103

M₁₆

—

284

—

104

M₁₆

—

284

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

284

—

110

C₂×C₈

C₄×Dic₇₁

yes

characteristic

111

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

112

C₂×C₈

Dic₇₁⋊C₄

yes

—

113

C₁₆

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

#14556053

yes

—

118

C₂³

#14556062

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₇₁⋊#95

yes

characteristic

126

C₂×C₄

#14556050

yes

—

127

C₂×C₄

#14556060

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

568

—

131

C₈

#14556043

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#17985762

yes

characteristic

139

C₂²

#17985829

yes

—

140

C₄

—

141

C₄

—

142

C₄

#17985756

yes

characteristic

143

C₄

#17985766

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#22586208

yes

characteristic

147

C₂

#22586434

yes

characteristic

148

C₂

#22586444

yes

characteristic

	subgroup	G/H
center	C₂²	#17985762
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂³×C₈	C₂×D₇₁
Fitting	C₂×C₂²⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂⁴×C₇₁	Q₁₆
cosocle	C₂³×C₈×C₇₁	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₇₁
hypocenter	C₇₁	#7231
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#17985762

commutator

C₂×C₈×C₇₁

C₄²

Frattini

C₂³×C₈

C₂×D₇₁

Fitting

C₂×C₂²⋊C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#340

prime radical

C₂⁴×C₇₁

Q₁₆

cosocle

C₂³×C₈×C₇₁

C₂²

hypercenter

C₂×C₂²⋊C₁₆

D₇₁

hypocenter

C₇₁

#7231

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

N	H ≅ G/N	type
77. C₇₁	#7231	semidirect

H ≅ G/N

type

77. C₇₁

#7231

semidirect

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

subgroup	index
#22588378	2
C₂×C₂²⋊C₁₆×C₇₁	2
#7231	71

subgroup

index

#22588378

C₂×C₂²⋊C₁₆×C₇₁

#7231

Subgroups — #31884528

Conjugacy classes of subgroups

Complements

Maximal subgroups