Group #31884524 — subgroups

Subgroups — #31884524

Order 18176.

244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22588390	C₂	1	2	yes	maximal
2	C₇₁×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556064	—	2	2	no	—
4	#17985983	—	2	2	no	—
5	C₇₁⋊#338	C₄	1	4	yes	—
6	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
9	C₂²⋊C₁₆×C₇₁	—	2	2	no	—
10	C₂×#11765849	—	2	4	no	—
11	#14556064	—	4	2	no	—
12	#14556064	—	4	2	no	—
13	#14556047	—	4	2	no	—
14	#14556045	—	4	2	no	—
15	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
16	#131×C₇₁	—	4	1	no	—
17	#131×C₇₁	—	4	1	no	—
18	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₇₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	D₄	1	2	yes	—
22	M₃₂×C₇₁	—	4	1	no	—
23	C₂×C₁₆×C₇₁	—	2	1	no	—
24	C₂×C₁₆×C₇₁	—	2	1	no	—
25	C₂×C₁₆×C₇₁	—	4	1	no	—
26	#11765849	—	4	4	no	—
27	#11765849	—	4	4	no	—
28	C₂×C₇₁⋊C₈	—	4	4	no	—
29	#41×C₇₁	—	4	1	no	—
30	#41×C₇₁	—	4	1	no	—
31	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
32	C₂×D₄×C₇₁	SD₁₆	1	2	yes	—
33	C₂×D₄×C₇₁	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	M₁₆×C₇₁	—	4	2	no	—
37	M₁₆×C₇₁	—	4	1	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
45	C₁₆×C₇₁	—	4	1	no	—
46	C₁₆×C₇₁	—	4	1	no	—
47	C₂³×C₇₁	#103	1	2	yes	—
48	C₂³×C₇₁	#106	1	1	yes	characteristic
49	Q₈×C₇₁	—	4	1	no	—
50	D₄×C₇₁	—	4	2	no	—
51	D₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#102	1	2	yes	—
58	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
59	C₈×C₇₁	—	2	1	no	—
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	#99	1	1	yes	characteristic
62	C₈×C₇₁	#99	1	1	yes	characteristic
63	C₇₁⋊C₈	—	8	4	no	—
64	C₂²×C₇₁	—	2	2	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	4	2	no	—
67	C₂²×C₇₁	—	4	1	no	—
68	C₂²×C₇₁	#340	1	1	yes	characteristic
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₇₁	#342	1	1	yes	characteristic
73	#7229	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	—	2	1	no	—
75	C₂×C₇₁	—	4	2	no	—
76	C₂×C₇₁	—	4	1	no	—
77	C₂×C₇₁	#1289	1	1	yes	characteristic
78	#2113	D₇₁	1	1	yes	characteristic
79	#1813	—	71	2	no	—
80	C₇₁	#7229	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
82	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
83	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
84	C₂×#108	—	142	2	no	—
85	#413	—	142	2	no	—
86	C₂²⋊C₁₆	—	2	2	no	—
87	#338	—	71	4	no	—
88	C₂×#41	D₂₈₄	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	142	4	no	—
92	C₂×M₁₆	Dic₁₄₂	1	1	yes	characteristic
93	C₂×M₁₆	C₄×D₇₁	1	2	yes	—
94	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	284	2	no	—
101	#108	—	284	2	no	—
102	#101	—	284	2	no	—
103	#100	—	284	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₅₆₈	1	1	yes	characteristic
107	C₂×D₄	#11765854	1	2	yes	—
108	C₂×D₄	Dic₂₈₄	1	1	yes	characteristic
109	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	284	4	no	—
114	M₁₆	—	284	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	284	4	no	—
120	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
122	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#14556053	1	2	yes	—
126	C₂³	#14556060	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
135	C₂×C₄	#14556050	1	2	yes	—
136	C₂×C₄	#14556062	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	568	4	no	—
140	C₈	#14556043	1	1	yes	characteristic
141	C₈	#14556043	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#17985762	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	#17985756	1	1	yes	characteristic
150	C₄	#17985766	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	#22586208	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588390

C₂

yes

maximal

C₇₁×#2113

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17985983

—

C₇₁⋊#338

C₄

yes

—

C₂×#131×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₄

yes

characteristic

C₂²⋊C₁₆×C₇₁

—

—

—

—

—

—

C₂×#41×C₇₁

D₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

—

C₂×M₁₆×C₇₁

Q₈

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

C₂×D₄×C₇₁

SD₁₆

yes

—

C₂×D₄×C₇₁

Q₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

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

—

C₁₆×C₇₁

—

C₂³×C₇₁

#103

yes

—

C₂³×C₇₁

#106

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×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

#107

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#340

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1289

yes

characteristic

#2113

D₇₁

yes

characteristic

#1813

—

C₇₁

#7229

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₁

yes

characteristic

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂²×C₁₆

Dic₇₁

yes

characteristic

C₂×#108

—

142

—

#413

—

142

—

C₂²⋊C₁₆

—

#338

—

C₂×#41

D₂₈₄

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

142

—

C₂×M₁₆

Dic₁₄₂

yes

characteristic

C₂×M₁₆

C₄×D₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²⋊D₇₁

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

284

—

101

#108

—

284

—

102

#101

—

284

—

103

#100

—

284

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₅₆₈

yes

characteristic

107

C₂×D₄

#11765854

yes

—

108

C₂×D₄

Dic₂₈₄

yes

characteristic

109

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

—

110

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

284

—

114

M₁₆

—

284

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

284

—

120

C₂×C₈

C₄×Dic₇₁

yes

characteristic

121

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

122

C₂×C₈

Dic₇₁⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#14556053

yes

—

126

C₂³

#14556060

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₇₁⋊#95

yes

characteristic

135

C₂×C₄

#14556050

yes

—

136

C₂×C₄

#14556062

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

568

—

140

C₈

#14556043

yes

characteristic

141

C₈

#14556043

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#17985762

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#17985756

yes

characteristic

150

C₄

#17985766

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#22586208

yes

characteristic

	subgroup	G/H
center	C₄	#17985756
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂×#131	C₂×D₇₁
Fitting	C₇₁×#2113	C₂
socle	C₂×C₇₁	#1289
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂×#131×C₇₁	C₂²
hypercenter	#2113	D₇₁
hypocenter	C₇₁	#7229
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#17985756

commutator

C₂×C₈×C₇₁

C₄²

Frattini

C₂×#131

C₂×D₇₁

Fitting

C₇₁×#2113

C₂

socle

C₂×C₇₁

#1289

prime radical

C₂×#41×C₇₁

D₄

cosocle

C₂×#131×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
80. C₇₁	#7229	semidirect

H ≅ G/N

type

80. C₇₁

#7229

semidirect

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

subgroup	index
#22588390	2
C₇₁×#2113	2
#7229	71

subgroup

index

#22588390

C₇₁×#2113

#7229

Subgroups — #31884524

Conjugacy classes of subgroups

Complements

Maximal subgroups