Group #31911728 — subgroups

Subgroups — #31911728

Order 18176.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22590248	C₂	1	1	yes	maximal, characteristic
2	#22587707	C₂	1	2	yes	maximal
3	#22586414	C₂	1	1	yes	maximal, characteristic
4	#22586389	C₂	1	2	yes	maximal
5	C₇₁×#2247	C₂	1	1	yes	maximal, characteristic
6	#17986281	—	2	2	no	—
7	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₇₁⋊C₁₆	—	2	1	no	—
10	#17985774	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#17985745	—	2	2	no	—
13	C₂×Q₃₂×C₇₁	—	2	1	no	—
14	C₂×Q₃₂×C₇₁	—	2	1	no	—
15	#494×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₁₆	—	2	1	no	—
19	#14556059	—	4	2	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₁⋊C₈	—	2	1	no	—
23	#14556051	—	4	2	no	—
24	C₈×Dic₇₁	—	2	1	no	—
25	C₂×Q₁₆×C₇₁	—	2	2	no	—
26	C₂×Q₁₆×C₇₁	—	2	2	no	—
27	C₄⋊Q₈×C₇₁	—	2	2	no	—
28	Q₃₂×C₇₁	—	4	2	no	—
29	Q₃₂×C₇₁	—	4	2	no	—
30	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
37	Dic₇₁⋊C₄	—	2	1	no	—
38	C₄×Dic₇₁	—	2	1	no	—
39	C₇₁⋊C₁₆	—	4	1	no	—
40	C₂×Q₈×C₇₁	—	4	2	no	—
41	Q₁₆×C₇₁	—	4	2	no	—
42	Q₁₆×C₇₁	—	4	2	no	—
43	Q₁₆×C₇₁	—	8	2	no	—
44	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₇₁	—	4	2	no	—
47	C₄⋊C₄×C₇₁	—	4	2	no	—
48	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₁	—	2	1	no	—
50	C₁₆×C₇₁	SD₁₆	1	2	yes	—
51	Q₈×C₇₁	—	8	2	no	—
52	Q₈×C₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	—	8	2	no	—
54	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
57	C₂×Dic₇₁	—	2	1	no	—
58	C₂×Dic₇₁	—	2	1	no	—
59	Dic₁₄₂	—	2	2	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₇₁⋊C₈	—	4	1	no	—
64	C₇₁⋊C₈	—	8	2	no	—
65	C₂²×C₇₁	#466	1	1	yes	characteristic
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	16	2	no	—
68	C₄×C₇₁	#460	1	1	yes	characteristic
69	C₄×C₇₁	#472	1	1	yes	characteristic
70	C₄×C₇₁	#510	1	2	yes	—
71	Dic₇₁	—	4	1	no	—
72	Dic₇₁	—	4	1	no	—
73	#12443	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	#1668	1	1	yes	characteristic
75	C₂×C₇₁	#2219	1	1	yes	characteristic
76	C₂×C₇₁	#2222	1	1	yes	characteristic
77	#2247	D₇₁	1	1	yes	characteristic
78	#2183	—	71	1	no	—
79	#1670	—	71	2	no	—
80	#1372	—	71	1	no	—
81	#1362	—	71	2	no	—
82	C₇₁	#12443	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	#494	C₂×D₇₁	1	2	yes	—
86	#475	—	142	2	no	—
87	C₈×Q₈	—	71	1	no	—
88	C₄⋊C₁₆	—	71	1	no	—
89	C₄⋊C₁₆	—	142	1	no	—
90	#346	—	142	1	no	—
91	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
92	#335	—	71	2	no	—
93	#332	—	142	2	no	—
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	2	2	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	71	1	no	—
98	Q₃₂	—	4	2	no	—
99	Q₃₂	—	4	2	no	—
100	C₂×C₁₆	—	142	1	no	—
101	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
103	#106	—	284	2	no	—
104	C₄⋊C₈	—	71	1	no	—
105	C₄⋊C₈	—	142	1	no	—
106	C₄⋊C₈	—	142	2	no	—
107	#103	—	284	2	no	—
108	C₄×C₈	—	142	1	no	—
109	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	71	1	no	—
112	Q₁₆	—	4	2	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	8	2	no	—
115	C₂×C₈	—	284	2	no	—
116	C₂×C₈	—	142	1	no	—
117	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
118	C₂×C₈	#11765864	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	71	1	no	—
122	C₄⋊C₄	—	142	1	no	—
123	C₄²	—	142	1	no	—
124	C₄²	D₄×D₇₁	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	284	1	no	—
127	C₁₆	#11765853	1	2	yes	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	—	142	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	142	1	no	—
133	C₂×C₄	—	142	1	no	—
134	C₂×C₄	#14556099	1	1	yes	characteristic
135	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
136	C₂×C₄	#14556147	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	284	1	no	—
139	C₈	—	568	2	no	—
140	C₈	C₂×#11765853	1	1	yes	characteristic
141	C₈	#14556154	1	1	yes	characteristic
142	C₂²	#17986232	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	16	2	no	—
145	C₄	—	284	1	no	—
146	C₄	—	284	1	no	—
147	C₄	#17986196	1	1	yes	characteristic
148	C₄	#17986266	1	1	yes	characteristic
149	C₄	#17986441	1	2	yes	—
150	C₂	#22587697	1	1	yes	characteristic
151	C₂	#22590445	1	1	yes	characteristic
152	C₂	#22590462	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590248

C₂

yes

maximal, characteristic

#22587707

C₂

yes

maximal

#22586414

C₂

yes

maximal, characteristic

#22586389

C₂

yes

maximal

C₇₁×#2247

C₂

yes

maximal, characteristic

#17986281

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

#17985774

—

#17985752

C₂²

yes

—

#17985745

—

C₂×Q₃₂×C₇₁

—

C₂×Q₃₂×C₇₁

—

#494×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556051

—

C₈×Dic₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

D₈

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

—

—

—

—

C₇₁

#12443

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₇₁

yes

—

#475

—

142

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

#346

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

#332

—

142

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

100

C₂×C₁₆

—

142

—

101

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

103

#106

—

284

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

142

—

106

C₄⋊C₈

—

142

—

107

#103

—

284

—

108

C₄×C₈

—

142

—

109

C₄×C₈

C₂²×D₇₁

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

284

—

116

C₂×C₈

—

142

—

117

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

118

C₂×C₈

#11765864

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

142

—

123

C₄²

—

142

—

124

C₄²

D₄×D₇₁

yes

characteristic

125

C₁₆

—

126

C₁₆

—

284

—

127

C₁₆

#11765853

yes

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

142

—

131

C₂×C₄

—

132

C₂×C₄

—

142

—

133

C₂×C₄

—

142

—

134

C₂×C₄

#14556099

yes

characteristic

135

C₂×C₄

D₈×D₇₁

yes

characteristic

136

C₂×C₄

#14556147

yes

characteristic

137

C₈

—

138

C₈

—

284

—

139

C₈

—

568

—

140

C₈

C₂×#11765853

yes

characteristic

141

C₈

#14556154

yes

characteristic

142

C₂²

#17986232

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

284

—

146

C₄

—

284

—

147

C₄

#17986196

yes

characteristic

148

C₄

#17986266

yes

characteristic

149

C₄

#17986441

yes

—

150

C₂

#22587697

yes

characteristic

151

C₂

#22590445

yes

characteristic

152

C₂

#22590462

yes

characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2247	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	#2247	D₇₁
hypocenter	C₇₁	#12443
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#17986232

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2247

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
82. C₇₁	#12443	semidirect

H ≅ G/N

type

82. C₇₁

#12443

semidirect

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

subgroup	index
#22590248	2
#22587707	2
#22586414	2
#22586389	2
C₇₁×#2247	2
#12443	71

subgroup

index

C₇₁×#2247

Subgroups — #31911728

Conjugacy classes of subgroups

Complements

Maximal subgroups