Group #31911715 — subgroups

Subgroups — #31911715

Order 18176.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22590593	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
3	#22587711	C₂	1	2	yes	maximal
4	#22586388	C₂	1	2	yes	maximal
5	C₇₁×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	—	2	1	no	—
7	C₂×Dic₅₆₈	—	2	1	no	—
8	#17986379	C₂²	1	2	yes	—
9	#17986285	—	2	2	no	—
10	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₇₁⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₇₁	—	2	1	no	—
14	#17985744	—	2	2	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#335×C₇₁	C₂²	1	2	yes	—
17	C₂×Dic₂₈₄	—	2	2	no	—
18	C₂×Dic₂₈₄	—	2	2	no	—
19	#14556128	—	2	2	no	—
20	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
21	Dic₅₆₈	—	4	2	no	—
22	Dic₅₆₈	—	4	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	#14556060	—	4	2	no	—
25	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₇₁⋊C₈	—	2	1	no	—
27	#14556053	—	4	2	no	—
28	C₈×Dic₇₁	—	2	1	no	—
29	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₂	—	4	2	no	—
34	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
35	Dic₂₈₄	—	4	2	no	—
36	Dic₂₈₄	—	4	2	no	—
37	Dic₂₈₄	—	8	2	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₄⋊Dic₇₁	—	4	2	no	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
42	Dic₇₁⋊C₄	—	2	1	no	—
43	C₄×Dic₇₁	—	2	1	no	—
44	C₇₁⋊C₁₆	—	4	1	no	—
45	C₂×C₈×C₇₁	—	4	2	no	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
48	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₁	—	2	1	no	—
50	C₁₆×C₇₁	SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
54	C₂×Dic₇₁	—	2	1	no	—
55	C₂×Dic₇₁	—	2	1	no	—
56	C₂×Dic₇₁	—	8	2	no	—
57	Dic₁₄₂	—	2	1	no	—
58	Dic₁₄₂	—	8	2	no	—
59	Dic₁₄₂	—	8	2	no	—
60	Dic₁₄₂	Q₃₂	1	2	yes	complemented
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	8	2	no	—
63	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₇₁⋊C₈	—	4	1	no	—
66	C₂²×C₇₁	#466	1	1	yes	characteristic
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	#460	1	1	yes	characteristic
69	C₄×C₇₁	#472	1	1	yes	characteristic
70	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₇₁	#508	1	1	yes	characteristic
72	Dic₇₁	—	2	2	no	—
73	Dic₇₁	—	4	1	no	—
74	Dic₇₁	—	16	2	no	—
75	#12440	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	#1668	1	1	yes	characteristic
77	C₂×C₇₁	#2216	1	1	yes	characteristic
78	C₂×C₇₁	#2222	1	1	yes	characteristic
79	#2247	—	71	1	no	—
80	Q₈×C₁₆	—	71	1	no	—
81	#1670	—	71	2	no	—
82	#1371	D₇₁	1	1	yes	characteristic
83	#1362	—	71	2	no	—
84	C₇₁	#12440	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₃₂	—	142	1	no	—
86	C₂×Q₃₂	—	142	1	no	—
87	#494	—	71	2	no	—
88	#475	—	142	2	no	—
89	C₈×Q₈	—	71	1	no	—
90	C₄⋊C₁₆	—	71	1	no	—
91	C₄⋊C₁₆	—	142	1	no	—
92	C₄×C₁₆	—	142	1	no	—
93	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
94	#335	C₂×D₇₁	1	2	yes	—
95	#332	—	142	2	no	—
96	C₂×Q₁₆	—	142	2	no	—
97	C₂×Q₁₆	—	142	2	no	—
98	C₄⋊Q₈	—	142	2	no	—
99	C₄×Q₈	—	71	1	no	—
100	Q₃₂	—	284	2	no	—
101	Q₃₂	—	284	2	no	—
102	C₂×C₁₆	—	142	1	no	—
103	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
104	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
105	#106	—	284	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	71	1	no	—
108	C₄⋊C₈	—	142	1	no	—
109	#103	—	284	2	no	—
110	C₄×C₈	—	142	1	no	—
111	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	C₂×Q₈	—	284	2	no	—
113	C₂×Q₈	—	71	1	no	—
114	Q₁₆	—	284	2	no	—
115	Q₁₆	—	284	2	no	—
116	Q₁₆	—	568	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	142	1	no	—
119	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
120	C₂×C₈	#11765866	1	1	yes	characteristic
121	C₄⋊C₄	—	284	2	no	—
122	C₄⋊C₄	—	284	2	no	—
123	C₄⋊C₄	—	71	1	no	—
124	C₄⋊C₄	—	142	1	no	—
125	C₄²	—	142	1	no	—
126	C₄²	D₄×D₇₁	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	284	1	no	—
129	C₁₆	#11765854	1	2	yes	—
130	Q₈	—	568	2	no	—
131	Q₈	—	568	2	no	—
132	Q₈	—	71	2	no	—
133	Q₈	—	142	1	no	—
134	C₂×C₄	—	568	2	no	—
135	C₂×C₄	—	142	1	no	—
136	C₂×C₄	—	142	1	no	—
137	C₂×C₄	#14556103	1	1	yes	characteristic
138	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
139	C₂×C₄	#14556151	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	284	1	no	—
143	C₈	C₂×#11765854	1	1	yes	characteristic
144	C₈	#14556156	1	1	yes	characteristic
145	C₂²	#17986236	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	284	1	no	—
148	C₄	—	1136	2	no	—
149	C₄	—	142	2	no	—
150	C₄	#17986200	1	1	yes	characteristic
151	C₄	#17986270	1	1	yes	characteristic
152	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
153	C₄	#17986428	1	1	yes	characteristic
154	C₂	#22587701	1	1	yes	characteristic
155	C₂	#22590432	1	1	yes	characteristic
156	C₂	#22590466	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590593

C₂

yes

maximal, characteristic

C₁₆×Dic₁₄₂

C₂

yes

maximal, characteristic

#22587711

C₂

yes

maximal

#22586388

C₂

yes

maximal

C₇₁×#1371

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

—

#17986379

C₂²

yes

—

#17986285

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

C₁₆×Dic₇₁

—

#17985744

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

C₄×Dic₁₄₂

D₄

yes

characteristic

Dic₅₆₈

—

Dic₅₆₈

—

C₂×C₇₁⋊C₁₆

—

#14556060

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

C₈×Dic₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₈

yes

characteristic

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

Q₃₂

yes

complemented

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

Q₈×C₁₆

—

#1670

—

#1371

D₇₁

yes

characteristic

#1362

—

C₇₁

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

142

—

C₂×Q₃₂

—

142

—

#494

—

#475

—

142

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₄×C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

C₂×D₇₁

yes

—

#332

—

142

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

142

—

C₄×Q₈

—

100

Q₃₂

—

284

—

101

Q₃₂

—

284

—

102

C₂×C₁₆

—

142

—

103

C₂×C₁₆

D₂₈₄

yes

characteristic

104

C₂×C₁₆

D₂₈₄

yes

characteristic

105

#106

—

284

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

142

—

109

#103

—

284

—

110

C₄×C₈

—

142

—

111

C₄×C₈

C₂²×D₇₁

yes

characteristic

112

C₂×Q₈

—

284

—

113

C₂×Q₈

—

114

Q₁₆

—

284

—

115

Q₁₆

—

284

—

116

Q₁₆

—

568

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

C₂×D₂₈₄

yes

characteristic

120

C₂×C₈

#11765866

yes

characteristic

121

C₄⋊C₄

—

284

—

122

C₄⋊C₄

—

284

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

142

—

125

C₄²

—

142

—

126

C₄²

D₄×D₇₁

yes

characteristic

127

C₁₆

—

128

C₁₆

—

284

—

129

C₁₆

#11765854

yes

—

130

Q₈

—

568

—

131

Q₈

—

568

—

132

Q₈

—

133

Q₈

—

142

—

134

C₂×C₄

—

568

—

135

C₂×C₄

—

142

—

136

C₂×C₄

—

142

—

137

C₂×C₄

#14556103

yes

characteristic

138

C₂×C₄

D₈×D₇₁

yes

characteristic

139

C₂×C₄

#14556151

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

284

—

143

C₈

C₂×#11765854

yes

characteristic

144

C₈

#14556156

yes

characteristic

145

C₂²

#17986236

yes

characteristic

146

C₄

—

147

C₄

—

284

—

148

C₄

—

1136

—

149

C₄

—

142

—

150

C₄

#17986200

yes

characteristic

151

C₄

#17986270

yes

characteristic

152

C₄

Q₃₂×D₇₁

yes

characteristic

153

C₄

#17986428

yes

characteristic

154

C₂

#22587701

yes

characteristic

155

C₂

#22590432

yes

characteristic

156

C₂

#22590466

yes

characteristic

	subgroup	G/H
center	C₂²	#17986236
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1371	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	#1371	D₇₁
hypocenter	C₇₁	#12440
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#17986236

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1371

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₁₄₂	Q₃₂	semidirect
84. C₇₁	#12440	semidirect

H ≅ G/N

type

60. Dic₁₄₂

Q₃₂

semidirect

84. C₇₁

#12440

semidirect

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

subgroup	index
#22590593	2
C₁₆×Dic₁₄₂	2
#22587711	2
#22586388	2
C₇₁×#1371	2
#12440	71

subgroup

index

C₁₆×Dic₁₄₂

C₇₁×#1371

Subgroups — #31911715

Conjugacy classes of subgroups

Complements

Maximal subgroups