Group #31911779 — subgroups

Subgroups — #31911779

Order 18176.

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

normal subgroups only

This group has 212 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	C₁₆×D₂₈₄	C₂	1	1	yes	maximal, characteristic
2	#22587736	C₂	1	2	yes	maximal
3	#22586412	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586398	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₁	—	2	1	no	—
7	#17986286	—	2	2	no	—
8	C₈×D₂₈₄	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
10	(C₂×D₇₁)⋊C₁₆	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#17985748	—	2	2	no	—
13	#500×C₇₁	C₂²	1	2	yes	—
14	#366×C₇₁	—	2	1	no	—
15	#366×C₇₁	—	2	1	no	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	—	2	1	no	—
18	C₄×D₂₈₄	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₁₆×D₇₁	—	4	2	no	—
21	#14556059	—	4	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
24	#14556048	—	4	2	no	—
25	(C₂×D₇₁)⋊C₈	—	2	1	no	—
26	C₄⋊Q₈×C₇₁	—	2	2	no	—
27	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
29	#107×C₇₁	—	2	2	no	—
30	#107×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×D₂₈₄	D₈	1	1	yes	characteristic
33	C₂×C₄×D₇₁	—	2	1	no	—
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₈×D₇₁	—	4	1	no	—
37	C₈×D₇₁	—	4	1	no	—
38	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
39	(C₂×D₇₁)⋊C₄	—	2	1	no	—
40	C₇₁⋊C₁₆	—	4	1	no	—
41	C₂×Q₈×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₇₁	—	4	2	no	—
45	C₄⋊C₄×C₇₁	—	4	2	no	—
46	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	SD₁₆	1	2	yes	—
49	C₂²×D₇₁	—	2	1	no	—
50	Q₈×C₇₁	—	16	2	no	—
51	C₂×C₄×C₇₁	—	8	2	no	—
52	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	1	no	—
56	D₂₈₄	—	2	1	no	—
57	D₂₈₄	Q₃₂	1	2	yes	—
58	C₄×D₇₁	—	4	1	no	—
59	C₄×D₇₁	—	4	1	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₇₁	#137	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	C₂×D₇₁	—	2	2	no	—
67	C₂×D₇₁	—	4	1	no	—
68	C₂×D₇₁	—	4	1	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	16	2	no	—
71	C₄×C₇₁	#461	1	1	yes	characteristic
72	C₄×C₇₁	#473	1	1	yes	characteristic
73	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₇₁	#508	1	1	yes	characteristic
75	Dic₇₁	—	4	1	no	—
76	#12452	—	71	1	no	maximal, Sylow
77	C₂×C₇₁	#1673	1	1	yes	characteristic
78	C₂×C₇₁	#2216	1	1	yes	characteristic
79	C₂×C₇₁	#2221	1	1	yes	characteristic
80	D₇₁	—	4	2	no	—
81	#2252	D₇₁	1	1	yes	characteristic, complemented
82	D₄×C₁₆	—	71	1	no	—
83	#1675	—	71	2	no	—
84	#1371	—	71	1	no	—
85	#1365	—	71	2	no	—
86	C₇₁	#12452	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	142	1	no	—
88	#500	C₂×D₇₁	1	2	yes	—
89	#476	—	142	2	no	—
90	C₈×D₄	—	71	1	no	—
91	#366	—	2	1	no	—
92	#366	—	2	1	no	—
93	C₄⋊C₁₆	—	71	1	no	—
94	C₂²⋊C₁₆	—	142	1	no	—
95	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
96	#335	—	71	2	no	—
97	#333	—	142	2	no	—
98	C₂²×C₈	—	142	1	no	—
99	C₄⋊Q₈	—	2	2	no	—
100	C₄×D₄	—	71	1	no	—
101	C₂×C₁₆	—	284	2	no	—
102	C₂×C₁₆	—	142	1	no	—
103	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
105	#107	—	2	2	no	—
106	#107	—	2	2	no	—
107	#106	—	284	2	no	—
108	C₄⋊C₈	—	71	1	no	—
109	C₄⋊C₈	—	142	2	no	—
110	#102	—	284	2	no	—
111	C₂²⋊C₈	—	142	1	no	—
112	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₂×Q₈	—	4	2	no	—
114	C₂×D₄	—	71	1	no	—
115	C₂²×C₄	—	142	1	no	—
116	C₂×C₈	—	284	1	no	—
117	C₂×C₈	—	284	1	no	—
118	C₂×C₈	—	284	2	no	—
119	C₂×C₈	—	142	1	no	—
120	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
121	C₂×C₈	#11765864	1	1	yes	characteristic
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	71	1	no	—
125	C₂²⋊C₄	—	142	1	no	—
126	C₄²	D₄×D₇₁	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	284	1	no	—
129	C₁₆	#11765852	1	2	yes	—
130	C₂³	—	142	1	no	—
131	Q₈	—	16	2	no	—
132	D₄	—	71	2	no	—
133	D₄	—	142	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	284	1	no	—
136	C₂×C₄	—	284	1	no	—
137	C₂×C₄	—	142	1	no	—
138	C₂×C₄	#14556099	1	1	yes	characteristic
139	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
140	C₂×C₄	#14556147	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	284	1	no	—
143	C₈	—	568	2	no	—
144	C₈	C₂×#11765852	1	1	yes	characteristic
145	C₈	#14556159	1	1	yes	characteristic
146	C₂²	—	284	1	no	—
147	C₂²	—	284	1	no	—
148	C₂²	—	142	2	no	—
149	C₂²	#17986232	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	16	2	no	—
152	C₄	—	284	1	no	—
153	C₄	#17986201	1	1	yes	characteristic
154	C₄	#17986271	1	1	yes	characteristic
155	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
156	C₄	#17986432	1	1	yes	characteristic
157	C₂	—	284	2	no	—
158	C₂	#22587726	1	1	yes	characteristic
159	C₂	#22590428	1	1	yes	characteristic
160	C₂	#22590457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×D₂₈₄

C₂

yes

maximal, characteristic

#22587736

C₂

yes

maximal

#22586412

C₂

yes

maximal, characteristic, complemented

#22586398

C₂

yes

maximal, complemented

C₇₁×#2252

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₁

—

#17986286

—

C₈×D₂₈₄

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₁₆

—

#17985752

C₂²

yes

—

#17985748

—

#500×C₇₁

C₂²

yes

—

#366×C₇₁

—

#366×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

—

C₄×D₂₈₄

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₁₆×D₇₁

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

#14556048

—

(C₂×D₇₁)⋊C₈

—

C₄⋊Q₈×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×D₂₈₄

D₈

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

D₈

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂×Q₈×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

—

C₂²×D₇₁

—

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

—

D₂₈₄

—

D₂₈₄

Q₃₂

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

#137

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#2252

D₇₁

yes

characteristic, complemented

D₄×C₁₆

—

#1675

—

#1371

—

#1365

—

C₇₁

#12452

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

142

—

#500

C₂×D₇₁

yes

—

#476

—

142

—

C₈×D₄

—

#366

—

#366

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

#333

—

142

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

100

C₄×D₄

—

101

C₂×C₁₆

—

284

—

102

C₂×C₁₆

—

142

—

103

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

105

#107

—

106

#107

—

107

#106

—

284

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

142

—

110

#102

—

284

—

111

C₂²⋊C₈

—

142

—

112

C₄×C₈

C₂²×D₇₁

yes

characteristic

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂²×C₄

—

142

—

116

C₂×C₈

—

284

—

117

C₂×C₈

—

284

—

118

C₂×C₈

—

284

—

119

C₂×C₈

—

142

—

120

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

121

C₂×C₈

#11765864

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

142

—

126

C₄²

D₄×D₇₁

yes

characteristic

127

C₁₆

—

128

C₁₆

—

284

—

129

C₁₆

#11765852

yes

—

130

C₂³

—

142

—

131

Q₈

—

132

D₄

—

133

D₄

—

142

—

134

C₂×C₄

—

135

C₂×C₄

—

284

—

136

C₂×C₄

—

284

—

137

C₂×C₄

—

142

—

138

C₂×C₄

#14556099

yes

characteristic

139

C₂×C₄

D₈×D₇₁

yes

characteristic

140

C₂×C₄

#14556147

yes

characteristic

141

C₈

—

142

C₈

—

284

—

143

C₈

—

568

—

144

C₈

C₂×#11765852

yes

characteristic

145

C₈

#14556159

yes

characteristic

146

C₂²

—

284

—

147

C₂²

—

284

—

148

C₂²

—

142

—

149

C₂²

#17986232

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

284

—

153

C₄

#17986201

yes

characteristic

154

C₄

#17986271

yes

characteristic

155

C₄

Q₃₂×D₇₁

yes

characteristic

156

C₄

#17986432

yes

characteristic

157

C₂

—

284

—

158

C₂

#22587726

yes

characteristic

159

C₂

#22590428

yes

characteristic

160

C₂

#22590457

yes

characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2252	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂×D₂₈₄	D₈
hypercenter	#2252	D₇₁
hypocenter	C₇₁	#12452
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#17986232

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2252

C₂

socle

C₂²×C₇₁

#466

prime radical

C₂×D₂₈₄

D₈

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
3. #22586412	C₂	semidirect
4. #22586398	C₂	semidirect
5. C₇₁×#2252	C₂	semidirect
81. #2252	D₇₁	semidirect
86. C₇₁	#12452	semidirect

H ≅ G/N

type

3. #22586412

C₂

semidirect

4. #22586398

C₂

semidirect

5. C₇₁×#2252

C₂

semidirect

81. #2252

D₇₁

semidirect

86. C₇₁

#12452

semidirect

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

subgroup	index
C₁₆×D₂₈₄	2
#22587736	2
#22586412	2
#22586398	2
C₇₁×#2252	2
#12452	71

subgroup

index

C₁₆×D₂₈₄

C₇₁×#2252

Subgroups — #31911779

Conjugacy classes of subgroups

Complements

Maximal subgroups