Group #21489995 — subgroups

Subgroups — C₃₁×#33236 #21489995

Order 7936.

910 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 318 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	C₂×C₃₁×#487	C₂	1	4	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1960	C₂	1	8	yes	maximal
5	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃₁×#487	—	2	16	no	—
10	C₂×C₃₁×#103	C₂²	1	16	yes	—
11	C₂×C₃₁×#97	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#391	—	2	4	no	—
14	C₃₁×#384	—	2	4	no	—
15	C₃₁×#382	C₂²	1	4	yes	—
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₃₁	—	2	2	no	—
18	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
19	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
20	C₃₁×C₄⋊Q₈	—	2	8	no	—
21	C₃₁×C₄⋊Q₈	—	2	8	no	—
22	C₃₁×C₄⋊Q₈	—	2	16	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
27	C₂×C₄²×C₃₁	D₄	1	4	yes	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
30	C₃₁×#103	—	2	32	no	—
31	C₃₁×#97	—	2	8	no	—
32	C₄×C₈×C₃₁	D₄	1	8	yes	—
33	C₃₁×#95	—	2	8	no	—
34	C₂×Q₈×C₃₁	—	2	8	no	—
35	C₂×Q₈×C₃₁	—	2	16	no	—
36	C₂×Q₈×C₃₁	—	4	4	no	—
37	C₂×Q₈×C₃₁	—	4	8	no	—
38	C₂²×C₄×C₃₁	—	2	4	no	—
39	C₂²×C₄×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	8	no	—
44	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
45	C₄⋊C₄×C₃₁	—	2	8	no	—
46	C₄⋊C₄×C₃₁	—	4	16	no	—
47	C₄⋊C₄×C₃₁	—	4	4	no	—
48	C₄⋊C₄×C₃₁	—	4	8	no	—
49	C₄²×C₃₁	—	2	8	no	—
50	C₄²×C₃₁	C₂×D₄	1	4	yes	—
51	C₄²×C₃₁	#41	1	8	yes	—
52	#33236	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
54	Q₈×C₃₁	—	4	16	no	—
55	Q₈×C₃₁	—	8	8	no	—
56	C₂×C₄×C₃₁	—	2	8	no	—
57	C₂×C₄×C₃₁	—	4	8	no	—
58	C₂×C₄×C₃₁	—	4	4	no	—
59	C₂×C₄×C₃₁	—	4	4	no	—
60	C₂×C₄×C₃₁	—	4	8	no	—
61	C₂×C₄×C₃₁	#124	1	8	yes	—
62	C₂×C₄×C₃₁	#127	1	4	yes	—
63	C₂×C₄×C₃₁	#137	1	8	yes	—
64	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
66	C₈×C₃₁	—	2	8	no	—
67	C₂×#487	C₂×C₃₁	1	4	yes	—
68	#2868	C₂×C₃₁	1	1	yes	characteristic
69	#2862	C₂×C₃₁	1	1	yes	characteristic
70	#1960	C₂×C₃₁	1	8	yes	—
71	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
72	C₂²×C₃₁	C₂×#124	1	2	yes	—
73	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
74	C₂²×C₃₁	C₂×#137	1	4	yes	—
75	C₄×C₃₁	—	2	8	no	—
76	C₄×C₃₁	—	8	4	no	—
77	C₄×C₃₁	—	8	4	no	—
78	C₄×C₃₁	#497	1	8	yes	—
79	C₄×C₃₁	#532	1	4	yes	—
80	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
82	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
83	#487	—	2	16	no	—
84	C₂×#103	C₂²×C₃₁	1	16	yes	—
85	C₂×#97	C₂²×C₃₁	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
87	#391	—	2	4	no	—
88	#384	—	2	4	no	—
89	#382	C₂²×C₃₁	1	4	yes	—
90	C₄³	C₂²×C₃₁	1	1	yes	characteristic
91	C₂×C₃₁	#2858	1	1	yes	characteristic
92	C₂×C₃₁	#3139	1	4	yes	—
93	C₂×C₃₁	C₂×#497	1	2	yes	—
94	C₂²×Q₈	—	2	2	no	—
95	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
96	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
97	C₄⋊Q₈	—	2	8	no	—
98	C₄⋊Q₈	—	2	8	no	—
99	C₄⋊Q₈	—	2	16	no	—
100	C₂×C₄⋊C₄	—	2	8	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
104	C₂×C₄²	D₄×C₃₁	1	4	yes	—
105	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
106	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
107	#103	—	2	32	no	—
108	#97	—	2	8	no	—
109	C₄×C₈	D₄×C₃₁	1	8	yes	—
110	#95	—	2	8	no	—
111	C₃₁	#33236	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×Q₈	—	2	8	no	—
113	C₂×Q₈	—	2	16	no	—
114	C₂×Q₈	—	4	4	no	—
115	C₂×Q₈	—	4	8	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
119	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
120	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
121	C₂×C₈	—	2	8	no	—
122	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
123	C₄⋊C₄	—	2	8	no	—
124	C₄⋊C₄	—	4	16	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	4	8	no	—
127	C₄²	—	2	8	no	—
128	C₄²	C₂×D₄×C₃₁	1	4	yes	—
129	C₄²	#41×C₃₁	1	8	yes	—
130	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
131	Q₈	—	4	16	no	—
132	Q₈	—	8	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	4	8	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	4	8	no	—
138	C₂×C₄	C₃₁×#124	1	8	yes	—
139	C₂×C₄	C₃₁×#127	1	4	yes	—
140	C₂×C₄	C₃₁×#137	1	8	yes	—
141	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
142	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
143	C₈	—	2	8	no	—
144	C₂²	C₂×C₃₁×#124	1	2	yes	—
145	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
146	C₂²	C₂×C₃₁×#137	1	4	yes	—
147	C₄	—	2	8	no	—
148	C₄	—	8	4	no	—
149	C₄	—	8	4	no	—
150	C₄	C₃₁×#497	1	8	yes	—
151	C₄	C₃₁×#532	1	4	yes	—
152	C₂	C₃₁×#2858	1	1	yes	characteristic
153	C₂	C₃₁×#3139	1	4	yes	—
154	C₂	C₂×C₃₁×#497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#487

C₂

yes

maximal

C₃₁×#2868

C₂

yes

maximal, characteristic

C₃₁×#2862

C₂

yes

maximal, characteristic

C₃₁×#1960

C₂

yes

maximal

C₄×C₃₁×#97

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#487

—

C₂×C₃₁×#103

C₂²

yes

—

C₂×C₃₁×#97

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×#391

—

C₃₁×#384

—

C₃₁×#382

C₂²

yes

—

C₄³×C₃₁

C₂²

yes

characteristic

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

C₂³

yes

—

C₂²×C₈×C₃₁

C₂³

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂³

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

C₂³

yes

—

C₃₁×#103

—

C₃₁×#97

—

C₄×C₈×C₃₁

D₄

yes

—

C₃₁×#95

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

—

#33236

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#124

yes

—

C₂×C₄×C₃₁

#127

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₂×C₄×C₃₁

C₂²×D₄

yes

—

C₂×C₄×C₃₁

C₂×#41

yes

—

C₈×C₃₁

—

C₂×#487

C₂×C₃₁

yes

—

#2868

C₂×C₃₁

yes

characteristic

#2862

C₂×C₃₁

yes

characteristic

#1960

C₂×C₃₁

yes

—

C₄×#97

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂×#124

yes

—

C₂²×C₃₁

C₂×#127

yes

characteristic

C₂²×C₃₁

C₂×#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#497

yes

—

C₄×C₃₁

#532

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

#487

—

C₂×#103

C₂²×C₃₁

yes

—

C₂×#97

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

#391

—

#384

—

#382

C₂²×C₃₁

yes

—

C₄³

C₂²×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

C₂×#497

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂³×C₃₁

yes

—

C₂²×C₈

C₂³×C₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

C₂³×C₃₁

yes

—

104

C₂×C₄²

D₄×C₃₁

yes

—

105

C₂×C₄²

C₂³×C₃₁

yes

characteristic

106

C₂×C₄²

C₂³×C₃₁

yes

—

107

#103

—

108

#97

—

109

C₄×C₈

D₄×C₃₁

yes

—

110

#95

—

111

C₃₁

#33236

yes

characteristic, direct product factor, Sylow

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂×D₄×C₃₁

yes

—

119

C₂²×C₄

C₂×D₄×C₃₁

yes

—

120

C₂²×C₄

C₂⁴×C₃₁

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

C₂×D₄×C₃₁

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

C₂×D₄×C₃₁

yes

—

129

C₄²

#41×C₃₁

yes

—

130

C₂³

C₂²×D₄×C₃₁

yes

characteristic

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₃₁×#124

yes

—

139

C₂×C₄

C₃₁×#127

yes

—

140

C₂×C₄

C₃₁×#137

yes

—

141

C₂×C₄

C₂²×D₄×C₃₁

yes

—

142

C₂×C₄

C₂×#41×C₃₁

yes

—

143

C₈

—

144

C₂²

C₂×C₃₁×#124

yes

—

145

C₂²

C₂×C₃₁×#127

yes

characteristic

146

C₂²

C₂×C₃₁×#137

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₃₁×#497

yes

—

151

C₄

C₃₁×#532

yes

—

152

C₂

C₃₁×#2858

yes

characteristic

153

C₂

C₃₁×#3139

yes

—

154

C₂

C₂×C₃₁×#497

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₃₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

N	H ≅ G/N	type
52. #33236	C₃₁	direct (N × H)
111. C₃₁	#33236	direct (N × H)

H ≅ G/N

type

52. #33236

C₃₁

direct (N × H)

111. C₃₁

#33236

direct (N × H)

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

subgroup	index
C₂×C₃₁×#487	2
C₃₁×#2868	2
C₃₁×#2862	2
C₃₁×#1960	2
C₄×C₃₁×#97	2
#33236	31

subgroup

index

C₂×C₃₁×#487

C₃₁×#2868

C₃₁×#2862

C₃₁×#1960

C₄×C₃₁×#97

#33236

Subgroups — C₃₁×#33236 #21489995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#33236 #21489995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33236 #21489995