Group #21467075 — subgroups

Subgroups — C₃₁×#10316 #21467075

Order 7936.

586 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 186 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×#498	C₂	1	3	yes	maximal
2	C₃₁×#1956	C₂	1	3	yes	maximal
3	C₃₁×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
5	C₃₁×#498	—	2	12	no	—
6	C₂×C₃₁×#106	—	2	6	no	—
7	C₂×C₃₁×#103	—	2	6	no	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
9	C₃₁×#384	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₃₁	—	2	3	no	—
11	C₂²×C₈×C₃₁	D₄	1	6	yes	—
12	C₃₁×C₄⋊Q₈	—	2	6	no	—
13	C₃₁×C₄⋊Q₈	—	2	3	no	—
14	C₃₁×C₄⋊Q₈	D₄	1	6	yes	—
15	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
16	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
17	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
18	C₃₁×#106	—	2	24	no	—
19	C₃₁×#103	—	4	12	no	—
20	C₄×C₈×C₃₁	—	2	6	no	—
21	C₂×Q₈×C₃₁	—	2	6	no	—
22	C₂×Q₈×C₃₁	—	4	3	no	—
23	C₂×Q₈×C₃₁	—	4	6	no	—
24	C₂×Q₈×C₃₁	—	4	6	no	—
25	C₂²×C₄×C₃₁	—	4	3	no	—
26	C₂²×C₄×C₃₁	—	4	1	no	—
27	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₃₁	—	2	6	no	—
29	C₂×C₈×C₃₁	—	2	6	no	—
30	C₂×C₈×C₃₁	SD₁₆	1	12	yes	—
31	C₄⋊C₄×C₃₁	—	2	12	no	—
32	C₄⋊C₄×C₃₁	—	4	6	no	—
33	C₄⋊C₄×C₃₁	—	4	6	no	—
34	C₄²×C₃₁	C₂×D₄	1	3	yes	—
35	C₄²×C₃₁	#41	1	1	yes	characteristic
36	#10316	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₃₁	#127	1	1	yes	characteristic
38	Q₈×C₃₁	—	4	12	no	—
39	Q₈×C₃₁	—	8	6	no	—
40	C₂×C₄×C₃₁	—	4	6	no	—
41	C₂×C₄×C₃₁	—	8	3	no	—
42	C₂×C₄×C₃₁	—	8	3	no	—
43	C₂×C₄×C₃₁	—	8	3	no	—
44	C₂×C₄×C₃₁	#120	1	3	yes	—
45	C₂×C₄×C₃₁	#121	1	3	yes	—
46	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
47	C₂×C₄×C₃₁	#137	1	6	yes	—
48	C₈×C₃₁	—	2	12	no	—
49	C₂×#498	C₂×C₃₁	1	3	yes	—
50	#1956	C₂×C₃₁	1	3	yes	—
51	#1273	C₂×C₃₁	1	1	yes	characteristic
52	C₂²×C₃₁	#392	1	1	yes	characteristic
53	C₂²×C₃₁	#492	1	3	yes	—
54	C₂²×C₃₁	#497	1	3	yes	—
55	C₄×C₃₁	—	8	6	no	—
56	C₄×C₃₁	—	16	1	no	—
57	C₄×C₃₁	#448	1	6	yes	—
58	C₄×C₃₁	#461	1	6	yes	—
59	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
60	#498	—	2	12	no	—
61	C₂×#106	—	2	6	no	—
62	C₂×#103	—	2	6	no	—
63	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
64	#384	C₂²×C₃₁	1	1	yes	characteristic
65	C₂×C₃₁	#1705	1	1	yes	characteristic
66	C₂×C₃₁	#1716	1	3	yes	—
67	C₂×C₃₁	#2013	1	3	yes	—
68	C₂²×Q₈	—	2	3	no	—
69	C₂²×C₈	D₄×C₃₁	1	6	yes	—
70	C₄⋊Q₈	—	2	6	no	—
71	C₄⋊Q₈	—	2	3	no	—
72	C₄⋊Q₈	D₄×C₃₁	1	6	yes	—
73	C₂×C₄⋊C₄	—	2	6	no	—
74	C₂×C₄⋊C₄	—	2	3	no	—
75	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
76	#106	—	2	24	no	—
77	#103	—	4	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₃₁	#10316	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×Q₈	—	2	6	no	—
81	C₂×Q₈	—	4	3	no	—
82	C₂×Q₈	—	4	6	no	—
83	C₂×Q₈	—	4	6	no	—
84	C₂²×C₄	—	4	3	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	SD₁₆×C₃₁	1	12	yes	—
90	C₄⋊C₄	—	2	12	no	—
91	C₄⋊C₄	—	4	6	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄²	C₂×D₄×C₃₁	1	3	yes	—
94	C₄²	#41×C₃₁	1	1	yes	characteristic
95	C₂³	C₃₁×#127	1	1	yes	characteristic
96	Q₈	—	4	12	no	—
97	Q₈	—	8	6	no	—
98	C₂×C₄	—	4	6	no	—
99	C₂×C₄	—	8	3	no	—
100	C₂×C₄	—	8	3	no	—
101	C₂×C₄	—	8	3	no	—
102	C₂×C₄	C₃₁×#120	1	3	yes	—
103	C₂×C₄	C₃₁×#121	1	3	yes	—
104	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
105	C₂×C₄	C₃₁×#137	1	6	yes	—
106	C₈	—	2	12	no	—
107	C₂²	C₃₁×#392	1	1	yes	characteristic
108	C₂²	C₃₁×#492	1	3	yes	—
109	C₂²	C₃₁×#497	1	3	yes	—
110	C₄	—	8	6	no	—
111	C₄	—	16	1	no	—
112	C₄	C₃₁×#448	1	6	yes	—
113	C₄	C₃₁×#461	1	6	yes	—
114	C₂	C₃₁×#1705	1	1	yes	characteristic
115	C₂	C₃₁×#1716	1	3	yes	—
116	C₂	C₃₁×#2013	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#498

C₂

yes

maximal

C₃₁×#1956

C₂

yes

maximal

C₃₁×#1273

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#498

—

C₂×C₃₁×#106

—

C₂×C₃₁×#103

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×#384

C₂²

yes

characteristic

C₂²×Q₈×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#106

—

C₃₁×#103

—

C₄×C₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

characteristic

#10316

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#127

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#120

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

C₂×#498

C₂×C₃₁

yes

—

#1956

C₂×C₃₁

yes

—

#1273

C₂×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#497

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#448

yes

—

C₄×C₃₁

#461

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

#498

—

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

#384

C₂²×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#2013

yes

—

C₂²×Q₈

—

C₂²×C₈

D₄×C₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄×C₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

#106

—

#103

—

C₄×C₈

—

C₃₁

#10316

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₃₁

yes

—

C₄²

#41×C₃₁

yes

characteristic

C₂³

C₃₁×#127

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₃₁×#120

yes

—

103

C₂×C₄

C₃₁×#121

yes

—

104

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

105

C₂×C₄

C₃₁×#137

yes

—

106

C₈

—

107

C₂²

C₃₁×#392

yes

characteristic

108

C₂²

C₃₁×#492

yes

—

109

C₂²

C₃₁×#497

yes

—

110

C₄

—

111

C₄

—

112

C₄

C₃₁×#448

yes

—

113

C₄

C₃₁×#461

yes

—

114

C₂

C₃₁×#1705

yes

characteristic

115

C₂

C₃₁×#1716

yes

—

116

C₂

C₃₁×#2013

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
36. #10316	C₃₁	direct (N × H)
79. C₃₁	#10316	direct (N × H)

H ≅ G/N

type

36. #10316

C₃₁

direct (N × H)

79. C₃₁

#10316

direct (N × H)

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

subgroup	index
C₂×C₃₁×#498	2
C₃₁×#1956	2
C₃₁×#1273	2
#10316	31

subgroup

index

C₂×C₃₁×#498

C₃₁×#1956

C₃₁×#1273

#10316

Subgroups — C₃₁×#10316 #21467075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10316 #21467075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10316 #21467075