Group #21465574 — subgroups

Subgroups — C₃₁×#8815 #21465574

Order 7936.

422 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 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	C₃₁×#1844	C₂	1	2	yes	maximal
2	C₃₁×#1834	C₂	1	4	yes	maximal
3	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#106	—	2	4	no	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#384	C₂²	1	4	yes	—
8	C₃₁×#340	—	2	8	no	—
9	C₈²×C₃₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₃₁	D₄	1	4	yes	—
11	C₂²×C₈×C₃₁	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
13	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
14	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
15	C₃₁×#106	—	4	8	no	—
16	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₃₁	Q₈	1	4	yes	—
19	C₄×C₈×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₁	—	4	4	no	—
21	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
23	C₂×C₈×C₃₁	—	2	8	no	—
24	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₃₁	SD₁₆	1	8	yes	—
27	C₄⋊C₄×C₃₁	—	4	8	no	—
28	C₄⋊C₄×C₃₁	—	4	16	no	—
29	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
32	#8815	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	#127	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	8	4	no	—
35	C₂×C₄×C₃₁	—	8	8	no	—
36	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
41	C₂×C₄×C₃₁	#135	1	8	yes	—
42	C₈×C₃₁	—	2	8	no	—
43	C₈×C₃₁	#106	1	8	yes	—
44	#1844	C₂×C₃₁	1	2	yes	—
45	#1834	C₂×C₃₁	1	4	yes	—
46	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
47	C₂²×C₃₁	#384	1	1	yes	characteristic
48	C₂²×C₃₁	#492	1	2	yes	—
49	C₂²×C₃₁	#495	1	4	yes	—
50	C₄×C₃₁	—	16	4	no	—
51	C₄×C₃₁	C₂×#106	1	2	yes	—
52	C₄×C₃₁	#427	1	4	yes	—
53	C₄×C₃₁	#498	1	2	yes	—
54	C₄×C₃₁	#499	1	4	yes	—
55	C₂×#106	—	2	4	no	—
56	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
57	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
58	#384	C₂²×C₃₁	1	4	yes	—
59	#340	—	2	8	no	—
60	C₈²	C₄×C₃₁	1	4	yes	complemented
61	C₂×C₃₁	#1708	1	4	yes	—
62	C₂×C₃₁	#1844	1	1	yes	characteristic
63	C₂×C₃₁	#1846	1	2	yes	—
64	C₂²×C₈	D₄×C₃₁	1	4	yes	—
65	C₂²×C₈	D₄×C₃₁	1	2	yes	—
66	C₂×C₄⋊C₄	—	2	8	no	—
67	C₂×C₄⋊C₄	—	2	4	no	—
68	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
69	#106	—	4	8	no	—
70	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
72	C₄×C₈	Q₈×C₃₁	1	4	yes	—
73	C₄×C₈	Q₈×C₃₁	1	2	yes	—
74	C₃₁	#8815	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	—	4	4	no	—
76	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
80	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
81	C₂×C₈	SD₁₆×C₃₁	1	8	yes	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	—	4	16	no	—
84	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
85	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
86	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
87	C₂³	C₃₁×#127	1	1	yes	characteristic
88	C₂×C₄	—	8	4	no	—
89	C₂×C₄	—	8	8	no	—
90	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
92	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
94	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
95	C₂×C₄	C₃₁×#135	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	C₃₁×#106	1	8	yes	—
98	C₂²	C₃₁×#384	1	1	yes	characteristic
99	C₂²	C₃₁×#492	1	2	yes	—
100	C₂²	C₃₁×#495	1	4	yes	—
101	C₄	—	16	4	no	—
102	C₄	C₂×C₃₁×#106	1	2	yes	—
103	C₄	C₃₁×#427	1	4	yes	—
104	C₄	C₃₁×#498	1	2	yes	—
105	C₄	C₃₁×#499	1	4	yes	—
106	C₂	C₃₁×#1708	1	4	yes	—
107	C₂	C₃₁×#1844	1	1	yes	characteristic
108	C₂	C₃₁×#1846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1844

C₂

yes

maximal

C₃₁×#1834

C₂

yes

maximal

C₂×C₈²×C₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×#106

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#384

C₂²

yes

—

C₃₁×#340

—

C₈²×C₃₁

C₄

yes

complemented

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#106

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×Q₈

yes

characteristic

C₄²×C₃₁

C₂×Q₈

yes

—

#8815

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#127

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₃₁

C₄⋊Q₈

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

#135

yes

—

C₈×C₃₁

—

C₈×C₃₁

#106

yes

—

#1844

C₂×C₃₁

yes

—

#1834

C₂×C₃₁

yes

—

C₂×C₈²

C₂×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#495

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×#106

yes

—

C₄×C₃₁

#427

yes

—

C₄×C₃₁

#498

yes

—

C₄×C₃₁

#499

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#384

C₂²×C₃₁

yes

—

#340

—

C₈²

C₄×C₃₁

yes

complemented

C₂×C₃₁

#1708

yes

—

C₂×C₃₁

#1844

yes

characteristic

C₂×C₃₁

#1846

yes

—

C₂²×C₈

D₄×C₃₁

yes

—

C₂²×C₈

D₄×C₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

#106

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

C₃₁

#8815

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

SD₁₆×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₃₁

yes

characteristic

C₄²

C₂×Q₈×C₃₁

yes

characteristic

C₄²

C₂×Q₈×C₃₁

yes

—

C₂³

C₃₁×#127

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

—

C₂×C₄

C₃₁×C₄⋊Q₈

yes

characteristic

C₂×C₄

C₃₁×C₄⋊Q₈

yes

—

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

C₂×C₄

C₃₁×#135

yes

—

C₈

—

C₈

C₃₁×#106

yes

—

C₂²

C₃₁×#384

yes

characteristic

C₂²

C₃₁×#492

yes

—

100

C₂²

C₃₁×#495

yes

—

101

C₄

—

102

C₄

C₂×C₃₁×#106

yes

—

103

C₄

C₃₁×#427

yes

—

104

C₄

C₃₁×#498

yes

—

105

C₄

C₃₁×#499

yes

—

106

C₂

C₃₁×#1708

yes

—

107

C₂

C₃₁×#1844

yes

characteristic

108

C₂

C₃₁×#1846

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

#127

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
9. C₈²×C₃₁	C₄	semidirect
32. #8815	C₃₁	direct (N × H)
60. C₈²	C₄×C₃₁	semidirect
74. C₃₁	#8815	direct (N × H)

H ≅ G/N

type

9. C₈²×C₃₁

C₄

semidirect

32. #8815

C₃₁

direct (N × H)

60. C₈²

C₄×C₃₁

semidirect

74. C₃₁

#8815

direct (N × H)

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

subgroup	index
C₃₁×#1844	2
C₃₁×#1834	2
C₂×C₈²×C₃₁	2
#8815	31

subgroup

index

C₃₁×#1844

C₃₁×#1834

C₂×C₈²×C₃₁

#8815

Subgroups — C₃₁×#8815 #21465574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8815 #21465574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8815 #21465574