Group #22630051 — subgroups

Subgroups — C₃₇×#8798 #22630051

Order 9472.

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₃₇×#1568	C₂	1	2	yes	maximal
2	C₃₇×#1452	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1287	C₂	1	4	yes	maximal
4	C₃₇×#432	C₂²	1	4	yes	—
5	C₃₇×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
7	C₃₇×#335	—	2	2	no	—
8	C₃₇×#334	—	2	2	no	—
9	C₃₇×#322	C₄	1	4	yes	—
10	C₂²×C₈×C₃₇	Q₈	1	2	yes	—
11	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
12	C₄⋊C₈×C₃₇	—	2	4	no	—
13	C₄⋊C₈×C₃₇	—	2	2	no	—
14	C₂²⋊C₈×C₃₇	—	2	2	no	—
15	C₂²⋊C₈×C₃₇	—	2	4	no	—
16	#97×C₃₇	—	2	4	no	—
17	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
20	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	#41	1	2	yes	—
23	C₂×C₈×C₃₇	—	2	4	no	—
24	C₂×C₈×C₃₇	—	4	2	no	—
25	C₂×C₈×C₃₇	—	4	4	no	—
26	C₂×C₈×C₃₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₃₇	C₄⋊C₄	1	2	yes	—
28	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
30	C₄²×C₃₇	#41	1	2	yes	—
31	C₂³×C₃₇	#125	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	#117	1	2	yes	—
36	C₂×C₄×C₃₇	#124	1	2	yes	—
37	C₂×C₄×C₃₇	#125	1	1	yes	characteristic
38	C₈×C₃₇	—	2	2	no	—
39	C₈×C₃₇	—	2	2	no	—
40	C₈×C₃₇	—	4	4	no	—
41	C₈×C₃₇	—	8	4	no	—
42	#8798	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	—	2	1	no	—
45	C₂²×C₃₇	#382	1	1	yes	characteristic
46	C₄×C₃₇	—	2	2	no	—
47	C₄×C₃₇	—	2	2	no	—
48	C₄×C₃₇	—	2	1	no	—
49	C₄×C₃₇	#430	1	2	yes	—
50	#1568	C₂×C₃₇	1	2	yes	—
51	#1452	C₂×C₃₇	1	1	yes	characteristic
52	#1287	C₂×C₃₇	1	4	yes	—
53	C₂×C₃₇	—	4	1	no	—
54	C₂×C₃₇	#1833	1	2	yes	—
55	C₂×C₃₇	#1838	1	1	yes	characteristic
56	#432	C₂²×C₃₇	1	4	yes	—
57	#431	C₂²×C₃₇	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
59	#335	—	2	2	no	—
60	#334	—	2	2	no	—
61	#322	C₄×C₃₇	1	4	yes	—
62	C₃₇	#8798	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₈	Q₈×C₃₇	1	2	yes	—
64	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	2	4	no	—
69	#97	—	2	4	no	—
70	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
72	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
73	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
75	C₂²×C₄	#41×C₃₇	1	2	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₄⋊C₄×C₃₇	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₃₇	1	2	yes	—
81	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
82	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
83	C₄²	#41×C₃₇	1	2	yes	—
84	C₂³	#125×C₃₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
88	C₂×C₄	#117×C₃₇	1	2	yes	—
89	C₂×C₄	#124×C₃₇	1	2	yes	—
90	C₂×C₄	#125×C₃₇	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	8	4	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₃₇×#382	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	C₃₇×#430	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	C₃₇×#1833	1	2	yes	—
104	C₂	C₃₇×#1838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#1568

C₂

yes

maximal

C₃₇×#1452

C₂

yes

maximal, characteristic

C₃₇×#1287

C₂

yes

maximal

C₃₇×#432

C₂²

yes

—

C₃₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₃₇

C₂²

yes

characteristic

C₃₇×#335

—

C₃₇×#334

—

C₃₇×#322

C₄

yes

—

C₂²×C₈×C₃₇

Q₈

yes

—

C₂×C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊C₈×C₃₇

—

C₄⋊C₈×C₃₇

—

C₂²⋊C₈×C₃₇

—

C₂²⋊C₈×C₃₇

—

#97×C₃₇

—

C₄×C₈×C₃₇

C₂×C₄

yes

—

C₄×C₈×C₃₇

C₂×C₄

yes

—

C₄×C₈×C₃₇

D₄

yes

characteristic

C₄×C₈×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₃₇

#41

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₄⋊C₄

yes

—

C₂×C₈×C₃₇

C₄⋊C₄

yes

—

C₄²×C₃₇

C₂²×C₄

yes

characteristic

C₄²×C₃₇

C₂×D₄

yes

characteristic

C₄²×C₃₇

#41

yes

—

C₂³×C₃₇

#125

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

#117

yes

—

C₂×C₄×C₃₇

#124

yes

—

C₂×C₄×C₃₇

#125

yes

characteristic

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

#8798

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#382

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#430

yes

—

#1568

C₂×C₃₇

yes

—

#1452

C₂×C₃₇

yes

characteristic

#1287

C₂×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

#1833

yes

—

C₂×C₃₇

#1838

yes

characteristic

#432

C₂²×C₃₇

yes

—

#431

C₂²×C₃₇

yes

—

C₂×C₄×C₈

C₂²×C₃₇

yes

—

—

C₄×C₃₇

yes

—

C₃₇

#8798

yes

characteristic, direct product factor, Sylow

C₂²×C₈

Q₈×C₃₇

yes

—

C₂×C₄²

C₂³×C₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₃₇

yes

—

C₄×C₈

C₂×C₄×C₃₇

yes

—

C₄×C₈

D₄×C₃₇

yes

characteristic

C₄×C₈

D₄×C₃₇

yes

characteristic

C₂²×C₄

C₂×Q₈×C₃₇

yes

characteristic

C₂²×C₄

#41×C₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₇

yes

—

C₂×C₈

C₄⋊C₄×C₃₇

yes

—

C₄²

C₂²×C₄×C₃₇

yes

characteristic

C₄²

C₂×D₄×C₃₇

yes

characteristic

C₄²

#41×C₃₇

yes

—

C₂³

#125×C₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₃₇

yes

characteristic

C₂×C₄

#117×C₃₇

yes

—

C₂×C₄

#124×C₃₇

yes

—

C₂×C₄

#125×C₃₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₃₇×#382

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

C₃₇×#430

yes

—

102

C₂

—

103

C₂

C₃₇×#1833

yes

—

104

C₂

C₃₇×#1838

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	#382
commutator	C₄²	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
prime radical	C₂³×C₃₇	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₇

#382

commutator

C₄²

C₂²×C₄×C₃₇

Frattini, cosocle

C₂×C₄²

C₂³×C₃₇

C₂³×C₃₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

N	H ≅ G/N	type
42. #8798	C₃₇	direct (N × H)
62. C₃₇	#8798	direct (N × H)

H ≅ G/N

type

42. #8798

C₃₇

direct (N × H)

62. C₃₇

#8798

direct (N × H)

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

subgroup	index
C₃₇×#1568	2
C₃₇×#1452	2
C₃₇×#1287	2
#8798	37

subgroup

index

C₃₇×#1568

C₃₇×#1452

C₃₇×#1287

#8798

Subgroups — C₃₇×#8798 #22630051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#8798 #22630051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8798 #22630051