Group #17994225 — subgroups

Subgroups — #17994225

Order 4736.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14557910	C₂	1	1	yes	maximal, characteristic
2	#14557907	C₂	1	2	yes	maximal, complemented
3	#14557881	C₂	1	2	yes	maximal, complemented
4	#14557254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596941	—	2	4	no	—
7	C₂×D₂₉₆	—	2	2	no	—
8	#11924533	C₂²	1	1	yes	characteristic
9	#11924533	C₂²	1	1	yes	characteristic
10	#11924456	—	2	4	no	—
11	C₄×#64436	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
13	#106×C₃₇	—	2	2	no	—
14	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	2	2	no	—
16	C₂×D₁₄₈	—	2	1	no	—
17	C₂×D₁₄₈	—	2	1	no	—
18	C₂×D₁₄₈	—	2	2	no	—
19	C₂×D₁₄₈	—	2	2	no	—
20	C₂×C₈×C₃₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₃₇	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄²×C₃₇	C₂³	1	1	yes	characteristic
24	#10596941	—	4	8	no	—
25	C₂×#64436	D₄	1	4	yes	—
26	D₂₉₆	—	4	4	no	—
27	C₂²×D₃₇	—	4	1	no	—
28	C₂²×D₃₇	—	4	1	no	—
29	Q₈×C₃₇	—	4	4	no	—
30	C₂×C₄×C₃₇	—	4	2	no	—
31	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
33	D₁₄₈	—	4	2	no	—
34	D₁₄₈	—	4	2	no	—
35	D₁₄₈	—	4	4	no	—
36	D₁₄₈	—	4	4	no	—
37	C₈×C₃₇	SD₁₆	1	4	yes	complemented
38	#64436	—	2	4	no	—
39	C₂²×C₃₇	#127	1	1	yes	characteristic
40	C₂×D₃₇	—	8	1	no	—
41	C₂×D₃₇	—	8	1	no	—
42	C₂×D₃₇	—	8	2	no	—
43	C₂×D₃₇	—	8	2	no	—
44	C₄×C₃₇	—	8	2	no	—
45	C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
46	C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
47	#1715	—	37	1	no	maximal, Sylow
48	C₂×C₃₇	#492	1	1	yes	characteristic
49	C₂×C₃₇	#496	1	2	yes	—
50	D₃₇	—	16	1	no	—
51	D₃₇	—	16	1	no	—
52	#498	D₃₇	1	1	yes	characteristic, complemented
53	#493	—	37	1	no	—
54	#492	—	37	2	no	—
55	#486	—	37	2	no	—
56	#322	—	37	1	no	—
57	C₃₇	#1715	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	74	4	no	—
59	C₂×D₈	—	74	2	no	—
60	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
61	#127	—	37	1	no	—
62	#127	—	37	1	no	—
63	#106	—	2	2	no	—
64	#102	—	74	4	no	—
65	C₄×C₈	—	37	2	no	—
66	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	74	2	no	—
69	C₂×D₄	—	74	2	no	—
70	C₂×D₄	—	74	1	no	—
71	C₂×D₄	—	74	1	no	—
72	SD₁₆	—	148	8	no	—
73	D₈	—	148	4	no	—
74	C₂×C₈	—	37	4	no	—
75	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₃₇	1	1	yes	characteristic
79	C₂³	—	148	1	no	—
80	C₂³	—	148	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	148	2	no	—
83	D₄	—	148	2	no	—
84	D₄	—	148	4	no	—
85	D₄	—	148	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₃₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
89	C₈	—	74	4	no	—
90	C₈	#10596941	1	4	yes	complemented
91	C₂²	—	296	1	no	—
92	C₂²	—	296	1	no	—
93	C₂²	—	296	2	no	—
94	C₂²	—	296	2	no	—
95	C₂²	#11924532	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#10596941	1	2	yes	—
98	C₄	#11924563	1	4	yes	—
99	C₂	—	592	1	no	—
100	C₂	—	592	1	no	—
101	C₂	#14557907	1	1	yes	characteristic
102	C₂	#14557919	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557910

C₂

yes

maximal, characteristic

#14557907

C₂

yes

maximal, complemented

#14557881

C₂

yes

maximal, complemented

#14557254

C₂

yes

maximal, characteristic, complemented

C₃₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×#10596941

—

C₂×D₂₉₆

—

#11924533

C₂²

yes

characteristic

#11924533

C₂²

yes

characteristic

#11924456

—

C₄×#64436

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

#106×C₃₇

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596941

—

C₂×#64436

D₄

yes

—

D₂₉₆

—

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

SD₁₆

yes

complemented

#64436

—

C₂²×C₃₇

#127

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

#498

D₃₇

yes

characteristic, complemented

—

—

—

—

C₃₇

#1715

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

148

—

D₈

—

148

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

148

—

C₂³

—

148

—

Q₈

—

D₄

—

148

—

D₄

—

148

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

D₄×D₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₈

—

C₈

#10596941

yes

complemented

C₂²

—

296

—

C₂²

—

296

—

C₂²

—

296

—

C₂²

—

296

—

C₂²

#11924532

yes

characteristic

C₄

—

C₄

C₂×#10596941

yes

—

C₄

#11924563

yes

—

C₂

—

592

—

100

C₂

—

592

—

101

C₂

#14557907

yes

characteristic

102

C₂

#14557919

yes

—

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#498	C₂
socle	C₂²×C₃₇	#127
prime radical	#14557910	C₂
hypercenter	#498	D₃₇
hypocenter	C₃₇	#1715
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#498

C₂

C₂²×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #14557907	C₂	semidirect
3. #14557881	C₂	semidirect
4. #14557254	C₂	semidirect
5. C₃₇×#498	C₂	semidirect
37. C₈×C₃₇	SD₁₆	semidirect
52. #498	D₃₇	semidirect
57. C₃₇	#1715	semidirect
90. C₈	#10596941	semidirect

H ≅ G/N

type

2. #14557907

C₂

semidirect

3. #14557881

C₂

semidirect

4. #14557254

C₂

semidirect

5. C₃₇×#498

C₂

semidirect

37. C₈×C₃₇

SD₁₆

semidirect

52. #498

D₃₇

semidirect

57. C₃₇

#1715

semidirect

90. C₈

#10596941

semidirect

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

subgroup	index
#14557910	2
#14557907	2
#14557881	2
#14557254	2
C₃₇×#498	2
#1715	37

subgroup

index

C₃₇×#498

Subgroups — #17994225

Conjugacy classes of subgroups

Complements

Maximal subgroups