Group #37560164 — subgroups

Subgroups — #37560164

Order 23872.

260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26045781	C₂	1	2	yes	maximal, complemented
2	#26045780	C₂	1	2	yes	maximal, complemented
3	#26045778	C₂	1	4	yes	maximal, complemented
4	#26045762	C₂	1	4	yes	maximal, complemented
5	#26045760	C₂	1	2	yes	maximal, complemented
6	#124×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₇₃	C₂²	1	2	yes	complemented
8	C₂×Dic₇₄₆	C₂²	1	2	yes	complemented
9	C₂×D₁₄₉₂	C₂²	1	2	yes	—
10	C₂×D₁₄₉₂	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₃₇₃	C₂²	1	4	yes	—
12	C₂²⋊Dic₃₇₃	C₂²	1	2	yes	complemented
13	#20238587	C₂²	1	4	yes	complemented
14	#20238587	C₂²	1	4	yes	complemented
15	#20238587	C₂²	1	4	yes	complemented
16	C₄×Dic₃₇₃	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₃₇₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₃₇₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₇₃	C₂²	1	4	yes	complemented
20	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₇₃	—	4	4	no	—
22	C₂²×D₃₇₃	—	4	2	no	—
23	C₂²×D₃₇₃	C₂³	1	4	yes	—
24	Dic₇₄₆	—	4	2	no	—
25	D₁₄₉₂	—	4	2	no	—
26	D₁₄₉₂	—	4	2	no	—
27	C₂²⋊D₃₇₃	—	4	4	no	—
28	C₂×Dic₃₇₃	C₂³	1	4	yes	—
29	C₂³×C₃₇₃	C₂³	1	2	yes	—
30	Q₈×C₃₇₃	—	4	1	no	—
31	D₄×C₃₇₃	—	4	1	no	—
32	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
33	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
34	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
35	C₂²×C₃₇₃	—	2	2	no	—
36	C₂²×C₃₇₃	—	2	4	no	—
37	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
38	C₂×D₃₇₃	—	2	4	no	—
39	C₂×D₃₇₃	—	2	4	no	—
40	C₂×D₃₇₃	—	2	4	no	—
41	C₂×D₃₇₃	—	4	8	no	—
42	C₄×C₃₇₃	—	2	1	no	—
43	C₄×C₃₇₃	—	2	2	no	—
44	C₄×C₃₇₃	—	2	2	no	—
45	Dic₃₇₃	—	2	4	no	—
46	C₂×C₃₇₃	—	4	2	no	—
47	C₂×C₃₇₃	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₃₇₃	H(5,𝔽₂)	1	2	yes	—
49	D₃₇₃	—	4	4	no	—
50	C₃₇₃	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	373	1	no	maximal, Sylow
52	#124	—	373	2	no	—
53	#124	—	373	4	no	—
54	#124	—	373	2	no	—
55	#124	D₃₇₃	1	1	yes	characteristic, complemented
56	#120	—	373	2	no	—
57	#120	—	373	4	no	—
58	C₂⁴	—	373	2	no	—
59	C₂×Q₈	—	373	2	no	—
60	C₂×Q₈	C₂×D₃₇₃	1	1	yes	characteristic, complemented
61	C₂×D₄	—	373	4	no	—
62	C₂×D₄	—	373	2	no	—
63	C₂×D₄	—	373	2	no	—
64	C₂×D₄	C₂×D₃₇₃	1	1	yes	characteristic
65	C₂²⋊C₄	—	373	2	no	—
66	C₂²⋊C₄	—	373	4	no	—
67	C₂²⋊C₄	—	373	4	no	—
68	C₂²⋊C₄	—	373	4	no	—
69	C₂²⋊C₄	C₂×D₃₇₃	1	4	yes	complemented
70	C₄²	—	373	2	no	—
71	C₄²	C₂×D₃₇₃	1	1	yes	characteristic, complemented
72	C₂³	—	373	4	no	—
73	C₂³	—	1492	2	no	—
74	C₂³	—	1492	4	no	—
75	C₂³	C₂²×D₃₇₃	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	1492	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	1492	2	no	—
80	D₄	—	1492	2	no	—
81	D₄	—	1492	4	no	—
82	C₂×C₄	—	373	4	no	—
83	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
84	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	1492	8	no	—
89	C₂²	—	746	4	no	—
90	C₂²	—	746	4	no	—
91	C₂²	—	746	4	no	—
92	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	746	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	1492	4	no	—
99	C₂	#26045837	1	1	yes	characteristic
100	C₂	#26045838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26045781

C₂

yes

maximal, complemented

#26045780

C₂

yes

maximal, complemented

#26045778

C₂

yes

maximal, complemented

#26045762

C₂

yes

maximal, complemented

#26045760

C₂

yes

maximal, complemented

#124×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇₃

C₂²

yes

complemented

C₂×Dic₇₄₆

C₂²

yes

complemented

C₂×D₁₄₉₂

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

—

C₂×C₂²⋊D₃₇₃

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

complemented

#20238587

C₂²

yes

complemented

#20238587

C₂²

yes

complemented

#20238587

C₂²

yes

complemented

C₄×Dic₃₇₃

C₂²

yes

complemented

C₂×Q₈×C₃₇₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₇₃

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇₃

C₂²

yes

complemented

C₄²×C₃₇₃

C₂²

yes

characteristic, complemented

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

C₂³

yes

—

Dic₇₄₆

—

D₁₄₉₂

—

D₁₄₉₂

—

C₂²⋊D₃₇₃

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂³×C₃₇₃

C₂³

yes

—

Q₈×C₃₇₃

—

D₄×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇₃

H(5,𝔽₂)

yes

—

D₃₇₃

—

C₃₇₃

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

373

maximal, Sylow

#124

—

373

—

#124

—

373

—

#124

—

373

—

#124

D₃₇₃

yes

characteristic, complemented

#120

—

373

—

#120

—

373

—

C₂⁴

—

373

—

C₂×Q₈

—

373

—

C₂×Q₈

C₂×D₃₇₃

yes

characteristic, complemented

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

C₂×D₃₇₃

yes

characteristic

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

C₂×D₃₇₃

yes

complemented

C₄²

—

373

—

C₄²

C₂×D₃₇₃

yes

characteristic, complemented

C₂³

—

373

—

C₂³

—

1492

—

C₂³

—

1492

—

C₂³

C₂²×D₃₇₃

yes

—

Q₈

—

Q₈

—

1492

—

D₄

—

D₄

—

1492

—

D₄

—

1492

—

D₄

—

1492

—

C₂×C₄

—

373

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂²

—

C₂²

—

C₂²

—

1492

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

C₂³×D₃₇₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

746

—

C₂

—

C₂

—

1492

—

C₂

#26045837

yes

characteristic

100

C₂

#26045838

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	#124×C₃₇₃	C₂
hypercenter	#124	D₃₇₃
hypocenter	C₃₇₃	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

#124×C₃₇₃

C₂

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #26045781	C₂	semidirect
2. #26045780	C₂	semidirect
3. #26045778	C₂	semidirect
4. #26045762	C₂	semidirect
5. #26045760	C₂	semidirect
6. #124×C₃₇₃	C₂	semidirect
7. C₂³×D₃₇₃	C₂²	semidirect
8. C₂×Dic₇₄₆	C₂²	semidirect
12. C₂²⋊Dic₃₇₃	C₂²	semidirect
13. #20238587	C₂²	semidirect
14. #20238587	C₂²	semidirect
15. #20238587	C₂²	semidirect
16. C₄×Dic₃₇₃	C₂²	semidirect
17. C₂×Q₈×C₃₇₃	C₂²	semidirect
19. C₂²⋊C₄×C₃₇₃	C₂²	semidirect
20. C₄²×C₃₇₃	C₂²	semidirect
50. C₃₇₃	H(3,𝔽₄)	semidirect
55. #124	D₃₇₃	semidirect
60. C₂×Q₈	C₂×D₃₇₃	semidirect
69. C₂²⋊C₄	C₂×D₃₇₃	semidirect
71. C₄²	C₂×D₃₇₃	semidirect

H ≅ G/N

type

1. #26045781

C₂

semidirect

2. #26045780

C₂

semidirect

3. #26045778

C₂

semidirect

4. #26045762

C₂

semidirect

5. #26045760

C₂

semidirect

6. #124×C₃₇₃

C₂

semidirect

7. C₂³×D₃₇₃

C₂²

semidirect

8. C₂×Dic₇₄₆

C₂²

semidirect

12. C₂²⋊Dic₃₇₃

C₂²

semidirect

13. #20238587

C₂²

semidirect

14. #20238587

C₂²

semidirect

15. #20238587

C₂²

semidirect

16. C₄×Dic₃₇₃

C₂²

semidirect

17. C₂×Q₈×C₃₇₃

C₂²

semidirect

19. C₂²⋊C₄×C₃₇₃

C₂²

semidirect

20. C₄²×C₃₇₃

C₂²

semidirect

50. C₃₇₃

H(3,𝔽₄)

semidirect

55. #124

D₃₇₃

semidirect

60. C₂×Q₈

C₂×D₃₇₃

semidirect

69. C₂²⋊C₄

C₂×D₃₇₃

semidirect

71. C₄²

C₂×D₃₇₃

semidirect

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

subgroup	index
#26045781	2
#26045780	2
#26045778	2
#26045762	2
#26045760	2
#124×C₃₇₃	2
H(3,𝔽₄)	373

subgroup

index

#124×C₃₇₃

373

Subgroups — #37560164

Conjugacy classes of subgroups

Complements

Maximal subgroups