Group #37554997 — subgroups

Subgroups — #37554997

Order 23488.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#26044308	C₂	1	1	yes	maximal, characteristic, complemented
2	#26044291	C₂	1	3	yes	maximal, complemented
3	#26044287	C₂	1	3	yes	maximal, complemented
4	#26044281	C₂	1	3	yes	maximal, complemented
5	C₂×#19154565	C₂	1	3	yes	maximal, complemented
6	#26044268	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19154577	C₂²	1	3	yes	complemented
9	#19154576	—	2	3	no	—
10	C₂×Dic₇₃₄	C₂²	1	3	yes	complemented
11	Q₈×D₃₆₇	—	2	1	no	—
12	#19154567	C₂²	1	3	yes	complemented
13	#19154565	C₂²	1	3	yes	complemented
14	#19154565	C₂²	1	6	yes	complemented
15	#19154564	C₂²	1	1	yes	characteristic, complemented
16	#19154562	C₂²	1	3	yes	complemented
17	#19154561	C₂²	1	3	yes	complemented
18	C₂×C₃₆₇⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₃₆₇	—	2	1	no	—
20	#41×C₃₆₇	C₂²	1	3	yes	—
21	#41×C₃₆₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₃₆₇	—	2	3	no	—
23	C₂×D₄×C₃₆₇	C₂²	1	3	yes	—
24	Dic₇₃₄	—	2	3	no	—
25	Dic₇₃₄	C₂³	1	3	yes	—
26	D₁₄₆₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₃₆₇	—	2	3	no	—
28	C₂×Dic₃₆₇	—	2	3	no	—
29	C₄×D₃₆₇	—	2	3	no	—
30	C₃₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₃₆₇⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₃₆₇	—	2	3	no	—
33	Q₈×C₃₆₇	D₄	1	1	yes	characteristic
34	Q₈×C₃₆₇	C₂³	1	1	yes	characteristic
35	D₄×C₃₆₇	—	2	3	no	—
36	D₄×C₃₆₇	—	2	3	no	—
37	D₄×C₃₆₇	D₄	1	3	yes	—
38	D₄×C₃₆₇	C₂³	1	3	yes	—
39	C₂×C₄×C₃₆₇	—	2	3	no	—
40	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
41	C₂²×C₃₆₇	—	2	3	no	—
42	C₂²×C₃₆₇	—	8	3	no	—
43	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
44	C₂×D₃₆₇	—	2	1	no	—
45	C₄×C₃₆₇	—	2	1	no	—
46	C₄×C₃₆₇	C₂×D₄	1	3	yes	—
47	C₄×C₃₆₇	C₂⁴	1	1	yes	characteristic
48	Dic₃₆₇	—	2	3	no	—
49	C₂×C₃₆₇	—	2	3	no	—
50	C₂×C₃₆₇	—	4	3	no	—
51	C₂×C₃₆₇	C₂²×D₄	1	1	yes	characteristic
52	D₃₆₇	—	4	1	no	—
53	C₃₆₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	367	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	367	1	no	—
56	H(5,𝔽₂)	D₃₆₇	1	1	yes	characteristic, complemented
57	#137	—	367	3	no	—
58	AGL(1,ℤ/8)	—	367	3	no	—
59	#135	—	367	3	no	—
60	C₂×SD₁₆	—	367	3	no	—
61	#131	—	367	1	no	—
62	#41	—	2	1	no	—
63	#41	—	367	3	no	—
64	#41	—	734	3	no	—
65	#41	C₂×D₃₆₇	1	3	yes	—
66	#41	C₂×D₃₆₇	1	1	yes	characteristic
67	C₂×Q₈	—	367	3	no	—
68	C₂×Q₈	—	734	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₃₆₇	1	3	yes	—
71	Q₁₆	—	367	3	no	—
72	SD₁₆	—	367	1	no	—
73	SD₁₆	—	367	3	no	—
74	SD₁₆	—	367	6	no	—
75	D₈	—	367	3	no	—
76	M₁₆	—	367	3	no	—
77	C₂×C₈	—	367	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	367	3	no	—
80	Q₈	—	734	3	no	—
81	Q₈	C₂²⋊D₃₆₇	1	1	yes	characteristic
82	Q₈	C₂²×D₃₆₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	367	1	no	—
86	D₄	—	734	3	no	—
87	D₄	C₂²⋊D₃₆₇	1	3	yes	—
88	D₄	C₂²×D₃₆₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	734	3	no	—
91	C₂×C₄	—	734	3	no	—
92	C₂×C₄	C₂²×D₃₆₇	1	3	yes	—
93	C₈	—	367	1	no	—
94	C₈	—	367	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	734	1	no	—
98	C₂²	C₂×C₂²⋊D₃₆₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	734	3	no	—
101	C₄	C₂×C₂²⋊D₃₆₇	1	3	yes	—
102	C₄	C₂³×D₃₆₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1468	1	no	—
106	C₂	C₂²×C₂²⋊D₃₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26044308

C₂

yes

maximal, characteristic, complemented

#26044291

C₂

yes

maximal, complemented

#26044287

C₂

yes

maximal, complemented

#26044281

C₂

yes

maximal, complemented

C₂×#19154565

C₂

yes

maximal, complemented

#26044268

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₆₇

C₂

yes

maximal, characteristic, complemented

#19154577

C₂²

yes

complemented

#19154576

—

C₂×Dic₇₃₄

C₂²

yes

complemented

Q₈×D₃₆₇

—

#19154567

C₂²

yes

complemented

#19154565

C₂²

yes

complemented

#19154565

C₂²

yes

complemented

#19154564

C₂²

yes

characteristic, complemented

#19154562

C₂²

yes

complemented

#19154561

C₂²

yes

complemented

C₂×C₃₆₇⋊C₈

C₂²

yes

complemented

#41×C₃₆₇

—

#41×C₃₆₇

C₂²

yes

—

#41×C₃₆₇

C₂²

yes

characteristic

C₂×D₄×C₃₆₇

—

C₂×D₄×C₃₆₇

C₂²

yes

—

Dic₇₃₄

—

Dic₇₃₄

C₂³

yes

—

D₁₄₆₈

C₂³

yes

characteristic

C₂²⋊D₃₆₇

—

C₂×Dic₃₆₇

—

C₄×D₃₆₇

—

C₃₆₇⋊C₈

C₂³

yes

characteristic

C₃₆₇⋊C₈

C₂³

yes

—

C₂³×C₃₆₇

—

Q₈×C₃₆₇

D₄

yes

characteristic

Q₈×C₃₆₇

C₂³

yes

characteristic

D₄×C₃₆₇

—

D₄×C₃₆₇

—

D₄×C₃₆₇

D₄

yes

—

D₄×C₃₆₇

C₂³

yes

—

C₂×C₄×C₃₆₇

—

C₂×C₄×C₃₆₇

C₂³

yes

—

C₂²×C₃₆₇

—

C₂²×C₃₆₇

—

C₂²×C₃₆₇

C₂×D₄

yes

—

C₂×D₃₆₇

—

C₄×C₃₆₇

—

C₄×C₃₆₇

C₂×D₄

yes

—

C₄×C₃₆₇

C₂⁴

yes

characteristic

Dic₃₆₇

—

C₂×C₃₆₇

—

C₂×C₃₆₇

—

C₂×C₃₆₇

C₂²×D₄

yes

characteristic

D₃₆₇

—

C₃₆₇

#577

yes

characteristic, complemented, Sylow

#577

—

367

maximal, Sylow

2₋¹⁺⁴

—

367

—

H(5,𝔽₂)

D₃₆₇

yes

characteristic, complemented

#137

—

367

—

AGL(1,ℤ/8)

—

367

—

#135

—

367

—

C₂×SD₁₆

—

367

—

#131

—

367

—

#41

—

#41

—

367

—

#41

—

734

—

#41

C₂×D₃₆₇

yes

—

#41

C₂×D₃₆₇

yes

characteristic

C₂×Q₈

—

367

—

C₂×Q₈

—

734

—

C₂×D₄

—

C₂×D₄

C₂×D₃₆₇

yes

—

Q₁₆

—

367

—

SD₁₆

—

367

—

SD₁₆

—

367

—

SD₁₆

—

367

—

D₈

—

367

—

M₁₆

—

367

—

C₂×C₈

—

367

—

C₂³

—

Q₈

—

367

—

Q₈

—

734

—

Q₈

C₂²⋊D₃₆₇

yes

characteristic

Q₈

C₂²×D₃₆₇

yes

characteristic

D₄

—

D₄

—

D₄

—

367

—

D₄

—

734

—

D₄

C₂²⋊D₃₆₇

yes

—

D₄

C₂²×D₃₆₇

yes

—

C₂×C₄

—

C₂×C₄

—

734

—

C₂×C₄

—

734

—

C₂×C₄

C₂²×D₃₆₇

yes

—

C₈

—

367

—

C₈

—

367

—

C₂²

—

C₂²

—

C₂²

—

734

—

C₂²

C₂×C₂²⋊D₃₆₇

yes

—

C₄

—

100

C₄

—

734

—

101

C₄

C₂×C₂²⋊D₃₆₇

yes

—

102

C₄

C₂³×D₃₆₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1468

—

106

C₂

C₂²×C₂²⋊D₃₆₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₆₇
commutator, cosocle	C₄×C₃₆₇	C₂⁴
Frattini	C₄	C₂³×D₃₆₇
Fitting	H(5,𝔽₂)×C₃₆₇	C₂
socle	C₂×C₃₆₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₆₇
hypocenter	C₃₆₇	#577
LCS	G ▷ C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₆₇

commutator, cosocle

C₄×C₃₆₇

C₂⁴

Frattini

C₄

C₂³×D₃₆₇

Fitting

H(5,𝔽₂)×C₃₆₇

C₂

socle

C₂×C₃₆₇

C₂²×D₄

D₃₆₇

C₃₆₇

G ▷ C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #26044308	C₂	semidirect
2. #26044291	C₂	semidirect
3. #26044287	C₂	semidirect
4. #26044281	C₂	semidirect
5. C₂×#19154565	C₂	semidirect
6. #26044268	C₂	semidirect
7. H(5,𝔽₂)×C₃₆₇	C₂	semidirect
8. #19154577	C₂²	semidirect
10. C₂×Dic₇₃₄	C₂²	semidirect
12. #19154567	C₂²	semidirect
13. #19154565	C₂²	semidirect
14. #19154565	C₂²	semidirect
15. #19154564	C₂²	semidirect
16. #19154562	C₂²	semidirect
17. #19154561	C₂²	semidirect
18. C₂×C₃₆₇⋊C₈	C₂²	semidirect
53. C₃₆₇	#577	semidirect
56. H(5,𝔽₂)	D₃₆₇	semidirect

H ≅ G/N

type

1. #26044308

C₂

semidirect

2. #26044291

C₂

semidirect

3. #26044287

C₂

semidirect

4. #26044281

C₂

semidirect

5. C₂×#19154565

C₂

semidirect

6. #26044268

C₂

semidirect

7. H(5,𝔽₂)×C₃₆₇

C₂

semidirect

8. #19154577

C₂²

semidirect

10. C₂×Dic₇₃₄

C₂²

semidirect

12. #19154567

C₂²

semidirect

13. #19154565

C₂²

semidirect

14. #19154565

C₂²

semidirect

15. #19154564

C₂²

semidirect

16. #19154562

C₂²

semidirect

17. #19154561

C₂²

semidirect

18. C₂×C₃₆₇⋊C₈

C₂²

semidirect

53. C₃₆₇

#577

semidirect

56. H(5,𝔽₂)

D₃₆₇

semidirect

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

subgroup	index
#26044308	2
#26044291	2
#26044287	2
#26044281	2
C₂×#19154565	2
#26044268	2
H(5,𝔽₂)×C₃₆₇	2
#577	367

subgroup

index

H(5,𝔽₂)×C₃₆₇

367

Subgroups — #37554997

Conjugacy classes of subgroups

Complements

Maximal subgroups