Group #26039250 — subgroups

Subgroups — #26039250

Order 11456.

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	#19153160	C₂	1	1	yes	maximal, characteristic, complemented
2	#19153143	C₂	1	3	yes	maximal, complemented
3	#19153139	C₂	1	3	yes	maximal, complemented
4	#19153133	C₂	1	3	yes	maximal, complemented
5	C₂×#15668247	C₂	1	3	yes	maximal, complemented
6	#19153120	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668259	C₂²	1	3	yes	complemented
9	#15668258	—	2	3	no	—
10	C₂×Dic₃₅₈	C₂²	1	3	yes	complemented
11	Q₈×D₁₇₉	—	2	1	no	—
12	#15668249	C₂²	1	3	yes	complemented
13	#15668247	C₂²	1	3	yes	complemented
14	#15668247	C₂²	1	6	yes	complemented
15	#15668246	C₂²	1	1	yes	characteristic, complemented
16	#15668244	C₂²	1	3	yes	complemented
17	#15668243	C₂²	1	3	yes	complemented
18	C₂×C₁₇₉⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₇₉	—	2	1	no	—
20	#41×C₁₇₉	C₂²	1	3	yes	—
21	#41×C₁₇₉	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₇₉	—	2	3	no	—
23	C₂×D₄×C₁₇₉	C₂²	1	3	yes	—
24	Dic₃₅₈	—	2	3	no	—
25	Dic₃₅₈	C₂³	1	3	yes	—
26	D₇₁₆	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₇₉	—	2	3	no	—
28	C₂×Dic₁₇₉	—	2	3	no	—
29	C₄×D₁₇₉	—	2	3	no	—
30	C₁₇₉⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₇₉⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₇₉	—	2	3	no	—
33	Q₈×C₁₇₉	D₄	1	1	yes	characteristic
34	Q₈×C₁₇₉	C₂³	1	1	yes	characteristic
35	D₄×C₁₇₉	—	2	3	no	—
36	D₄×C₁₇₉	—	2	3	no	—
37	D₄×C₁₇₉	D₄	1	3	yes	—
38	D₄×C₁₇₉	C₂³	1	3	yes	—
39	C₂×C₄×C₁₇₉	—	2	3	no	—
40	C₂×C₄×C₁₇₉	C₂³	1	3	yes	—
41	C₂²×C₁₇₉	—	2	3	no	—
42	C₂²×C₁₇₉	—	8	3	no	—
43	C₂²×C₁₇₉	C₂×D₄	1	3	yes	—
44	C₂×D₁₇₉	—	2	1	no	—
45	C₄×C₁₇₉	—	2	1	no	—
46	C₄×C₁₇₉	C₂×D₄	1	3	yes	—
47	C₄×C₁₇₉	C₂⁴	1	1	yes	characteristic
48	Dic₁₇₉	—	2	3	no	—
49	C₂×C₁₇₉	—	2	3	no	—
50	C₂×C₁₇₉	—	4	3	no	—
51	C₂×C₁₇₉	C₂²×D₄	1	1	yes	characteristic
52	D₁₇₉	—	4	1	no	—
53	C₁₇₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	179	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	179	1	no	—
56	H(5,𝔽₂)	D₁₇₉	1	1	yes	characteristic, complemented
57	#137	—	179	3	no	—
58	AGL(1,ℤ/8)	—	179	3	no	—
59	#135	—	179	3	no	—
60	C₂×SD₁₆	—	179	3	no	—
61	#131	—	179	1	no	—
62	#41	—	2	1	no	—
63	#41	—	358	3	no	—
64	#41	—	179	3	no	—
65	#41	C₂×D₁₇₉	1	3	yes	—
66	#41	C₂×D₁₇₉	1	1	yes	characteristic
67	C₂×Q₈	—	358	1	no	—
68	C₂×Q₈	—	179	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₇₉	1	3	yes	—
71	Q₁₆	—	179	3	no	—
72	SD₁₆	—	179	1	no	—
73	SD₁₆	—	179	3	no	—
74	SD₁₆	—	179	6	no	—
75	D₈	—	179	3	no	—
76	M₁₆	—	179	3	no	—
77	C₂×C₈	—	179	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	358	3	no	—
80	Q₈	—	179	3	no	—
81	Q₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
82	Q₈	C₂²×D₁₇₉	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	358	3	no	—
86	D₄	—	179	1	no	—
87	D₄	C₂²⋊D₁₇₉	1	3	yes	—
88	D₄	C₂²×D₁₇₉	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	358	3	no	—
91	C₂×C₄	—	358	3	no	—
92	C₂×C₄	C₂²×D₁₇₉	1	3	yes	—
93	C₈	—	179	1	no	—
94	C₈	—	179	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	358	1	no	—
98	C₂²	C₂×C₂²⋊D₁₇₉	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	358	3	no	—
101	C₄	C₂×C₂²⋊D₁₇₉	1	3	yes	—
102	C₄	C₂³×D₁₇₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	716	1	no	—
106	C₂	C₂²×C₂²⋊D₁₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19153160

C₂

yes

maximal, characteristic, complemented

#19153143

C₂

yes

maximal, complemented

#19153139

C₂

yes

maximal, complemented

#19153133

C₂

yes

maximal, complemented

C₂×#15668247

C₂

yes

maximal, complemented

#19153120

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₇₉

C₂

yes

maximal, characteristic, complemented

#15668259

C₂²

yes

complemented

#15668258

—

C₂×Dic₃₅₈

C₂²

yes

complemented

Q₈×D₁₇₉

—

#15668249

C₂²

yes

complemented

#15668247

C₂²

yes

complemented

#15668247

C₂²

yes

complemented

#15668246

C₂²

yes

characteristic, complemented

#15668244

C₂²

yes

complemented

#15668243

C₂²

yes

complemented

C₂×C₁₇₉⋊C₈

C₂²

yes

complemented

#41×C₁₇₉

—

#41×C₁₇₉

C₂²

yes

—

#41×C₁₇₉

C₂²

yes

characteristic

C₂×D₄×C₁₇₉

—

C₂×D₄×C₁₇₉

C₂²

yes

—

Dic₃₅₈

—

Dic₃₅₈

C₂³

yes

—

D₇₁₆

C₂³

yes

characteristic

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

C₂³

yes

characteristic

C₁₇₉⋊C₈

C₂³

yes

—

C₂³×C₁₇₉

—

Q₈×C₁₇₉

D₄

yes

characteristic

Q₈×C₁₇₉

C₂³

yes

characteristic

D₄×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

D₄

yes

—

D₄×C₁₇₉

C₂³

yes

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂³

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×D₄

yes

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×D₄

yes

—

C₄×C₁₇₉

C₂⁴

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

C₂²×D₄

yes

characteristic

D₁₇₉

—

C₁₇₉

#577

yes

characteristic, complemented, Sylow

#577

—

179

maximal, Sylow

2₋¹⁺⁴

—

179

—

H(5,𝔽₂)

D₁₇₉

yes

characteristic, complemented

#137

—

179

—

AGL(1,ℤ/8)

—

179

—

#135

—

179

—

C₂×SD₁₆

—

179

—

#131

—

179

—

#41

—

#41

—

358

—

#41

—

179

—

#41

C₂×D₁₇₉

yes

—

#41

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂×D₄

—

C₂×D₄

C₂×D₁₇₉

yes

—

Q₁₆

—

179

—

SD₁₆

—

179

—

SD₁₆

—

179

—

SD₁₆

—

179

—

D₈

—

179

—

M₁₆

—

179

—

C₂×C₈

—

179

—

C₂³

—

Q₈

—

358

—

Q₈

—

179

—

Q₈

C₂²⋊D₁₇₉

yes

characteristic

Q₈

C₂²×D₁₇₉

yes

characteristic

D₄

—

D₄

—

D₄

—

358

—

D₄

—

179

—

D₄

C₂²⋊D₁₇₉

yes

—

D₄

C₂²×D₁₇₉

yes

—

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

C₂²×D₁₇₉

yes

—

C₈

—

179

—

C₈

—

179

—

C₂²

—

C₂²

—

C₂²

—

358

—

C₂²

C₂×C₂²⋊D₁₇₉

yes

—

C₄

—

100

C₄

—

358

—

101

C₄

C₂×C₂²⋊D₁₇₉

yes

—

102

C₄

C₂³×D₁₇₉

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

716

—

106

C₂

C₂²×C₂²⋊D₁₇₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₇₉
commutator, cosocle	C₄×C₁₇₉	C₂⁴
Frattini	C₄	C₂³×D₁₇₉
Fitting	H(5,𝔽₂)×C₁₇₉	C₂
socle	C₂×C₁₇₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₇₉
hypocenter	C₁₇₉	#577
LCS	G ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₇₉

commutator, cosocle

C₄×C₁₇₉

C₂⁴

Frattini

C₄

C₂³×D₁₇₉

Fitting

H(5,𝔽₂)×C₁₇₉

C₂

socle

C₂×C₁₇₉

C₂²×D₄

D₁₇₉

C₁₇₉

G ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #19153160	C₂	semidirect
2. #19153143	C₂	semidirect
3. #19153139	C₂	semidirect
4. #19153133	C₂	semidirect
5. C₂×#15668247	C₂	semidirect
6. #19153120	C₂	semidirect
7. H(5,𝔽₂)×C₁₇₉	C₂	semidirect
8. #15668259	C₂²	semidirect
10. C₂×Dic₃₅₈	C₂²	semidirect
12. #15668249	C₂²	semidirect
13. #15668247	C₂²	semidirect
14. #15668247	C₂²	semidirect
15. #15668246	C₂²	semidirect
16. #15668244	C₂²	semidirect
17. #15668243	C₂²	semidirect
18. C₂×C₁₇₉⋊C₈	C₂²	semidirect
53. C₁₇₉	#577	semidirect
56. H(5,𝔽₂)	D₁₇₉	semidirect

H ≅ G/N

type

1. #19153160

C₂

semidirect

2. #19153143

C₂

semidirect

3. #19153139

C₂

semidirect

4. #19153133

C₂

semidirect

5. C₂×#15668247

C₂

semidirect

6. #19153120

C₂

semidirect

7. H(5,𝔽₂)×C₁₇₉

C₂

semidirect

8. #15668259

C₂²

semidirect

10. C₂×Dic₃₅₈

C₂²

semidirect

12. #15668249

C₂²

semidirect

13. #15668247

C₂²

semidirect

14. #15668247

C₂²

semidirect

15. #15668246

C₂²

semidirect

16. #15668244

C₂²

semidirect

17. #15668243

C₂²

semidirect

18. C₂×C₁₇₉⋊C₈

C₂²

semidirect

53. C₁₇₉

#577

semidirect

56. H(5,𝔽₂)

D₁₇₉

semidirect

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

subgroup	index
#19153160	2
#19153143	2
#19153139	2
#19153133	2
C₂×#15668247	2
#19153120	2
H(5,𝔽₂)×C₁₇₉	2
#577	179

subgroup

index

H(5,𝔽₂)×C₁₇₉

179

Subgroups — #26039250

Conjugacy classes of subgroups

Complements

Maximal subgroups