Group #26028847 — subgroups

Subgroups — #26028847

Order 11392.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#19152081	C₂	1	2	yes	maximal
2	#19151640	C₂	1	6	yes	maximal, complemented
3	#19151636	C₂	1	3	yes	maximal, complemented
4	C₂×#15667944	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₈₉	—	2	2	no	—
7	D₄×Dic₈₉	—	2	6	no	—
8	#15667971	C₂²	1	6	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	6	yes	—
10	#15667946	C₂²	1	6	yes	—
11	#15667944	C₂²	1	3	yes	—
12	#15667944	C₂²	1	6	yes	—
13	#15667944	C₂²	1	1	yes	characteristic
14	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
15	#131×C₈₉	C₄	1	8	yes	complemented
16	C₂×M₁₆×C₈₉	C₂²	1	3	yes	—
17	C₂²×C₈×C₈₉	C₂²	1	3	yes	—
18	C₂²×Dic₈₉	—	2	6	no	—
19	C₄⋊Dic₈₉	—	2	6	no	—
20	C₄⋊Dic₈₉	C₂³	1	2	yes	—
21	C₄⋊Dic₈₉	C₂³	1	6	yes	—
22	C₂²⋊Dic₈₉	—	2	6	no	—
23	C₄×Dic₈₉	—	2	6	no	—
24	#41×C₈₉	C₂×C₄	1	4	yes	—
25	#41×C₈₉	Q₈	1	4	yes	—
26	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₈₉	D₄	1	3	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
29	M₁₆×C₈₉	C₂×C₄	1	12	yes	complemented
30	C₂×C₈×C₈₉	C₂×C₄	1	12	yes	complemented
31	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₂³	1	3	yes	—
33	C₂³×C₈₉	C₂×D₄	1	3	yes	—
34	Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
35	D₄×C₈₉	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₈₉	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₈₉	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₈₉	—	2	2	no	—
41	C₂×Dic₈₉	—	2	6	no	—
42	C₂×Dic₈₉	—	8	6	no	—
43	C₈×C₈₉	C₂²×C₄	1	2	yes	—
44	C₈×C₈₉	C₂²×C₄	1	6	yes	—
45	C₂²×C₈₉	—	2	6	no	—
46	C₂²×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
47	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
49	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
50	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
51	Dic₈₉	—	4	2	no	—
52	Dic₈₉	—	4	6	no	—
53	C₂×C₈₉	—	2	6	no	—
54	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈₉	#577	1	2	yes	—
56	#2912	—	89	1	no	maximal, Sylow
57	C₈₉	#2912	1	1	yes	characteristic, complemented, Sylow
58	C₂×#131	D₈₉	1	1	yes	characteristic
59	#520	—	89	2	no	—
60	#428	—	89	6	no	—
61	#427	—	89	3	no	—
62	C₂×#106	—	89	3	no	—
63	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
64	#131	Dic₈₉	1	8	yes	complemented
65	C₂×M₁₆	C₂×D₈₉	1	3	yes	—
66	C₂²×C₈	C₂×D₈₉	1	3	yes	—
67	C₄×Q₈	—	178	2	no	—
68	C₄×D₄	—	178	6	no	—
69	#117	—	89	6	no	—
70	C₂×C₄⋊C₄	—	89	6	no	—
71	#107	—	89	6	no	—
72	#106	—	89	3	no	—
73	#106	—	89	6	no	—
74	#106	—	89	1	no	—
75	#41	Dic₁₇₈	1	4	yes	—
76	#41	C₂×Dic₈₉	1	4	yes	—
77	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
78	C₂×D₄	D₃₅₆	1	3	yes	—
79	C₂²×C₄	—	178	6	no	—
80	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
81	M₁₆	C₂×Dic₈₉	1	12	yes	complemented
82	C₂×C₈	C₂×Dic₈₉	1	12	yes	complemented
83	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₈₉	1	3	yes	—
85	C₄⋊C₄	—	89	6	no	—
86	C₄⋊C₄	—	89	2	no	—
87	C₄⋊C₄	—	178	6	no	—
88	C₂²⋊C₄	—	178	6	no	—
89	C₄²	—	178	6	no	—
90	C₂³	C₂×D₃₅₆	1	3	yes	—
91	Q₈	C₄⋊Dic₈₉	1	4	yes	—
92	D₄	C₄⋊Dic₈₉	1	12	yes	—
93	C₂×C₄	—	178	2	no	—
94	C₂×C₄	—	178	6	no	—
95	C₂×C₄	—	712	6	no	—
96	C₂×C₄	C₂²×Dic₈₉	1	6	yes	—
97	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
98	C₂×C₄	C₂×Dic₁₇₈	1	6	yes	—
99	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
100	C₈	C₂²×Dic₈₉	1	2	yes	—
101	C₈	C₂²×Dic₈₉	1	6	yes	—
102	C₂²	—	2	6	no	—
103	C₂²	C₂×C₄⋊Dic₈₉	1	6	yes	—
104	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
105	C₄	—	356	6	no	—
106	C₄	—	356	2	no	—
107	C₄	C₂×C₄⋊Dic₈₉	1	6	yes	—
108	C₄	C₂³×Dic₈₉	1	1	yes	characteristic
109	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
110	C₂	—	2	6	no	—
111	C₂	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
112	C₂	#19152410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152081

C₂

yes

maximal

#19151640

C₂

yes

maximal, complemented

#19151636

C₂

yes

maximal, complemented

C₂×#15667944

C₂

yes

maximal, complemented

C₂×#131×C₈₉

C₂

yes

maximal, characteristic

Q₈×Dic₈₉

—

D₄×Dic₈₉

—

#15667971

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667944

C₂²

yes

—

#15667944

C₂²

yes

characteristic

C₂×#41×C₈₉

C₂²

yes

characteristic

#131×C₈₉

C₄

yes

complemented

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#41×C₈₉

C₂×C₄

yes

—

#41×C₈₉

Q₈

yes

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

M₁₆×C₈₉

C₂×C₄

yes

complemented

C₂×C₈×C₈₉

C₂×C₄

yes

complemented

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

—

Q₈×C₈₉

C₄⋊C₄

yes

—

D₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

C₂²×C₄

yes

—

C₈×C₈₉

C₂²×C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₈₉

#577

yes

—

#2912

—

maximal, Sylow

C₈₉

#2912

yes

characteristic, complemented, Sylow

C₂×#131

D₈₉

yes

—

—

—

—

C₂×#41

C₂×D₈₉

yes

characteristic

#131

Dic₈₉

yes

complemented

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄×Q₈

—

178

—

C₄×D₄

—

178

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

Dic₁₇₈

yes

—

#41

C₂×Dic₈₉

yes

—

C₂×Q₈

D₃₅₆

yes

characteristic

C₂×D₄

D₃₅₆

yes

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

—

M₁₆

C₂×Dic₈₉

yes

complemented

C₂×C₈

C₂×Dic₈₉

yes

complemented

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₂³

C₂×D₃₅₆

yes

—

Q₈

C₄⋊Dic₈₉

yes

—

D₄

C₄⋊Dic₈₉

yes

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

712

—

C₂×C₄

C₂²×Dic₈₉

yes

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

100

C₈

C₂²×Dic₈₉

yes

—

101

C₈

C₂²×Dic₈₉

yes

—

102

C₂²

—

103

C₂²

C₂×C₄⋊Dic₈₉

yes

—

104

C₂²

C₂²×D₃₅₆

yes

characteristic

105

C₄

—

356

—

106

C₄

—

356

—

107

C₄

C₂×C₄⋊Dic₈₉

yes

—

108

C₄

C₂³×Dic₈₉

yes

characteristic

109

C₄

C₂²×Dic₁₇₈

yes

characteristic

110

C₂

—

111

C₂

C₂²×C₄⋊Dic₈₉

yes

characteristic

112

C₂

#19152410

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#131×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×#131	D₈₉
hypocenter	C₈₉	#2912
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator

C₄×C₈₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#131×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×#41×C₈₉

C₂²

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #19151640	C₂	semidirect
3. #19151636	C₂	semidirect
4. C₂×#15667944	C₂	semidirect
15. #131×C₈₉	C₄	semidirect
29. M₁₆×C₈₉	C₂×C₄	semidirect
30. C₂×C₈×C₈₉	C₂×C₄	semidirect
57. C₈₉	#2912	semidirect
64. #131	Dic₈₉	semidirect
81. M₁₆	C₂×Dic₈₉	semidirect
82. C₂×C₈	C₂×Dic₈₉	semidirect

H ≅ G/N

type

2. #19151640

C₂

semidirect

3. #19151636

C₂

semidirect

4. C₂×#15667944

C₂

semidirect

15. #131×C₈₉

C₄

semidirect

29. M₁₆×C₈₉

C₂×C₄

semidirect

30. C₂×C₈×C₈₉

C₂×C₄

semidirect

57. C₈₉

#2912

semidirect

64. #131

Dic₈₉

semidirect

81. M₁₆

C₂×Dic₈₉

semidirect

82. C₂×C₈

C₂×Dic₈₉

semidirect

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

subgroup	index
#19152081	2
#19151640	2
#19151636	2
C₂×#15667944	2
C₂×#131×C₈₉	2
#2912	89

subgroup

index

C₂×#131×C₈₉

Subgroups — #26028847

Conjugacy classes of subgroups

Complements

Maximal subgroups