Group #22578205 — subgroups

Subgroups — #22578205

Order 8768.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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	#17984134	C₂	1	2	yes	maximal, complemented
2	#17984115	C₂	1	4	yes	maximal, complemented
3	#17984113	C₂	1	2	yes	maximal, complemented
4	#17984111	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
7	#17984094	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14555511	C₂²	1	4	yes	complemented
10	#14555510	—	2	2	no	—
11	#14555510	C₂²	1	1	yes	characteristic
12	C₂×D₅₄₈	—	2	2	no	—
13	C₂×D₅₄₈	C₂²	1	2	yes	complemented
14	D₄×D₁₃₇	—	2	4	no	—
15	D₄×D₁₃₇	C₂²	1	4	yes	complemented
16	#14555497	C₂²	1	4	yes	complemented
17	D₁₀₉₆	C₂²	1	2	yes	complemented
18	D₁₀₉₆	C₂²	1	2	yes	complemented
19	#14555495	C₂²	1	4	yes	complemented
20	#14555494	C₂²	1	1	yes	characteristic, complemented
21	#14555493	C₂²	1	2	yes	complemented
22	C₈×D₁₃₇	C₂²	1	2	yes	complemented
23	#41×C₁₃₇	C₂²	1	2	yes	complemented
24	Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
25	SD₁₆×C₁₃₇	C₂²	1	2	yes	complemented
26	D₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
28	C₂²×D₁₃₇	—	2	2	no	—
29	C₂²×D₁₃₇	—	2	4	no	—
30	Dic₂₇₄	D₄	1	1	yes	characteristic
31	D₅₄₈	—	2	4	no	—
32	D₅₄₈	—	2	2	no	—
33	D₅₄₈	D₄	1	1	yes	characteristic
34	D₅₄₈	C₂³	1	4	yes	—
35	C₂²⋊D₁₃₇	—	2	4	no	—
36	C₂²⋊D₁₃₇	D₄	1	2	yes	—
37	C₄×D₁₃₇	—	2	4	no	—
38	C₄×D₁₃₇	C₂³	1	2	yes	—
39	#11764040	C₂³	1	2	yes	—
40	Q₈×C₁₃₇	C₂³	1	2	yes	—
41	D₄×C₁₃₇	—	2	2	no	—
42	D₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₃₇	—	2	2	no	—
48	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₃₇	—	2	4	no	—
50	C₂×D₁₃₇	—	8	4	no	—
51	C₂×D₁₃₇	—	8	2	no	—
52	C₂×D₁₃₇	C₂×D₄	1	2	yes	—
53	C₄×C₁₃₇	—	2	2	no	—
54	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
56	Dic₁₃₇	C₂×D₄	1	2	yes	—
57	C₂×C₁₃₇	—	2	1	no	—
58	C₂×C₁₃₇	—	4	2	no	—
59	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
60	D₁₃₇	—	2	2	no	—
61	D₁₃₇	—	4	4	no	—
62	C₁₃₇	#576	1	1	yes	characteristic, complemented, Sylow
63	#576	—	137	1	no	maximal, Sylow
64	H(5,𝔽₂)	—	137	2	no	—
65	AGL(1,ℤ/8)	—	137	2	no	—
66	AGL(1,ℤ/8)	—	137	4	no	—
67	#135	—	137	2	no	—
68	#135	D₁₃₇	1	1	yes	characteristic, complemented
69	C₂×D₈	—	137	2	no	—
70	C₂×D₈	—	137	1	no	—
71	#131	—	137	1	no	—
72	#41	—	274	2	no	—
73	#41	—	137	4	no	—
74	#41	—	137	1	no	—
75	#41	C₂×D₁₃₇	1	2	yes	complemented
76	C₂×D₄	—	274	2	no	—
77	C₂×D₄	—	274	4	no	—
78	C₂×D₄	—	137	4	no	—
79	C₂×D₄	—	137	2	no	—
80	Q₁₆	C₂×D₁₃₇	1	1	yes	characteristic, complemented
81	SD₁₆	—	137	4	no	—
82	SD₁₆	C₂×D₁₃₇	1	2	yes	complemented
83	D₈	—	137	4	no	—
84	D₈	—	137	2	no	—
85	D₈	—	137	2	no	—
86	D₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
87	M₁₆	—	137	1	no	—
88	M₁₆	—	137	2	no	—
89	C₂×C₈	—	137	2	no	—
90	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
91	C₂³	—	274	4	no	—
92	C₂³	—	274	2	no	—
93	Q₈	—	137	1	no	—
94	Q₈	C₂²×D₁₃₇	1	2	yes	—
95	D₄	—	2	2	no	—
96	D₄	—	274	4	no	—
97	D₄	—	274	4	no	—
98	D₄	—	274	2	no	—
99	D₄	—	137	2	no	—
100	D₄	—	137	1	no	—
101	D₄	—	137	4	no	—
102	D₄	C₂²×D₁₃₇	1	2	yes	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	274	4	no	—
105	C₂×C₄	—	137	2	no	—
106	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
107	C₈	—	137	2	no	—
108	C₈	C₂²×D₁₃₇	1	1	yes	characteristic
109	C₈	C₂²×D₁₃₇	1	1	yes	characteristic
110	C₂²	—	2	2	no	—
111	C₂²	—	274	4	no	—
112	C₂²	—	1096	2	no	—
113	C₂²	—	1096	4	no	—
114	C₂²	—	137	2	no	—
115	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	137	2	no	—
118	C₄	D₄×D₁₃₇	1	1	yes	characteristic
119	C₄	C₂³×D₁₃₇	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	4	2	no	—
122	C₂	—	274	2	no	—
123	C₂	—	548	4	no	—
124	C₂	C₂×D₄×D₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984134

C₂

yes

maximal, complemented

#17984115

C₂

yes

maximal, complemented

#17984113

C₂

yes

maximal, complemented

#17984111

C₂

yes

maximal, complemented

C₂×D₁₀₉₆

C₂

yes

maximal, characteristic, complemented

D₈×D₁₃₇

C₂

yes

maximal, complemented

#17984094

C₂

yes

maximal, characteristic, complemented

#135×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#14555511

C₂²

yes

complemented

#14555510

—

#14555510

C₂²

yes

characteristic

C₂×D₅₄₈

—

C₂×D₅₄₈

C₂²

yes

complemented

D₄×D₁₃₇

—

D₄×D₁₃₇

C₂²

yes

complemented

#14555497

C₂²

yes

complemented

D₁₀₉₆

C₂²

yes

complemented

D₁₀₉₆

C₂²

yes

complemented

#14555495

C₂²

yes

complemented

#14555494

C₂²

yes

characteristic, complemented

#14555493

C₂²

yes

complemented

C₈×D₁₃₇

C₂²

yes

complemented

#41×C₁₃₇

C₂²

yes

complemented

Q₁₆×C₁₃₇

C₂²

yes

characteristic, complemented

SD₁₆×C₁₃₇

C₂²

yes

complemented

D₈×C₁₃₇

C₂²

yes

characteristic, complemented

C₂×C₈×C₁₃₇

C₂²

yes

characteristic, complemented

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

Dic₂₇₄

D₄

yes

characteristic

D₅₄₈

—

D₅₄₈

—

D₅₄₈

D₄

yes

characteristic

D₅₄₈

C₂³

yes

—

C₂²⋊D₁₃₇

—

C₂²⋊D₁₃₇

D₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂³

yes

—

#11764040

C₂³

yes

—

Q₈×C₁₃₇

C₂³

yes

—

D₄×C₁₃₇

—

D₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×D₄

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₄×C₁₃₇

C₂⁴

yes

characteristic

Dic₁₃₇

C₂×D₄

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#576

yes

characteristic, complemented, Sylow

#576

—

137

maximal, Sylow

H(5,𝔽₂)

—

137

—

AGL(1,ℤ/8)

—

137

—

AGL(1,ℤ/8)

—

137

—

#135

—

137

—

#135

D₁₃₇

yes

characteristic, complemented

C₂×D₈

—

137

—

C₂×D₈

—

137

—

#131

—

137

—

#41

—

274

—

#41

—

137

—

#41

—

137

—

#41

C₂×D₁₃₇

yes

complemented

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

Q₁₆

C₂×D₁₃₇

yes

characteristic, complemented

SD₁₆

—

137

—

SD₁₆

C₂×D₁₃₇

yes

complemented

D₈

—

137

—

D₈

—

137

—

D₈

—

137

—

D₈

C₂×D₁₃₇

yes

characteristic, complemented

M₁₆

—

137

—

M₁₆

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×D₁₃₇

yes

characteristic, complemented

C₂³

—

274

—

C₂³

—

274

—

Q₈

—

137

—

Q₈

C₂²×D₁₃₇

yes

—

D₄

—

D₄

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

137

—

100

D₄

—

137

—

101

D₄

—

137

—

102

D₄

C₂²×D₁₃₇

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

274

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

107

C₈

—

137

—

108

C₈

C₂²×D₁₃₇

yes

characteristic

109

C₈

C₂²×D₁₃₇

yes

characteristic

110

C₂²

—

111

C₂²

—

274

—

112

C₂²

—

1096

—

113

C₂²

—

1096

—

114

C₂²

—

137

—

115

C₂²

D₄×D₁₃₇

yes

characteristic

116

C₄

—

117

C₄

—

137

—

118

C₄

D₄×D₁₃₇

yes

characteristic

119

C₄

C₂³×D₁₃₇

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

—

274

—

123

C₂

—

548

—

124

C₂

C₂×D₄×D₁₃₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂⁴
Frattini	C₄	C₂³×D₁₃₇
Fitting	#135×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂²×D₄
hypercenter	#135	D₁₃₇
hypocenter	C₁₃₇	#576
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₁₃₇

commutator

C₄×C₁₃₇

C₂⁴

Frattini

C₄

C₂³×D₁₃₇

Fitting

#135×C₁₃₇

C₂

socle

C₂×C₁₃₇

C₂²×D₄

D₁₃₇

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #17984134	C₂	semidirect
2. #17984115	C₂	semidirect
3. #17984113	C₂	semidirect
4. #17984111	C₂	semidirect
5. C₂×D₁₀₉₆	C₂	semidirect
6. D₈×D₁₃₇	C₂	semidirect
7. #17984094	C₂	semidirect
8. #135×C₁₃₇	C₂	semidirect
9. #14555511	C₂²	semidirect
13. C₂×D₅₄₈	C₂²	semidirect
15. D₄×D₁₃₇	C₂²	semidirect
16. #14555497	C₂²	semidirect
17. D₁₀₉₆	C₂²	semidirect
18. D₁₀₉₆	C₂²	semidirect
19. #14555495	C₂²	semidirect
20. #14555494	C₂²	semidirect
21. #14555493	C₂²	semidirect
22. C₈×D₁₃₇	C₂²	semidirect
23. #41×C₁₃₇	C₂²	semidirect
24. Q₁₆×C₁₃₇	C₂²	semidirect
25. SD₁₆×C₁₃₇	C₂²	semidirect
26. D₈×C₁₃₇	C₂²	semidirect
27. C₂×C₈×C₁₃₇	C₂²	semidirect
62. C₁₃₇	#576	semidirect
68. #135	D₁₃₇	semidirect
75. #41	C₂×D₁₃₇	semidirect
80. Q₁₆	C₂×D₁₃₇	semidirect
82. SD₁₆	C₂×D₁₃₇	semidirect
86. D₈	C₂×D₁₃₇	semidirect
90. C₂×C₈	C₂×D₁₃₇	semidirect

H ≅ G/N

type

1. #17984134

C₂

semidirect

2. #17984115

C₂

semidirect

3. #17984113

C₂

semidirect

4. #17984111

C₂

semidirect

5. C₂×D₁₀₉₆

C₂

semidirect

6. D₈×D₁₃₇

C₂

semidirect

7. #17984094

C₂

semidirect

8. #135×C₁₃₇

C₂

semidirect

9. #14555511

C₂²

semidirect

13. C₂×D₅₄₈

C₂²

semidirect

15. D₄×D₁₃₇

C₂²

semidirect

16. #14555497

C₂²

semidirect

17. D₁₀₉₆

C₂²

semidirect

18. D₁₀₉₆

C₂²

semidirect

19. #14555495

C₂²

semidirect

20. #14555494

C₂²

semidirect

21. #14555493

C₂²

semidirect

22. C₈×D₁₃₇

C₂²

semidirect

23. #41×C₁₃₇

C₂²

semidirect

24. Q₁₆×C₁₃₇

C₂²

semidirect

25. SD₁₆×C₁₃₇

C₂²

semidirect

26. D₈×C₁₃₇

C₂²

semidirect

27. C₂×C₈×C₁₃₇

C₂²

semidirect

62. C₁₃₇

#576

semidirect

68. #135

D₁₃₇

semidirect

75. #41

C₂×D₁₃₇

semidirect

80. Q₁₆

C₂×D₁₃₇

semidirect

82. SD₁₆

C₂×D₁₃₇

semidirect

86. D₈

C₂×D₁₃₇

semidirect

90. C₂×C₈

C₂×D₁₃₇

semidirect

Subgroups — #22578205

Conjugacy classes of subgroups

Complements