Group #22550678 — subgroups

Subgroups — #22550678

Order 8128.

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	#16861255	C₂	1	2	yes	maximal, complemented
2	#16861236	C₂	1	4	yes	maximal, complemented
3	#16861234	C₂	1	2	yes	maximal, complemented
4	#16861232	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
7	#16861215	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523249	C₂²	1	4	yes	complemented
10	#14523248	—	2	2	no	—
11	#14523248	C₂²	1	1	yes	characteristic
12	C₂×D₅₀₈	—	2	2	no	—
13	C₂×D₅₀₈	C₂²	1	2	yes	complemented
14	D₄×D₁₂₇	—	2	4	no	—
15	D₄×D₁₂₇	C₂²	1	4	yes	complemented
16	#14523235	C₂²	1	4	yes	complemented
17	D₁₀₁₆	C₂²	1	2	yes	complemented
18	D₁₀₁₆	C₂²	1	2	yes	complemented
19	#14523233	C₂²	1	4	yes	complemented
20	#14523232	C₂²	1	1	yes	characteristic, complemented
21	#14523231	C₂²	1	2	yes	complemented
22	C₈×D₁₂₇	C₂²	1	2	yes	complemented
23	#41×C₁₂₇	C₂²	1	2	yes	complemented
24	Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
25	SD₁₆×C₁₂₇	C₂²	1	2	yes	complemented
26	D₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
28	C₂²×D₁₂₇	—	2	2	no	—
29	C₂²×D₁₂₇	—	2	4	no	—
30	Dic₂₅₄	D₄	1	1	yes	characteristic
31	D₅₀₈	—	2	4	no	—
32	D₅₀₈	—	2	2	no	—
33	D₅₀₈	D₄	1	1	yes	characteristic
34	D₅₀₈	C₂³	1	4	yes	—
35	C₂²⋊D₁₂₇	—	2	4	no	—
36	C₂²⋊D₁₂₇	D₄	1	2	yes	—
37	C₄×D₁₂₇	—	2	4	no	—
38	C₄×D₁₂₇	C₂³	1	2	yes	—
39	C₁₂₇⋊C₈	C₂³	1	2	yes	—
40	Q₈×C₁₂₇	C₂³	1	2	yes	—
41	D₄×C₁₂₇	—	2	2	no	—
42	D₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₂₇	—	2	4	no	—
50	C₂×D₁₂₇	—	8	4	no	—
51	C₂×D₁₂₇	—	8	2	no	—
52	C₂×D₁₂₇	C₂×D₄	1	2	yes	—
53	C₄×C₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
56	Dic₁₂₇	C₂×D₄	1	2	yes	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	—	4	2	no	—
59	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
60	D₁₂₇	—	2	2	no	—
61	D₁₂₇	—	4	4	no	—
62	C₁₂₇	#576	1	1	yes	characteristic, complemented, Sylow
63	#576	—	127	1	no	maximal, Sylow
64	H(5,𝔽₂)	—	127	2	no	—
65	AGL(1,ℤ/8)	—	127	2	no	—
66	AGL(1,ℤ/8)	—	127	4	no	—
67	#135	—	127	2	no	—
68	#135	D₁₂₇	1	1	yes	characteristic, complemented
69	C₂×D₈	—	127	2	no	—
70	C₂×D₈	—	127	1	no	—
71	#131	—	127	1	no	—
72	#41	—	127	1	no	—
73	#41	—	127	4	no	—
74	#41	—	254	2	no	—
75	#41	C₂×D₁₂₇	1	2	yes	complemented
76	C₂×D₄	—	127	4	no	—
77	C₂×D₄	—	127	2	no	—
78	C₂×D₄	—	254	2	no	—
79	C₂×D₄	—	254	4	no	—
80	Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic, complemented
81	SD₁₆	—	127	4	no	—
82	SD₁₆	C₂×D₁₂₇	1	2	yes	complemented
83	D₈	—	127	4	no	—
84	D₈	—	127	2	no	—
85	D₈	—	127	2	no	—
86	D₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
87	M₁₆	—	127	1	no	—
88	M₁₆	—	127	2	no	—
89	C₂×C₈	—	127	2	no	—
90	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
91	C₂³	—	254	4	no	—
92	C₂³	—	254	2	no	—
93	Q₈	—	127	1	no	—
94	Q₈	C₂²×D₁₂₇	1	2	yes	—
95	D₄	—	2	2	no	—
96	D₄	—	127	2	no	—
97	D₄	—	127	1	no	—
98	D₄	—	127	4	no	—
99	D₄	—	254	4	no	—
100	D₄	—	254	4	no	—
101	D₄	—	254	2	no	—
102	D₄	C₂²×D₁₂₇	1	2	yes	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	127	2	no	—
105	C₂×C₄	—	254	4	no	—
106	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
107	C₈	—	127	2	no	—
108	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
110	C₂²	—	2	2	no	—
111	C₂²	—	127	2	no	—
112	C₂²	—	1016	2	no	—
113	C₂²	—	1016	4	no	—
114	C₂²	—	254	4	no	—
115	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	127	2	no	—
118	C₄	D₄×D₁₂₇	1	1	yes	characteristic
119	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	4	2	no	—
122	C₂	—	508	4	no	—
123	C₂	—	254	2	no	—
124	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861255

C₂

yes

maximal, complemented

#16861236

C₂

yes

maximal, complemented

#16861234

C₂

yes

maximal, complemented

#16861232

C₂

yes

maximal, complemented

C₂×D₁₀₁₆

C₂

yes

maximal, characteristic, complemented

D₈×D₁₂₇

C₂

yes

maximal, complemented

#16861215

C₂

yes

maximal, characteristic, complemented

#135×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

C₂²

yes

complemented

#14523248

—

#14523248

C₂²

yes

characteristic

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

complemented

D₄×D₁₂₇

—

D₄×D₁₂₇

C₂²

yes

complemented

#14523235

C₂²

yes

complemented

D₁₀₁₆

C₂²

yes

complemented

D₁₀₁₆

C₂²

yes

complemented

#14523233

C₂²

yes

complemented

#14523232

C₂²

yes

characteristic, complemented

#14523231

C₂²

yes

complemented

C₈×D₁₂₇

C₂²

yes

complemented

#41×C₁₂₇

C₂²

yes

complemented

Q₁₆×C₁₂₇

C₂²

yes

characteristic, complemented

SD₁₆×C₁₂₇

C₂²

yes

complemented

D₈×C₁₂₇

C₂²

yes

characteristic, complemented

C₂×C₈×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

D₄

yes

characteristic

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

characteristic

D₅₀₈

C₂³

yes

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

D₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

C₂³

yes

—

Q₈×C₁₂₇

C₂³

yes

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#576

yes

characteristic, complemented, Sylow

#576

—

127

maximal, Sylow

H(5,𝔽₂)

—

127

—

AGL(1,ℤ/8)

—

127

—

AGL(1,ℤ/8)

—

127

—

#135

—

127

—

#135

D₁₂₇

yes

characteristic, complemented

C₂×D₈

—

127

—

C₂×D₈

—

127

—

#131

—

127

—

#41

—

127

—

#41

—

127

—

#41

—

254

—

#41

C₂×D₁₂₇

yes

complemented

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

Q₁₆

C₂×D₁₂₇

yes

characteristic, complemented

SD₁₆

—

127

—

SD₁₆

C₂×D₁₂₇

yes

complemented

D₈

—

127

—

D₈

—

127

—

D₈

—

127

—

D₈

C₂×D₁₂₇

yes

characteristic, complemented

M₁₆

—

127

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic, complemented

C₂³

—

254

—

C₂³

—

254

—

Q₈

—

127

—

Q₈

C₂²×D₁₂₇

yes

—

D₄

—

D₄

—

127

—

D₄

—

127

—

D₄

—

127

—

D₄

—

254

—

100

D₄

—

254

—

101

D₄

—

254

—

102

D₄

C₂²×D₁₂₇

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

127

—

105

C₂×C₄

—

254

—

106

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

107

C₈

—

127

—

108

C₈

C₂²×D₁₂₇

yes

characteristic

109

C₈

C₂²×D₁₂₇

yes

characteristic

110

C₂²

—

111

C₂²

—

127

—

112

C₂²

—

1016

—

113

C₂²

—

1016

—

114

C₂²

—

254

—

115

C₂²

D₄×D₁₂₇

yes

characteristic

116

C₄

—

117

C₄

—

127

—

118

C₄

D₄×D₁₂₇

yes

characteristic

119

C₄

C₂³×D₁₂₇

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

—

508

—

123

C₂

—

254

—

124

C₂

C₂×D₄×D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	#135×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₄
hypercenter	#135	D₁₂₇
hypocenter	C₁₂₇	#576
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂⁴

Frattini

C₄

C₂³×D₁₂₇

Fitting

#135×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #16861255	C₂	semidirect
2. #16861236	C₂	semidirect
3. #16861234	C₂	semidirect
4. #16861232	C₂	semidirect
5. C₂×D₁₀₁₆	C₂	semidirect
6. D₈×D₁₂₇	C₂	semidirect
7. #16861215	C₂	semidirect
8. #135×C₁₂₇	C₂	semidirect
9. #14523249	C₂²	semidirect
13. C₂×D₅₀₈	C₂²	semidirect
15. D₄×D₁₂₇	C₂²	semidirect
16. #14523235	C₂²	semidirect
17. D₁₀₁₆	C₂²	semidirect
18. D₁₀₁₆	C₂²	semidirect
19. #14523233	C₂²	semidirect
20. #14523232	C₂²	semidirect
21. #14523231	C₂²	semidirect
22. C₈×D₁₂₇	C₂²	semidirect
23. #41×C₁₂₇	C₂²	semidirect
24. Q₁₆×C₁₂₇	C₂²	semidirect
25. SD₁₆×C₁₂₇	C₂²	semidirect
26. D₈×C₁₂₇	C₂²	semidirect
27. C₂×C₈×C₁₂₇	C₂²	semidirect
62. C₁₂₇	#576	semidirect
68. #135	D₁₂₇	semidirect
75. #41	C₂×D₁₂₇	semidirect
80. Q₁₆	C₂×D₁₂₇	semidirect
82. SD₁₆	C₂×D₁₂₇	semidirect
86. D₈	C₂×D₁₂₇	semidirect
90. C₂×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

1. #16861255

C₂

semidirect

2. #16861236

C₂

semidirect

3. #16861234

C₂

semidirect

4. #16861232

C₂

semidirect

5. C₂×D₁₀₁₆

C₂

semidirect

6. D₈×D₁₂₇

C₂

semidirect

7. #16861215

C₂

semidirect

8. #135×C₁₂₇

C₂

semidirect

9. #14523249

C₂²

semidirect

13. C₂×D₅₀₈

C₂²

semidirect

15. D₄×D₁₂₇

C₂²

semidirect

16. #14523235

C₂²

semidirect

17. D₁₀₁₆

C₂²

semidirect

18. D₁₀₁₆

C₂²

semidirect

19. #14523233

C₂²

semidirect

20. #14523232

C₂²

semidirect

21. #14523231

C₂²

semidirect

22. C₈×D₁₂₇

C₂²

semidirect

23. #41×C₁₂₇

C₂²

semidirect

24. Q₁₆×C₁₂₇

C₂²

semidirect

25. SD₁₆×C₁₂₇

C₂²

semidirect

26. D₈×C₁₂₇

C₂²

semidirect

27. C₂×C₈×C₁₂₇

C₂²

semidirect

62. C₁₂₇

#576

semidirect

68. #135

D₁₂₇

semidirect

75. #41

C₂×D₁₂₇

semidirect

80. Q₁₆

C₂×D₁₂₇

semidirect

82. SD₁₆

C₂×D₁₂₇

semidirect

86. D₈

C₂×D₁₂₇

semidirect

90. C₂×C₈

C₂×D₁₂₇

semidirect

Subgroups — #22550678

Conjugacy classes of subgroups

Complements