Group #22550690 — subgroups

Subgroups — #22550690

Order 8128.

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

normal subgroups only

This group has 248 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	#16861256	C₂	1	2	yes	maximal, complemented
2	#16861239	C₂	1	2	yes	maximal, complemented
3	#16861238	C₂	1	4	yes	maximal, complemented
4	#16861228	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×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	#14523248	—	2	2	no	—
10	#14523248	C₂²	1	1	yes	characteristic
11	#14523247	—	2	4	no	—
12	#14523247	C₂²	1	4	yes	—
13	C₂×Dic₂₅₄	—	2	2	no	—
14	C₂×Dic₂₅₄	C₂²	1	2	yes	—
15	Q₈×D₁₂₇	C₂²	1	4	yes	—
16	Dic₅₀₈	C₂²	1	2	yes	—
17	Dic₅₀₈	C₂²	1	2	yes	—
18	#14523238	C₂²	1	4	yes	—
19	#14523236	C₂²	1	4	yes	—
20	#14523232	C₂²	1	1	yes	characteristic
21	#14523231	C₂²	1	2	yes	—
22	C₈×D₁₂₇	C₂²	1	2	yes	—
23	#41×C₁₂₇	C₂²	1	2	yes	—
24	Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
25	SD₁₆×C₁₂₇	C₂²	1	2	yes	—
26	D₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	Dic₂₅₄	—	2	2	no	—
29	Dic₂₅₄	—	2	4	no	—
30	Dic₂₅₄	D₄	1	1	yes	characteristic
31	Dic₂₅₄	C₂³	1	4	yes	—
32	D₅₀₈	D₄	1	1	yes	characteristic
33	C₂²⋊D₁₂₇	—	2	4	no	—
34	C₂²⋊D₁₂₇	D₄	1	2	yes	—
35	C₂×Dic₁₂₇	—	2	2	no	—
36	C₂×Dic₁₂₇	—	2	4	no	—
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	C₂×D₄	1	2	yes	—
50	C₄×C₁₂₇	—	2	2	no	—
51	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₂₇	—	2	4	no	—
54	Dic₁₂₇	C₂×D₄	1	2	yes	—
55	C₂×C₁₂₇	—	2	1	no	—
56	C₂×C₁₂₇	—	4	2	no	—
57	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	D₁₂₇	—	2	2	no	—
59	C₁₂₇	#578	1	1	yes	characteristic, complemented, Sylow
60	#578	—	127	1	no	maximal, Sylow
61	2₋¹⁺⁴	—	127	2	no	—
62	#137	—	127	2	no	—
63	#137	—	127	4	no	—
64	#135	—	127	2	no	—
65	#135	D₁₂₇	1	1	yes	characteristic, complemented
66	C₂×Q₁₆	—	127	2	no	—
67	C₂×Q₁₆	—	127	1	no	—
68	#131	—	127	1	no	—
69	#41	—	127	1	no	—
70	#41	—	127	4	no	—
71	#41	—	254	2	no	—
72	#41	—	254	4	no	—
73	#41	C₂×D₁₂₇	1	2	yes	—
74	C₂×Q₈	—	127	4	no	—
75	C₂×Q₈	—	127	2	no	—
76	C₂×Q₈	—	254	2	no	—
77	Q₁₆	—	127	4	no	—
78	Q₁₆	—	127	2	no	—
79	Q₁₆	—	127	2	no	—
80	Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
81	SD₁₆	—	127	4	no	—
82	SD₁₆	C₂×D₁₂₇	1	2	yes	—
83	D₈	C₂×D₁₂₇	1	1	yes	characteristic
84	M₁₆	—	127	1	no	—
85	M₁₆	—	127	2	no	—
86	C₂×C₈	—	127	2	no	—
87	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
88	Q₈	—	127	1	no	—
89	Q₈	—	127	4	no	—
90	Q₈	—	254	4	no	—
91	Q₈	—	254	2	no	—
92	Q₈	C₂²×D₁₂₇	1	2	yes	—
93	D₄	—	2	2	no	—
94	D₄	—	127	2	no	—
95	D₄	—	127	1	no	—
96	D₄	—	254	4	no	—
97	D₄	C₂²×D₁₂₇	1	2	yes	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	127	2	no	—
100	C₂×C₄	—	254	4	no	—
101	C₂×C₄	—	254	4	no	—
102	C₂×C₄	—	254	2	no	—
103	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
104	C₈	—	127	2	no	—
105	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
106	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	127	2	no	—
109	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	127	2	no	—
112	C₄	—	254	4	no	—
113	C₄	D₄×D₁₂₇	1	1	yes	characteristic
114	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	4	2	no	—
117	C₂	—	254	2	no	—
118	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861256

C₂

yes

maximal, complemented

#16861239

C₂

yes

maximal, complemented

#16861238

C₂

yes

maximal, complemented

#16861228

C₂

yes

maximal, complemented

C₂×Dic₅₀₈

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₁₂₇

C₂

yes

maximal, complemented

#16861215

C₂

yes

maximal, characteristic, complemented

#135×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

C₂²

yes

characteristic

#14523247

—

#14523247

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂²

yes

—

Q₈×D₁₂₇

C₂²

yes

—

Dic₅₀₈

C₂²

yes

—

Dic₅₀₈

C₂²

yes

—

#14523238

C₂²

yes

—

#14523236

C₂²

yes

—

#14523232

C₂²

yes

characteristic

#14523231

C₂²

yes

—

C₈×D₁₂₇

C₂²

yes

—

#41×C₁₂₇

C₂²

yes

—

Q₁₆×C₁₂₇

C₂²

yes

characteristic

SD₁₆×C₁₂₇

C₂²

yes

—

D₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

D₄

yes

characteristic

Dic₂₅₄

C₂³

yes

—

D₅₀₈

D₄

yes

characteristic

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

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×D₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#578

yes

characteristic, complemented, Sylow

#578

—

127

maximal, Sylow

2₋¹⁺⁴

—

127

—

#137

—

127

—

#137

—

127

—

#135

—

127

—

#135

D₁₂₇

yes

characteristic, complemented

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

#131

—

127

—

#41

—

127

—

#41

—

127

—

#41

—

254

—

#41

—

254

—

#41

C₂×D₁₂₇

yes

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

Q₁₆

—

127

—

Q₁₆

—

127

—

Q₁₆

—

127

—

Q₁₆

C₂×D₁₂₇

yes

characteristic

SD₁₆

—

127

—

SD₁₆

C₂×D₁₂₇

yes

—

D₈

C₂×D₁₂₇

yes

characteristic

M₁₆

—

127

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

Q₈

—

127

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

—

254

—

Q₈

C₂²×D₁₂₇

yes

—

D₄

—

D₄

—

127

—

D₄

—

127

—

D₄

—

254

—

D₄

C₂²×D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

127

—

100

C₂×C₄

—

254

—

101

C₂×C₄

—

254

—

102

C₂×C₄

—

254

—

103

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

104

C₈

—

127

—

105

C₈

C₂²×D₁₂₇

yes

characteristic

106

C₈

C₂²×D₁₂₇

yes

characteristic

107

C₂²

—

108

C₂²

—

127

—

109

C₂²

D₄×D₁₂₇

yes

characteristic

110

C₄

—

111

C₄

—

127

—

112

C₄

—

254

—

113

C₄

D₄×D₁₂₇

yes

characteristic

114

C₄

C₂³×D₁₂₇

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

254

—

118

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	#578
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. #16861256	C₂	semidirect
2. #16861239	C₂	semidirect
3. #16861238	C₂	semidirect
4. #16861228	C₂	semidirect
5. C₂×Dic₅₀₈	C₂	semidirect
6. Q₁₆×D₁₂₇	C₂	semidirect
7. #16861215	C₂	semidirect
8. #135×C₁₂₇	C₂	semidirect
59. C₁₂₇	#578	semidirect
65. #135	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861256

C₂

semidirect

2. #16861239

C₂

semidirect

3. #16861238

C₂

semidirect

4. #16861228

C₂

semidirect

5. C₂×Dic₅₀₈

C₂

semidirect

6. Q₁₆×D₁₂₇

C₂

semidirect

7. #16861215

C₂

semidirect

8. #135×C₁₂₇

C₂

semidirect

59. C₁₂₇

#578

semidirect

65. #135

D₁₂₇

semidirect

Subgroups — #22550690

Conjugacy classes of subgroups

Complements