Group #22550137 — subgroups

Subgroups — #22550137

Order 8128.

104 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 37 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	#16861239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523236	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861216	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861144	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861127	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861126	C₂	1	1	yes	maximal, characteristic, complemented
7	AGL(1,ℤ/8)×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	#14523238	—	2	1	no	—
10	#14523236	—	2	1	no	—
11	#14523236	—	2	1	no	—
12	#14523236	—	2	1	no	—
13	#14523232	—	2	1	no	—
14	#14523232	C₂²	1	1	yes	characteristic
15	#14523232	C₂²	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	—	2	1	no	—
17	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	#41×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	SD₁₆×C₁₂₇	—	2	1	no	—
21	D₈×C₁₂₇	—	2	1	no	—
22	M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
23	Dic₂₅₄	—	2	1	no	—
24	Dic₂₅₄	—	2	1	no	—
25	C₂×Dic₁₂₇	—	2	1	no	—
26	C₁₂₇⋊C₈	—	2	1	no	—
27	C₁₂₇⋊C₈	—	2	1	no	—
28	C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
29	C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂³×C₁₂₇	—	2	1	no	—
31	Q₈×C₁₂₇	D₄	1	1	yes	characteristic
32	D₄×C₁₂₇	—	2	1	no	—
33	D₄×C₁₂₇	—	2	1	no	—
34	D₄×C₁₂₇	—	2	1	no	—
35	D₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	2	1	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	—	8	1	no	—
42	C₂²×C₁₂₇	#41	1	1	yes	characteristic
43	C₄×C₁₂₇	—	2	1	no	—
44	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	Dic₁₂₇	—	4	1	no	—
47	C₂×C₁₂₇	—	2	1	no	—
48	C₂×C₁₂₇	—	4	1	no	—
49	C₂×C₁₂₇	—	8	1	no	—
50	C₂×C₁₂₇	#121	1	1	yes	characteristic
51	C₁₂₇	#471	1	1	yes	characteristic, complemented, Sylow
52	#471	—	127	1	no	maximal, Sylow
53	#137	—	127	1	no	—
54	AGL(1,ℤ/8)	D₁₂₇	1	1	yes	characteristic
55	C₂×SD₁₆	—	127	1	no	—
56	#131	—	127	1	no	—
57	#108	—	127	1	no	—
58	#101	—	127	1	no	—
59	#100	—	127	1	no	—
60	#41	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×Q₈	—	127	1	no	—
62	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
63	Q₁₆	—	254	1	no	—
64	SD₁₆	—	2	1	no	—
65	SD₁₆	—	254	1	no	—
66	SD₁₆	—	254	1	no	—
67	SD₁₆	—	254	1	no	—
68	D₈	—	2	1	no	—
69	M₁₆	—	127	1	no	—
70	M₁₆	—	127	1	no	—
71	M₁₆	—	254	1	no	—
72	M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
73	C₂×C₈	—	127	1	no	—
74	C₂×C₈	—	254	1	no	—
75	C₂³	—	2	1	no	—
76	Q₈	—	254	1	no	—
77	Q₈	—	254	1	no	—
78	Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
79	D₄	—	2	1	no	—
80	D₄	—	2	1	no	—
81	D₄	—	2	1	no	—
82	D₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	254	1	no	—
85	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	127	1	no	—
88	C₈	—	127	1	no	—
89	C₈	—	254	1	no	—
90	C₈	—	254	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	—	8	1	no	—
94	C₂²	#14523247	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	508	1	no	—
97	C₄	D₄×D₁₂₇	1	1	yes	characteristic
98	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
99	C₂	—	2	1	no	—
100	C₂	—	4	1	no	—
101	C₂	—	8	1	no	—
102	C₂	#16861181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861239

C₂

yes

maximal, characteristic, complemented

C₂×#14523236

C₂

yes

maximal, characteristic, complemented

#16861216

C₂

yes

maximal, characteristic, complemented

#16861144

C₂

yes

maximal, characteristic, complemented

#16861127

C₂

yes

maximal, characteristic, complemented

#16861126

C₂

yes

maximal, characteristic, complemented

AGL(1,ℤ/8)×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#14523232

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

#41×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

SD₁₆×C₁₂₇

—

D₈×C₁₂₇

—

M₁₆×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

D₄

yes

characteristic

C₁₂₇⋊C₈

D₄

yes

characteristic

C₂³×C₁₂₇

—

Q₈×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#41

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₁₂₇

#471

yes

characteristic, complemented, Sylow

#471

—

127

maximal, Sylow

#137

—

127

—

AGL(1,ℤ/8)

D₁₂₇

yes

characteristic

C₂×SD₁₆

—

127

—

#131

—

127

—

#108

—

127

—

#101

—

127

—

#100

—

127

—

#41

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

254

—

SD₁₆

—

SD₁₆

—

254

—

SD₁₆

—

254

—

SD₁₆

—

254

—

D₈

—

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

C₂×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂³

—

Q₈

—

254

—

Q₈

—

254

—

Q₈

C₂²⋊D₁₂₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

127

—

C₈

—

127

—

C₈

—

254

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523247

yes

characteristic

C₄

—

C₄

—

508

—

C₄

D₄×D₁₂₇

yes

characteristic

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂

—

100

C₂

—

101

C₂

—

102

C₂

#16861181

yes

characteristic

	subgroup	G/H
center	C₂	#16861181
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	AGL(1,ℤ/8)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#121
hypercenter	AGL(1,ℤ/8)	D₁₂₇
hypocenter	C₁₂₇	#471
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

subgroup

G/H

center

C₂

#16861181

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

AGL(1,ℤ/8)×C₁₂₇

C₂

C₂×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

N	H ≅ G/N	type
1. #16861239	C₂	semidirect
2. C₂×#14523236	C₂	semidirect
3. #16861216	C₂	semidirect
4. #16861144	C₂	semidirect
5. #16861127	C₂	semidirect
6. #16861126	C₂	semidirect
51. C₁₂₇	#471	semidirect

H ≅ G/N

type

1. #16861239

C₂

semidirect

2. C₂×#14523236

C₂

semidirect

3. #16861216

C₂

semidirect

4. #16861144

C₂

semidirect

5. #16861127

C₂

semidirect

6. #16861126

C₂

semidirect

51. C₁₂₇

#471

semidirect

Subgroups — #22550137

Conjugacy classes of subgroups

Complements