Group #22549996 — subgroups

Subgroups — #22549996

Order 8128.

104 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16861229	C₂	1	1	yes	maximal, characteristic
2	#16861215	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861144	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861136	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861135	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861119	C₂	1	1	yes	maximal, characteristic, complemented
7	#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	C₂²	1	1	yes	characteristic
9	#14523238	C₄	1	1	yes	characteristic, complemented
10	#14523236	C₄	1	1	yes	characteristic, complemented
11	#14523235	C₄	1	1	yes	characteristic, complemented
12	#14523233	C₄	1	1	yes	characteristic, complemented
13	#14523232	C₂²	1	1	yes	characteristic
14	#14523231	—	2	1	no	—
15	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₈×D₁₂₇	—	2	1	no	—
17	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	#41×C₁₂₇	C₂²	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	2	1	no	—
20	M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	—	2	1	no	—
22	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
24	D₅₀₈	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊D₁₂₇	—	2	1	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₄×D₁₂₇	—	2	1	no	—
28	C₁₂₇⋊C₈	—	2	1	no	—
29	C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	D₄×C₁₂₇	—	2	1	no	—
33	D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₂₇	—	2	1	no	—
37	C₈×C₁₂₇	D₄	1	1	yes	characteristic
38	C₈×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	2	1	no	—
40	C₂²×C₁₂₇	#41	1	1	yes	characteristic
41	C₂×D₁₂₇	—	2	1	no	—
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	Dic₁₂₇	—	2	1	no	—
46	Dic₁₂₇	—	4	1	no	—
47	C₂×C₁₂₇	—	2	1	no	—
48	C₂×C₁₂₇	—	4	1	no	—
49	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
50	D₁₂₇	—	4	1	no	—
51	C₁₂₇	#444	1	1	yes	characteristic, complemented, Sylow
52	#444	—	127	1	no	maximal, Sylow
53	#135	—	127	1	no	—
54	#131	—	127	1	no	—
55	#131	D₁₂₇	1	1	yes	characteristic, complemented
56	#108	—	127	1	no	—
57	#104	—	127	1	no	—
58	#104	—	127	1	no	—
59	#97	—	127	1	no	—
60	#41	—	127	1	no	—
61	#41	C₂×D₁₂₇	1	1	yes	characteristic
62	Q₁₆	—	127	1	no	—
63	SD₁₆	—	127	1	no	—
64	SD₁₆	—	127	1	no	—
65	D₈	—	127	1	no	—
66	M₁₆	—	2	1	no	—
67	M₁₆	—	127	1	no	—
68	M₁₆	—	254	1	no	—
69	M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	127	1	no	—
72	C₂×C₈	—	254	1	no	—
73	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₄²	—	127	1	no	—
75	Q₈	—	127	1	no	—
76	Q₈	C₄×D₁₂₇	1	1	yes	characteristic
77	D₄	—	2	1	no	—
78	D₄	—	127	1	no	—
79	D₄	—	254	1	no	—
80	D₄	C₄×D₁₂₇	1	1	yes	characteristic
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	254	1	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₈	—	2	1	no	—
86	C₈	—	127	1	no	—
87	C₈	—	127	1	no	—
88	C₈	—	254	1	no	—
89	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
91	C₂²	—	2	1	no	—
92	C₂²	—	254	1	no	—
93	C₂²	#14523248	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	508	1	no	—
96	C₄	—	254	1	no	—
97	C₄	C₂×C₄×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	—	508	1	no	—
102	C₂	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861229

C₂

yes

maximal, characteristic

#16861215

C₂

yes

maximal, characteristic, complemented

#16861144

C₂

yes

maximal, characteristic, complemented

#16861136

C₂

yes

maximal, characteristic, complemented

#16861135

C₂

yes

maximal, characteristic, complemented

#16861119

C₂

yes

maximal, characteristic, complemented

#131×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

C₂²

yes

characteristic

#14523238

C₄

yes

characteristic, complemented

#14523236

C₄

yes

characteristic, complemented

#14523235

C₄

yes

characteristic, complemented

#14523233

C₄

yes

characteristic, complemented

#14523232

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×D₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

#41×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

C₂×C₄

yes

characteristic

D₅₀₈

C₂×C₄

yes

characteristic

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

Q₈×C₁₂₇

C₂×C₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

D₄

yes

characteristic

C₈×C₁₂₇

D₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#41

yes

characteristic

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#444

yes

characteristic, complemented, Sylow

#444

—

127

maximal, Sylow

#135

—

127

—

#131

—

127

—

#131

D₁₂₇

yes

characteristic, complemented

#108

—

127

—

#104

—

127

—

#104

—

127

—

#97

—

127

—

#41

—

127

—

#41

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

127

—

D₈

—

127

—

M₁₆

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

C₂×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄²

—

127

—

Q₈

—

127

—

Q₈

C₄×D₁₂₇

yes

characteristic

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

127

—

C₈

—

127

—

C₈

—

254

—

C₈

C₂²⋊D₁₂₇

yes

characteristic

C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

254

—

C₂²

#14523248

yes

characteristic

C₄

—

C₄

—

508

—

C₄

—

254

—

C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂

—

100

C₂

—

101

C₂

—

508

—

102

C₂

C₄×C₂²⋊D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₄×D₄
hypercenter	#131	D₁₂₇
hypocenter	C₁₂₇	#444
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #131

subgroup

G/H

center

C₄

C₂×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#131×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #131

N	H ≅ G/N	type
2. #16861215	C₂	semidirect
3. #16861144	C₂	semidirect
4. #16861136	C₂	semidirect
5. #16861135	C₂	semidirect
6. #16861119	C₂	semidirect
7. #131×C₁₂₇	C₂	semidirect
9. #14523238	C₄	semidirect
10. #14523236	C₄	semidirect
11. #14523235	C₄	semidirect
12. #14523233	C₄	semidirect
51. C₁₂₇	#444	semidirect
55. #131	D₁₂₇	semidirect

H ≅ G/N

type

2. #16861215

C₂

semidirect

3. #16861144

C₂

semidirect

4. #16861136

C₂

semidirect

5. #16861135

C₂

semidirect

6. #16861119

C₂

semidirect

7. #131×C₁₂₇

C₂

semidirect

9. #14523238

C₄

semidirect

10. #14523236

C₄

semidirect

11. #14523235

C₄

semidirect

12. #14523233

C₄

semidirect

51. C₁₂₇

#444

semidirect

55. #131

D₁₂₇

semidirect

Subgroups — #22549996

Conjugacy classes of subgroups

Complements