Group #22550079 — subgroups

Subgroups — #22550079

Order 8128.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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	C₂×#14523236	C₂	1	1	yes	maximal, characteristic
2	#16861207	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861134	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861133	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₂₇	—	2	1	no	—
10	#14523236	—	2	2	no	—
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
13	Dic₁₂₇⋊C₄	—	2	1	no	—
14	Dic₁₂₇⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₂₇	—	2	1	no	—
16	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₂₇	—	2	2	no	—
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	Dic₂₅₄	—	2	1	no	—
22	Dic₂₅₄	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₂×Dic₁₂₇	—	4	2	no	—
26	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
27	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
28	C₁₂₇⋊C₈	—	2	1	no	—
29	C₂³×C₁₂₇	—	2	1	no	—
30	Q₈×C₁₂₇	—	2	1	no	—
31	Q₈×C₁₂₇	—	2	1	no	—
32	D₄×C₁₂₇	—	2	1	no	—
33	D₄×C₁₂₇	D₄	1	2	yes	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	—	2	2	no	—
38	C₂²×C₁₂₇	—	4	1	no	—
39	C₂²×C₁₂₇	—	4	1	no	—
40	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	#41	1	1	yes	characteristic
44	Dic₁₂₇	—	2	1	no	—
45	Dic₁₂₇	—	4	1	no	—
46	Dic₁₂₇	—	4	1	no	—
47	Dic₁₂₇	SD₁₆	1	2	yes	complemented
48	C₂×C₁₂₇	—	4	2	no	—
49	C₂×C₁₂₇	#121	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₂×C₁₂₇	#137	1	1	yes	characteristic
52	C₁₂₇	#461	1	1	yes	characteristic, complemented, Sylow
53	#461	—	127	1	no	maximal, Sylow
54	C₂×SD₁₆	—	127	1	no	—
55	C₂×SD₁₆	D₁₂₇	1	1	yes	characteristic
56	C₄⋊Q₈	—	127	1	no	—
57	C₄×D₄	—	127	1	no	—
58	C₄⋊C₈	—	127	1	no	—
59	#103	—	127	1	no	—
60	#103	—	127	1	no	—
61	C₂×Q₈	—	127	1	no	—
62	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
64	C₂²×C₄	—	254	1	no	—
65	SD₁₆	—	2	2	no	—
66	SD₁₆	—	254	2	no	—
67	C₂×C₈	—	127	1	no	—
68	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
69	C₄⋊C₄	—	127	1	no	—
70	C₄⋊C₄	—	254	1	no	—
71	C₄⋊C₄	—	254	1	no	—
72	C₂²⋊C₄	—	254	1	no	—
73	C₄²	—	127	1	no	—
74	C₂³	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	2	1	no	—
77	Q₈	—	254	1	no	—
78	Q₈	—	254	1	no	—
79	D₄	—	2	1	no	—
80	D₄	C₂²⋊D₁₂₇	1	2	yes	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	127	1	no	—
83	C₂×C₄	—	127	1	no	—
84	C₂×C₄	—	508	2	no	—
85	C₂×C₄	—	254	1	no	—
86	C₂×C₄	—	254	1	no	—
87	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	254	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
94	C₄	—	4	1	no	—
95	C₄	—	127	2	no	—
96	C₄	—	508	1	no	—
97	C₄	—	508	1	no	—
98	C₄	—	254	1	no	—
99	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	C₄	#14523247	1	1	yes	characteristic
101	C₂	—	4	2	no	—
102	C₂	#16861181	1	1	yes	characteristic
103	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
104	C₂	#16861238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523236

C₂

yes

maximal, characteristic

#16861207

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₂₇

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16861134

C₂

yes

maximal, characteristic, complemented

#16861133

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

#14523236

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

SD₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

SD₁₆

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#461

—

127

maximal, Sylow

C₂×SD₁₆

—

127

—

C₂×SD₁₆

D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

C₄×D₄

—

127

—

C₄⋊C₈

—

127

—

#103

—

127

—

#103

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

254

—

SD₁₆

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

D₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

127

—

C₄

—

508

—

C₄

—

508

—

C₄

—

254

—

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

C₄

#14523247

yes

characteristic

101

C₂

—

102

C₂

#16861181

yes

characteristic

103

C₂

SD₁₆×D₁₂₇

yes

characteristic

104

C₂

#16861238

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×SD₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₂₇
hypocenter	C₁₂₇	#461
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂×SD₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₂₇

hypocenter

C₁₂₇

#461

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
2. #16861207	C₂	semidirect
4. Dic₁₂₇⋊C₈	C₂	semidirect
5. #16861134	C₂	semidirect
6. #16861133	C₂	semidirect
47. Dic₁₂₇	SD₁₆	semidirect
52. C₁₂₇	#461	semidirect

H ≅ G/N

type

2. #16861207

C₂

semidirect

4. Dic₁₂₇⋊C₈

C₂

semidirect

5. #16861134

C₂

semidirect

6. #16861133

C₂

semidirect

47. Dic₁₂₇

SD₁₆

semidirect

52. C₁₂₇

#461

semidirect

Subgroups — #22550079

Conjugacy classes of subgroups

Complements