Group #22549774 — subgroups

Subgroups — #22549774

Order 8128.

170 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 87 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×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal
4	C₁₂₇⋊#95	C₂	1	2	yes	maximal
5	C₂×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
7	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
8	C₄⋊Dic₁₂₇	—	2	2	no	—
9	C₄⋊Dic₁₂₇	C₄	1	4	yes	complemented
10	Dic₁₂₇⋊C₄	—	2	4	no	—
11	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄²×C₁₂₇	—	2	2	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	—	2	2	no	—
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₂×Dic₁₂₇	—	2	2	no	—
20	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	—
21	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	—	2	1	no	—
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	—	2	1	no	—
25	C₂×C₄×C₁₂₇	—	2	1	no	—
26	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	Q₈	1	2	yes	—
30	C₂×C₄×C₁₂₇	Q₈	1	2	yes	—
31	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₂₇	#41	1	1	yes	characteristic
37	C₂²×C₁₂₇	#41	1	1	yes	characteristic
38	C₄×C₁₂₇	—	2	2	no	—
39	C₄×C₁₂₇	—	2	2	no	—
40	C₄×C₁₂₇	C₄⋊C₄	1	4	yes	complemented
41	Dic₁₂₇	—	2	4	no	—
42	Dic₁₂₇	—	4	2	no	—
43	C₂×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
45	C₂×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₁₂₇	#121	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₁₂₇	#125	1	1	yes	characteristic
49	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₁₂₇	#389	1	1	yes	characteristic, complemented, Sylow
51	#389	—	127	1	no	maximal, Sylow
52	C₂×C₄⋊C₄	—	127	1	no	—
53	C₂×C₄⋊C₄	—	127	2	no	—
54	C₂×C₄⋊C₄	—	127	1	no	—
55	C₂×C₄²	D₁₂₇	1	1	yes	characteristic
56	#95	—	127	2	no	—
57	C₂²×C₄	—	127	2	no	—
58	C₂²×C₄	—	127	2	no	—
59	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	127	4	no	—
63	C₄⋊C₄	—	254	2	no	—
64	C₄⋊C₄	—	254	4	no	—
65	C₄²	—	2	2	no	—
66	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	127	4	no	—
72	C₂×C₄	—	254	2	no	—
73	C₂×C₄	—	254	2	no	—
74	C₂×C₄	—	254	2	no	—
75	C₂×C₄	—	254	2	no	—
76	C₂×C₄	—	254	2	no	—
77	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
78	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
79	C₂×C₄	D₅₀₈	1	2	yes	—
80	C₂×C₄	Dic₂₅₄	1	2	yes	—
81	C₂×C₄	Dic₂₅₄	1	2	yes	—
82	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
83	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
84	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
85	C₂²	C₂×Dic₂₅₄	1	1	yes	characteristic
86	C₂²	C₂×Dic₂₅₄	1	1	yes	characteristic
87	C₂²	#14523248	1	1	yes	characteristic
88	C₂²	#14523248	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	508	2	no	—
92	C₄	—	254	4	no	—
93	C₄	Dic₁₂₇⋊C₄	1	4	yes	complemented
94	C₂	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
95	C₂	C₄×D₅₀₈	1	1	yes	characteristic
96	C₂	C₄×Dic₂₅₄	1	1	yes	characteristic
97	C₂	#16861183	1	1	yes	characteristic
98	C₂	#16861188	1	1	yes	characteristic
99	C₂	#16861201	1	1	yes	characteristic
100	C₂	#16861209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal

C₁₂₇⋊#95

C₂

yes

maximal

C₂×C₄²×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₄

yes

complemented

Dic₁₂₇⋊C₄

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

Q₈

yes

—

C₂×C₄×C₁₂₇

Q₈

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊C₄

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₄×Q₈

yes

characteristic

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₂₇

#125

yes

characteristic

C₂×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₁₂₇

#389

yes

characteristic, complemented, Sylow

#389

—

127

maximal, Sylow

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

D₁₂₇

yes

characteristic

#95

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

D₅₀₈

yes

—

C₂×C₄

Dic₂₅₄

yes

—

C₂×C₄

Dic₂₅₄

yes

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂²

C₂×D₅₀₈

yes

characteristic

C₂²

C₂×Dic₂₅₄

yes

characteristic

C₂²

C₂×Dic₂₅₄

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

508

—

C₄

—

254

—

C₄

Dic₁₂₇⋊C₄

yes

complemented

C₂

C₂×Dic₁₂₇⋊C₄

yes

characteristic

C₂

C₄×D₅₀₈

yes

characteristic

C₂

C₄×Dic₂₅₄

yes

C₂

yes

C₂

yes

C₂

yes

100

C₂

#16861209

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂×C₄²	D₁₂₇
hypocenter	C₁₂₇	#389
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂×C₄²×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₄²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
9. C₄⋊Dic₁₂₇	C₄	semidirect
40. C₄×C₁₂₇	C₄⋊C₄	semidirect
50. C₁₂₇	#389	semidirect
93. C₄	Dic₁₂₇⋊C₄	semidirect

H ≅ G/N

type

9. C₄⋊Dic₁₂₇

C₄

semidirect

40. C₄×C₁₂₇

C₄⋊C₄

semidirect

50. C₁₂₇

#389

semidirect

93. C₄

Dic₁₂₇⋊C₄

semidirect

Subgroups — #22549774

Conjugacy classes of subgroups

Complements