Group #22550374 — subgroups

Subgroups — C₂×#16861188 #22550374

Order 8128.

322 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 143 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	C₂²×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861188	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Dic₁₂₇	C₂	1	2	yes	maximal
6	C₂³×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₅₄	—	2	4	no	—
8	C₂×Dic₂₅₄	C₂²	1	4	yes	complemented
9	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
10	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	4	yes	complemented
12	Dic₁₂₇⋊C₄	C₂²	1	8	yes	complemented
13	C₂²⋊Dic₁₂₇	C₂²	1	8	yes	—
14	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	—	2	4	no	—
16	C₂²×C₄×C₁₂₇	C₂²	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	Dic₂₅₄	—	2	8	no	—
20	C₂×Dic₁₂₇	—	2	4	no	—
21	C₂×Dic₁₂₇	—	2	4	no	—
22	C₂×Dic₁₂₇	C₂³	1	4	yes	—
23	C₂×Dic₁₂₇	C₂³	1	4	yes	—
24	C₂³×C₁₂₇	—	2	2	no	—
25	C₂³×C₁₂₇	—	2	2	no	—
26	C₂³×C₁₂₇	Q₈	1	4	yes	complemented
27	C₂³×C₁₂₇	C₂³	1	2	yes	—
28	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	8	no	—
31	C₂×C₄×C₁₂₇	D₄	1	4	yes	—
32	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×C₁₂₇	—	2	8	no	—
35	C₂²×C₁₂₇	—	2	2	no	—
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
38	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₂₇	C₂×Q₈	1	4	yes	complemented
40	C₂²×C₁₂₇	#41	1	2	yes	—
41	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₁₂₇	—	2	2	no	—
43	C₄×C₁₂₇	C₂×D₄	1	4	yes	—
44	Dic₁₂₇	—	2	4	no	—
45	Dic₁₂₇	—	2	4	no	—
46	C₂×C₁₂₇	—	2	4	no	—
47	C₂×C₁₂₇	C₂²⋊Q₈	1	4	yes	complemented
48	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
51	C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₂²⋊Q₈	—	127	1	no	maximal, Sylow
53	C₂²×Q₈	—	127	1	no	—
54	C₂³×C₄	D₁₂₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	127	8	no	—
56	C₂×C₄⋊C₄	—	127	2	no	—
57	C₂×C₄⋊C₄	—	127	1	no	—
58	C₂×C₂²⋊C₄	—	127	2	no	—
59	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	—	127	4	no	—
61	C₂×Q₈	—	254	4	no	—
62	C₂²×C₄	—	2	4	no	—
63	C₂²×C₄	—	127	2	no	—
64	C₂²×C₄	—	127	2	no	—
65	C₂²×C₄	C₂×D₁₂₇	1	4	yes	—
66	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	127	8	no	—
69	C₄⋊C₄	—	127	4	no	—
70	C₂²⋊C₄	—	127	8	no	—
71	C₂³	—	2	2	no	—
72	C₂³	—	2	2	no	—
73	C₂³	Dic₂₅₄	1	4	yes	complemented
74	C₂³	C₂²×D₁₂₇	1	2	yes	—
75	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
76	Q₈	—	254	8	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	8	no	—
79	C₂×C₄	—	127	4	no	—
80	C₂×C₄	—	127	4	no	—
81	C₂×C₄	—	254	4	no	—
82	C₂×C₄	—	254	4	no	—
83	C₂×C₄	C₂²⋊D₁₂₇	1	4	yes	—
84	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
85	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
86	C₂²	—	2	8	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	2	no	—
89	C₂²	C₂×C₂²⋊D₁₂₇	1	2	yes	—
90	C₂²	C₂×Dic₂₅₄	1	2	yes	—
91	C₂²	C₂×Dic₂₅₄	1	4	yes	complemented
92	C₂²	#14523248	1	2	yes	—
93	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	254	4	no	—
96	C₄	—	254	4	no	—
97	C₄	C₂×C₂²⋊D₁₂₇	1	4	yes	—
98	C₂	—	2	4	no	—
99	C₂	#16861188	1	4	yes	direct product factor
100	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
101	C₂	C₂²×Dic₂₅₄	1	1	yes	characteristic
102	C₂	C₂×#14523248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal

C₂³×C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂²

yes

complemented

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

complemented

Dic₁₂₇⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

Q₈

yes

complemented

C₂³×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂×Q₈

yes

complemented

C₂²×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²⋊Q₈

yes

complemented

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₁₂₇

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

127

maximal, Sylow

C₂²×Q₈

—

127

—

C₂³×C₄

D₁₂₇

yes

characteristic

C₂²⋊Q₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂²×C₄

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂³

—

C₂³

—

C₂³

Dic₂₅₄

yes

complemented

C₂³

C₂²×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

C₂²

C₂×Dic₂₅₄

yes

—

C₂²

C₂×Dic₂₅₄

yes

complemented

C₂²

#14523248

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂×C₂²⋊D₁₂₇

yes

—

C₂

—

C₂

#16861188

yes

direct product factor

100

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

101

C₂

C₂²×Dic₂₅₄

yes

characteristic

102

C₂

C₂×#14523248

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂⁴
Frattini	C₂²	C₂³×D₁₂₇
Fitting	C₂³×C₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂³×C₄	D₁₂₇
hypocenter	C₁₂₇	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂⁴

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂³×C₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂³×C₄

D₁₂₇

hypocenter

C₁₂₇

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂²×Dic₂₅₄	C₂	semidirect
2. #16861188	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
4. C₂×Dic₁₂₇⋊C₄	C₂	semidirect
8. C₂×Dic₂₅₄	C₂²	semidirect
11. C₄⋊Dic₁₂₇	C₂²	semidirect
12. Dic₁₂₇⋊C₄	C₂²	semidirect
26. C₂³×C₁₂₇	Q₈	semidirect
39. C₂²×C₁₂₇	C₂×Q₈	semidirect
47. C₂×C₁₂₇	C₂²⋊Q₈	semidirect
51. C₁₂₇	C₂×C₂²⋊Q₈	semidirect
73. C₂³	Dic₂₅₄	semidirect
91. C₂²	C₂×Dic₂₅₄	semidirect
99. C₂	#16861188	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₂₅₄

C₂

semidirect

2. #16861188

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

4. C₂×Dic₁₂₇⋊C₄

C₂

semidirect

8. C₂×Dic₂₅₄

C₂²

semidirect

11. C₄⋊Dic₁₂₇

C₂²

semidirect

12. Dic₁₂₇⋊C₄

C₂²

semidirect

26. C₂³×C₁₂₇

Q₈

semidirect

39. C₂²×C₁₂₇

C₂×Q₈

semidirect

47. C₂×C₁₂₇

C₂²⋊Q₈

semidirect

51. C₁₂₇

C₂×C₂²⋊Q₈

semidirect

73. C₂³

Dic₂₅₄

semidirect

91. C₂²

C₂×Dic₂₅₄

semidirect

99. C₂

#16861188

direct (N × H)

Subgroups — C₂×#16861188 #22550374

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16861188 #22550374

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16861188 #22550374