Group #22581408 — subgroups

Subgroups — C₂×#17984800 #22581408

Order 8896.

330 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 167 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×C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#17984800	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₁₃₉)⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₃₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₃₉	C₄	1	8	yes	complemented
11	C₂×C₄×D₁₃₉	C₂²	1	4	yes	—
12	C₄⋊Dic₁₃₉	C₂²	1	4	yes	complemented
13	(C₂×D₁₃₉)⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₁₃₉	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₁₃₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	complemented
18	C₂²×D₁₃₉	—	2	2	no	—
19	C₂²×D₁₃₉	—	2	2	no	—
20	C₂²×D₁₃₉	C₂×C₄	1	4	yes	—
21	C₂²×D₁₃₉	C₂³	1	2	yes	—
22	C₂×Dic₁₃₉	—	2	4	no	—
23	C₂×Dic₁₃₉	C₂×C₄	1	4	yes	—
24	C₂×Dic₁₃₉	C₂³	1	4	yes	—
25	C₂×Dic₁₃₉	C₂³	1	2	yes	—
26	C₄×D₁₃₉	C₂×C₄	1	16	yes	complemented
27	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₃₉	—	2	4	no	—
29	C₂×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
30	C₂×C₄×C₁₃₉	C₂³	1	4	yes	—
31	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
32	C₂²×C₁₃₉	C₂²×C₄	1	2	yes	—
33	C₂²×C₁₃₉	#41	1	2	yes	—
34	C₂²×C₁₃₉	#41	1	2	yes	—
35	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₃₉	—	2	8	no	—
37	C₂×D₁₃₉	—	2	2	no	—
38	C₂×D₁₃₉	—	2	2	no	—
39	C₂×D₁₃₉	C₂²×C₄	1	4	yes	—
40	C₄×C₁₃₉	—	2	4	no	—
41	C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
42	Dic₁₃₉	—	2	4	no	—
43	Dic₁₃₉	C₂²×C₄	1	4	yes	—
44	C₂×C₁₃₉	#117	1	4	yes	complemented
45	C₂×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
46	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
47	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
48	D₁₃₉	—	2	4	no	—
49	C₁₃₉	C₂×#117	1	1	yes	characteristic, complemented, Sylow
50	C₂×#117	—	139	1	no	maximal, Sylow
51	C₂³×C₄	—	139	1	no	—
52	#117	—	139	8	no	—
53	C₂×C₄⋊C₄	—	139	1	no	—
54	C₂×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic, complemented
55	C₂×C₂²⋊C₄	—	139	2	no	—
56	C₂×C₄²	—	139	2	no	—
57	C₂⁴	—	139	1	no	—
58	C₂²×C₄	—	139	2	no	—
59	C₂²×C₄	—	139	1	no	—
60	C₂²×C₄	—	139	8	no	—
61	C₂²×C₄	—	139	4	no	—
62	C₂²×C₄	C₂×D₁₃₉	1	2	yes	—
63	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
64	C₄⋊C₄	—	139	4	no	—
65	C₄⋊C₄	C₂×D₁₃₉	1	4	yes	complemented
66	C₂²⋊C₄	—	139	8	no	—
67	C₄²	—	139	8	no	—
68	C₂³	—	278	2	no	—
69	C₂³	—	278	2	no	—
70	C₂³	—	139	4	no	—
71	C₂³	—	139	2	no	—
72	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	278	4	no	—
75	C₂×C₄	—	139	4	no	—
76	C₂×C₄	—	139	4	no	—
77	C₂×C₄	—	139	2	no	—
78	C₂×C₄	—	139	16	no	—
79	C₂×C₄	C₄×D₁₃₉	1	4	yes	—
80	C₂×C₄	C₂²×D₁₃₉	1	4	yes	—
81	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
82	C₂²	—	278	8	no	—
83	C₂²	—	278	2	no	—
84	C₂²	—	278	2	no	—
85	C₂²	—	139	4	no	—
86	C₂²	C₂×C₄×D₁₃₉	1	2	yes	—
87	C₂²	#14555784	1	2	yes	—
88	C₂²	#14555786	1	2	yes	—
89	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	278	4	no	—
92	C₄	—	139	4	no	—
93	C₄	C₂×C₄×D₁₃₉	1	4	yes	—
94	C₂	—	278	4	no	—
95	C₂	#17984800	1	4	yes	direct product factor
96	C₂	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
97	C₂	C₂×#14555784	1	1	yes	characteristic
98	C₂	C₂×#14555786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₃₉

C₂

yes

maximal, characteristic

#17984800

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₁₃₉)⋊C₄

C₂

yes

maximal

C₂×C₄×Dic₁₃₉

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₄

yes

complemented

C₂×C₄×D₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

complemented

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

complemented

C₂²×C₄×C₁₃₉

C₂²

yes

—

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂×C₄

yes

—

C₂²×D₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₄×D₁₃₉

C₂×C₄

yes

complemented

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₄

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₁₃₉

C₂²×C₄

yes

—

C₂²×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂²×C₄

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×C₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

C₂²×C₄

yes

—

C₂×C₁₃₉

#117

yes

complemented

C₂×C₁₃₉

C₂³×C₄

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

D₁₃₉

—

C₁₃₉

C₂×#117

yes

characteristic, complemented, Sylow

C₂×#117

—

139

maximal, Sylow

C₂³×C₄

—

139

—

#117

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

D₁₃₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂⁴

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

complemented

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

139

—

C₂³

—

139

—

C₂³

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₄×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

139

—

C₂²

C₂×C₄×D₁₃₉

yes

—

C₂²

#14555784

yes

—

C₂²

#14555786

yes

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

—

139

—

C₄

C₂×C₄×D₁₃₉

yes

—

C₂

—

278

—

C₂

#17984800

yes

direct product factor

C₂

C₂²×C₄×D₁₃₉

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂³	C₂²×D₁₃₉
commutator	C₂×C₁₃₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₃₉
Fitting	C₂×C₄⋊C₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₂×#117
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₃₉

commutator

C₂×C₁₃₉

C₂³×C₄

Frattini

C₂²

C₂³×D₁₃₉

Fitting

C₂×C₄⋊C₄×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #17984800	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₃₉	C₂	semidirect
5. C₂×C₄×Dic₁₃₉	C₂	semidirect
6. C₂×C₄⋊C₄×C₁₃₉	C₂	semidirect
10. C₂×C₄×D₁₃₉	C₄	semidirect
12. C₄⋊Dic₁₃₉	C₂²	semidirect
14. C₄×Dic₁₃₉	C₂²	semidirect
17. C₄⋊C₄×C₁₃₉	C₂²	semidirect
26. C₄×D₁₃₉	C₂×C₄	semidirect
44. C₂×C₁₃₉	#117	semidirect
49. C₁₃₉	C₂×#117	semidirect
54. C₂×C₄⋊C₄	D₁₃₉	semidirect
65. C₄⋊C₄	C₂×D₁₃₉	semidirect
95. C₂	#17984800	direct (N × H)

H ≅ G/N

type

2. #17984800

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₃₉

C₂

semidirect

5. C₂×C₄×Dic₁₃₉

C₂

semidirect

6. C₂×C₄⋊C₄×C₁₃₉

C₂

semidirect

10. C₂×C₄×D₁₃₉

C₄

semidirect

12. C₄⋊Dic₁₃₉

C₂²

semidirect

14. C₄×Dic₁₃₉

C₂²

semidirect

17. C₄⋊C₄×C₁₃₉

C₂²

semidirect

26. C₄×D₁₃₉

C₂×C₄

semidirect

44. C₂×C₁₃₉

#117

semidirect

49. C₁₃₉

C₂×#117

semidirect

54. C₂×C₄⋊C₄

D₁₃₉

semidirect

65. C₄⋊C₄

C₂×D₁₃₉

semidirect

95. C₂

#17984800

direct (N × H)

Subgroups — C₂×#17984800 #22581408

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17984800 #22581408

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17984800 #22581408