Group #22581581 — subgroups

Subgroups — #22581581

Order 8896.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	#17984842	C₂	1	2	yes	maximal, complemented
2	#17984841	C₂	1	2	yes	maximal, complemented
3	#17984834	C₂	1	2	yes	maximal, complemented
4	#17984833	C₂	1	2	yes	maximal, complemented
5	#17984819	C₂	1	4	yes	maximal
6	#17984801	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₇₈	C₂²	1	4	yes	—
10	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
11	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
12	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
13	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
16	Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
17	Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
19	C₄×Dic₁₃₉	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₄²×C₁₃₉	C₂²	1	2	yes	—
25	Dic₂₇₈	—	4	4	no	—
26	C₂×Dic₁₃₉	—	4	2	no	—
27	C₂×Dic₁₃₉	C₂³	1	2	yes	—
28	C₂×Dic₁₃₉	C₂³	1	2	yes	—
29	C₂×Dic₁₃₉	C₂³	1	4	yes	—
30	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₃₉	Q₈	1	4	yes	complemented
32	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
33	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
34	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂²×C₁₃₉	—	2	1	no	—
37	C₂²×C₁₃₉	—	2	1	no	—
38	C₂²×C₁₃₉	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₃₉	—	2	2	no	—
41	C₄×C₁₃₉	—	2	2	no	—
42	C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
43	C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
44	Dic₁₃₉	—	2	4	no	—
45	Dic₁₃₉	—	2	2	no	—
46	Dic₁₃₉	—	2	2	no	—
47	C₂×C₁₃₉	—	2	2	no	—
48	C₂×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
51	C₁₃₉	#544	1	1	yes	characteristic, complemented, Sylow
52	#544	—	139	1	no	maximal, Sylow
53	C₄⋊Q₈	—	139	2	no	—
54	C₄⋊Q₈	—	139	2	no	—
55	#125	—	139	2	no	—
56	#125	—	139	2	no	—
57	C₂²⋊Q₈	—	139	4	no	—
58	#117	—	139	1	no	—
59	#117	D₁₃₉	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	139	1	no	—
61	C₂×Q₈	—	139	4	no	—
62	C₂²×C₄	—	139	2	no	—
63	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
64	C₄⋊C₄	—	139	2	no	—
65	C₄⋊C₄	—	139	2	no	—
66	C₄⋊C₄	—	139	4	no	—
67	C₄⋊C₄	—	139	4	no	—
68	C₄⋊C₄	—	139	1	no	—
69	C₄⋊C₄	—	139	1	no	—
70	C₄⋊C₄	—	139	4	no	—
71	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
73	C₂²⋊C₄	—	139	2	no	—
74	C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
75	C₄²	—	139	2	no	—
76	C₄²	C₂×D₁₃₉	1	2	yes	—
77	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
78	Q₈	—	556	4	no	—
79	C₂×C₄	—	556	2	no	—
80	C₂×C₄	—	139	2	no	—
81	C₂×C₄	—	139	2	no	—
82	C₂×C₄	—	139	4	no	—
83	C₂×C₄	Dic₂₇₈	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
85	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
86	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₂₇₈	1	2	yes	—
91	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	278	4	no	—
95	C₄	—	278	2	no	—
96	C₄	—	278	2	no	—
97	C₄	C₂×Dic₂₇₈	1	2	yes	—
98	C₄	C₂×Dic₂₇₈	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₂₇₈	1	1	yes	characteristic
101	C₂	#17984888	1	1	yes	characteristic
102	C₂	#17984889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984842

C₂

yes

maximal, complemented

#17984841

C₂

yes

maximal, complemented

#17984834

C₂

yes

maximal, complemented

#17984833

C₂

yes

maximal, complemented

#17984819

C₂

yes

maximal

#17984801

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₃₉

C₂

yes

maximal, characteristic

#117×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

C₂²

yes

—

C₂²×Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₃₉

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

—

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

Q₈

yes

complemented

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₈

yes

—

C₄×C₁₃₉

C₂×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₃₉

2₋¹⁺⁴

yes

characteristic

C₁₃₉

#544

yes

characteristic, complemented, Sylow

#544

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

139

—

#125

—

139

—

C₂²⋊Q₈

—

139

—

#117

—

139

—

#117

D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

139

—

C₂×Q₈

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

139

—

C₂²⋊C₄

C₂×D₁₃₉

yes

—

C₄²

—

139

—

C₄²

C₂×D₁₃₉

yes

—

C₂³

C₂²×D₁₃₉

yes

characteristic

Q₈

—

556

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

Dic₂₇₈

yes

complemented

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₂₇₈

yes

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

278

—

C₄

—

278

—

C₄

—

278

—

C₄

C₂×Dic₂₇₈

yes

—

C₄

C₂×Dic₂₇₈

yes

—

C₂

—

100

C₂

C₂²×Dic₂₇₈

yes

characteristic

101

C₂

#17984888

yes

characteristic

102

C₂

#17984889

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	#117×C₁₃₉	C₂
hypercenter	#117	D₁₃₉
hypocenter	C₁₃₉	#544
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

#117×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #17984842	C₂	semidirect
2. #17984841	C₂	semidirect
3. #17984834	C₂	semidirect
4. #17984833	C₂	semidirect
31. C₂×C₄×C₁₃₉	Q₈	semidirect
51. C₁₃₉	#544	semidirect
83. C₂×C₄	Dic₂₇₈	semidirect

H ≅ G/N

type

1. #17984842

C₂

semidirect

2. #17984841

C₂

semidirect

3. #17984834

C₂

semidirect

4. #17984833

C₂

semidirect

31. C₂×C₄×C₁₃₉

Q₈

semidirect

51. C₁₃₉

#544

semidirect

83. C₂×C₄

Dic₂₇₈

semidirect

Subgroups — #22581581

Conjugacy classes of subgroups

Complements