Group #22581663 — subgroups

Subgroups — #22581663

Order 8896.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	1	yes	maximal, characteristic, complemented
2	#17984834	C₂	1	2	yes	maximal, complemented
3	#17984820	C₂	1	4	yes	maximal, complemented
4	#17984818	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₅₆	C₂	1	2	yes	maximal
6	C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×D₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₇₈	C₂²	1	2	yes	—
10	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₃₉	C₂²	1	2	yes	—
12	C₂×C₄×D₁₃₉	C₂²	1	4	yes	—
13	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
14	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
16	Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
17	Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₄²×C₁₃₉	C₂²	1	2	yes	—
24	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂²×D₁₃₉	C₂³	1	2	yes	—
26	Dic₂₇₈	—	4	2	no	—
27	D₅₅₆	Q₈	1	4	yes	complemented
28	C₂×Dic₁₃₉	C₂³	1	2	yes	—
29	C₂×Dic₁₃₉	C₂³	1	4	yes	—
30	C₄×D₁₃₉	—	2	4	no	—
31	C₄×D₁₃₉	—	4	4	no	—
32	Q₈×C₁₃₉	—	2	2	no	—
33	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
35	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
36	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₃₉	—	2	2	no	—
40	C₂×D₁₃₉	—	2	2	no	—
41	C₂×D₁₃₉	C₂×Q₈	1	4	yes	—
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	—	2	2	no	—
44	C₄×C₁₃₉	—	2	1	no	—
45	C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
46	C₄×C₁₃₉	#41	1	2	yes	—
47	Dic₁₃₉	—	2	2	no	—
48	Dic₁₃₉	—	2	4	no	—
49	C₂×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
52	D₁₃₉	—	2	4	no	—
53	C₁₃₉	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	139	1	no	maximal, Sylow
55	C₄⋊Q₈	—	139	1	no	—
56	#125	—	139	2	no	—
57	C₂²⋊Q₈	—	139	2	no	—
58	C₂²⋊Q₈	—	139	4	no	—
59	C₄×Q₈	D₁₃₉	1	1	yes	characteristic, complemented
60	C₄×D₄	—	139	2	no	—
61	C₄×D₄	—	139	1	no	—
62	C₂×C₄⋊C₄	—	139	2	no	—
63	C₂×Q₈	—	139	2	no	—
64	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
65	C₂×D₄	—	139	1	no	—
66	C₂²×C₄	—	139	2	no	—
67	C₂²×C₄	—	139	4	no	—
68	C₄⋊C₄	—	139	4	no	—
69	C₄⋊C₄	—	139	2	no	—
70	C₄⋊C₄	—	139	4	no	—
71	C₄⋊C₄	—	139	1	no	—
72	C₄⋊C₄	—	139	2	no	—
73	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
75	C₂²⋊C₄	—	139	2	no	—
76	C₂²⋊C₄	—	139	4	no	—
77	C₄²	C₂×D₁₃₉	1	2	yes	—
78	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
79	C₂³	—	139	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	556	2	no	—
82	D₄	—	139	4	no	—
83	C₂×C₄	—	278	4	no	—
84	C₂×C₄	—	556	4	no	—
85	C₂×C₄	—	139	2	no	—
86	C₂×C₄	—	139	4	no	—
87	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
88	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
90	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
92	C₂²	—	278	2	no	—
93	C₂²	—	278	2	no	—
94	C₂²	—	139	4	no	—
95	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	278	2	no	—
100	C₄	—	278	4	no	—
101	C₄	Q₈×D₁₃₉	1	2	yes	—
102	C₄	#14555785	1	2	yes	—
103	C₂	—	278	4	no	—
104	C₂	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
105	C₂	C₂×#14555785	1	1	yes	characteristic
106	C₂	#17984888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984842

C₂

yes

maximal, characteristic, complemented

#17984834

C₂

yes

maximal, complemented

#17984820

C₂

yes

maximal, complemented

#17984818

C₂

yes

maximal, complemented

C₄×D₅₅₆

C₂

yes

maximal

C₄×D₅₅₆

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₃₉

C₂

yes

maximal, complemented

C₄×Q₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

—

Dic₂₇₈

—

D₅₅₆

Q₈

yes

complemented

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₈

yes

—

C₄×C₁₃₉

#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

H(5,𝔽₂)

yes

characteristic

D₁₃₉

—

C₁₃₉

#554

yes

characteristic, complemented, Sylow

#554

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

#125

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

C₄×Q₈

D₁₃₉

yes

characteristic, complemented

C₄×D₄

—

139

—

C₄×D₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₂²×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

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

C₂×D₁₃₉

yes

—

C₄²

C₂×D₁₃₉

yes

characteristic

C₂³

—

139

—

Q₈

—

Q₈

—

556

—

D₄

—

139

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

139

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

278

—

100

C₄

—

278

—

101

C₄

Q₈×D₁₃₉

yes

—

102

C₄

#14555785

yes

—

103

C₂

—

278

—

104

C₂

C₂×Q₈×D₁₃₉

yes

characteristic

105

C₂

C₂×#14555785

yes

characteristic

106

C₂

#17984888

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	C₄×Q₈×C₁₃₉	C₂
hypercenter	C₄×Q₈	D₁₃₉
hypocenter	C₁₃₉	#554
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

C₄×Q₈×C₁₃₉

C₂

hypercenter

C₄×Q₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #17984842	C₂	semidirect
2. #17984834	C₂	semidirect
3. #17984820	C₂	semidirect
4. #17984818	C₂	semidirect
7. C₄⋊C₄×D₁₃₉	C₂	semidirect
8. C₄×Q₈×C₁₃₉	C₂	semidirect
27. D₅₅₆	Q₈	semidirect
53. C₁₃₉	#554	semidirect
59. C₄×Q₈	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984842

C₂

semidirect

2. #17984834

C₂

semidirect

3. #17984820

C₂

semidirect

4. #17984818

C₂

semidirect

7. C₄⋊C₄×D₁₃₉

C₂

semidirect

8. C₄×Q₈×C₁₃₉

C₂

semidirect

27. D₅₅₆

Q₈

semidirect

53. C₁₃₉

#554

semidirect

59. C₄×Q₈

D₁₃₉

semidirect

Subgroups — #22581663

Conjugacy classes of subgroups

Complements