Group #22581226 — subgroups

Subgroups — #22581226

Order 8896.

100 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 37 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	#17984872	C₂	1	1	yes	maximal, characteristic
2	C₂×#14555775	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984849	C₂	1	1	yes	maximal, characteristic
4	#17984777	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984761	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984760	C₂	1	1	yes	maximal, characteristic, complemented
7	#137×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
9	#14555775	—	2	1	no	—
10	#14555775	—	2	1	no	—
11	#14555775	—	2	1	no	—
12	#14555773	—	2	1	no	—
13	#14555769	—	2	1	no	—
14	#14555769	C₂²	1	1	yes	characteristic
15	#14555769	C₂²	1	1	yes	characteristic
16	C₂×C₁₃₉⋊C₈	—	2	1	no	—
17	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
18	#41×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	Q₁₆×C₁₃₉	—	2	1	no	—
21	SD₁₆×C₁₃₉	—	2	1	no	—
22	M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
23	Dic₂₇₈	—	2	1	no	—
24	Dic₂₇₈	—	2	1	no	—
25	C₂×Dic₁₃₉	—	2	1	no	—
26	C₁₃₉⋊C₈	—	2	1	no	—
27	C₁₃₉⋊C₈	—	2	1	no	—
28	C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
29	C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
30	Q₈×C₁₃₉	—	2	1	no	—
31	Q₈×C₁₃₉	—	2	1	no	—
32	Q₈×C₁₃₉	D₄	1	1	yes	characteristic
33	D₄×C₁₃₉	—	2	1	no	—
34	D₄×C₁₃₉	D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	—	2	1	no	—
36	C₂×C₄×C₁₃₉	—	2	1	no	—
37	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₈×C₁₃₉	—	2	1	no	—
39	C₂²×C₁₃₉	—	2	1	no	—
40	C₂²×C₁₃₉	#41	1	1	yes	characteristic
41	C₄×C₁₃₉	—	2	1	no	—
42	C₄×C₁₃₉	—	4	1	no	—
43	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	Dic₁₃₉	—	4	1	no	—
46	C₂×C₁₃₉	—	2	1	no	—
47	C₂×C₁₃₉	—	4	1	no	—
48	C₂×C₁₃₉	#121	1	1	yes	characteristic
49	C₁₃₉	#473	1	1	yes	characteristic, complemented, Sylow
50	#473	—	139	1	no	maximal, Sylow
51	#137	—	139	1	no	—
52	#137	D₁₃₉	1	1	yes	characteristic
53	C₂×Q₁₆	—	139	1	no	—
54	#131	—	139	1	no	—
55	#108	—	139	1	no	—
56	#101	—	139	1	no	—
57	#101	—	139	1	no	—
58	#41	C₂×D₁₃₉	1	1	yes	characteristic
59	C₂×Q₈	—	139	1	no	—
60	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
61	Q₁₆	—	2	1	no	—
62	Q₁₆	—	278	1	no	—
63	Q₁₆	—	278	1	no	—
64	Q₁₆	—	278	1	no	—
65	SD₁₆	—	2	1	no	—
66	SD₁₆	—	278	1	no	—
67	M₁₆	—	278	1	no	—
68	M₁₆	—	139	1	no	—
69	M₁₆	—	139	1	no	—
70	M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
71	C₂×C₈	—	278	1	no	—
72	C₂×C₈	—	139	1	no	—
73	Q₈	—	2	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	278	1	no	—
76	Q₈	—	278	1	no	—
77	Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
78	D₄	—	2	1	no	—
79	D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	278	1	no	—
83	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	278	1	no	—
86	C₈	—	278	1	no	—
87	C₈	—	139	1	no	—
88	C₈	—	139	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	#14555784	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	4	1	no	—
93	C₄	—	556	1	no	—
94	C₄	D₄×D₁₃₉	1	1	yes	characteristic
95	C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	1	no	—
98	C₂	#17984814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984872

C₂

yes

maximal, characteristic

C₂×#14555775

C₂

yes

maximal, characteristic, complemented

#17984849

C₂

yes

maximal, characteristic

#17984777

C₂

yes

maximal, characteristic, complemented

#17984761

C₂

yes

maximal, characteristic, complemented

#17984760

C₂

yes

maximal, characteristic, complemented

#137×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#14555769

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

#41×C₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

Q₁₆×C₁₃₉

—

SD₁₆×C₁₃₉

—

M₁₆×C₁₃₉

C₂²

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

D₄

yes

characteristic

C₁₃₉⋊C₈

D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

D₄

yes

characteristic

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#41

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#121

yes

characteristic

C₁₃₉

#473

yes

characteristic, complemented, Sylow

#473

—

139

maximal, Sylow

#137

—

139

—

#137

D₁₃₉

yes

characteristic

C₂×Q₁₆

—

139

—

#131

—

139

—

#108

—

139

—

#101

—

139

—

#101

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

Q₁₆

—

Q₁₆

—

278

—

Q₁₆

—

278

—

Q₁₆

—

278

—

SD₁₆

—

SD₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

139

—

M₁₆

—

139

—

M₁₆

C₂×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

139

—

Q₈

—

Q₈

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

C₂²⋊D₁₃₉

yes

characteristic

D₄

—

D₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

C₈

—

278

—

C₈

—

278

—

C₈

—

139

—

C₈

—

139

—

C₂²

—

C₂²

#14555784

yes

characteristic

C₄

—

C₄

—

C₄

—

556

—

C₄

D₄×D₁₃₉

yes

characteristic

C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂

—

C₂

—

C₂

#17984814

yes

characteristic

	subgroup	G/H
center	C₂	#17984814
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#137×C₁₃₉	C₂
socle	C₂×C₁₃₉	#121
hypercenter	#137	D₁₃₉
hypocenter	C₁₃₉	#473
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

subgroup

G/H

center

C₂

#17984814

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#137×C₁₃₉

C₂

C₂×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

N	H ≅ G/N	type
2. C₂×#14555775	C₂	semidirect
4. #17984777	C₂	semidirect
5. #17984761	C₂	semidirect
6. #17984760	C₂	semidirect
49. C₁₃₉	#473	semidirect

H ≅ G/N

type

2. C₂×#14555775

C₂

semidirect

4. #17984777

C₂

semidirect

5. #17984761

C₂

semidirect

6. #17984760

C₂

semidirect

49. C₁₃₉

#473

semidirect

Subgroups — #22581226

Conjugacy classes of subgroups

Complements