Group #22580865 — subgroups

Subgroups — #22580865

Order 8896.

202 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 63 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×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₃₉	C₂	1	2	yes	maximal
3	C₁₃₉⋊#95	C₂	1	2	yes	maximal, complemented
4	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₇₈	—	2	4	no	—
7	C₂×Dic₂₇₈	—	2	2	no	—
8	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
10	C₂²⋊Dic₁₃₉	—	2	4	no	—
11	C₂⁴×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
15	Dic₂₇₈	—	2	8	no	—
16	C₂×Dic₁₃₉	—	2	2	no	—
17	C₂×Dic₁₃₉	—	2	2	no	—
18	C₂×Dic₁₃₉	—	2	2	no	—
19	C₂×Dic₁₃₉	—	2	2	no	—
20	C₂×Dic₁₃₉	—	2	2	no	—
21	C₂×Dic₁₃₉	D₄	1	4	yes	—
22	C₂³×C₁₃₉	—	2	1	no	—
23	C₂³×C₁₃₉	—	2	2	no	—
24	C₂³×C₁₃₉	—	2	2	no	—
25	C₂³×C₁₃₉	—	2	1	no	—
26	C₂³×C₁₃₉	Q₈	1	2	yes	complemented
27	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₃₉	—	2	2	no	—
29	C₂×C₄×C₁₃₉	—	2	1	no	—
30	C₂×C₄×C₁₃₉	—	2	2	no	—
31	C₂×C₄×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₁₃₉	—	2	4	no	—
33	C₂²×C₁₃₉	—	2	2	no	—
34	C₂²×C₁₃₉	—	4	1	no	—
35	C₂²×C₁₃₉	—	4	2	no	—
36	C₂²×C₁₃₉	—	4	1	no	—
37	C₂²×C₁₃₉	C₂×D₄	1	2	yes	—
38	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₃₉	#41	1	2	yes	—
41	C₂²×C₁₃₉	#41	1	1	yes	characteristic
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	—	4	1	no	—
44	Dic₁₃₉	—	2	4	no	—
45	Dic₁₃₉	—	4	2	no	—
46	C₂×C₁₃₉	—	4	2	no	—
47	C₂×C₁₃₉	#120	1	1	yes	characteristic
48	C₂×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
49	C₂×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₃₉	#124	1	2	yes	—
51	C₂×C₁₃₉	#124	1	1	yes	characteristic
52	C₁₃₉	#393	1	1	yes	characteristic, complemented, Sylow
53	#393	—	139	1	no	maximal, Sylow
54	C₂²×Q₈	—	139	1	no	—
55	C₂×C₂²⋊C₄	—	139	2	no	—
56	C₂×C₂²⋊C₄	D₁₃₉	1	1	yes	characteristic
57	#95	—	139	1	no	—
58	#95	—	139	2	no	—
59	C₂⁴	C₂×D₁₃₉	1	1	yes	characteristic
60	C₂×Q₈	—	278	4	no	—
61	C₂×Q₈	—	278	2	no	—
62	C₂²×C₄	—	139	2	no	—
63	C₂²×C₄	—	139	2	no	—
64	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
66	C₂²⋊C₄	—	2	2	no	—
67	C₂²⋊C₄	—	278	4	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	2	1	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	1	no	—
72	C₂³	Dic₂₇₈	1	2	yes	complemented
73	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
74	Q₈	—	278	8	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	278	2	no	—
79	C₂×C₄	—	278	2	no	—
80	C₂×C₄	—	278	2	no	—
81	C₂×C₄	—	278	2	no	—
82	C₂×C₄	—	278	2	no	—
83	C₂×C₄	—	139	4	no	—
84	C₂×C₄	C₂²⋊D₁₃₉	1	2	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	1	no	—
88	C₂²	—	4	2	no	—
89	C₂²	—	4	1	no	—
90	C₂²	D₄×D₁₃₉	1	2	yes	—
91	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
92	C₂²	C₂×Dic₂₇₈	1	1	yes	characteristic
93	C₂²	#14555784	1	2	yes	—
94	C₂²	#14555785	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	4	1	no	—
97	C₄	—	278	4	no	—
98	C₄	—	556	2	no	—
99	C₂	—	4	2	no	—
100	C₂	#17984810	1	1	yes	characteristic
101	C₂	#17984819	1	2	yes	—
102	C₂	#17984821	1	1	yes	characteristic
103	C₂	#17984828	1	2	yes	—
104	C₂	#17984829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₇₈

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₃₉

C₂

yes

maximal

C₁₃₉⋊#95

C₂

yes

maximal, complemented

C₁₃₉⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂²×Dic₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

C₂²

yes

—

C₂²⋊Dic₁₃₉

—

C₂⁴×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

Q₈

yes

complemented

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×Q₈

yes

characteristic

C₂²×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

#41

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

C₂×C₁₃₉

C₂²⋊Q₈

yes

—

C₂×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₃₉

#124

yes

—

C₂×C₁₃₉

#124

yes

characteristic

C₁₃₉

#393

yes

characteristic, complemented, Sylow

#393

—

139

maximal, Sylow

C₂²×Q₈

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

D₁₃₉

yes

characteristic

#95

—

139

—

#95

—

139

—

C₂⁴

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Dic₂₇₈

yes

complemented

C₂³

C₂²×D₁₃₉

yes

characteristic

Q₈

—

278

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

C₂²⋊D₁₃₉

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₃₉

yes

—

C₂²

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂²

C₂×Dic₂₇₈

yes

characteristic

C₂²

#14555784

yes

—

C₂²

#14555785

yes

characteristic

C₄

—

C₄

—

C₄

—

278

—

C₄

—

556

—

C₂

—

100

C₂

#17984810

yes

characteristic

101

C₂

#17984819

yes

—

102

C₂

#17984821

yes

characteristic

103

C₂

#17984828

yes

—

104

C₂

#17984829

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₃₉
commutator, socle	C₂³×C₁₃₉	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₃₉	C₂
hypercenter	C₂×C₂²⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#393
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₃₉

commutator, socle

C₂³×C₁₃₉

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₃₉

C₂

hypercenter

C₂×C₂²⋊C₄

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂²×Dic₂₇₈	C₂	semidirect
3. C₁₃₉⋊#95	C₂	semidirect
4. C₁₃₉⋊#95	C₂	semidirect
26. C₂³×C₁₃₉	Q₈	semidirect
52. C₁₃₉	#393	semidirect
72. C₂³	Dic₂₇₈	semidirect

H ≅ G/N

type

1. C₂²×Dic₂₇₈

C₂

semidirect

3. C₁₃₉⋊#95

C₂

semidirect

4. C₁₃₉⋊#95

C₂

semidirect

26. C₂³×C₁₃₉

Q₈

semidirect

52. C₁₃₉

#393

semidirect

72. C₂³

Dic₂₇₈

semidirect

Subgroups — #22580865

Conjugacy classes of subgroups

Complements