Group #22581117 — subgroups

Subgroups — #22581117

Order 8896.

152 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 43 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	C₂×#14555785	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984821	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984767	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984764	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555785	—	2	2	no	—
9	#14555785	—	2	2	no	—
10	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
12	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
14	C₂×C₄×D₁₃₉	—	2	1	no	—
15	Dic₅₅₆	—	2	2	no	—
16	#14555774	—	2	2	no	—
17	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	Dic₁₃₉⋊C₄	—	2	1	no	—
19	C₂²⋊Dic₁₃₉	—	2	1	no	—
20	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂²×D₁₃₉	—	2	1	no	—
24	Dic₂₇₈	—	2	1	no	—
25	Dic₂₇₈	—	2	1	no	—
26	Dic₂₇₈	—	2	1	no	—
27	Dic₂₇₈	D₄	1	2	yes	—
28	D₅₅₆	—	2	1	no	—
29	D₅₅₆	D₄	1	2	yes	—
30	C₂²⋊D₁₃₉	—	2	4	no	—
31	C₂×Dic₁₃₉	—	2	1	no	—
32	C₂×Dic₁₃₉	—	2	1	no	—
33	C₂×Dic₁₃₉	—	2	1	no	—
34	C₄×D₁₃₉	—	2	4	no	—
35	C₂³×C₁₃₉	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	—	2	1	no	—
37	C₂×C₄×C₁₃₉	—	2	1	no	—
38	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
40	C₈×C₁₃₉	—	2	1	no	—
41	C₈×C₁₃₉	—	2	1	no	—
42	C₂²×C₁₃₉	—	2	1	no	—
43	C₂²×C₁₃₉	—	2	1	no	—
44	C₂²×C₁₃₉	—	2	1	no	—
45	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×D₁₃₉	—	2	2	no	—
47	C₂×D₁₃₉	—	4	1	no	—
48	C₂×D₁₃₉	—	4	1	no	—
49	C₄×C₁₃₉	—	2	1	no	—
50	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	Dic₁₃₉	—	2	2	no	—
53	Dic₁₃₉	—	4	1	no	—
54	Dic₁₃₉	—	4	1	no	—
55	C₂×C₁₃₉	—	4	1	no	—
56	C₂×C₁₃₉	#120	1	1	yes	characteristic
57	C₂×C₁₃₉	#135	1	1	yes	characteristic
58	C₂×C₁₃₉	#137	1	1	yes	characteristic
59	D₁₃₉	—	4	2	no	—
60	C₁₃₉	#452	1	1	yes	characteristic, complemented, Sylow
61	#452	—	139	1	no	maximal, Sylow
62	C₂×#41	—	139	1	no	—
63	C₂×Q₁₆	—	139	1	no	—
64	C₂×SD₁₆	—	139	1	no	—
65	C₂²⋊Q₈	—	139	1	no	—
66	#103	—	139	1	no	—
67	#102	—	139	1	no	—
68	C₂²⋊C₈	D₁₃₉	1	1	yes	characteristic, complemented
69	#41	—	278	2	no	—
70	#41	—	278	2	no	—
71	C₂×Q₈	—	139	1	no	—
72	C₂×Q₈	—	139	1	no	—
73	C₂×D₄	—	278	1	no	—
74	C₂×D₄	—	139	1	no	—
75	C₂²×C₄	—	278	1	no	—
76	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
77	Q₁₆	—	278	2	no	—
78	SD₁₆	—	278	2	no	—
79	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
80	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
81	C₄⋊C₄	—	278	1	no	—
82	C₄⋊C₄	—	139	1	no	—
83	C₂²⋊C₄	—	278	1	no	—
84	C₂³	—	278	1	no	—
85	C₂³	D₅₅₆	1	1	yes	characteristic
86	Q₈	—	278	1	no	—
87	Q₈	—	278	1	no	—
88	Q₈	—	278	1	no	—
89	Q₈	—	139	2	no	—
90	D₄	—	278	4	no	—
91	D₄	—	278	1	no	—
92	D₄	—	139	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	278	4	no	—
96	C₂×C₄	—	278	1	no	—
97	C₂×C₄	—	278	1	no	—
98	C₂×C₄	—	278	1	no	—
99	C₂×C₄	D₅₅₆	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	278	2	no	—
107	C₂²	—	556	1	no	—
108	C₂²	—	556	1	no	—
109	C₂²	C₂×D₅₅₆	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	278	2	no	—
112	C₄	—	556	1	no	—
113	C₄	—	556	1	no	—
114	C₄	D₄×D₁₃₉	1	1	yes	characteristic
115	C₄	D₄×D₁₃₉	1	1	yes	characteristic
116	C₂	—	4	1	no	—
117	C₂	—	556	2	no	—
118	C₂	#17984812	1	1	yes	characteristic
119	C₂	#17984864	1	1	yes	characteristic
120	C₂	#17984875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555785

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

C₂

yes

maximal, characteristic, complemented

C₂×#14555774

C₂

yes

maximal, characteristic, complemented

#17984821

C₂

yes

maximal, characteristic, complemented

#17984767

C₂

yes

maximal, characteristic, complemented

#17984764

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555785

—

#14555785

—

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

C₂×C₄×D₁₃₉

—

Dic₅₅₆

—

#14555774

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

D₄

yes

—

D₅₅₆

—

D₅₅₆

D₄

yes

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#452

yes

characteristic, complemented, Sylow

#452

—

139

maximal, Sylow

C₂×#41

—

139

—

C₂×Q₁₆

—

139

—

C₂×SD₁₆

—

139

—

C₂²⋊Q₈

—

139

—

#103

—

139

—

#102

—

139

—

C₂²⋊C₈

D₁₃₉

yes

characteristic, complemented

#41

—

278

—

#41

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

Q₁₆

—

278

—

SD₁₆

—

278

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₂³

—

278

—

C₂³

D₅₅₆

yes

characteristic

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

139

—

D₄

—

278

—

D₄

—

278

—

D₄

—

139

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

D₅₅₆

yes

characteristic

100

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

278

—

107

C₂²

—

556

—

108

C₂²

—

556

—

109

C₂²

C₂×D₅₅₆

yes

characteristic

110

C₄

—

111

C₄

—

278

—

112

C₄

—

556

—

113

C₄

—

556

—

114

C₄

D₄×D₁₃₉

yes

characteristic

115

C₄

D₄×D₁₃₉

yes

characteristic

116

C₂

—

117

C₂

—

556

—

118

C₂

#17984812

yes

characteristic

119

C₂

#17984864

yes

characteristic

120

C₂

#17984875

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₅₆
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	C₂²⋊C₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#452
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂×D₅₅₆

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

C₂²⋊C₈×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

hypercenter

C₂²⋊C₈

D₁₃₉

hypocenter

C₁₃₉

#452

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₅₅₆	C₂	semidirect
3. C₂×#14555774	C₂	semidirect
4. #17984821	C₂	semidirect
5. #17984767	C₂	semidirect
6. #17984764	C₂	semidirect
7. C₂²⋊C₈×C₁₃₉	C₂	semidirect
60. C₁₃₉	#452	semidirect
68. C₂²⋊C₈	D₁₃₉	semidirect

H ≅ G/N

type

2. C₂×Dic₅₅₆

C₂

semidirect

3. C₂×#14555774

C₂

semidirect

4. #17984821

C₂

semidirect

5. #17984767

C₂

semidirect

6. #17984764

C₂

semidirect

7. C₂²⋊C₈×C₁₃₉

C₂

semidirect

60. C₁₃₉

#452

semidirect

68. C₂²⋊C₈

D₁₃₉

semidirect

Subgroups — #22581117

Conjugacy classes of subgroups

Complements