Group #22581134 — subgroups

Subgroups — #22581134

Order 8896.

142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 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	#17984889	C₂	1	1	yes	maximal, characteristic
2	#17984873	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984840	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984769	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984768	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, complemented
8	#14555785	—	2	1	no	—
9	#14555785	C₂²	1	1	yes	characteristic
10	#14555784	—	2	2	no	—
11	#14555784	—	2	1	no	—
12	C₂×Dic₂₇₈	—	2	1	no	—
13	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
14	Q₈×D₁₃₉	—	2	1	no	—
15	#14555775	—	2	1	no	—
16	#14555772	—	2	1	no	—
17	#14555769	C₂²	1	1	yes	characteristic
18	Dic₁₃₉⋊C₄	—	2	2	no	—
19	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
20	#41×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
22	Q₁₆×C₁₃₉	—	2	1	no	—
23	SD₁₆×C₁₃₉	—	2	1	no	—
24	M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
25	Dic₂₇₈	—	2	2	no	—
26	Dic₂₇₈	—	2	1	no	—
27	Dic₂₇₈	—	2	1	no	—
28	Dic₂₇₈	D₄	1	1	yes	characteristic
29	D₅₅₆	D₄	1	1	yes	characteristic
30	C₂²⋊D₁₃₉	—	2	2	no	—
31	C₂²⋊D₁₃₉	—	2	1	no	—
32	C₂×Dic₁₃₉	—	2	2	no	—
33	C₂×Dic₁₃₉	—	2	1	no	—
34	C₂×Dic₁₃₉	D₄	1	2	yes	—
35	C₄×D₁₃₉	—	2	2	no	—
36	C₄×D₁₃₉	—	2	1	no	—
37	C₁₃₉⋊C₈	—	2	1	no	—
38	Q₈×C₁₃₉	—	2	1	no	—
39	Q₈×C₁₃₉	—	2	1	no	—
40	Q₈×C₁₃₉	D₄	1	1	yes	characteristic
41	D₄×C₁₃₉	—	2	1	no	—
42	D₄×C₁₃₉	D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	—	2	1	no	—
44	C₂×C₄×C₁₃₉	—	2	1	no	—
45	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₈×C₁₃₉	—	2	1	no	—
47	C₂²×C₁₃₉	—	2	1	no	—
48	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₃₉	—	2	1	no	—
50	C₄×C₁₃₉	—	2	1	no	—
51	C₄×C₁₃₉	—	4	1	no	—
52	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
54	Dic₁₃₉	—	2	2	no	—
55	Dic₁₃₉	—	2	1	no	—
56	Dic₁₃₉	—	2	2	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	—	4	1	no	—
59	C₂×C₁₃₉	#120	1	1	yes	characteristic
60	D₁₃₉	—	4	1	no	—
61	C₁₃₉	#456	1	1	yes	characteristic, complemented, Sylow
62	#456	—	139	1	no	maximal, Sylow
63	2₋¹⁺⁴	—	139	1	no	—
64	#137	—	139	1	no	—
65	#137	D₁₃₉	1	1	yes	characteristic, complemented
66	C₄⋊Q₈	—	139	1	no	—
67	#104	—	139	1	no	—
68	#104	—	139	1	no	—
69	#101	—	139	1	no	—
70	#41	—	278	1	no	—
71	#41	—	278	1	no	—
72	#41	—	278	2	no	—
73	#41	—	139	1	no	—
74	#41	C₂×D₁₃₉	1	1	yes	characteristic
75	C₂×Q₈	—	278	1	no	—
76	C₂×Q₈	—	278	1	no	—
77	C₂×Q₈	—	139	1	no	—
78	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
79	Q₁₆	—	2	1	no	—
80	Q₁₆	—	278	1	no	—
81	SD₁₆	—	2	1	no	—
82	SD₁₆	—	278	1	no	—
83	M₁₆	—	139	1	no	—
84	M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
85	C₄⋊C₄	—	278	2	no	—
86	C₄²	—	139	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	2	1	no	—
89	Q₈	—	278	2	no	—
90	Q₈	—	278	1	no	—
91	Q₈	—	278	1	no	—
92	Q₈	—	139	1	no	—
93	Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
94	D₄	—	2	1	no	—
95	D₄	—	278	2	no	—
96	D₄	—	278	1	no	—
97	D₄	—	139	1	no	—
98	D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	278	2	no	—
102	C₂×C₄	—	278	2	no	—
103	C₂×C₄	—	278	1	no	—
104	C₂×C₄	—	278	1	no	—
105	C₂×C₄	—	139	2	no	—
106	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	278	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	278	1	no	—
111	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	1	no	—
114	C₄	—	278	2	no	—
115	C₄	—	278	1	no	—
116	C₄	—	278	2	no	—
117	C₄	D₄×D₁₃₉	1	1	yes	characteristic
118	C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4	1	no	—
121	C₂	—	556	1	no	—
122	C₂	#17984810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984889

C₂

yes

maximal, characteristic

#17984873

C₂

yes

maximal, characteristic, complemented

#17984840

C₂

yes

maximal, characteristic, complemented

#17984769

C₂

yes

maximal, characteristic, complemented

#17984768

C₂

yes

maximal, characteristic, complemented

#17984760

C₂

yes

maximal, characteristic, complemented

#137×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555785

—

#14555785

C₂²

yes

characteristic

#14555784

—

#14555784

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂²

yes

characteristic

Q₈×D₁₃₉

—

#14555775

—

#14555772

—

#14555769

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

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

—

Dic₂₇₈

—

Dic₂₇₈

D₄

yes

characteristic

D₅₅₆

D₄

yes

characteristic

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

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

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

D₁₃₉

—

C₁₃₉

#456

yes

characteristic, complemented, Sylow

#456

—

139

maximal, Sylow

2₋¹⁺⁴

—

139

—

#137

—

139

—

#137

D₁₃₉

yes

characteristic, complemented

C₄⋊Q₈

—

139

—

#104

—

139

—

#104

—

139

—

#101

—

139

—

#41

—

278

—

#41

—

278

—

#41

—

278

—

#41

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

Q₁₆

—

Q₁₆

—

278

—

SD₁₆

—

SD₁₆

—

278

—

M₁₆

—

139

—

M₁₆

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄²

—

139

—

Q₈

—

Q₈

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

139

—

Q₈

C₂²⋊D₁₃₉

yes

characteristic

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

139

—

D₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

278

—

104

C₂×C₄

—

278

—

105

C₂×C₄

—

139

—

106

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

107

C₈

—

108

C₈

—

278

—

109

C₂²

—

110

C₂²

—

278

—

111

C₂²

D₄×D₁₃₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

278

—

115

C₄

—

278

—

116

C₄

—

278

—

117

C₄

D₄×D₁₃₉

yes

characteristic

118

C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

556

—

122

C₂

#17984810

yes

characteristic

	subgroup	G/H
center	C₂	#17984810
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#137×C₁₃₉	C₂
socle	C₂×C₁₃₉	#120
hypercenter	#137	D₁₃₉
hypocenter	C₁₃₉	#456
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

subgroup

G/H

center

C₂

#17984810

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. #17984873	C₂	semidirect
3. #17984840	C₂	semidirect
4. #17984769	C₂	semidirect
5. #17984768	C₂	semidirect
6. #17984760	C₂	semidirect
7. #137×C₁₃₉	C₂	semidirect
61. C₁₃₉	#456	semidirect
65. #137	D₁₃₉	semidirect

H ≅ G/N

type

2. #17984873

C₂

semidirect

3. #17984840

C₂

semidirect

4. #17984769

C₂

semidirect

5. #17984768

C₂

semidirect

6. #17984760

C₂

semidirect

7. #137×C₁₃₉

C₂

semidirect

61. C₁₃₉

#456

semidirect

65. #137

D₁₃₉

semidirect

Subgroups — #22581134

Conjugacy classes of subgroups

Complements