Group #22581105 — subgroups

Subgroups — #22581105

Order 8896.

188 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 45 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×D₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14555773	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984822	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984762	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₃₉	—	2	1	no	—
9	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
12	D₄×D₁₃₉	—	2	2	no	—
13	D₄×D₁₃₉	—	2	2	no	—
14	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
15	#14555774	—	2	2	no	—
16	#14555773	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
18	C₄⋊Dic₁₃₉	—	2	1	no	—
19	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
20	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₁₃₉	—	2	4	no	—
24	C₂²×D₁₃₉	—	2	1	no	—
25	C₂²×D₁₃₉	—	4	2	no	—
26	C₂²×D₁₃₉	—	4	2	no	—
27	C₂²×D₁₃₉	D₄	1	1	yes	characteristic
28	Dic₂₇₈	—	2	1	no	—
29	Dic₂₇₈	—	2	1	no	—
30	D₅₅₆	—	2	1	no	—
31	D₅₅₆	D₄	1	2	yes	—
32	C₂²⋊D₁₃₉	—	2	4	no	—
33	C₂×Dic₁₃₉	—	2	1	no	—
34	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
35	C₄×D₁₃₉	—	2	1	no	—
36	C₄×D₁₃₉	—	2	1	no	—
37	C₁₃₉⋊C₈	—	2	1	no	—
38	C₂³×C₁₃₉	—	2	1	no	—
39	D₄×C₁₃₉	—	2	1	no	—
40	D₄×C₁₃₉	D₄	1	2	yes	—
41	C₂×C₄×C₁₃₉	—	2	1	no	—
42	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
43	C₈×C₁₃₉	—	2	1	no	—
44	C₂²×C₁₃₉	—	2	2	no	—
45	C₂²×C₁₃₉	—	4	1	no	—
46	C₂²×C₁₃₉	—	4	1	no	—
47	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₃₉	—	2	1	no	—
49	C₂×D₁₃₉	—	2	1	no	—
50	C₂×D₁₃₉	—	2	2	no	—
51	C₂×D₁₃₉	—	4	8	no	—
52	C₂×D₁₃₉	—	4	1	no	—
53	C₂×D₁₃₉	—	4	1	no	—
54	C₂×D₁₃₉	SD₁₆	1	2	yes	complemented
55	C₄×C₁₃₉	—	4	1	no	—
56	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	Dic₁₃₉	—	2	1	no	—
59	Dic₁₃₉	—	4	1	no	—
60	C₂×C₁₃₉	—	4	2	no	—
61	C₂×C₁₃₉	#120	1	1	yes	characteristic
62	C₂×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	D₁₃₉	—	2	2	no	—
65	D₁₃₉	—	4	2	no	—
66	C₁₃₉	#450	1	1	yes	characteristic, complemented, Sylow
67	#450	—	139	1	no	maximal, Sylow
68	C₂²×D₄	—	139	1	no	—
69	C₂×SD₁₆	—	139	1	no	—
70	C₂×SD₁₆	—	139	1	no	—
71	C₂²⋊Q₈	—	139	1	no	—
72	#102	—	139	1	no	—
73	#102	D₁₃₉	1	1	yes	characteristic, complemented
74	C₂²⋊C₈	—	139	1	no	—
75	C₂⁴	—	278	1	no	—
76	C₂×Q₈	—	139	1	no	—
77	C₂×D₄	—	278	2	no	—
78	C₂×D₄	—	278	2	no	—
79	C₂×D₄	—	278	1	no	—
80	C₂×D₄	—	139	1	no	—
81	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
82	C₂²×C₄	—	139	1	no	—
83	SD₁₆	—	278	2	no	—
84	SD₁₆	—	278	2	no	—
85	C₂×C₈	—	139	1	no	—
86	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
87	C₄⋊C₄	—	278	1	no	—
88	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic, complemented
89	C₂²⋊C₄	—	278	1	no	—
90	C₂³	—	2	1	no	—
91	C₂³	—	278	4	no	—
92	C₂³	—	278	1	no	—
93	C₂³	—	556	2	no	—
94	C₂³	—	556	2	no	—
95	C₂³	—	139	1	no	—
96	Q₈	—	278	1	no	—
97	Q₈	—	278	1	no	—
98	D₄	—	2	1	no	—
99	D₄	—	278	4	no	—
100	D₄	—	278	1	no	—
101	D₄	—	139	2	no	—
102	D₄	D₅₅₆	1	2	yes	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	278	1	no	—
105	C₂×C₄	—	278	1	no	—
106	C₂×C₄	—	278	1	no	—
107	C₂×C₄	—	139	1	no	—
108	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	278	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	278	1	no	—
115	C₂²	—	278	1	no	—
116	C₂²	—	278	2	no	—
117	C₂²	—	556	8	no	—
118	C₂²	—	556	1	no	—
119	C₂²	—	556	1	no	—
120	C₂²	—	139	2	no	—
121	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
122	C₄	—	4	1	no	—
123	C₄	—	278	1	no	—
124	C₄	—	556	1	no	—
125	C₄	D₄×D₁₃₉	1	1	yes	characteristic
126	C₄	C₂×D₅₅₆	1	1	yes	characteristic
127	C₂	—	4	2	no	—
128	C₂	—	278	2	no	—
129	C₂	—	556	2	no	—
130	C₂	#17984812	1	1	yes	characteristic
131	C₂	SD₁₆×D₁₃₉	1	1	yes	characteristic
132	C₂	#17984866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555774

C₂

yes

maximal, characteristic, complemented

C₂×#14555773

C₂

yes

maximal, characteristic, complemented

#17984822

C₂

yes

maximal, characteristic, complemented

#17984762

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₃₉)⋊C₈

C₂

yes

maximal, characteristic, complemented

#102×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₉

—

C₂×Dic₂₇₈

C₂²

yes

characteristic, complemented

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

D₄×D₁₃₉

—

D₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

#14555774

—

#14555773

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic, complemented

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic, complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

D₄

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

D₄

yes

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

SD₁₆

yes

complemented

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

C₂×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₉

AGL(1,ℤ/8)

yes

characteristic

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#450

yes

characteristic, complemented, Sylow

#450

—

139

maximal, Sylow

C₂²×D₄

—

139

—

C₂×SD₁₆

—

139

—

C₂×SD₁₆

—

139

—

C₂²⋊Q₈

—

139

—

#102

—

139

—

#102

D₁₃₉

yes

characteristic, complemented

C₂²⋊C₈

—

139

—

C₂⁴

—

278

—

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

SD₁₆

—

278

—

SD₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic, complemented

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic, complemented

C₂²⋊C₄

—

278

—

C₂³

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

556

—

C₂³

—

556

—

C₂³

—

139

—

Q₈

—

278

—

Q₈

—

278

—

D₄

—

D₄

—

278

—

100

D₄

—

278

—

101

D₄

—

139

—

102

D₄

D₅₅₆

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

278

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

139

—

108

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

109

C₈

—

110

C₈

—

278

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

278

—

115

C₂²

—

278

—

116

C₂²

—

278

—

117

C₂²

—

556

—

118

C₂²

—

556

—

119

C₂²

—

556

—

120

C₂²

—

139

—

121

C₂²

D₄×D₁₃₉

yes

characteristic

122

C₄

—

123

C₄

—

278

—

124

C₄

—

556

—

125

C₄

D₄×D₁₃₉

yes

characteristic

126

C₄

C₂×D₅₅₆

yes

characteristic

127

C₂

—

128

C₂

—

278

—

129

C₂

—

556

—

130

C₂

#17984812

yes

characteristic

131

C₂

SD₁₆×D₁₃₉

yes

characteristic

132

C₂

#17984866

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#102×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#102	D₁₃₉
hypocenter	C₁₃₉	#450
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#102×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. C₂×#14555774	C₂	semidirect
3. C₂×#14555773	C₂	semidirect
4. #17984822	C₂	semidirect
5. #17984762	C₂	semidirect
6. (C₂×D₁₃₉)⋊C₈	C₂	semidirect
7. #102×C₁₃₉	C₂	semidirect
9. C₂×Dic₂₇₈	C₂²	semidirect
17. C₂×C₁₃₉⋊C₈	C₂²	semidirect
21. C₂×C₈×C₁₃₉	C₂²	semidirect
22. C₄⋊C₄×C₁₃₉	C₂²	semidirect
54. C₂×D₁₃₉	SD₁₆	semidirect
66. C₁₃₉	#450	semidirect
73. #102	D₁₃₉	semidirect
86. C₂×C₈	C₂×D₁₃₉	semidirect
88. C₄⋊C₄	C₂×D₁₃₉	semidirect

H ≅ G/N

type

2. C₂×#14555774

C₂

semidirect

3. C₂×#14555773

C₂

semidirect

4. #17984822

C₂

semidirect

5. #17984762

C₂

semidirect

6. (C₂×D₁₃₉)⋊C₈

C₂

semidirect

7. #102×C₁₃₉

C₂

semidirect

9. C₂×Dic₂₇₈

C₂²

semidirect

17. C₂×C₁₃₉⋊C₈

C₂²

semidirect

21. C₂×C₈×C₁₃₉

C₂²

semidirect

22. C₄⋊C₄×C₁₃₉

C₂²

semidirect

54. C₂×D₁₃₉

SD₁₆

semidirect

66. C₁₃₉

#450

semidirect

73. #102

D₁₃₉

semidirect

86. C₂×C₈

C₂×D₁₃₉

semidirect

88. C₄⋊C₄

C₂×D₁₃₉

semidirect

Subgroups — #22581105

Conjugacy classes of subgroups

Complements