Group #22581129 — subgroups

Subgroups — #22581129

Order 8896.

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

normal subgroups only

This group has 152 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	#17984888	C₂	1	1	yes	maximal, characteristic
2	#17984873	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984831	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	#17984758	C₂	1	1	yes	maximal, characteristic, complemented
7	#137×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555786	—	2	1	no	—
9	#14555785	—	2	1	no	—
10	#14555785	C₂²	1	1	yes	characteristic
11	C₂×D₅₅₆	—	2	1	no	—
12	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
13	D₄×D₁₃₉	—	2	2	no	—
14	D₄×D₁₃₉	—	2	1	no	—
15	#14555775	—	2	1	no	—
16	#14555772	—	2	1	no	—
17	#14555769	C₂²	1	1	yes	characteristic, complemented
18	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
19	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
20	#41×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
22	Q₁₆×C₁₃₉	—	2	1	no	—
23	SD₁₆×C₁₃₉	—	2	1	no	—
24	M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₃₉	—	2	2	no	—
26	C₂²×D₁₃₉	D₄	1	2	yes	—
27	Dic₂₇₈	D₄	1	1	yes	characteristic
28	D₅₅₆	—	2	2	no	—
29	D₅₅₆	—	2	1	no	—
30	D₅₅₆	—	2	1	no	—
31	D₅₅₆	D₄	1	1	yes	characteristic
32	C₂²⋊D₁₃₉	—	2	2	no	—
33	C₂²⋊D₁₃₉	—	2	1	no	—
34	C₂×Dic₁₃₉	—	2	1	no	—
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	2	no	—
50	C₂×D₁₃₉	—	2	1	no	—
51	C₂×D₁₃₉	—	4	2	no	—
52	C₂×D₁₃₉	—	8	2	no	—
53	C₄×C₁₃₉	—	2	1	no	—
54	C₄×C₁₃₉	—	4	1	no	—
55	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	Dic₁₃₉	—	2	1	no	—
58	Dic₁₃₉	—	4	1	no	—
59	C₂×C₁₃₉	—	2	1	no	—
60	C₂×C₁₃₉	—	4	1	no	—
61	C₂×C₁₃₉	#120	1	1	yes	characteristic
62	D₁₃₉	—	4	2	no	—
63	D₁₃₉	—	4	1	no	—
64	C₁₃₉	#455	1	1	yes	characteristic, complemented, Sylow
65	#455	—	139	1	no	maximal, Sylow
66	H(5,𝔽₂)	—	139	1	no	—
67	#137	—	139	1	no	—
68	#137	D₁₃₉	1	1	yes	characteristic, complemented
69	#124	—	139	1	no	—
70	#104	—	139	1	no	—
71	#104	—	139	1	no	—
72	#100	—	139	1	no	—
73	#41	—	278	1	no	—
74	#41	—	278	1	no	—
75	#41	—	139	1	no	—
76	#41	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
78	C₂×D₄	—	278	1	no	—
79	C₂×D₄	—	278	2	no	—
80	C₂×D₄	—	278	1	no	—
81	C₂×D₄	—	139	1	no	—
82	Q₁₆	—	2	1	no	—
83	Q₁₆	—	278	1	no	—
84	SD₁₆	—	2	1	no	—
85	SD₁₆	—	278	1	no	—
86	M₁₆	—	139	1	no	—
87	M₁₆	C₂×D₁₃₉	1	1	yes	characteristic, complemented
88	C₂²⋊C₄	—	278	2	no	—
89	C₄²	—	139	1	no	—
90	C₂³	—	278	2	no	—
91	C₂³	—	139	2	no	—
92	Q₈	—	2	1	no	—
93	Q₈	—	2	1	no	—
94	Q₈	—	139	1	no	—
95	Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
96	D₄	—	2	1	no	—
97	D₄	—	278	2	no	—
98	D₄	—	278	2	no	—
99	D₄	—	278	1	no	—
100	D₄	—	278	1	no	—
101	D₄	—	278	1	no	—
102	D₄	—	139	1	no	—
103	D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	278	2	no	—
107	C₂×C₄	—	278	1	no	—
108	C₂×C₄	—	278	1	no	—
109	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	278	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	278	2	no	—
114	C₂²	—	278	1	no	—
115	C₂²	—	556	2	no	—
116	C₂²	—	1112	2	no	—
117	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	278	1	no	—
121	C₄	—	556	1	no	—
122	C₄	D₄×D₁₃₉	1	1	yes	characteristic
123	C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	1	no	—
126	C₂	—	556	2	no	—
127	C₂	—	556	1	no	—
128	C₂	#17984810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984888

C₂

yes

maximal, characteristic

#17984873

C₂

yes

maximal, characteristic, complemented

#17984831

C₂

yes

maximal, characteristic, complemented

#17984769

C₂

yes

maximal, characteristic, complemented

#17984768

C₂

yes

maximal, characteristic, complemented

#17984758

C₂

yes

maximal, characteristic, complemented

#137×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555786

—

#14555785

—

#14555785

C₂²

yes

characteristic

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂²

yes

characteristic

D₄×D₁₃₉

—

D₄×D₁₃₉

—

#14555775

—

#14555772

—

#14555769

C₂²

yes

characteristic, complemented

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

C₂²

yes

characteristic, complemented

#41×C₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic, complemented

Q₁₆×C₁₃₉

—

SD₁₆×C₁₃₉

—

M₁₆×C₁₃₉

C₂²

yes

characteristic, complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

D₄

yes

—

Dic₂₇₈

D₄

yes

characteristic

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

D₄

yes

characteristic

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

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

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#455

yes

characteristic, complemented, Sylow

#455

—

139

maximal, Sylow

H(5,𝔽₂)

—

139

—

#137

—

139

—

#137

D₁₃₉

yes

characteristic, complemented

#124

—

139

—

#104

—

139

—

#104

—

139

—

#100

—

139

—

#41

—

278

—

#41

—

278

—

#41

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

C₂×D₁₃₉

yes

characteristic, complemented

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

Q₁₆

—

Q₁₆

—

278

—

SD₁₆

—

SD₁₆

—

278

—

M₁₆

—

139

—

M₁₆

C₂×D₁₃₉

yes

characteristic, complemented

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₂³

—

278

—

C₂³

—

139

—

Q₈

—

Q₈

—

Q₈

—

139

—

Q₈

C₂²⋊D₁₃₉

yes

characteristic

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

278

—

100

D₄

—

278

—

101

D₄

—

278

—

102

D₄

—

139

—

103

D₄

C₂²⋊D₁₃₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

278

—

108

C₂×C₄

—

278

—

109

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

110

C₈

—

111

C₈

—

278

—

112

C₂²

—

113

C₂²

—

278

—

114

C₂²

—

278

—

115

C₂²

—

556

—

116

C₂²

—

1112

—

117

C₂²

D₄×D₁₃₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

278

—

121

C₄

—

556

—

122

C₄

D₄×D₁₃₉

yes

characteristic

123

C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

556

—

127

C₂

—

556

—

128

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	#455
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. #17984831	C₂	semidirect
4. #17984769	C₂	semidirect
5. #17984768	C₂	semidirect
6. #17984758	C₂	semidirect
7. #137×C₁₃₉	C₂	semidirect
17. #14555769	C₂²	semidirect
19. C₄×Dic₁₃₉	C₂²	semidirect
21. C₂×Q₈×C₁₃₉	C₂²	semidirect
24. M₁₆×C₁₃₉	C₂²	semidirect
64. C₁₃₉	#455	semidirect
68. #137	D₁₃₉	semidirect
77. C₂×Q₈	C₂×D₁₃₉	semidirect
87. M₁₆	C₂×D₁₃₉	semidirect

H ≅ G/N

type

2. #17984873

C₂

semidirect

3. #17984831

C₂

semidirect

4. #17984769

C₂

semidirect

5. #17984768

C₂

semidirect

6. #17984758

C₂

semidirect

7. #137×C₁₃₉

C₂

semidirect

17. #14555769

C₂²

semidirect

19. C₄×Dic₁₃₉

C₂²

semidirect

21. C₂×Q₈×C₁₃₉

C₂²

semidirect

24. M₁₆×C₁₃₉

C₂²

semidirect

64. C₁₃₉

#455

semidirect

68. #137

D₁₃₉

semidirect

77. C₂×Q₈

C₂×D₁₃₉

semidirect

87. M₁₆

C₂×D₁₃₉

semidirect

Subgroups — #22581129

Conjugacy classes of subgroups

Complements