Group #22581099 — subgroups

Subgroups — #22581099

Order 8896.

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

normal subgroups only

This group has 162 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×#14555784	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14555770	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984817	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984765	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	#14555784	—	2	2	no	—
9	#14555784	—	2	2	no	—
10	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
11	C₂×D₅₅₆	—	2	1	no	—
12	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
14	C₂²×Dic₁₃₉	—	2	1	no	—
15	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
16	#14555774	—	2	2	no	—
17	#14555770	—	2	2	no	—
18	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
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
22	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂²×D₁₃₉	—	2	1	no	—
24	C₂²×D₁₃₉	D₄	1	1	yes	characteristic
25	Dic₂₇₈	—	2	1	no	—
26	Dic₂₇₈	D₄	1	2	yes	complemented
27	D₅₅₆	—	2	1	no	—
28	D₅₅₆	—	2	1	no	—
29	D₅₅₆	—	4	2	no	—
30	C₂²⋊D₁₃₉	—	2	4	no	—
31	C₂×Dic₁₃₉	—	2	4	no	—
32	C₂×Dic₁₃₉	—	2	1	no	—
33	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
34	C₄×D₁₃₉	—	2	1	no	—
35	C₄×D₁₃₉	—	2	1	no	—
36	C₁₃₉⋊C₈	—	2	1	no	—
37	C₂³×C₁₃₉	—	2	1	no	—
38	D₄×C₁₃₉	—	2	1	no	—
39	D₄×C₁₃₉	D₄	1	2	yes	complemented
40	C₂×C₄×C₁₃₉	—	2	1	no	—
41	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₈×C₁₃₉	—	2	1	no	—
43	C₂²×C₁₃₉	—	2	2	no	—
44	C₂²×C₁₃₉	—	4	1	no	—
45	C₂²×C₁₃₉	—	4	1	no	—
46	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₂×D₁₃₉	—	2	1	no	—
48	C₂×D₁₃₉	—	2	1	no	—
49	C₂×D₁₃₉	—	2	1	no	—
50	C₂×D₁₃₉	—	4	1	no	—
51	C₂×D₁₃₉	—	4	1	no	—
52	C₂×D₁₃₉	—	4	1	no	—
53	C₄×C₁₃₉	—	4	1	no	—
54	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	Dic₁₃₉	—	2	1	no	—
57	Dic₁₃₉	—	2	2	no	—
58	C₂×C₁₃₉	—	4	2	no	—
59	C₂×C₁₃₉	#120	1	1	yes	characteristic
60	C₂×C₁₃₉	#135	1	1	yes	characteristic
61	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	D₁₃₉	—	4	1	no	—
63	D₁₃₉	—	8	1	no	—
64	C₁₃₉	#449	1	1	yes	characteristic, complemented, Sylow
65	#449	—	139	1	no	maximal, Sylow
66	C₂×#41	—	139	1	no	—
67	C₂×SD₁₆	—	139	1	no	—
68	C₂×D₈	—	139	1	no	—
69	#121	—	139	1	no	—
70	#103	—	139	1	no	—
71	#102	D₁₃₉	1	1	yes	characteristic, complemented
72	C₂²⋊C₈	—	139	1	no	—
73	#41	—	278	2	no	—
74	#41	—	278	2	no	—
75	C₂×Q₈	—	139	1	no	—
76	C₂×D₄	—	278	1	no	—
77	C₂×D₄	—	278	1	no	—
78	C₂×D₄	—	139	1	no	—
79	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
80	C₂²×C₄	—	278	1	no	—
81	C₂²×C₄	—	139	1	no	—
82	SD₁₆	—	278	2	no	—
83	D₈	—	278	2	no	—
84	C₂×C₈	—	139	1	no	—
85	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
86	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂²⋊C₄	—	278	1	no	—
88	C₂³	—	2	1	no	—
89	C₂³	—	278	1	no	—
90	C₂³	—	139	1	no	—
91	Q₈	—	278	1	no	—
92	Q₈	—	139	2	no	—
93	D₄	—	2	1	no	—
94	D₄	—	278	4	no	—
95	D₄	—	278	1	no	—
96	D₄	—	278	1	no	—
97	D₄	—	556	2	no	—
98	D₄	D₅₅₆	1	2	yes	complemented
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	278	4	no	—
101	C₂×C₄	—	278	1	no	—
102	C₂×C₄	—	278	1	no	—
103	C₂×C₄	—	278	1	no	—
104	C₂×C₄	—	139	1	no	—
105	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	278	1	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	1	no	—
111	C₂²	—	278	1	no	—
112	C₂²	—	278	1	no	—
113	C₂²	—	278	1	no	—
114	C₂²	—	556	1	no	—
115	C₂²	—	556	1	no	—
116	C₂²	—	556	1	no	—
117	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
118	C₄	—	4	1	no	—
119	C₄	—	278	1	no	—
120	C₄	—	278	2	no	—
121	C₄	D₄×D₁₃₉	1	1	yes	characteristic
122	C₄	C₂×D₅₅₆	1	1	yes	characteristic
123	C₂	—	4	2	no	—
124	C₂	—	556	1	no	—
125	C₂	—	1112	1	no	—
126	C₂	#17984812	1	1	yes	characteristic
127	C₂	#17984863	1	1	yes	characteristic
128	C₂	#17984866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555784

C₂

yes

maximal, characteristic, complemented

C₂×#14555774

C₂

yes

maximal, characteristic, complemented

C₂×#14555770

C₂

yes

maximal, characteristic, complemented

#17984817

C₂

yes

maximal, characteristic, complemented

#17984765

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₃₉)⋊C₈

C₂

yes

maximal, characteristic, complemented

#102×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555784

—

#14555784

—

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

#14555774

—

#14555770

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

—

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

D₄

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

D₄

yes

complemented

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

complemented

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

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#449

yes

characteristic, complemented, Sylow

#449

—

139

maximal, Sylow

C₂×#41

—

139

—

C₂×SD₁₆

—

139

—

C₂×D₈

—

139

—

#121

—

139

—

#103

—

139

—

#102

D₁₃₉

yes

characteristic, complemented

C₂²⋊C₈

—

139

—

#41

—

278

—

#41

—

278

—

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

SD₁₆

—

278

—

D₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

278

—

C₂³

—

C₂³

—

278

—

C₂³

—

139

—

Q₈

—

278

—

Q₈

—

139

—

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

278

—

D₄

—

556

—

D₄

D₅₅₆

yes

complemented

C₂×C₄

—

100

C₂×C₄

—

278

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

278

—

104

C₂×C₄

—

139

—

105

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

106

C₈

—

107

C₈

—

278

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

278

—

112

C₂²

—

278

—

113

C₂²

—

278

—

114

C₂²

—

556

—

115

C₂²

—

556

—

116

C₂²

—

556

—

117

C₂²

D₄×D₁₃₉

yes

characteristic

118

C₄

—

119

C₄

—

278

—

120

C₄

—

278

—

121

C₄

D₄×D₁₃₉

yes

characteristic

122

C₄

C₂×D₅₅₆

yes

characteristic

123

C₂

—

124

C₂

—

556

—

125

C₂

—

1112

—

126

C₂

#17984812

yes

characteristic

127

C₂

#17984863

yes

characteristic

128

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	#449
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
1. C₂×#14555784	C₂	semidirect
2. C₂×#14555774	C₂	semidirect
3. C₂×#14555770	C₂	semidirect
4. #17984817	C₂	semidirect
5. #17984765	C₂	semidirect
6. (C₂×D₁₃₉)⋊C₈	C₂	semidirect
7. #102×C₁₃₉	C₂	semidirect
26. Dic₂₇₈	D₄	semidirect
39. D₄×C₁₃₉	D₄	semidirect
64. C₁₃₉	#449	semidirect
71. #102	D₁₃₉	semidirect
98. D₄	D₅₅₆	semidirect

H ≅ G/N

type

1. C₂×#14555784

C₂

semidirect

2. C₂×#14555774

C₂

semidirect

3. C₂×#14555770

C₂

semidirect

4. #17984817

C₂

semidirect

5. #17984765

C₂

semidirect

6. (C₂×D₁₃₉)⋊C₈

C₂

semidirect

7. #102×C₁₃₉

C₂

semidirect

26. Dic₂₇₈

D₄

semidirect

39. D₄×C₁₃₉

D₄

semidirect

64. C₁₃₉

#449

semidirect

71. #102

D₁₃₉

semidirect

98. D₄

D₅₅₆

semidirect

Subgroups — #22581099

Conjugacy classes of subgroups

Complements