Group #22581119 — subgroups

Subgroups — #22581119

Order 8896.

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

normal subgroups only

This group has 168 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, complemented
2	#17984868	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984838	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	AGL(1,ℤ/8)×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	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
14	D₄×D₁₃₉	—	2	2	no	—
15	D₄×D₁₃₉	—	2	1	no	—
16	#14555773	—	2	1	no	—
17	#14555770	—	2	1	no	—
18	#14555769	C₂²	1	1	yes	characteristic, complemented
19	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
20	#41×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
22	SD₁₆×C₁₃₉	—	2	1	no	—
23	D₈×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, complemented
28	D₅₅₆	—	2	2	no	—
29	D₅₅₆	—	2	1	no	—
30	D₅₅₆	—	2	1	no	—
31	D₅₅₆	D₄	1	1	yes	characteristic, complemented
32	C₂²⋊D₁₃₉	—	2	2	no	—
33	C₂²⋊D₁₃₉	—	2	1	no	—
34	C₂²⋊D₁₃₉	—	4	4	no	—
35	C₂×Dic₁₃₉	—	2	1	no	—
36	C₄×D₁₃₉	—	2	2	no	—
37	C₄×D₁₃₉	—	2	1	no	—
38	C₁₃₉⋊C₈	—	2	1	no	—
39	C₂³×C₁₃₉	—	2	1	no	—
40	Q₈×C₁₃₉	D₄	1	1	yes	characteristic, complemented
41	D₄×C₁₃₉	—	2	1	no	—
42	D₄×C₁₃₉	—	2	1	no	—
43	D₄×C₁₃₉	—	2	1	no	—
44	D₄×C₁₃₉	D₄	1	1	yes	characteristic, complemented
45	C₂×C₄×C₁₃₉	—	2	1	no	—
46	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₈×C₁₃₉	—	2	1	no	—
48	C₂²×C₁₃₉	—	2	1	no	—
49	C₂²×C₁₃₉	—	4	2	no	—
50	C₂²×C₁₃₉	—	4	1	no	—
51	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×D₁₃₉	—	2	2	no	—
53	C₂×D₁₃₉	—	2	1	no	—
54	C₂×D₁₃₉	—	4	2	no	—
55	C₂×D₁₃₉	—	8	2	no	—
56	C₄×C₁₃₉	—	2	1	no	—
57	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	Dic₁₃₉	—	2	2	no	—
60	Dic₁₃₉	—	2	1	no	—
61	C₂×C₁₃₉	—	2	1	no	—
62	C₂×C₁₃₉	—	4	1	no	—
63	C₂×C₁₃₉	—	8	1	no	—
64	C₂×C₁₃₉	#120	1	1	yes	characteristic
65	D₁₃₉	—	4	2	no	—
66	D₁₃₉	—	4	1	no	—
67	C₁₃₉	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	139	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	139	1	no	—
70	AGL(1,ℤ/8)	—	139	1	no	—
71	AGL(1,ℤ/8)	D₁₃₉	1	1	yes	characteristic, complemented
72	#127	—	139	1	no	—
73	#104	—	139	1	no	—
74	#104	—	139	1	no	—
75	#100	—	139	1	no	—
76	#41	—	278	1	no	—
77	#41	—	278	1	no	—
78	#41	—	139	1	no	—
79	#41	C₂×D₁₃₉	1	1	yes	characteristic
80	C₂×D₄	—	278	1	no	—
81	C₂×D₄	—	278	2	no	—
82	C₂×D₄	—	278	1	no	—
83	C₂×D₄	—	278	2	no	—
84	C₂×D₄	—	139	1	no	—
85	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic, complemented
86	SD₁₆	—	2	1	no	—
87	SD₁₆	—	278	1	no	—
88	D₈	—	2	1	no	—
89	D₈	—	278	1	no	—
90	M₁₆	—	139	1	no	—
91	M₁₆	C₂×D₁₃₉	1	1	yes	characteristic, complemented
92	C₄²	—	139	1	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	278	2	no	—
95	C₂³	—	139	2	no	—
96	Q₈	—	139	1	no	—
97	Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	2	1	no	—
100	D₄	—	2	1	no	—
101	D₄	—	278	2	no	—
102	D₄	—	278	2	no	—
103	D₄	—	278	1	no	—
104	D₄	—	278	1	no	—
105	D₄	—	278	1	no	—
106	D₄	—	556	4	no	—
107	D₄	—	139	1	no	—
108	D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	278	2	no	—
111	C₂×C₄	—	278	1	no	—
112	C₂×C₄	—	278	1	no	—
113	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	278	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	278	2	no	—
120	C₂²	—	278	1	no	—
121	C₂²	—	556	2	no	—
122	C₂²	—	1112	2	no	—
123	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	278	2	no	—
126	C₄	—	278	1	no	—
127	C₄	D₄×D₁₃₉	1	1	yes	characteristic
128	C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	8	1	no	—
132	C₂	—	556	2	no	—
133	C₂	—	556	1	no	—
134	C₂	#17984810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984888

C₂

yes

maximal, characteristic, complemented

#17984868

C₂

yes

maximal, characteristic, complemented

#17984838

C₂

yes

maximal, characteristic, complemented

#17984769

C₂

yes

maximal, characteristic, complemented

#17984768

C₂

yes

maximal, characteristic, complemented

#17984758

C₂

yes

maximal, characteristic, complemented

AGL(1,ℤ/8)×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555786

—

#14555785

—

#14555785

C₂²

yes

characteristic

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

D₄×D₁₃₉

—

D₄×D₁₃₉

—

#14555773

—

#14555770

—

#14555769

C₂²

yes

characteristic, complemented

C₄×Dic₁₃₉

C₂²

yes

characteristic, complemented

#41×C₁₃₉

C₂²

yes

characteristic

C₂×D₄×C₁₃₉

C₂²

yes

characteristic, complemented

SD₁₆×C₁₃₉

—

D₈×C₁₃₉

—

M₁₆×C₁₃₉

C₂²

yes

characteristic, complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

D₄

yes

—

Dic₂₇₈

D₄

yes

characteristic, complemented

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

D₄

yes

characteristic, complemented

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

D₄

yes

characteristic, complemented

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

characteristic, 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

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#453

yes

characteristic, complemented, Sylow

#453

—

139

maximal, Sylow

H(5,𝔽₂)

—

139

—

AGL(1,ℤ/8)

—

139

—

AGL(1,ℤ/8)

D₁₃₉

yes

characteristic, complemented

#127

—

139

—

#104

—

139

—

#104

—

139

—

#100

—

139

—

#41

—

278

—

#41

—

278

—

#41

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic, complemented

SD₁₆

—

SD₁₆

—

278

—

D₈

—

D₈

—

278

—

M₁₆

—

139

—

M₁₆

C₂×D₁₃₉

yes

characteristic, complemented

C₄²

—

139

—

C₂³

—

C₂³

—

278

—

C₂³

—

139

—

Q₈

—

139

—

Q₈

C₂²⋊D₁₃₉

yes

characteristic, complemented

D₄

—

D₄

—

100

D₄

—

101

D₄

—

278

—

102

D₄

—

278

—

103

D₄

—

278

—

104

D₄

—

278

—

105

D₄

—

278

—

106

D₄

—

556

—

107

D₄

—

139

—

108

D₄

C₂²⋊D₁₃₉

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

278

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

278

—

113

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

114

C₈

—

115

C₈

—

278

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

278

—

120

C₂²

—

278

—

121

C₂²

—

556

—

122

C₂²

—

1112

—

123

C₂²

D₄×D₁₃₉

yes

characteristic

124

C₄

—

125

C₄

—

278

—

126

C₄

—

278

—

127

C₄

D₄×D₁₃₉

yes

characteristic

128

C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

—

556

—

133

C₂

—

556

—

134

C₂

#17984810

yes

characteristic

	subgroup	G/H
center	C₂	#17984810
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	AGL(1,ℤ/8)×C₁₃₉	C₂
socle	C₂×C₁₃₉	#120
hypercenter	AGL(1,ℤ/8)	D₁₃₉
hypocenter	C₁₃₉	#453
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

subgroup

G/H

center

C₂

#17984810

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

AGL(1,ℤ/8)×C₁₃₉

C₂

C₂×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

N	H ≅ G/N	type
1. #17984888	C₂	semidirect
2. #17984868	C₂	semidirect
3. #17984838	C₂	semidirect
4. #17984769	C₂	semidirect
5. #17984768	C₂	semidirect
6. #17984758	C₂	semidirect
7. AGL(1,ℤ/8)×C₁₃₉	C₂	semidirect
18. #14555769	C₂²	semidirect
19. C₄×Dic₁₃₉	C₂²	semidirect
21. C₂×D₄×C₁₃₉	C₂²	semidirect
24. M₁₆×C₁₃₉	C₂²	semidirect
27. Dic₂₇₈	D₄	semidirect
31. D₅₅₆	D₄	semidirect
40. Q₈×C₁₃₉	D₄	semidirect
44. D₄×C₁₃₉	D₄	semidirect
67. C₁₃₉	#453	semidirect
71. AGL(1,ℤ/8)	D₁₃₉	semidirect
85. C₂×D₄	C₂×D₁₃₉	semidirect
91. M₁₆	C₂×D₁₃₉	semidirect
97. Q₈	C₂²⋊D₁₃₉	semidirect
108. D₄	C₂²⋊D₁₃₉	semidirect

H ≅ G/N

type

1. #17984888

C₂

semidirect

2. #17984868

C₂

semidirect

3. #17984838

C₂

semidirect

4. #17984769

C₂

semidirect

5. #17984768

C₂

semidirect

6. #17984758

C₂

semidirect

7. AGL(1,ℤ/8)×C₁₃₉

C₂

semidirect

18. #14555769

C₂²

semidirect

19. C₄×Dic₁₃₉

C₂²

semidirect

21. C₂×D₄×C₁₃₉

C₂²

semidirect

24. M₁₆×C₁₃₉

C₂²

semidirect

27. Dic₂₇₈

D₄

semidirect

31. D₅₅₆

D₄

semidirect

40. Q₈×C₁₃₉

D₄

semidirect

44. D₄×C₁₃₉

D₄

semidirect

67. C₁₃₉

#453

semidirect

71. AGL(1,ℤ/8)

D₁₃₉

semidirect

85. C₂×D₄

C₂×D₁₃₉

semidirect

91. M₁₆

C₂×D₁₃₉

semidirect

97. Q₈

C₂²⋊D₁₃₉

semidirect

108. D₄

C₂²⋊D₁₃₉

semidirect

Subgroups — #22581119

Conjugacy classes of subgroups

Complements