Group #22550045 — subgroups

Subgroups — #22550045

Order 8128.

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	#16861255	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861237	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861234	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861206	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861137	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861125	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	—	2	1	no	—
9	#14523248	—	2	1	no	—
10	#14523248	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	—	2	1	no	—
12	C₂×D₅₀₈	—	2	2	no	—
13	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
14	C₂×D₅₀₈	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₁₂₇	—	2	2	no	—
16	D₄×D₁₂₇	—	2	1	no	—
17	#14523237	—	2	1	no	—
18	#14523235	—	2	1	no	—
19	D₁₀₁₆	—	2	1	no	—
20	#14523233	—	2	1	no	—
21	#14523232	C₂²	1	1	yes	characteristic, complemented
22	#41×C₁₂₇	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₂₇	—	2	2	no	—
26	C₂²×D₁₂₇	—	2	1	no	—
27	C₂²×D₁₂₇	D₄	1	2	yes	—
28	Dic₂₅₄	D₄	1	1	yes	characteristic, complemented
29	D₅₀₈	—	2	2	no	—
30	D₅₀₈	—	2	1	no	—
31	D₅₀₈	—	2	1	no	—
32	D₅₀₈	—	2	1	no	—
33	D₅₀₈	—	2	1	no	—
34	D₅₀₈	—	4	4	no	—
35	D₅₀₈	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₁₂₇	—	2	2	no	—
37	C₂²⋊D₁₂₇	—	2	1	no	—
38	C₄×D₁₂₇	—	2	2	no	—
39	C₄×D₁₂₇	—	2	1	no	—
40	C₁₂₇⋊C₈	—	2	1	no	—
41	Q₈×C₁₂₇	D₄	1	1	yes	characteristic, complemented
42	D₄×C₁₂₇	—	2	1	no	—
43	D₄×C₁₂₇	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₁₂₇	—	2	1	no	—
45	C₂×C₄×C₁₂₇	—	2	1	no	—
46	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₈×C₁₂₇	—	2	1	no	—
48	C₂²×C₁₂₇	—	2	1	no	—
49	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₂₇	—	2	2	no	—
51	C₂×D₁₂₇	—	2	1	no	—
52	C₂×D₁₂₇	—	4	2	no	—
53	C₂×D₁₂₇	—	4	2	no	—
54	C₂×D₁₂₇	—	4	1	no	—
55	C₂×D₁₂₇	—	8	2	no	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	—	2	1	no	—
58	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	Dic₁₂₇	—	2	1	no	—
61	C₂×C₁₂₇	—	2	1	no	—
62	C₂×C₁₂₇	—	4	1	no	—
63	C₂×C₁₂₇	#120	1	1	yes	characteristic
64	D₁₂₇	—	4	2	no	—
65	D₁₂₇	—	4	1	no	—
66	D₁₂₇	—	8	1	no	—
67	C₁₂₇	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	127	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	127	1	no	—
70	AGL(1,ℤ/8)	—	127	1	no	—
71	AGL(1,ℤ/8)	—	127	1	no	—
72	#127	—	127	1	no	—
73	#104	—	127	1	no	—
74	#104	D₁₂₇	1	1	yes	characteristic, complemented
75	#100	—	127	1	no	—
76	#41	—	127	1	no	—
77	#41	—	254	1	no	—
78	#41	—	254	1	no	—
79	#41	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂×D₄	—	127	1	no	—
81	C₂×D₄	—	127	1	no	—
82	C₂×D₄	—	254	1	no	—
83	C₂×D₄	—	254	2	no	—
84	C₂×D₄	—	254	2	no	—
85	C₂×D₄	—	254	1	no	—
86	SD₁₆	—	254	1	no	—
87	SD₁₆	—	254	1	no	—
88	D₈	—	254	1	no	—
89	D₈	—	254	1	no	—
90	M₁₆	—	127	1	no	—
91	M₁₆	C₂×D₁₂₇	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₁₂₇	1	1	yes	characteristic, complemented
93	C₂³	—	127	2	no	—
94	C₂³	—	254	2	no	—
95	C₂³	—	254	1	no	—
96	Q₈	—	127	1	no	—
97	Q₈	D₅₀₈	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	127	1	no	—
100	D₄	—	508	4	no	—
101	D₄	—	254	2	no	—
102	D₄	—	254	2	no	—
103	D₄	—	254	1	no	—
104	D₄	—	254	1	no	—
105	D₄	—	254	1	no	—
106	D₄	—	254	1	no	—
107	D₄	—	254	1	no	—
108	D₄	D₅₀₈	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	254	2	no	—
112	C₂×C₄	—	254	1	no	—
113	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	254	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	1016	2	no	—
118	C₂²	—	508	2	no	—
119	C₂²	—	508	2	no	—
120	C₂²	—	508	1	no	—
121	C₂²	—	254	1	no	—
122	C₂²	—	254	2	no	—
123	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	254	1	no	—
127	C₄	D₄×D₁₂₇	1	1	yes	characteristic
128	C₄	C₂×D₅₀₈	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	1016	1	no	—
132	C₂	—	508	1	no	—
133	C₂	—	508	2	no	—
134	C₂	#16861179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861255

C₂

yes

maximal, characteristic, complemented

#16861237

C₂

yes

maximal, characteristic, complemented

#16861234

C₂

yes

maximal, characteristic, complemented

#16861206

C₂

yes

maximal, characteristic, complemented

#16861137

C₂

yes

maximal, characteristic, complemented

#16861125

C₂

yes

maximal, characteristic, complemented

#104×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

#14523248

—

#14523248

C₂²

yes

characteristic

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic, complemented

D₄×D₁₂₇

—

D₄×D₁₂₇

—

#14523237

—

#14523235

—

D₁₀₁₆

—

#14523233

—

#14523232

C₂²

yes

characteristic, complemented

#41×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

characteristic, complemented

C₄²×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

—

Dic₂₅₄

D₄

yes

characteristic, complemented

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

characteristic, complemented

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

D₄

yes

characteristic, complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

D₄

yes

characteristic, complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#120

yes

characteristic

D₁₂₇

—

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#453

yes

characteristic, complemented, Sylow

#453

—

127

maximal, Sylow

H(5,𝔽₂)

—

127

—

AGL(1,ℤ/8)

—

127

—

AGL(1,ℤ/8)

—

127

—

#127

—

127

—

#104

—

127

—

#104

D₁₂₇

yes

characteristic, complemented

#100

—

127

—

#41

—

127

—

#41

—

254

—

#41

—

254

—

#41

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

SD₁₆

—

254

—

SD₁₆

—

254

—

D₈

—

254

—

D₈

—

254

—

M₁₆

—

127

—

M₁₆

C₂×D₁₂₇

yes

characteristic, complemented

C₄²

C₂×D₁₂₇

yes

characteristic, complemented

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

Q₈

—

127

—

Q₈

D₅₀₈

yes

characteristic, complemented

D₄

—

D₄

—

127

—

100

D₄

—

508

—

101

D₄

—

254

—

102

D₄

—

254

—

103

D₄

—

254

—

104

D₄

—

254

—

105

D₄

—

254

—

106

D₄

—

254

—

107

D₄

—

254

—

108

D₄

D₅₀₈

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

114

C₈

—

115

C₈

—

254

—

116

C₂²

—

117

C₂²

—

1016

—

118

C₂²

—

508

—

119

C₂²

—

508

—

120

C₂²

—

508

—

121

C₂²

—

254

—

122

C₂²

—

254

—

123

C₂²

D₄×D₁₂₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

254

—

127

C₄

D₄×D₁₂₇

yes

characteristic

128

C₄

C₂×D₅₀₈

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

1016

—

132

C₂

—

508

—

133

C₂

—

508

—

134

C₂

#16861179

yes

characteristic

	subgroup	G/H
center	C₂	#16861179
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#104×C₁₂₇	C₂
socle	C₂×C₁₂₇	#120
hypercenter	#104	D₁₂₇
hypocenter	C₁₂₇	#453
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#16861179

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#104×C₁₂₇

C₂

C₂×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #16861255	C₂	semidirect
2. #16861237	C₂	semidirect
3. #16861234	C₂	semidirect
4. #16861206	C₂	semidirect
5. #16861137	C₂	semidirect
6. #16861125	C₂	semidirect
7. #104×C₁₂₇	C₂	semidirect
14. C₂×D₅₀₈	C₂²	semidirect
21. #14523232	C₂²	semidirect
23. M₁₆×C₁₂₇	C₂²	semidirect
24. C₄²×C₁₂₇	C₂²	semidirect
28. Dic₂₅₄	D₄	semidirect
35. D₅₀₈	D₄	semidirect
41. Q₈×C₁₂₇	D₄	semidirect
43. D₄×C₁₂₇	D₄	semidirect
67. C₁₂₇	#453	semidirect
74. #104	D₁₂₇	semidirect
91. M₁₆	C₂×D₁₂₇	semidirect
92. C₄²	C₂×D₁₂₇	semidirect
97. Q₈	D₅₀₈	semidirect
108. D₄	D₅₀₈	semidirect

H ≅ G/N

type

1. #16861255

C₂

semidirect

2. #16861237

C₂

semidirect

3. #16861234

C₂

semidirect

4. #16861206

C₂

semidirect

5. #16861137

C₂

semidirect

6. #16861125

C₂

semidirect

7. #104×C₁₂₇

C₂

semidirect

14. C₂×D₅₀₈

C₂²

semidirect

21. #14523232

C₂²

semidirect

23. M₁₆×C₁₂₇

C₂²

semidirect

24. C₄²×C₁₂₇

C₂²

semidirect

28. Dic₂₅₄

D₄

semidirect

35. D₅₀₈

D₄

semidirect

41. Q₈×C₁₂₇

D₄

semidirect

43. D₄×C₁₂₇

D₄

semidirect

67. C₁₂₇

#453

semidirect

74. #104

D₁₂₇

semidirect

91. M₁₆

C₂×D₁₂₇

semidirect

92. C₄²

C₂×D₁₂₇

semidirect

97. Q₈

D₅₀₈

semidirect

108. D₄

D₅₀₈

semidirect

Subgroups — #22550045

Conjugacy classes of subgroups

Complements