Group #22550036 — subgroups

Subgroups — #22550036

Order 8128.

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

normal subgroups only

This group has 152 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×#14523248	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523238	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523235	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	—	2	2	no	—
9	#14523248	—	2	2	no	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
13	C₂×C₄×D₁₂₇	—	2	1	no	—
14	#14523238	—	2	2	no	—
15	#14523235	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	1	no	—
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
23	C₂²×D₁₂₇	—	2	1	no	—
24	Dic₂₅₄	—	2	1	no	—
25	Dic₂₅₄	D₄	1	2	yes	—
26	D₅₀₈	—	2	1	no	—
27	D₅₀₈	D₄	1	2	yes	—
28	C₂²⋊D₁₂₇	—	2	4	no	—
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₄×D₁₂₇	—	2	4	no	—
31	C₁₂₇⋊C₈	—	2	1	no	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
34	Q₈×C₁₂₇	—	2	1	no	—
35	Q₈×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	—	2	1	no	—
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	—	2	1	no	—
40	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×C₁₂₇	—	2	1	no	—
43	C₂²×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	1	no	—
45	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂×D₁₂₇	—	2	2	no	—
47	C₂×D₁₂₇	—	4	1	no	—
48	C₂×D₁₂₇	—	4	1	no	—
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	—	4	1	no	—
51	C₄×C₁₂₇	—	4	1	no	—
52	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	Dic₁₂₇	—	2	2	no	—
55	C₂×C₁₂₇	—	4	1	no	—
56	C₂×C₁₂₇	#120	1	1	yes	characteristic
57	C₂×C₁₂₇	#135	1	1	yes	characteristic
58	C₂×C₁₂₇	#137	1	1	yes	characteristic
59	D₁₂₇	—	4	2	no	—
60	C₁₂₇	#452	1	1	yes	characteristic, complemented, Sylow
61	#452	—	127	1	no	maximal, Sylow
62	C₂×#41	—	127	1	no	—
63	C₂×Q₁₆	—	127	1	no	—
64	C₂×SD₁₆	—	127	1	no	—
65	C₂²⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
66	#103	—	127	1	no	—
67	#102	—	127	1	no	—
68	C₂²⋊C₈	—	127	1	no	—
69	#41	—	254	2	no	—
70	#41	—	254	2	no	—
71	C₂×Q₈	—	127	1	no	—
72	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₂×D₄	—	127	1	no	—
74	C₂×D₄	—	254	1	no	—
75	C₂²×C₄	—	254	1	no	—
76	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
77	Q₁₆	—	254	2	no	—
78	SD₁₆	—	254	2	no	—
79	C₂×C₈	—	127	1	no	—
80	C₂×C₈	—	127	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
83	C₂²⋊C₄	—	2	1	no	—
84	C₂³	—	254	1	no	—
85	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
86	Q₈	—	2	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	127	2	no	—
89	Q₈	—	254	1	no	—
90	D₄	—	127	2	no	—
91	D₄	—	254	4	no	—
92	D₄	—	254	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	254	4	no	—
98	C₂×C₄	—	254	1	no	—
99	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₈	—	254	1	no	—
102	C₈	—	254	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	508	1	no	—
107	C₂²	—	508	1	no	—
108	C₂²	—	254	2	no	—
109	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	4	1	no	—
113	C₄	—	254	2	no	—
114	C₄	D₄×D₁₂₇	1	1	yes	characteristic
115	C₄	D₄×D₁₂₇	1	1	yes	characteristic
116	C₂	—	4	1	no	—
117	C₂	—	508	2	no	—
118	C₂	#16861177	1	1	yes	characteristic
119	C₂	#16861229	1	1	yes	characteristic
120	C₂	#16861240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

C₂×#14523238

C₂

yes

maximal, characteristic, complemented

C₂×#14523235

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523238

—

#14523235

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

D₄

yes

—

D₅₀₈

—

D₅₀₈

D₄

yes

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#452

yes

characteristic, complemented, Sylow

#452

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×Q₁₆

—

127

—

C₂×SD₁₆

—

127

—

C₂²⋊Q₈

D₁₂₇

yes

characteristic, complemented

#103

—

127

—

#102

—

127

—

C₂²⋊C₈

—

127

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

254

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

127

—

Q₈

—

254

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₈

—

254

—

102

C₈

—

254

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

508

—

107

C₂²

—

508

—

108

C₂²

—

254

—

109

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

254

—

114

C₄

D₄×D₁₂₇

yes

characteristic

115

C₄

D₄×D₁₂₇

yes

characteristic

116

C₂

—

117

C₂

—

508

—

118

C₂

#16861177

yes

characteristic

119

C₂

#16861229

yes

characteristic

120

C₂

#16861240

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂²⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂²⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#452
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂²⋊Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂²⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#452

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#14523238	C₂	semidirect
3. C₂×#14523235	C₂	semidirect
4. #16861132	C₂	semidirect
5. #16861129	C₂	semidirect
6. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
7. C₂²⋊Q₈×C₁₂₇	C₂	semidirect
60. C₁₂₇	#452	semidirect
65. C₂²⋊Q₈	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523238

C₂

semidirect

3. C₂×#14523235

C₂

semidirect

4. #16861132

C₂

semidirect

5. #16861129

C₂

semidirect

6. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

7. C₂²⋊Q₈×C₁₂₇

C₂

semidirect

60. C₁₂₇

#452

semidirect

65. C₂²⋊Q₈

D₁₂₇

semidirect

Subgroups — #22550036

Conjugacy classes of subgroups

Complements