Group #22550451 — subgroups

Subgroups — #22550451

Order 8128.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#14523249	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861207	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861198	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861197	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861194	C₂	1	4	yes	maximal, complemented
7	#16861187	C₂	1	4	yes	maximal, complemented
8	#16861166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#14523249	—	2	4	no	—
11	C₂×Dic₂₅₄	C₂²	1	2	yes	—
12	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
13	Q₈×D₁₂₇	—	2	4	no	—
14	C₂×D₅₀₈	C₂²	1	2	yes	—
15	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
19	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
21	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
22	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
23	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
28	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂²×D₁₂₇	C₂³	1	2	yes	—
30	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
31	Dic₂₅₄	—	2	2	no	—
32	Dic₂₅₄	—	2	4	no	—
33	D₅₀₈	—	2	4	no	—
34	D₅₀₈	—	2	2	no	—
35	C₂×Dic₁₂₇	C₂³	1	2	yes	—
36	C₂×Dic₁₂₇	C₂³	1	2	yes	—
37	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
38	C₄×D₁₂₇	—	2	4	no	—
39	C₄×D₁₂₇	—	2	4	no	—
40	C₄×D₁₂₇	D₄	1	4	yes	—
41	Q₈×C₁₂₇	—	2	2	no	—
42	Q₈×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
48	C₂×D₁₂₇	—	2	2	no	—
49	C₂×D₁₂₇	—	2	2	no	—
50	C₂×D₁₂₇	—	2	4	no	—
51	C₂×D₁₂₇	C₂×D₄	1	2	yes	—
52	C₄×C₁₂₇	—	2	2	no	—
53	C₄×C₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	—	2	2	no	—
55	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
56	Dic₁₂₇	—	2	2	no	—
57	Dic₁₂₇	—	2	2	no	—
58	Dic₁₂₇	C₂×D₄	1	2	yes	—
59	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
61	D₁₂₇	—	2	2	no	—
62	D₁₂₇	—	4	2	no	—
63	C₁₂₇	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	127	1	no	maximal, Sylow
65	C₂×#41	—	127	1	no	—
66	C₂²×Q₈	—	127	1	no	—
67	C₄⋊Q₈	—	127	1	no	—
68	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
69	#124	—	127	1	no	—
70	#124	—	127	1	no	—
71	#123	—	127	4	no	—
72	C₂²⋊Q₈	—	127	4	no	—
73	#117	—	127	1	no	—
74	#41	—	254	4	no	—
75	C₂×Q₈	—	127	2	no	—
76	C₂×Q₈	—	127	1	no	—
77	C₂×Q₈	—	254	4	no	—
78	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂×D₄	—	127	2	no	—
81	C₂×D₄	—	127	1	no	—
82	C₂²×C₄	—	127	1	no	—
83	C₂²×C₄	—	127	2	no	—
84	C₂²×C₄	—	127	2	no	—
85	C₄⋊C₄	—	127	2	no	—
86	C₄⋊C₄	—	127	4	no	—
87	C₄⋊C₄	C₂×D₁₂₇	1	4	yes	—
88	C₂²⋊C₄	—	127	2	no	—
89	C₂²⋊C₄	—	127	4	no	—
90	C₂²⋊C₄	—	127	4	no	—
91	C₄²	—	127	1	no	—
92	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
93	C₂³	—	127	1	no	—
94	C₂³	—	127	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	254	4	no	—
98	Q₈	—	254	2	no	—
99	D₄	—	254	4	no	—
100	D₄	—	254	2	no	—
101	C₂×C₄	—	127	1	no	—
102	C₂×C₄	—	127	4	no	—
103	C₂×C₄	—	127	2	no	—
104	C₂×C₄	—	127	2	no	—
105	C₂×C₄	—	254	4	no	—
106	C₂×C₄	—	254	4	no	—
107	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
108	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
109	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
110	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₂²	—	127	2	no	—
112	C₂²	—	254	2	no	—
113	C₂²	—	254	4	no	—
114	C₂²	—	254	2	no	—
115	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	127	2	no	—
120	C₄	—	254	2	no	—
121	C₄	—	254	2	no	—
122	C₄	D₄×D₁₂₇	1	2	yes	—
123	C₂	—	508	2	no	—
124	C₂	—	254	2	no	—
125	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
126	C₂	#16861257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523249

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861207

C₂

yes

maximal, characteristic, complemented

#16861198

C₂

yes

maximal, characteristic, complemented

#16861197

C₂

yes

maximal, characteristic, complemented

#16861194

C₂

yes

maximal, complemented

#16861187

C₂

yes

maximal, complemented

#16861166

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

C₂×Dic₂₅₄

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

Q₈×D₁₂₇

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

D₄

yes

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

—

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#536

yes

characteristic, complemented, Sylow

#536

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

D₁₂₇

yes

characteristic, complemented

#124

—

127

—

#124

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

#117

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

—

127

—

Q₈

—

Q₈

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

254

—

100

D₄

—

254

—

101

C₂×C₄

—

127

—

102

C₂×C₄

—

127

—

103

C₂×C₄

—

127

—

104

C₂×C₄

—

127

—

105

C₂×C₄

—

254

—

106

C₂×C₄

—

254

—

107

C₂×C₄

C₂²×D₁₂₇

yes

—

108

C₂×C₄

C₂²×D₁₂₇

yes

—

109

C₂×C₄

C₂²×D₁₂₇

yes

—

110

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

111

C₂²

—

127

—

112

C₂²

—

254

—

113

C₂²

—

254

—

114

C₂²

—

254

—

115

C₂²

C₂³×D₁₂₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

127

—

120

C₄

—

254

—

121

C₄

—

254

—

122

C₄

D₄×D₁₂₇

yes

—

123

C₂

—

508

—

124

C₂

—

254

—

125

C₂

C₂×D₄×D₁₂₇

yes

characteristic

126

C₂

#16861257

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₂₇	C₂
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#536
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₂₇

C₂

hypercenter

C₄⋊Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₁₂₇	C₂	semidirect
3. #16861207	C₂	semidirect
4. #16861198	C₂	semidirect
5. #16861197	C₂	semidirect
6. #16861194	C₂	semidirect
7. #16861187	C₂	semidirect
8. #16861166	C₂	semidirect
9. C₄⋊Q₈×C₁₂₇	C₂	semidirect
63. C₁₂₇	#536	semidirect
68. C₄⋊Q₈	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₂₇

C₂

semidirect

3. #16861207

C₂

semidirect

4. #16861198

C₂

semidirect

5. #16861197

C₂

semidirect

6. #16861194

C₂

semidirect

7. #16861187

C₂

semidirect

8. #16861166

C₂

semidirect

9. C₄⋊Q₈×C₁₂₇

C₂

semidirect

63. C₁₂₇

#536

semidirect

68. C₄⋊Q₈

D₁₂₇

semidirect

Subgroups — #22550451

Conjugacy classes of subgroups

Complements