Group #22549780 — subgroups

Subgroups — #22549780

Order 8128.

186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂²×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
7	#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	—	2	2	no	—
9	C₂×Dic₂₅₄	—	2	2	no	—
10	C₂×Dic₂₅₄	C₄	1	4	yes	complemented
11	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
15	Dic₁₂₇⋊C₄	—	2	2	no	—
16	C₄×Dic₁₂₇	—	2	2	no	—
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	Dic₂₅₄	—	2	8	no	—
21	C₂×Dic₁₂₇	—	2	1	no	—
22	C₂×Dic₁₂₇	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₂×Dic₁₂₇	—	2	1	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	—	2	1	no	—
28	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₂₇	D₄	1	2	yes	—
31	C₂×Dic₁₂₇	D₄	1	2	yes	—
32	C₂×Dic₁₂₇	Q₈	1	2	yes	complemented
33	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×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	—	2	1	no	—
41	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	Q₈	1	2	yes	complemented
43	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₁₂₇	#41	1	1	yes	characteristic
49	C₂²×C₁₂₇	#41	1	1	yes	characteristic
50	C₄×C₁₂₇	—	2	2	no	—
51	C₄×C₁₂₇	—	2	2	no	—
52	C₄×C₁₂₇	—	4	1	no	—
53	Dic₁₂₇	—	2	2	no	—
54	Dic₁₂₇	—	2	2	no	—
55	Dic₁₂₇	—	4	1	no	—
56	Dic₁₂₇	C₂²⋊C₄	1	4	yes	—
57	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₂₇	#124	1	1	yes	characteristic
63	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₁₂₇	#391	1	1	yes	characteristic, complemented, Sylow
65	#391	—	127	1	no	maximal, Sylow
66	C₂²×Q₈	—	127	1	no	—
67	C₂×C₄⋊C₄	—	127	1	no	—
68	C₂×C₄²	—	127	1	no	—
69	#95	—	127	1	no	—
70	#95	—	127	1	no	—
71	#95	—	127	1	no	—
72	#95	D₁₂₇	1	1	yes	characteristic
73	C₂×Q₈	—	127	4	no	—
74	C₂×Q₈	—	254	2	no	—
75	C₂×Q₈	—	254	2	no	—
76	C₂²×C₄	—	127	1	no	—
77	C₂²×C₄	—	127	1	no	—
78	C₂²×C₄	—	127	1	no	—
79	C₂²×C₄	—	127	1	no	—
80	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
83	C₄⋊C₄	—	254	2	no	—
84	C₄²	—	254	2	no	—
85	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
86	Q₈	—	254	8	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	127	2	no	—
95	C₂×C₄	—	127	2	no	—
96	C₂×C₄	—	127	2	no	—
97	C₂×C₄	—	127	2	no	—
98	C₂×C₄	—	127	2	no	—
99	C₂×C₄	—	254	1	no	—
100	C₂×C₄	—	254	1	no	—
101	C₂×C₄	—	254	1	no	—
102	C₂×C₄	—	254	1	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	—	254	1	no	—
105	C₂×C₄	—	254	1	no	—
106	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
107	C₂×C₄	Dic₂₅₄	1	2	yes	complemented
108	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
109	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
110	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
111	C₂²	Q₈×D₁₂₇	1	1	yes	characteristic
112	C₂²	C₂×Dic₂₅₄	1	1	yes	characteristic
113	C₂²	#14523247	1	1	yes	characteristic
114	C₂²	#14523248	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	1	no	—
118	C₄	—	127	4	no	—
119	C₄	—	508	1	no	—
120	C₄	—	254	2	no	—
121	C₄	—	254	2	no	—
122	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
123	C₂	#16861174	1	1	yes	characteristic
124	C₂	C₄×Dic₂₅₄	1	1	yes	characteristic
125	C₂	#16861186	1	1	yes	characteristic
126	C₂	#16861187	1	1	yes	characteristic
127	C₂	#16861195	1	1	yes	characteristic
128	C₂	#16861208	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₅₄

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic

C₁₂₇⋊#95

C₂

yes

maximal, characteristic

C₁₂₇⋊#95

C₂

yes

maximal, characteristic

C₁₂₇⋊#95

C₂

yes

maximal, characteristic

#95×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₄

yes

complemented

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

Q₈

yes

complemented

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

Q₈

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₂₇

#124

yes

characteristic

C₂×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₁₂₇

#391

yes

characteristic, complemented, Sylow

#391

—

127

maximal, Sylow

C₂²×Q₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

—

127

—

#95

—

127

—

#95

—

127

—

#95

—

127

—

#95

D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

100

C₂×C₄

—

254

—

101

C₂×C₄

—

254

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

—

254

—

105

C₂×C₄

—

254

—

106

C₂×C₄

C₄×D₁₂₇

yes

—

107

C₂×C₄

Dic₂₅₄

yes

complemented

108

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

109

C₂²

D₄×D₁₂₇

yes

characteristic

110

C₂²

D₄×D₁₂₇

yes

characteristic

111

C₂²

Q₈×D₁₂₇

yes

characteristic

112

C₂²

C₂×Dic₂₅₄

yes

characteristic

113

C₂²

#14523247

yes

characteristic

114

C₂²

#14523248

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

127

—

119

C₄

—

508

—

120

C₄

—

254

—

121

C₄

—

254

—

122

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

123

C₂

#16861174

yes

characteristic

124

C₂

C₄×Dic₂₅₄

yes

characteristic

125

C₂

#16861186

yes

characteristic

126

C₂

#16861187

yes

characteristic

127

C₂

#16861195

yes

characteristic

128

C₂

#16861208

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	#95×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	#95	D₁₂₇
hypocenter	C₁₂₇	#391
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

#95×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
10. C₂×Dic₂₅₄	C₄	semidirect
32. C₂×Dic₁₂₇	Q₈	semidirect
42. C₂×C₄×C₁₂₇	Q₈	semidirect
64. C₁₂₇	#391	semidirect
107. C₂×C₄	Dic₂₅₄	semidirect

H ≅ G/N

type

10. C₂×Dic₂₅₄

C₄

semidirect

32. C₂×Dic₁₂₇

Q₈

semidirect

42. C₂×C₄×C₁₂₇

Q₈

semidirect

64. C₁₂₇

#391

semidirect

107. C₂×C₄

Dic₂₅₄

semidirect

Subgroups — #22549780

Conjugacy classes of subgroups

Complements