Group #22549841 — subgroups

Subgroups — #22549841

Order 8128.

158 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	—	2	2	no	—
9	#14523248	—	2	2	no	—
10	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
11	C₂×D₅₀₈	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₂₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
16	#14523231	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×D₁₂₇	—	2	2	no	—
20	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
25	Dic₂₅₄	—	2	2	no	—
26	D₅₀₈	—	2	2	no	—
27	C₂²⋊D₁₂₇	—	2	2	no	—
28	C₂²⋊D₁₂₇	—	2	2	no	—
29	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄×D₁₂₇	—	2	1	no	—
32	C₄×D₁₂₇	—	2	1	no	—
33	C₄×D₁₂₇	D₄	1	2	yes	—
34	C₄×D₁₂₇	D₄	1	2	yes	—
35	C₁₂₇⋊C₈	—	2	1	no	—
36	C₁₂₇⋊C₈	—	2	1	no	—
37	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	—	2	1	no	—
40	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₈×C₁₂₇	—	2	1	no	—
43	C₈×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	1	no	—
45	C₂²×C₁₂₇	—	2	1	no	—
46	C₂²×C₁₂₇	—	2	1	no	—
47	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×D₁₂₇	—	2	1	no	—
49	C₂×D₁₂₇	—	2	1	no	—
50	C₂×D₁₂₇	—	2	1	no	—
51	C₂×D₁₂₇	—	2	1	no	—
52	C₂×D₁₂₇	—	2	1	no	—
53	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₄×C₁₂₇	—	2	1	no	—
55	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	Dic₁₂₇	—	2	1	no	—
58	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₁₂₇	—	4	1	no	—
60	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₁₂₇	#131	1	1	yes	characteristic
62	C₂×C₁₂₇	#131	1	1	yes	characteristic
63	D₁₂₇	—	2	2	no	—
64	D₁₂₇	—	4	1	no	—
65	C₁₂₇	#408	1	1	yes	characteristic, complemented, Sylow
66	#408	—	127	1	no	maximal, Sylow
67	C₂×#41	—	127	1	no	—
68	C₂×M₁₆	—	127	1	no	—
69	C₂²×C₈	—	127	1	no	—
70	C₂²⋊C₈	—	127	1	no	—
71	C₂²⋊C₈	—	127	1	no	—
72	C₂²⋊C₈	—	127	1	no	—
73	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
74	#41	—	254	2	no	—
75	#41	—	254	2	no	—
76	C₂×Q₈	—	127	1	no	—
77	C₂×D₄	—	127	1	no	—
78	C₂×D₄	—	127	1	no	—
79	C₂×D₄	—	127	1	no	—
80	C₂²×C₄	—	127	1	no	—
81	C₂²×C₄	—	127	1	no	—
82	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
83	M₁₆	—	254	2	no	—
84	C₂×C₈	—	127	1	no	—
85	C₂×C₈	—	127	1	no	—
86	C₂×C₈	—	254	2	no	—
87	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
88	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂³	—	127	1	no	—
90	C₂³	—	127	1	no	—
91	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
92	Q₈	—	254	2	no	—
93	D₄	—	254	2	no	—
94	D₄	—	254	2	no	—
95	D₄	—	254	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	127	1	no	—
99	C₂×C₄	—	127	2	no	—
100	C₂×C₄	—	127	2	no	—
101	C₂×C₄	—	127	1	no	—
102	C₂×C₄	—	254	1	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	254	1	no	—
109	C₈	—	254	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	127	2	no	—
114	C₂²	—	254	1	no	—
115	C₂²	—	254	1	no	—
116	C₂²	—	254	1	no	—
117	C₂²	—	254	1	no	—
118	C₂²	—	254	1	no	—
119	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	127	2	no	—
122	C₄	—	254	1	no	—
123	C₄	D₄×D₁₂₇	1	1	yes	characteristic
124	C₄	D₄×D₁₂₇	1	1	yes	characteristic
125	C₂	—	4	1	no	—
126	C₂	—	508	1	no	—
127	C₂	—	254	2	no	—
128	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
129	C₂	#16861215	1	1	yes	characteristic
130	C₂	#16861217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

—

C₂×Dic₂₅₄

C₄

yes

characteristic

C₂×D₅₀₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×D₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

D₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#408

yes

characteristic, complemented, Sylow

#408

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

D₁₂₇

yes

characteristic, complemented

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

—

127

—

C₂³

C₄×D₁₂₇

yes

characteristic

Q₈

—

254

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

100

C₂×C₄

—

127

—

101

C₂×C₄

—

127

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

C₄×D₁₂₇

yes

characteristic

105

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

254

—

109

C₈

—

254

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

127

—

114

C₂²

—

254

—

115

C₂²

—

254

—

116

C₂²

—

254

—

117

C₂²

—

254

—

118

C₂²

—

254

—

119

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

120

C₄

—

121

C₄

—

127

—

122

C₄

—

254

—

123

C₄

D₄×D₁₂₇

yes

characteristic

124

C₄

D₄×D₁₂₇

yes

characteristic

125

C₂

—

126

C₂

—

508

—

127

C₂

—

254

—

128

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

129

C₂

#16861215

yes

characteristic

130

C₂

#16861217

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₂₇	C₂
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#408
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₂₇

C₂

hypercenter

C₂²⋊C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#14523231	C₂	semidirect
3. C₂×C₈×D₁₂₇	C₂	semidirect
4. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
6. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
7. C₂²⋊C₈×C₁₂₇	C₂	semidirect
65. C₁₂₇	#408	semidirect
73. C₂²⋊C₈	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523231

C₂

semidirect

3. C₂×C₈×D₁₂₇

C₂

semidirect

4. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

5. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

6. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

7. C₂²⋊C₈×C₁₂₇

C₂

semidirect

65. C₁₂₇

#408

semidirect

73. C₂²⋊C₈

D₁₂₇

semidirect

Subgroups — #22549841

Conjugacy classes of subgroups

Complements