Group #22550454 — subgroups

Subgroups — #22550454

Order 8128.

294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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, complemented
2	#16861188	C₂	1	2	yes	maximal, complemented
3	#16861185	C₂	1	2	yes	maximal, complemented
4	#16861183	C₂	1	2	yes	maximal, complemented
5	#16861181	C₂	1	4	yes	maximal, complemented
6	#16861180	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523248	—	2	4	no	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
14	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
18	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
19	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
21	#41×C₁₂₇	—	2	4	no	—
22	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₁₂₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₂₇	C₂³	1	2	yes	—
28	Dic₂₅₄	—	2	2	no	—
29	D₅₀₈	—	2	2	no	—
30	C₂²⋊D₁₂₇	—	2	4	no	—
31	C₂²⋊D₁₂₇	—	4	4	no	—
32	C₂×Dic₁₂₇	—	4	2	no	—
33	C₂×Dic₁₂₇	C₂³	1	4	yes	—
34	C₂×Dic₁₂₇	C₂³	1	2	yes	—
35	C₄×D₁₂₇	—	2	4	no	—
36	C₂³×C₁₂₇	C₂³	1	2	yes	—
37	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
38	Q₈×C₁₂₇	—	2	2	no	—
39	D₄×C₁₂₇	—	2	4	no	—
40	D₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	D₄	1	4	yes	complemented
43	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂²×C₁₂₇	—	2	1	no	—
47	C₂²×C₁₂₇	—	2	1	no	—
48	C₂²×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	2	2	no	—
50	C₂²×C₁₂₇	—	2	2	no	—
51	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₂₇	—	2	2	no	—
54	C₂×D₁₂₇	—	2	2	no	—
55	C₂×D₁₂₇	—	2	2	no	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
59	Dic₁₂₇	—	2	2	no	—
60	Dic₁₂₇	—	2	4	no	—
61	C₂×C₁₂₇	—	2	2	no	—
62	C₂×C₁₂₇	—	4	2	no	—
63	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₂₇	—	4	2	no	—
67	C₁₂₇	#537	1	1	yes	characteristic, complemented, Sylow
68	#537	—	127	1	no	maximal, Sylow
69	C₂×#41	—	127	1	no	—
70	C₂×#41	D₁₂₇	1	1	yes	characteristic, complemented
71	C₂²⋊Q₈	—	127	2	no	—
72	C₂²⋊Q₈	—	127	2	no	—
73	#121	—	127	2	no	—
74	#121	—	127	4	no	—
75	#121	—	127	2	no	—
76	C₂×C₄⋊C₄	—	127	1	no	—
77	#41	—	2	4	no	—
78	#41	—	254	4	no	—
79	C₂×Q₈	—	127	1	no	—
80	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂×D₄	—	127	4	no	—
82	C₂×D₄	—	127	2	no	—
83	C₂×D₄	—	127	1	no	—
84	C₂×D₄	C₂×D₁₂₇	1	2	yes	—
85	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂²×C₄	—	127	2	no	—
87	C₂²×C₄	—	127	2	no	—
88	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
89	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
90	C₄⋊C₄	—	127	4	no	—
91	C₄⋊C₄	—	127	2	no	—
92	C₄⋊C₄	—	127	2	no	—
93	C₂²⋊C₄	—	127	4	no	—
94	C₂²⋊C₄	—	127	4	no	—
95	C₂³	—	127	2	no	—
96	C₂³	C₂²×D₁₂₇	1	2	yes	—
97	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	Q₈	—	254	2	no	—
100	D₄	—	2	4	no	—
101	D₄	—	2	2	no	—
102	D₄	—	508	4	no	—
103	D₄	—	254	4	no	—
104	D₄	—	254	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	127	4	no	—
107	C₂×C₄	—	127	2	no	—
108	C₂×C₄	—	508	2	no	—
109	C₂×C₄	—	254	4	no	—
110	C₂×C₄	C₂²⋊D₁₂₇	1	4	yes	complemented
111	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
112	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	254	2	no	—
120	C₂²	—	254	2	no	—
121	C₂²	—	254	2	no	—
122	C₂²	C₂×C₂²⋊D₁₂₇	1	2	yes	—
123	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	254	2	no	—
126	C₄	—	254	4	no	—
127	C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
128	C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	508	2	no	—
132	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
133	C₂	#16861255	1	1	yes	characteristic
134	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

#16861183

C₂

yes

maximal, complemented

#16861181

C₂

yes

maximal, complemented

#16861180

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#537

yes

characteristic, complemented, Sylow

#537

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×#41

D₁₂₇

yes

characteristic, complemented

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

#121

—

127

—

#121

—

127

—

C₂×C₄⋊C₄

—

127

—

#41

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

254

—

100

D₄

—

101

D₄

—

102

D₄

—

508

—

103

D₄

—

254

—

104

D₄

—

254

—

105

C₂×C₄

—

106

C₂×C₄

—

127

—

107

C₂×C₄

—

127

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

254

—

110

C₂×C₄

C₂²⋊D₁₂₇

yes

complemented

111

C₂×C₄

C₂²×D₁₂₇

yes

—

112

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

113

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

254

—

120

C₂²

—

254

—

121

C₂²

—

254

—

122

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

123

C₂²

C₂³×D₁₂₇

yes

characteristic

124

C₄

—

125

C₄

—

254

—

126

C₄

—

254

—

127

C₄

C₂×C₂²⋊D₁₂₇

yes

—

128

C₄

C₂×C₂²⋊D₁₂₇

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

508

—

132

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

133

C₂

#16861255

yes

characteristic

134

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₂×#41×C₁₂₇	C₂
hypercenter	C₂×#41	D₁₂₇
hypocenter	C₁₂₇	#537
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₂×#41×C₁₂₇

C₂

hypercenter

C₂×#41

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14523248	C₂	semidirect
2. #16861188	C₂	semidirect
3. #16861185	C₂	semidirect
4. #16861183	C₂	semidirect
5. #16861181	C₂	semidirect
6. #16861180	C₂	semidirect
7. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
8. C₂×#41×C₁₂₇	C₂	semidirect
42. C₂×C₄×C₁₂₇	D₄	semidirect
67. C₁₂₇	#537	semidirect
70. C₂×#41	D₁₂₇	semidirect
110. C₂×C₄	C₂²⋊D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523248

C₂

semidirect

2. #16861188

C₂

semidirect

3. #16861185

C₂

semidirect

4. #16861183

C₂

semidirect

5. #16861181

C₂

semidirect

6. #16861180

C₂

semidirect

7. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

8. C₂×#41×C₁₂₇

C₂

semidirect

42. C₂×C₄×C₁₂₇

D₄

semidirect

67. C₁₂₇

#537

semidirect

70. C₂×#41

D₁₂₇

semidirect

110. C₂×C₄

C₂²⋊D₁₂₇

semidirect

Subgroups — #22550454

Conjugacy classes of subgroups

Complements