Group #30640337 — subgroups

Subgroups — C₂×(C₂×D₁₂₇)⋊C₁₆ #30640337

Order 16256.

272 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₂₇)⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	—	2	2	no	—
8	C₂×C₈×D₁₂₇	—	2	2	no	—
9	C₂×C₈×D₁₂₇	C₄	1	2	yes	—
10	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
11	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
12	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	—	2	2	no	—
15	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	complemented
16	C₂³×D₁₂₇	C₈	1	1	yes	characteristic
17	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₂₇	—	2	2	no	—
19	C₂×C₄×D₁₂₇	—	2	2	no	—
20	C₂×C₄×D₁₂₇	C₈	1	2	yes	—
21	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₈×D₁₂₇	—	2	4	no	—
25	C₈×D₁₂₇	—	2	2	no	—
26	C₈×D₁₂₇	—	2	2	no	—
27	C₁₂₇⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₁₆×C₁₂₇	—	2	2	no	—
34	C₂²×D₁₂₇	—	2	2	no	—
35	C₂²×D₁₂₇	—	2	2	no	—
36	C₂²×D₁₂₇	C₁₆	1	4	yes	complemented
37	C₂²×D₁₂₇	C₂×C₈	1	2	yes	—
38	C₂×Dic₁₂₇	—	2	2	no	—
39	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
40	C₄×D₁₂₇	—	2	4	no	—
41	C₄×D₁₂₇	—	2	2	no	—
42	C₄×D₁₂₇	—	2	2	no	—
43	C₁₂₇⋊C₈	—	2	2	no	—
44	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
50	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₁₂₇	C₂×C₁₆	1	2	yes	—
54	C₂²×C₁₂₇	M₃₂	1	2	yes	—
55	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×D₁₂₇	—	2	8	no	—
57	C₂×D₁₂₇	—	2	2	no	—
58	C₂×D₁₂₇	—	2	2	no	—
59	C₂×D₁₂₇	C₂×C₁₆	1	4	yes	complemented
60	C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
61	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
63	Dic₁₂₇	—	2	2	no	—
64	C₂×C₁₂₇	C₂²⋊C₁₆	1	4	yes	complemented
65	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
68	D₁₂₇	—	2	4	no	—
69	C₂×C₂²⋊C₁₆	—	127	1	no	maximal, Sylow
70	C₁₂₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
71	C₂³×C₈	—	127	1	no	—
72	C₂²×C₁₆	—	127	1	no	—
73	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic, complemented
74	C₂²⋊C₁₆	—	127	4	no	—
75	C₂³×C₄	—	127	1	no	—
76	C₂²×C₈	—	127	1	no	—
77	C₂²×C₈	—	127	2	no	—
78	C₂²×C₈	—	127	2	no	—
79	C₂²×C₈	—	254	2	no	—
80	C₂²×C₈	—	254	2	no	—
81	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	127	2	no	—
84	C₂×C₁₆	—	254	2	no	—
85	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	complemented
86	C₂⁴	—	127	1	no	—
87	C₂²×C₄	—	127	1	no	—
88	C₂²×C₄	—	127	2	no	—
89	C₂²×C₄	—	127	2	no	—
90	C₂²×C₄	—	254	2	no	—
91	C₂²×C₄	—	254	2	no	—
92	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
93	C₂×C₈	—	127	2	no	—
94	C₂×C₈	—	254	2	no	—
95	C₂×C₈	—	254	4	no	—
96	C₂×C₈	—	254	2	no	—
97	C₂×C₈	—	254	2	no	—
98	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
100	C₂×C₈	D₅₀₈	1	2	yes	—
101	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
102	C₁₆	—	2	2	no	—
103	C₁₆	—	254	2	no	—
104	C₂³	—	127	2	no	—
105	C₂³	—	127	4	no	—
106	C₂³	—	254	2	no	—
107	C₂³	—	254	2	no	—
108	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
109	C₂×C₄	—	127	2	no	—
110	C₂×C₄	—	254	2	no	—
111	C₂×C₄	—	254	2	no	—
112	C₂×C₄	—	254	4	no	—
113	C₂×C₄	—	254	2	no	—
114	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
115	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	#14523231	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
118	C₈	—	254	2	no	—
119	C₈	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
120	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
121	C₈	C₂×D₅₀₈	1	1	yes	characteristic
122	C₂²	—	127	4	no	—
123	C₂²	—	254	2	no	—
124	C₂²	—	254	2	no	—
125	C₂²	—	254	8	no	—
126	C₂²	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
127	C₂²	C₁₆×D₁₂₇	1	2	yes	—
128	C₂²	#16861148	1	2	yes	—
129	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
130	C₄	—	254	2	no	—
131	C₄	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
132	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
133	C₄	C₂×#14523231	1	1	yes	characteristic
134	C₂	—	254	4	no	—
135	C₂	(C₂×D₁₂₇)⋊C₁₆	1	4	yes	direct product factor
136	C₂	C₂×(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
137	C₂	C₂×C₁₆×D₁₂₇	1	1	yes	characteristic
138	C₂	C₂×#16861148	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₂₇

C₄

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₂×C₈×D₁₂₇

—

C₂×C₈×D₁₂₇

—

C₂×C₈×D₁₂₇

C₄

yes

—

C₂×C₈×D₁₂₇

C₂²

yes

—

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂²

yes

complemented

C₂³×D₁₂₇

C₈

yes

characteristic

C₂²×Dic₁₂₇

C₈

yes

characteristic

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₈

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₈×D₁₂₇

—

C₈×D₁₂₇

—

C₈×D₁₂₇

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₁₆

yes

complemented

C₂²×D₁₂₇

C₂×C₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊C₈

yes

—

C₂²×C₁₂₇

C₂×C₁₆

yes

—

C₂²×C₁₂₇

M₃₂

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×C₁₆

yes

complemented

C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

C₂²⋊C₁₆

yes

complemented

C₂×C₁₂₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₂₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

D₁₂₇

—

C₂×C₂²⋊C₁₆

—

127

maximal, Sylow

C₁₂₇

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

C₂³×C₈

—

127

—

C₂²×C₁₆

—

127

—

C₂²×C₁₆

D₁₂₇

yes

characteristic, complemented

C₂²⋊C₁₆

—

127

—

C₂³×C₄

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

254

—

C₂²×C₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

C₂×C₁₆

C₂×D₁₂₇

yes

complemented

C₂⁴

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

—

100

C₂×C₈

D₅₀₈

yes

—

101

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

102

C₁₆

—

103

C₁₆

—

254

—

104

C₂³

—

127

—

105

C₂³

—

127

—

106

C₂³

—

254

—

107

C₂³

—

254

—

108

C₂³

C₈×D₁₂₇

yes

characteristic

109

C₂×C₄

—

127

—

110

C₂×C₄

—

254

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

—

254

—

114

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

115

C₂×C₄

C₈×D₁₂₇

yes

characteristic

116

C₂×C₄

#14523231

yes

—

117

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

118

C₈

—

254

—

119

C₈

(C₂×D₁₂₇)⋊C₄

yes

—

120

C₈

C₂×C₂²⋊D₁₂₇

yes

characteristic

121

C₈

C₂×D₅₀₈

yes

characteristic

122

C₂²

—

127

—

123

C₂²

—

254

—

124

C₂²

—

254

—

125

C₂²

—

254

—

126

C₂²

(C₂×D₁₂₇)⋊C₈

yes

—

127

C₂²

C₁₆×D₁₂₇

yes

—

128

C₂²

#16861148

yes

—

129

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

130

C₄

—

254

—

131

C₄

(C₂×D₁₂₇)⋊C₈

yes

—

132

C₄

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

133

C₄

C₂×#14523231

yes

characteristic

134

C₂

—

254

—

135

C₂

(C₂×D₁₂₇)⋊C₁₆

yes

direct product factor

136

C₂

C₂×(C₂×D₁₂₇)⋊C₈

yes

characteristic

137

C₂

C₂×C₁₆×D₁₂₇

yes

characteristic

138

C₂

C₂×#16861148

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×C₈
prime radical	C₂³×D₁₂₇	C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

C₂²×C₁₆×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂×C₈

prime radical

C₂³×D₁₂₇

C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂²×C₁₆

D₁₂₇

hypocenter

C₁₂₇

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂²×C₁₂₇⋊C₁₆	C₂	semidirect
3. (C₂×D₁₂₇)⋊C₁₆	C₂	direct (N × H)
4. C₂²×C₁₆×C₁₂₇	C₂	semidirect
12. C₂×C₁₂₇⋊C₁₆	C₂²	semidirect
15. C₂×C₁₆×C₁₂₇	C₂²	semidirect
36. C₂²×D₁₂₇	C₁₆	semidirect
59. C₂×D₁₂₇	C₂×C₁₆	semidirect
64. C₂×C₁₂₇	C₂²⋊C₁₆	semidirect
70. C₁₂₇	C₂×C₂²⋊C₁₆	semidirect
73. C₂²×C₁₆	D₁₂₇	semidirect
85. C₂×C₁₆	C₂×D₁₂₇	semidirect
135. C₂	(C₂×D₁₂₇)⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂²×C₁₂₇⋊C₁₆

C₂

semidirect

3. (C₂×D₁₂₇)⋊C₁₆

C₂

direct (N × H)

4. C₂²×C₁₆×C₁₂₇

C₂

semidirect

12. C₂×C₁₂₇⋊C₁₆

C₂²

semidirect

15. C₂×C₁₆×C₁₂₇

C₂²

semidirect

36. C₂²×D₁₂₇

C₁₆

semidirect

59. C₂×D₁₂₇

C₂×C₁₆

semidirect

64. C₂×C₁₂₇

C₂²⋊C₁₆

semidirect

70. C₁₂₇

C₂×C₂²⋊C₁₆

semidirect

73. C₂²×C₁₆

D₁₂₇

semidirect

85. C₂×C₁₆

C₂×D₁₂₇

semidirect

135. C₂

(C₂×D₁₂₇)⋊C₁₆

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₈×D₁₂₇	2
C₂²×C₁₂₇⋊C₁₆	2
(C₂×D₁₂₇)⋊C₁₆	2
C₂²×C₁₆×C₁₂₇	2
C₂×C₂²⋊C₁₆	127

subgroup

index

C₂²×C₈×D₁₂₇

C₂²×C₁₂₇⋊C₁₆

(C₂×D₁₂₇)⋊C₁₆

C₂²×C₁₆×C₁₂₇

C₂×C₂²⋊C₁₆

127

Subgroups — C₂×(C₂×D₁₂₇)⋊C₁₆ #30640337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×(C2×D127)⋊C16 #30640337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂×D₁₂₇)⋊C₁₆ #30640337