Group #30647654 — subgroups

Subgroups — C₂×#22550003 #30647654

Order 16256.

400 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	#22550003	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal
4	C₂×#16861119	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₄	1	1	yes	characteristic
8	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₅₄	C₄	1	4	yes	—
11	C₄×Dic₂₅₄	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
13	C₂×Dic₁₂₇⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
15	Dic₁₂₇⋊C₈	C₂²	1	8	yes	—
16	#16861119	C₂²	1	8	yes	—
17	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
21	C₂×Dic₂₅₄	—	2	4	no	—
22	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
23	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	complemented
27	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
30	Dic₁₂₇⋊C₄	C₂×C₄	1	8	yes	—
31	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₂₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	—	2	4	no	—
37	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
38	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₄²×C₁₂₇	C₂³	1	2	yes	—
40	Dic₂₅₄	—	2	8	no	—
41	C₂×Dic₁₂₇	—	2	4	no	—
42	C₂×Dic₁₂₇	—	2	4	no	—
43	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
44	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
45	C₁₂₇⋊C₈	—	2	2	no	—
46	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	complemented
47	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
50	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₂₇	#41	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
56	C₈×C₁₂₇	—	2	4	no	—
57	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
59	C₂²×C₁₂₇	#131	1	2	yes	—
60	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
63	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
64	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
66	Dic₁₂₇	—	2	4	no	—
67	Dic₁₂₇	—	2	4	no	—
68	C₂×C₁₂₇	#446	1	4	yes	complemented
69	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
72	C₂×#446	—	127	1	no	maximal, Sylow
73	C₁₂₇	C₂×#446	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	127	1	no	—
75	#446	—	127	8	no	—
76	C₂×C₄⋊C₈	—	127	2	no	—
77	C₂×C₄⋊C₈	D₁₂₇	1	1	yes	characteristic
78	C₂×#97	—	127	2	no	—
79	C₂×C₄×C₈	—	127	1	no	—
80	C₂²×Q₈	—	127	1	no	—
81	C₂²×C₈	—	127	1	no	—
82	C₂²×C₈	—	127	1	no	—
83	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
84	C₄×Q₈	—	127	4	no	—
85	C₄×Q₈	—	127	4	no	—
86	C₂×C₄⋊C₄	—	127	2	no	—
87	C₂×C₄⋊C₄	—	127	1	no	—
88	C₂×C₄²	—	127	2	no	—
89	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
90	C₄⋊C₈	—	127	8	no	—
91	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	—
92	#97	—	127	8	no	—
93	C₄×C₈	—	127	4	no	—
94	C₂×Q₈	—	127	4	no	—
95	C₂×Q₈	—	254	4	no	—
96	C₂²×C₄	—	127	2	no	—
97	C₂²×C₄	—	127	2	no	—
98	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	127	4	no	—
103	C₂×C₈	—	127	2	no	—
104	C₂×C₈	—	127	2	no	—
105	C₂×C₈	—	254	2	no	—
106	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
107	C₄⋊C₄	—	127	8	no	—
108	C₄⋊C₄	—	127	4	no	—
109	C₄²	—	127	4	no	—
110	C₄²	—	127	4	no	—
111	C₄²	C₄×D₁₂₇	1	2	yes	—
112	C₄²	C₂²×D₁₂₇	1	2	yes	—
113	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
114	Q₈	—	254	8	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	127	4	no	—
117	C₂×C₄	—	127	4	no	—
118	C₂×C₄	—	254	4	no	—
119	C₂×C₄	—	254	4	no	—
120	C₂×C₄	#14523231	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
125	C₂×C₄	#14523247	1	2	yes	—
126	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	127	4	no	—
129	C₈	—	254	2	no	—
130	C₂²	#16861174	1	2	yes	—
131	C₂²	C₂×#14523231	1	2	yes	—
132	C₂²	#16861217	1	2	yes	—
133	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	254	4	no	—
136	C₄	—	254	4	no	—
137	C₄	#16861174	1	2	yes	—
138	C₄	C₂×#14523231	1	4	yes	—
139	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
140	C₄	C₂×#14523247	1	1	yes	characteristic
141	C₂	#22550003	1	4	yes	direct product factor
142	C₂	C₂×#16861174	1	1	yes	characteristic
143	C₂	C₂²×#14523231	1	1	yes	characteristic
144	C₂	C₂×#16861217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₅₄

C₂

yes

maximal, characteristic

#22550003

C₂

yes

maximal, direct product factor

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal

C₂×#16861119

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₄

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×Dic₂₅₄

C₄

yes

—

C₄×Dic₂₅₄

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₄

yes

characteristic

C₂×Dic₁₂₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

—

#16861119

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂×Q₈

yes

complemented

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₂²×C₁₂₇

#131

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

C₂×M₁₆

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#446

yes

complemented

C₂×C₁₂₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₂²×M₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

C₂×#446

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

127

—

#446

—

127

—

C₂×C₄⋊C₈

—

127

—

C₂×C₄⋊C₈

D₁₂₇

yes

characteristic

C₂×#97

—

127

—

C₂×C₄×C₈

—

127

—

C₂²×Q₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#97

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

100

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

127

—

103

C₂×C₈

—

127

—

104

C₂×C₈

—

127

—

105

C₂×C₈

—

254

—

106

C₂×C₈

C₂²×D₁₂₇

yes

—

107

C₄⋊C₄

—

127

—

108

C₄⋊C₄

—

127

—

109

C₄²

—

127

—

110

C₄²

—

127

—

111

C₄²

C₄×D₁₂₇

yes

—

112

C₄²

C₂²×D₁₂₇

yes

—

113

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

114

Q₈

—

254

—

115

C₂×C₄

—

116

C₂×C₄

—

127

—

117

C₂×C₄

—

127

—

118

C₂×C₄

—

254

—

119

C₂×C₄

—

254

—

120

C₂×C₄

#14523231

yes

—

121

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

122

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

123

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

124

C₂×C₄

Q₈×D₁₂₇

yes

—

125

C₂×C₄

#14523247

yes

—

126

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

127

C₈

—

128

C₈

—

127

—

129

C₈

—

254

—

130

C₂²

#16861174

yes

—

131

C₂²

C₂×#14523231

yes

—

132

C₂²

#16861217

yes

—

133

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

134

C₄

—

135

C₄

—

254

—

136

C₄

—

254

—

137

C₄

#16861174

yes

—

138

C₄

C₂×#14523231

yes

—

139

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

140

C₄

C₂×#14523247

yes

characteristic

141

C₂

#22550003

yes

direct product factor

142

C₂

C₂×#16861174

yes

characteristic

143

C₂

C₂²×#14523231

yes

characteristic

144

C₂

C₂×#16861217

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄⋊C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#446
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄⋊C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₂×C₄⋊C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #22550003	C₂	direct (N × H)
26. C₂×C₁₂₇⋊C₈	Q₈	semidirect
46. C₁₂₇⋊C₈	C₂×Q₈	semidirect
68. C₂×C₁₂₇	#446	semidirect
73. C₁₂₇	C₂×#446	semidirect
141. C₂	#22550003	direct (N × H)

H ≅ G/N

type

2. #22550003

C₂

direct (N × H)

26. C₂×C₁₂₇⋊C₈

Q₈

semidirect

46. C₁₂₇⋊C₈

C₂×Q₈

semidirect

68. C₂×C₁₂₇

#446

semidirect

73. C₁₂₇

C₂×#446

semidirect

141. C₂

#22550003

direct (N × H)

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

subgroup	index
C₂×C₄×Dic₂₅₄	2
#22550003	2
C₂×Dic₁₂₇⋊C₈	2
C₂×#16861119	2
C₂×C₄×C₁₂₇⋊C₈	2
C₂×C₄⋊C₈×C₁₂₇	2
C₂×#446	127

subgroup

index

C₂×C₄×Dic₂₅₄

#22550003

C₂×Dic₁₂₇⋊C₈

C₂×#16861119

C₂×C₄×C₁₂₇⋊C₈

C₂×C₄⋊C₈×C₁₂₇

C₂×#446

127

Subgroups — C₂×#22550003 #30647654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550003 #30647654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550003 #30647654