Group #30647518 — subgroups

Subgroups — C₂×#22549984 #30647518

Order 16256.

480 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 205 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×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₅₄	C₂	1	2	yes	maximal
3	#22549984	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861141	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861134	C₂	1	2	yes	maximal
6	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
8	C₂×Dic₅₀₈	C₄	1	8	yes	complemented
9	C₂×Dic₅₀₈	C₂²	1	4	yes	—
10	C₄×Dic₂₅₄	—	2	4	no	—
11	C₄×Dic₂₅₄	C₂²	1	8	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	C₂×Dic₁₂₇⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₁₂₇	—	2	2	no	—
15	#16861141	C₂²	1	4	yes	—
16	#16861134	C₂²	1	8	yes	—
17	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
20	#97×C₁₂₇	C₂²	1	4	yes	—
21	C₂×Dic₂₅₄	—	2	2	no	—
22	C₂×Dic₂₅₄	—	2	4	no	—
23	C₂×Dic₂₅₄	C₂×C₄	1	8	yes	—
24	C₂×Dic₂₅₄	C₂³	1	4	yes	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	C₂²×Dic₁₂₇	—	2	2	no	—
27	Dic₅₀₈	C₂×C₄	1	16	yes	complemented
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
30	Dic₁₂₇⋊C₄	—	2	8	no	—
31	C₄×Dic₁₂₇	—	2	8	no	—
32	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₄²×C₁₂₇	D₄	1	2	yes	—
39	C₄²×C₁₂₇	C₂³	1	2	yes	—
40	Dic₂₅₄	—	2	8	no	—
41	Dic₂₅₄	—	2	4	no	—
42	Dic₂₅₄	C₂²×C₄	1	8	yes	—
43	C₂×Dic₁₂₇	—	2	8	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	2	4	no	—
46	C₂×Dic₁₂₇	—	4	4	no	—
47	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	—	2	4	no	—
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	#41	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₂₇	—	2	2	no	—
55	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
57	C₂²×C₁₂₇	#137	1	4	yes	—
58	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₂₇	—	2	4	no	—
60	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
61	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₂₇	—	2	8	no	—
64	Dic₁₂₇	—	4	4	no	—
65	C₂×C₁₂₇	#441	1	4	yes	complemented
66	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#137	1	2	yes	—
68	C₂×#441	—	127	1	no	maximal, Sylow
69	C₁₂₇	C₂×#441	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₁₆	—	127	1	no	—
71	C₂×C₄×Q₈	—	127	2	no	—
72	#441	—	127	8	no	—
73	C₂×#106	—	127	1	no	—
74	C₂×#103	—	127	2	no	—
75	C₂×#97	D₁₂₇	1	1	yes	characteristic
76	C₂²×Q₈	—	127	2	no	—
77	C₂×Q₁₆	—	127	8	no	—
78	C₂×Q₁₆	—	127	4	no	—
79	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
81	C₄×Q₈	—	127	8	no	—
82	C₄×Q₈	—	254	4	no	—
83	C₂×C₄⋊C₄	—	127	2	no	—
84	C₂×C₄⋊C₄	—	254	2	no	—
85	C₂×C₄²	—	254	2	no	—
86	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
87	#106	—	127	4	no	—
88	#103	—	127	8	no	—
89	#97	C₂×D₁₂₇	1	4	yes	—
90	C₂×Q₈	—	127	8	no	—
91	C₂×Q₈	—	127	4	no	—
92	C₂×Q₈	—	254	2	no	—
93	C₂×Q₈	—	254	4	no	—
94	C₂²×C₄	—	254	2	no	—
95	C₂²×C₄	—	254	2	no	—
96	C₂²×C₄	D₅₀₈	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	Q₁₆	—	127	16	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
101	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
102	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
103	C₄⋊C₄	—	127	4	no	—
104	C₄⋊C₄	—	254	8	no	—
105	C₄⋊C₄	—	254	2	no	—
106	C₄²	—	254	8	no	—
107	C₄²	D₅₀₈	1	2	yes	—
108	C₄²	C₂²×D₁₂₇	1	2	yes	—
109	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
110	Q₈	—	127	8	no	—
111	Q₈	—	254	4	no	—
112	Q₈	—	254	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	508	4	no	—
115	C₂×C₄	—	254	4	no	—
116	C₂×C₄	—	254	4	no	—
117	C₂×C₄	—	254	8	no	—
118	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
119	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
120	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
121	C₂×C₄	#14523248	1	2	yes	—
122	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
125	C₂²	C₄×D₅₀₈	1	2	yes	—
126	C₂²	#16861242	1	4	yes	—
127	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	—	508	4	no	—
130	C₄	—	254	8	no	—
131	C₄	C₄×D₅₀₈	1	2	yes	—
132	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
133	C₄	C₂×#14523248	1	1	yes	characteristic
134	C₂	#22549984	1	4	yes	direct product factor
135	C₂	C₂×C₄×D₅₀₈	1	1	yes	characteristic
136	C₂	C₂×#16861242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₅₄

C₂

yes

maximal

#22549984

C₂

yes

maximal, direct product factor

C₂×#16861141

C₂

yes

maximal, characteristic

C₂×#16861134

C₂

yes

maximal

C₂×#97×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×Dic₅₀₈

C₄

yes

complemented

C₂×Dic₅₀₈

C₂²

yes

—

C₄×Dic₂₅₄

—

C₄×Dic₂₅₄

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

—

C₂×C₄×Dic₁₂₇

—

#16861141

C₂²

yes

—

#16861134

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

C₂×C₄

yes

complemented

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

C₂²×C₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₄×D₄

yes

—

C₂²×C₁₂₇

#137

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#441

yes

complemented

C₂×C₁₂₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#137

yes

—

C₂×#441

—

127

maximal, Sylow

C₁₂₇

C₂×#441

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

C₂×C₄×Q₈

—

127

—

#441

—

127

—

C₂×#106

—

127

—

C₂×#103

—

127

—

C₂×#97

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#106

—

127

—

#103

—

127

—

#97

C₂×D₁₂₇

yes

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

Q₁₆

—

127

—

C₂×C₈

—

100

C₂×C₈

C₄×D₁₂₇

yes

—

101

C₂×C₈

C₂²×D₁₂₇

yes

—

102

C₂×C₈

C₂²×D₁₂₇

yes

—

103

C₄⋊C₄

—

127

—

104

C₄⋊C₄

—

254

—

105

C₄⋊C₄

—

254

—

106

C₄²

—

254

—

107

C₄²

D₅₀₈

yes

—

108

C₄²

C₂²×D₁₂₇

yes

—

109

C₂³

C₂×D₅₀₈

yes

characteristic

110

Q₈

—

127

—

111

Q₈

—

254

—

112

Q₈

—

254

—

113

C₂×C₄

—

114

C₂×C₄

—

508

—

115

C₂×C₄

—

254

—

116

C₂×C₄

—

254

—

117

C₂×C₄

—

254

—

118

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

119

C₂×C₄

C₂×D₅₀₈

yes

—

120

C₂×C₄

C₂×D₅₀₈

yes

characteristic

121

C₂×C₄

#14523248

yes

—

122

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

123

C₈

—

124

C₈

C₂×C₄×D₁₂₇

yes

—

125

C₂²

C₄×D₅₀₈

yes

—

126

C₂²

#16861242

yes

—

127

C₂²

C₂²×D₅₀₈

yes

characteristic

128

C₄

—

129

C₄

—

508

—

130

C₄

—

254

—

131

C₄

C₄×D₅₀₈

yes

—

132

C₄

C₂²×C₄×D₁₂₇

yes

characteristic

133

C₄

C₂×#14523248

yes

characteristic

134

C₂

#22549984

yes

direct product factor

135

C₂

C₂×C₄×D₅₀₈

yes

characteristic

136

C₂

C₂×#16861242

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	C₂×#441
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₄×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×#97×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₂×#97

D₁₂₇

hypocenter

C₁₂₇

C₂×#441

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #22549984	C₂	direct (N × H)
8. C₂×Dic₅₀₈	C₄	semidirect
27. Dic₅₀₈	C₂×C₄	semidirect
65. C₂×C₁₂₇	#441	semidirect
69. C₁₂₇	C₂×#441	semidirect
134. C₂	#22549984	direct (N × H)

H ≅ G/N

type

3. #22549984

C₂

direct (N × H)

8. C₂×Dic₅₀₈

C₄

semidirect

27. Dic₅₀₈

C₂×C₄

semidirect

65. C₂×C₁₂₇

#441

semidirect

69. C₁₂₇

C₂×#441

semidirect

134. C₂

#22549984

direct (N × H)

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

subgroup	index
C₂²×Dic₅₀₈	2
C₂×C₄×Dic₂₅₄	2
#22549984	2
C₂×#16861141	2
C₂×#16861134	2
C₂×#97×C₁₂₇	2
C₂×#441	127

subgroup

index

C₂²×Dic₅₀₈

C₂×C₄×Dic₂₅₄

C₂×#97×C₁₂₇

127

Subgroups — C₂×#22549984 #30647518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22549984 #30647518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549984 #30647518