Group #30640273 — subgroups

Subgroups — #30640273

Order 16256.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16861140	C₂	1	2	yes	maximal
2	C₂×#16861133	C₂	1	2	yes	maximal
3	#22549746	C₂	1	1	yes	maximal, characteristic
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	#16861140	—	2	4	no	—
10	#16861133	—	2	4	no	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	#95×C₁₂₇	—	2	2	no	—
15	C₂²×Dic₁₂₇	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	4	4	no	—
22	C₂×Q₈×C₁₂₇	—	2	1	no	—
23	C₂×Q₈×C₁₂₇	—	2	2	no	—
24	C₂×Q₈×C₁₂₇	—	2	2	no	—
25	C₂×Q₈×C₁₂₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₂₇	—	2	1	no	—
27	C₂²×C₄×C₁₂₇	—	2	1	no	—
28	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	C₂×Dic₁₂₇	—	4	2	no	—
33	C₂×Dic₁₂₇	—	4	4	no	—
34	C₁₂₇⋊C₈	—	2	4	no	—
35	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₁₂₇	—	2	4	no	—
37	Q₈×C₁₂₇	—	4	2	no	—
38	C₂×C₄×C₁₂₇	—	2	2	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	4	1	no	—
42	C₂×C₄×C₁₂₇	—	4	1	no	—
43	C₂×C₄×C₁₂₇	—	4	1	no	—
44	C₂×C₄×C₁₂₇	—	4	2	no	—
45	C₂×C₄×C₁₂₇	SD₁₆	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#41	1	2	yes	—
49	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	#123	1	1	yes	characteristic
52	C₂²×C₁₂₇	#127	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₁₂₇	#137	1	2	yes	—
55	C₄×C₁₂₇	—	2	4	no	—
56	C₄×C₁₂₇	—	4	2	no	—
57	C₄×C₁₂₇	—	8	1	no	—
58	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₂₇	#123	1	2	yes	—
60	Dic₁₂₇	—	8	2	no	—
61	C₂×C₁₂₇	#399	1	1	yes	characteristic
62	C₂×C₁₂₇	#475	1	2	yes	—
63	C₂×C₁₂₇	#483	1	2	yes	—
64	C₂×C₁₂₇	#492	1	1	yes	characteristic
65	C₂×C₁₂₇	#497	1	1	yes	characteristic
66	#2096	—	127	1	no	maximal, Sylow
67	C₁₂₇	#2096	1	1	yes	characteristic, complemented, Sylow
68	C₂×#106	—	127	2	no	—
69	C₂×#103	—	127	2	no	—
70	#391	D₁₂₇	1	1	yes	characteristic
71	#384	—	127	1	no	—
72	#336	—	127	1	no	—
73	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂²×C₈	—	127	2	no	—
75	C₂×C₄⋊C₄	—	127	2	no	—
76	C₂×C₄⋊C₄	—	254	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
79	#106	—	254	4	no	—
80	#103	—	254	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₂₅₄	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	254	2	no	—
89	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	127	4	no	—
92	C₂×C₈	—	254	2	no	—
93	C₂×C₈	—	254	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	508	4	no	—
96	C₄⋊C₄	—	254	2	no	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	508	2	no	—
110	C₂×C₄	—	508	4	no	—
111	C₂×C₄	#14523235	1	4	yes	—
112	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
113	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
114	C₂×C₄	#14523247	1	2	yes	—
115	C₂×C₄	#14523248	1	1	yes	characteristic
116	C₈	—	254	4	no	—
117	C₂²	#16861188	1	1	yes	characteristic
118	C₂²	#16861190	1	1	yes	characteristic
119	C₂²	#16861205	1	1	yes	characteristic
120	C₂²	C₂×#14523235	1	2	yes	—
121	C₂²	#16861239	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	1016	2	no	—
126	C₄	#16861186	1	2	yes	—
127	C₄	#16861191	1	2	yes	—
128	C₂	#22549812	1	1	yes	characteristic
129	C₂	#22550161	1	2	yes	—
130	C₂	#22550204	1	2	yes	—
131	C₂	#22550250	1	1	yes	characteristic
132	C₂	#22550271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861140

C₂

yes

maximal

C₂×#16861133

C₂

yes

maximal

#22549746

C₂

yes

maximal, characteristic

#22549631

C₂

yes

maximal, characteristic

#391×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861140

—

#16861133

—

C₂²×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×SD₁₆

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#123

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#399

yes

characteristic

C₂×C₁₂₇

#475

yes

—

C₂×C₁₂₇

#483

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2096

yes

characteristic, complemented, Sylow

C₂×#106

—

127

—

C₂×#103

—

127

—

#391

D₁₂₇

yes

characteristic

#384

—

127

—

#336

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#106

—

254

—

#103

—

254

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₅₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

508

—

111

C₂×C₄

#14523235

yes

—

112

C₂×C₄

D₄×D₁₂₇

yes

—

113

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

114

C₂×C₄

#14523247

yes

—

115

C₂×C₄

#14523248

yes

characteristic

116

C₈

—

254

—

117

C₂²

#16861188

yes

characteristic

118

C₂²

#16861190

yes

characteristic

119

C₂²

#16861205

yes

characteristic

120

C₂²

C₂×#14523235

yes

—

121

C₂²

#16861239

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

1016

—

126

C₄

#16861186

yes

—

127

C₄

#16861191

yes

—

128

C₂

#22549812

yes

characteristic

129

C₂

#22550161

yes

—

130

C₂

#22550204

yes

—

131

C₂

#22550250

yes

characteristic

132

C₂

#22550271

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#391×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#391	D₁₂₇
hypocenter	C₁₂₇	#2096
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#391×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
67. C₁₂₇	#2096	semidirect

H ≅ G/N

type

67. C₁₂₇

#2096

semidirect

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

subgroup	index
C₂×#16861140	2
C₂×#16861133	2
#22549746	2
#22549631	2
#391×C₁₂₇	2
#2096	127

subgroup

index

#391×C₁₂₇

127

Subgroups — #30640273

Conjugacy classes of subgroups

Complements

Maximal subgroups