Group #30638857 — subgroups

Subgroups — #30638857

Order 16256.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22549746	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal
4	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
7	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₂₇	—	2	4	no	—
19	C₄⋊Dic₁₂₇	—	4	2	no	—
20	C₄⋊Dic₁₂₇	—	4	2	no	—
21	C₂²×C₄×C₁₂₇	—	2	2	no	—
22	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₂₇	—	2	4	no	—
26	C₄⋊C₄×C₁₂₇	—	4	2	no	—
27	C₄⋊C₄×C₁₂₇	—	4	2	no	—
28	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₂₇	Q₈	1	2	yes	—
30	C₂×Dic₁₂₇	—	4	2	no	—
31	C₂×Dic₁₂₇	—	4	2	no	—
32	C₂×Dic₁₂₇	—	4	2	no	—
33	C₁₂₇⋊C₈	—	2	4	no	—
34	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	—	4	2	no	—
36	C₂×C₄×C₁₂₇	—	4	2	no	—
37	C₂×C₄×C₁₂₇	—	4	2	no	—
38	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
41	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₂×C₄×C₁₂₇	#41	1	2	yes	—
48	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₂₇	#124	1	1	yes	characteristic
50	C₂²×C₁₂₇	#124	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₂₇	—	8	2	no	—
56	C₄×C₁₂₇	#106	1	4	yes	—
57	C₄×C₁₂₇	#107	1	4	yes	—
58	C₄×C₁₂₇	#117	1	2	yes	—
59	C₄×C₁₂₇	#125	1	2	yes	—
60	Dic₁₂₇	—	8	2	no	—
61	C₂×C₁₂₇	#382	1	1	yes	characteristic
62	C₂×C₁₂₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₂₇	#486	1	1	yes	characteristic
65	C₂×C₁₂₇	#486	1	1	yes	characteristic
66	C₂×C₁₂₇	#487	1	1	yes	characteristic
67	C₂×C₁₂₇	#487	1	1	yes	characteristic
68	#1834	—	127	1	no	maximal, Sylow
69	C₁₂₇	#1834	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄×C₈	—	127	1	no	—
71	#384	—	127	1	no	—
72	#384	D₁₂₇	1	1	yes	characteristic
73	#340	—	127	4	no	—
74	C₂²×C₈	—	127	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	127	2	no	—
78	C₂×C₄⋊C₄	—	254	1	no	—
79	C₂×C₄⋊C₄	—	254	1	no	—
80	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
81	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄×C₈	—	127	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	254	2	no	—
85	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₂×C₈	—	127	4	no	—
89	C₂×C₈	—	254	2	no	—
90	C₂×C₈	—	254	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	508	2	no	—
95	C₄⋊C₄	—	508	2	no	—
96	C₄⋊C₄	—	254	4	no	—
97	C₄²	C₄×D₁₂₇	1	2	yes	—
98	C₄²	Dic₂₅₄	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	508	2	no	—
104	C₂×C₄	—	508	2	no	—
105	C₂×C₄	—	508	2	no	—
106	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
108	C₂×C₄	#14523233	1	2	yes	—
109	C₂×C₄	#14523235	1	2	yes	—
110	C₂×C₄	#14523236	1	2	yes	—
111	C₂×C₄	#14523238	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
114	C₂×C₄	#14523247	1	2	yes	—
115	C₂×C₄	#14523249	1	2	yes	—
116	C₈	—	254	4	no	—
117	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
118	C₂²	#16861196	1	1	yes	characteristic
119	C₂²	#16861198	1	1	yes	characteristic
120	C₂²	C₂×#14523233	1	1	yes	characteristic
121	C₂²	C₂×#14523235	1	1	yes	characteristic
122	C₂²	C₂×#14523236	1	1	yes	characteristic
123	C₂²	C₂×#14523238	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1016	2	no	—
126	C₄	#16861140	1	4	yes	—
127	C₄	#16861142	1	4	yes	—
128	C₄	#16861167	1	2	yes	—
129	C₄	#16861200	1	2	yes	—
130	C₂	#22549741	1	1	yes	characteristic
131	C₂	C₂×#16861140	1	1	yes	characteristic
132	C₂	C₂×#16861142	1	1	yes	characteristic
133	C₂	#22550222	1	1	yes	characteristic
134	C₂	#22550224	1	1	yes	characteristic
135	C₂	#22550226	1	1	yes	characteristic
136	C₂	#22550228	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549746

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

#384×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#106

yes

—

C₄×C₁₂₇

#107

yes

—

C₄×C₁₂₇

#117

yes

—

C₄×C₁₂₇

#125

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1834

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

127

—

#384

—

127

—

#384

D₁₂₇

yes

characteristic

#340

—

127

—

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄×C₈

—

127

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

Dic₂₅₄

yes

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

508

—

106

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

107

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

108

C₂×C₄

#14523233

yes

—

109

C₂×C₄

#14523235

yes

—

110

C₂×C₄

#14523236

yes

—

111

C₂×C₄

#14523238

yes

—

112

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

113

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

114

C₂×C₄

#14523247

yes

—

115

C₂×C₄

#14523249

yes

—

116

C₈

—

254

—

117

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

118

C₂²

#16861196

yes

characteristic

119

C₂²

#16861198

yes

characteristic

120

C₂²

C₂×#14523233

yes

characteristic

121

C₂²

C₂×#14523235

yes

characteristic

122

C₂²

C₂×#14523236

yes

characteristic

123

C₂²

C₂×#14523238

yes

characteristic

124

C₄

—

125

C₄

—

1016

—

126

C₄

#16861140

yes

—

127

C₄

#16861142

yes

—

128

C₄

#16861167

yes

—

129

C₄

#16861200

yes

—

130

C₂

#22549741

yes

characteristic

131

C₂

C₂×#16861140

yes

characteristic

132

C₂

C₂×#16861142

yes

characteristic

133

C₂

#22550222

yes

characteristic

134

C₂

#22550224

yes

characteristic

135

C₂

#22550226

yes

characteristic

136

C₂

#22550228

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#384×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#384	D₁₂₇
hypocenter	C₁₂₇	#1834
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#384×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₁₂₇⋊C₈	C₄	semidirect
69. C₁₂₇	#1834	semidirect

H ≅ G/N

type

9. C₄×C₁₂₇⋊C₈

C₄

semidirect

69. C₁₂₇

#1834

semidirect

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

subgroup	index
C₂×C₄×C₁₂₇⋊C₈	2
#22549746	2
#22549638	2
#384×C₁₂₇	2
#1834	127

subgroup

index

C₂×C₄×C₁₂₇⋊C₈

#22549746

#22549638

#384×C₁₂₇

#1834

127

Subgroups — #30638857

Conjugacy classes of subgroups

Complements

Maximal subgroups