Group #30638848 — subgroups

Subgroups — #30638848

Order 16256.

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

normal subgroups only

This group has 216 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×Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	#22549722	C₂	1	1	yes	maximal, characteristic
3	#22549631	C₂	1	1	yes	maximal, characteristic
4	#22549630	C₂	1	2	yes	maximal
5	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	Dic₁₂₇⋊C₈	—	2	4	no	—
10	C₁₂₇⋊#95	C₄	1	2	yes	—
11	C₁₂₇⋊#95	C₄	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₄×Dic₁₂₇	—	2	2	no	—
21	C₄×Dic₁₂₇	D₄	1	2	yes	—
22	C₄×Dic₁₂₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	—	2	2	no	—
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₄²×C₁₂₇	—	2	2	no	—
29	C₂×Dic₁₂₇	—	2	2	no	—
30	C₂×Dic₁₂₇	—	2	2	no	—
31	C₂×Dic₁₂₇	—	2	1	no	—
32	C₂×Dic₁₂₇	—	2	1	no	—
33	C₂×Dic₁₂₇	—	2	2	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	C₄⋊C₄	1	4	yes	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	#41	1	2	yes	—
44	C₂×C₄×C₁₂₇	#41	1	2	yes	—
45	C₈×C₁₂₇	—	4	2	no	—
46	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₂₇	#117	1	1	yes	characteristic
48	C₂²×C₁₂₇	#117	1	1	yes	characteristic
49	C₂²×C₁₂₇	#131	1	2	yes	—
50	C₂²×C₁₂₇	#131	1	2	yes	—
51	C₄×C₁₂₇	—	4	2	no	—
52	C₄×C₁₂₇	#124	1	2	yes	—
53	C₄×C₁₂₇	#125	1	2	yes	—
54	Dic₁₂₇	—	2	4	no	—
55	Dic₁₂₇	—	4	1	no	—
56	Dic₁₂₇	—	4	1	no	—
57	C₂×C₁₂₇	#382	1	1	yes	characteristic
58	C₂×C₁₂₇	#424	1	2	yes	—
59	C₂×C₁₂₇	#433	1	2	yes	—
60	C₂×C₁₂₇	#433	1	1	yes	characteristic
61	C₂×C₁₂₇	#433	1	1	yes	characteristic
62	#1832	—	127	1	no	maximal, Sylow
63	C₁₂₇	#1832	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	127	2	no	—
65	#376	—	127	1	no	—
66	#336	—	127	2	no	—
67	#336	—	127	1	no	—
68	#336	D₁₂₇	1	1	yes	characteristic
69	C₂²×C₈	—	127	2	no	—
70	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
71	C₂×C₄²	—	127	1	no	—
72	C₂×C₄²	—	127	1	no	—
73	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
74	C₄⋊C₈	—	254	4	no	—
75	#95	—	127	2	no	—
76	#95	—	127	2	no	—
77	C₂²×C₄	—	127	2	no	—
78	C₂²×C₄	—	127	1	no	—
79	C₂²×C₄	—	127	1	no	—
80	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	254	2	no	—
86	C₂×C₈	—	254	2	no	—
87	C₂×C₈	—	254	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	127	2	no	—
90	C₄²	—	127	2	no	—
91	C₄²	—	254	2	no	—
92	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	127	4	no	—
97	C₂×C₄	—	254	2	no	—
98	C₂×C₄	—	254	2	no	—
99	C₂×C₄	—	254	1	no	—
100	C₂×C₄	—	254	1	no	—
101	C₂×C₄	—	254	2	no	—
102	C₂×C₄	—	254	2	no	—
103	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	#14523247	1	2	yes	—
106	C₂×C₄	#14523248	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	508	2	no	—
109	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
110	C₂²	#16861165	1	1	yes	characteristic
111	C₂²	#16861166	1	1	yes	characteristic
112	C₂²	#16861215	1	2	yes	—
113	C₂²	#16861217	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	508	1	no	—
116	C₄	—	508	1	no	—
117	C₄	—	254	4	no	—
118	C₄	#16861195	1	2	yes	—
119	C₄	#16861199	1	2	yes	—
120	C₂	#22549740	1	1	yes	characteristic
121	C₂	#22549901	1	2	yes	—
122	C₂	#22549934	1	2	yes	—
123	C₂	#22549935	1	1	yes	characteristic
124	C₂	#22549938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal

#22549722

C₂

yes

maximal, characteristic

#22549631

C₂

yes

maximal, characteristic

#22549630

C₂

yes

maximal

#336×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

C₁₂₇⋊#95

C₄

yes

—

C₁₂₇⋊#95

C₄

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₄⋊C₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂²×C₁₂₇

#131

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

—

C₄×C₁₂₇

#125

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#382

yes

characteristic

C₂×C₁₂₇

#424

yes

—

C₂×C₁₂₇

#433

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

#376

—

127

—

#336

—

127

—

#336

—

127

—

#336

D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

#95

—

127

—

#95

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄²

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

254

—

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

100

C₂×C₄

—

254

—

101

C₂×C₄

—

254

—

102

C₂×C₄

—

254

—

103

C₂×C₄

D₄×D₁₂₇

yes

characteristic

104

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

105

C₂×C₄

#14523247

yes

—

106

C₂×C₄

#14523248

yes

—

107

C₈

—

108

C₈

—

508

—

109

C₂²

C₄⋊C₄×D₁₂₇

yes

characteristic

110

C₂²

#16861165

yes

characteristic

111

C₂²

#16861166

yes

characteristic

112

C₂²

#16861215

yes

—

113

C₂²

#16861217

yes

—

114

C₄

—

115

C₄

—

508

—

116

C₄

—

508

—

117

C₄

—

254

—

118

C₄

#16861195

yes

—

119

C₄

#16861199

yes

—

120

C₂

#22549740

yes

characteristic

121

C₂

#22549901

yes

—

122

C₂

#22549934

yes

—

123

C₂

#22549935

yes

characteristic

124

C₂

#22549938

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator, socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
Fitting	#336×C₁₂₇	C₂
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#336	D₁₂₇
hypocenter	C₁₂₇	#1832
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator, socle, prime radical

C₂³×C₁₂₇

C₂²×C₄

Fitting

#336×C₁₂₇

C₂

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
63. C₁₂₇	#1832	semidirect

H ≅ G/N

type

63. C₁₂₇

#1832

semidirect

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

subgroup	index
C₂×Dic₁₂₇⋊C₈	2
#22549722	2
#22549631	2
#22549630	2
#336×C₁₂₇	2
#1832	127

subgroup

index

C₂×Dic₁₂₇⋊C₈

#336×C₁₂₇

127

Subgroups — #30638848

Conjugacy classes of subgroups

Complements

Maximal subgroups