Group #30638376 — subgroups

Subgroups — #30638376

Order 16256.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22550281	C₂	1	1	yes	maximal, characteristic
2	#22550274	C₂	1	1	yes	maximal, characteristic
3	#22550227	C₂	1	1	yes	maximal, characteristic
4	#22550227	C₂	1	1	yes	maximal, characteristic
5	#22550227	C₂	1	1	yes	maximal, characteristic
6	#22550227	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861209	C₂²	1	1	yes	characteristic
10	#16861209	C₂²	1	1	yes	characteristic
11	#16861209	C₂²	1	1	yes	characteristic
12	#16861143	—	2	2	no	—
13	#16861141	—	2	2	no	—
14	#16861134	—	2	2	no	—
15	#16861134	—	2	2	no	—
16	#16861134	—	2	2	no	—
17	#16861134	—	2	2	no	—
18	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₅₄	—	2	1	no	—
22	C₂×Dic₂₅₄	—	2	1	no	—
23	C₂×Dic₂₅₄	—	2	1	no	—
24	C₂×Dic₂₅₄	—	2	1	no	—
25	C₄⋊Dic₁₂₇	—	2	1	no	—
26	C₄⋊Dic₁₂₇	—	2	1	no	—
27	C₄⋊Dic₁₂₇	—	2	1	no	—
28	C₄⋊Dic₁₂₇	—	2	1	no	—
29	C₄⋊Dic₁₂₇	—	2	1	no	—
30	C₄⋊Dic₁₂₇	—	2	1	no	—
31	C₄⋊Dic₁₂₇	—	2	1	no	—
32	C₄⋊Dic₁₂₇	—	2	1	no	—
33	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
34	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
35	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
36	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
37	Dic₂₅₄	—	4	2	no	—
38	Dic₂₅₄	—	4	2	no	—
39	Dic₂₅₄	—	4	2	no	—
40	Dic₂₅₄	—	4	2	no	—
41	C₂×Dic₁₂₇	—	4	1	no	—
42	C₂×Dic₁₂₇	—	4	1	no	—
43	C₂×Dic₁₂₇	—	4	1	no	—
44	C₂×Dic₁₂₇	—	4	1	no	—
45	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₂₇	—	2	2	no	—
49	C₈×C₁₂₇	—	2	2	no	—
50	C₈×C₁₂₇	—	2	2	no	—
51	C₂²×C₁₂₇	#127	1	1	yes	characteristic
52	C₄×C₁₂₇	#135	1	2	yes	—
53	C₄×C₁₂₇	#137	1	2	yes	—
54	C₄×C₁₂₇	#137	1	2	yes	—
55	Dic₁₂₇	—	8	1	no	—
56	Dic₁₂₇	—	8	1	no	—
57	Dic₁₂₇	—	8	1	no	—
58	Dic₁₂₇	—	8	1	no	—
59	C₂×C₁₂₇	#495	1	1	yes	characteristic
60	C₂×C₁₂₇	#497	1	1	yes	characteristic
61	C₂×C₁₂₇	#497	1	1	yes	characteristic
62	#1718	—	127	1	no	maximal, Sylow
63	C₁₂₇	#1718	1	1	yes	characteristic, complemented, Sylow
64	#500	—	127	1	no	—
65	#498	—	127	1	no	—
66	#487	—	127	1	no	—
67	#487	—	127	1	no	—
68	#487	—	127	1	no	—
69	#487	—	127	1	no	—
70	#322	D₁₂₇	1	1	yes	characteristic
71	C₄⋊Q₈	—	127	1	no	—
72	C₄⋊Q₈	—	127	1	no	—
73	C₄⋊Q₈	—	127	1	no	—
74	C₄⋊Q₈	—	127	1	no	—
75	#107	—	254	2	no	—
76	#106	—	254	2	no	—
77	#103	—	254	2	no	—
78	#103	—	254	2	no	—
79	#103	—	254	2	no	—
80	#103	—	254	2	no	—
81	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
84	C₂×Q₈	—	254	1	no	—
85	C₂×Q₈	—	254	1	no	—
86	C₂×Q₈	—	254	1	no	—
87	C₂×Q₈	—	254	1	no	—
88	C₂×C₈	D₅₀₈	1	2	yes	—
89	C₂×C₈	D₅₀₈	1	2	yes	—
90	C₂×C₈	D₅₀₈	1	2	yes	—
91	C₄⋊C₄	—	254	1	no	—
92	C₄⋊C₄	—	254	1	no	—
93	C₄⋊C₄	—	254	1	no	—
94	C₄⋊C₄	—	254	1	no	—
95	C₄⋊C₄	—	254	1	no	—
96	C₄⋊C₄	—	254	1	no	—
97	C₄⋊C₄	—	254	1	no	—
98	C₄⋊C₄	—	254	1	no	—
99	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
100	Q₈	—	508	2	no	—
101	Q₈	—	508	2	no	—
102	Q₈	—	508	2	no	—
103	Q₈	—	508	2	no	—
104	C₂×C₄	—	508	1	no	—
105	C₂×C₄	—	508	1	no	—
106	C₂×C₄	—	508	1	no	—
107	C₂×C₄	—	508	1	no	—
108	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
110	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₂²	#16861206	1	1	yes	characteristic
115	C₄	—	1016	1	no	—
116	C₄	—	1016	1	no	—
117	C₄	—	1016	1	no	—
118	C₄	—	1016	1	no	—
119	C₄	#16861231	1	2	yes	—
120	C₄	#16861242	1	2	yes	—
121	C₄	#16861242	1	2	yes	—
122	C₂	#22550260	1	1	yes	characteristic
123	C₂	#22550270	1	1	yes	characteristic
124	C₂	#22550270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550281

C₂

yes

maximal, characteristic

#22550274

C₂

yes

maximal, characteristic

#22550227

C₂

yes

maximal, characteristic

#22550227

C₂

yes

maximal, characteristic

#22550227

C₂

yes

maximal, characteristic

#22550227

C₂

yes

maximal, characteristic

#322×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₂²

yes

characteristic

#16861209

C₂²

yes

characteristic

#16861209

C₂²

yes

characteristic

#16861209

C₂²

yes

—

—

—

—

—

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

#127

yes

characteristic

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

#137

yes

—

C₄×C₁₂₇

#137

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1718

yes

characteristic, complemented, Sylow

#500

—

127

—

#498

—

127

—

#487

—

127

—

#487

—

127

—

#487

—

127

—

#487

—

127

—

#322

D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#107

—

254

—

#106

—

254

—

#103

—

254

—

#103

—

254

—

#103

—

254

—

#103

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

100

Q₈

—

508

—

101

Q₈

—

508

—

102

Q₈

—

508

—

103

Q₈

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

508

—

106

C₂×C₄

—

508

—

107

C₂×C₄

—

508

—

108

C₂×C₄

C₂×D₅₀₈

yes

characteristic

109

C₂×C₄

C₂×D₅₀₈

yes

characteristic

110

C₂×C₄

C₂×D₅₀₈

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

#16861206

yes

characteristic

115

C₄

—

1016

—

116

C₄

—

1016

—

117

C₄

—

1016

—

118

C₄

—

1016

—

119

C₄

#16861231

yes

—

120

C₄

#16861242

yes

—

121

C₄

#16861242

yes

—

122

C₂

#22550260

yes

characteristic

123

C₂

#22550270

yes

characteristic

124

C₂

#22550270

yes

characteristic

	subgroup	G/H
center	C₂²	#16861206
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#322×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#127
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	#1718
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16861206

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#322×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
63. C₁₂₇	#1718	semidirect

H ≅ G/N

type

63. C₁₂₇

#1718

semidirect

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

subgroup	index
#22550281	2
#22550274	2
#22550227	2
#322×C₁₂₇	2
#1718	127

subgroup

index

#322×C₁₂₇

127

Subgroups — #30638376

Conjugacy classes of subgroups

Complements

Maximal subgroups