Group #30638583 — subgroups

Subgroups — #30638583

Order 16256.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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	2	yes	maximal
2	C₂×#16861120	C₂	1	2	yes	maximal
3	#22549631	C₂	1	2	yes	maximal
4	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
6	C₄⋊(C₁₂₇⋊C₈)	C₄	1	8	yes	complemented
7	#16861120	—	2	4	no	—
8	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₂₇⋊C₈	—	2	4	no	—
14	C₂×C₁₂₇⋊C₈	—	2	4	no	—
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₂₇	—	2	4	no	—
22	C₄²×C₁₂₇	—	2	4	no	—
23	C₄²×C₁₂₇	—	2	2	no	—
24	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	Q₈	1	2	yes	—
27	C₁₂₇⋊C₈	—	2	8	no	—
28	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	2	no	—
31	C₂×C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄×C₁₂₇	—	2	2	no	—
34	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
36	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	#41	1	2	yes	—
43	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₂₇	#117	1	1	yes	characteristic
46	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
47	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₁₂₇	—	2	2	no	—
49	C₄×C₁₂₇	—	2	4	no	—
50	C₄×C₁₂₇	—	2	4	no	—
51	C₄×C₁₂₇	#97	1	4	yes	complemented
52	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
53	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
54	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₁₂₇	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₂₇	#423	1	2	yes	—
58	C₂×C₁₂₇	#432	1	2	yes	—
59	#1767	—	127	1	no	maximal, Sylow
60	C₁₂₇	#1767	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄⋊C₈	—	127	2	no	—
62	C₂×#97	—	127	2	no	—
63	C₄³	D₁₂₇	1	1	yes	characteristic
64	#336	—	127	2	no	—
65	C₂²×C₈	—	127	4	no	—
66	C₂×C₄²	Dic₁₂₇	1	2	yes	—
67	C₂×C₄²	Dic₁₂₇	1	2	yes	—
68	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
71	C₄⋊C₈	—	127	8	no	—
72	#97	—	254	4	no	—
73	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
75	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
76	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
78	C₂×C₈	—	127	8	no	—
79	C₂×C₈	—	254	4	no	—
80	C₂×C₈	—	254	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₄×D₁₂₇	1	4	yes	—
85	C₄²	D₅₀₈	1	2	yes	—
86	C₄²	Dic₂₅₄	1	2	yes	—
87	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
94	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
95	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
96	C₂×C₄	#14523232	1	4	yes	—
97	C₂×C₄	#14523232	1	4	yes	—
98	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
99	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
101	C₂×C₄	#14523248	1	2	yes	—
102	C₈	—	254	8	no	—
103	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
105	C₂²	#16861168	1	1	yes	characteristic
106	C₂²	C₂×#14523232	1	2	yes	—
107	C₂²	C₂×#14523232	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#16861120	1	4	yes	complemented
112	C₄	C₄×D₅₀₈	1	2	yes	—
113	C₄	C₄×Dic₂₅₄	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₁₂₇	1	1	yes	characteristic
115	C₂	C₂×#16861120	1	1	yes	characteristic
116	C₂	C₄×#14523232	1	1	yes	characteristic
117	C₂	#22549900	1	2	yes	—
118	C₂	#22549933	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

C₂×#16861120

C₂

yes

maximal

#22549631

C₂

yes

maximal

C₄³×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

complemented

#16861120

—

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

Q₈

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₄²

yes

characteristic

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

#117

yes

characteristic

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#97

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₂×C₁₂₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂×#97

yes

characteristic

C₂×C₁₂₇

C₄×M₁₆

yes

characteristic

C₂×C₁₂₇

#423

yes

—

C₂×C₁₂₇

#432

yes

—

#1767

—

127

maximal, Sylow

C₁₂₇

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

C₂×#97

—

127

—

C₄³

D₁₂₇

yes

characteristic

#336

—

127

—

C₂²×C₈

—

127

—

C₂×C₄²

Dic₁₂₇

yes

—

C₂×C₄²

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

#97

—

254

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

D₅₀₈

yes

—

C₄²

Dic₂₅₄

yes

—

C₂³

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₂₇

yes

—

C₂×C₄

C₄×Dic₁₂₇

yes

—

C₂×C₄

C₄⋊Dic₁₂₇

yes

—

C₂×C₄

#14523232

yes

—

C₂×C₄

#14523232

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

100

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

101

C₂×C₄

#14523248

yes

—

102

C₈

—

254

—

103

C₂²

C₂×C₄×Dic₁₂₇

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₁₂₇

yes

characteristic

105

C₂²

#16861168

yes

characteristic

106

C₂²

C₂×#14523232

yes

—

107

C₂²

C₂×#14523232

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#16861120

yes

complemented

112

C₄

C₄×D₅₀₈

yes

—

113

C₄

C₄×Dic₂₅₄

yes

—

114

C₂

C₄×C₄⋊Dic₁₂₇

yes

characteristic

115

C₂

C₂×#16861120

yes

characteristic

116

C₂

C₄×#14523232

yes

characteristic

117

C₂

#22549900

yes

—

118

C₂

#22549933

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₄³×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#1767
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂×C₄²

Fitting

C₄³×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄³

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊(C₁₂₇⋊C₈)	C₄	semidirect
51. C₄×C₁₂₇	#97	semidirect
60. C₁₂₇	#1767	semidirect
111. C₄	#16861120	semidirect

H ≅ G/N

type

6. C₄⋊(C₁₂₇⋊C₈)

C₄

semidirect

51. C₄×C₁₂₇

#97

semidirect

60. C₁₂₇

#1767

semidirect

111. C₄

#16861120

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
C₂×#16861120	2
#22549631	2
C₄³×C₁₂₇	2
#1767	127

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×#16861120

#22549631

C₄³×C₁₂₇

#1767

127

Subgroups — #30638583

Conjugacy classes of subgroups

Complements

Maximal subgroups