Group #30647280 — subgroups

Subgroups — C₂²×Dic₁₂₇⋊C₈ #30647280

Order 16256.

640 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 389 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	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₁₂₇⋊C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₂₇	C₄	1	1	yes	characteristic
6	C₂³×Dic₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	—	2	4	no	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	6	yes	—
9	C₂×C₄×Dic₁₂₇	C₄	1	6	yes	—
10	C₂×C₄×Dic₁₂₇	C₂²	1	6	yes	—
11	Dic₁₂₇⋊C₈	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	—	2	4	no	—
14	C₂²×C₈×C₁₂₇	C₂²	1	6	yes	—
15	C₂²×Dic₁₂₇	—	2	4	no	—
16	C₂²×Dic₁₂₇	C₈	1	8	yes	complemented
17	C₂²×Dic₁₂₇	C₂×C₄	1	6	yes	—
18	C₂²×Dic₁₂₇	C₂×C₄	1	6	yes	—
19	C₂×C₁₂₇⋊C₈	—	2	12	no	—
20	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
21	C₄×Dic₁₂₇	C₂×C₄	1	12	yes	—
22	C₄×Dic₁₂₇	C₂³	1	4	yes	—
23	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
26	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
28	C₂×C₈×C₁₂₇	—	2	12	no	—
29	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
30	C₂×Dic₁₂₇	—	2	12	no	—
31	C₂×Dic₁₂₇	C₂×C₈	1	24	yes	complemented
32	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
33	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
34	C₁₂₇⋊C₈	—	2	4	no	—
35	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₂₇	C₂×C₈	1	4	yes	—
37	C₂³×C₁₂₇	M₁₆	1	4	yes	—
38	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₁₂₇	C₄⋊C₄	1	12	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
43	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
44	C₈×C₁₂₇	—	2	4	no	—
45	C₂²×C₁₂₇	C₄⋊C₈	1	16	yes	complemented
46	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
47	C₂²×C₁₂₇	C₂²×C₈	1	6	yes	—
48	C₂²×C₁₂₇	C₂×M₁₆	1	6	yes	—
49	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
50	C₄×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
51	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
53	Dic₁₂₇	—	2	4	no	—
54	Dic₁₂₇	C₂²×C₈	1	8	yes	complemented
55	C₂×C₁₂₇	C₂×C₄⋊C₈	1	12	yes	complemented
56	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
59	C₂²×C₄⋊C₈	—	127	1	no	maximal, Sylow
60	C₁₂₇	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₈	—	127	1	no	—
62	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
63	C₂²×C₄²	—	127	1	no	—
64	C₂×C₄⋊C₈	—	127	12	no	—
65	C₂³×C₄	—	127	1	no	—
66	C₂³×C₄	—	127	1	no	—
67	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	C₂²×C₈	—	2	4	no	—
69	C₂²×C₈	—	127	6	no	—
70	C₂²×C₈	—	254	4	no	—
71	C₂²×C₈	C₂×D₁₂₇	1	6	yes	—
72	C₂×C₄²	—	127	6	no	—
73	C₂×C₄²	—	127	6	no	—
74	C₄⋊C₈	—	127	16	no	—
75	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
76	C₂²×C₄	—	127	8	no	—
77	C₂²×C₄	—	127	6	no	—
78	C₂²×C₄	—	127	6	no	—
79	C₂²×C₄	—	254	4	no	—
80	C₂²×C₄	C₄×D₁₂₇	1	3	yes	—
81	C₂²×C₄	C₂²⋊D₁₂₇	1	4	yes	—
82	C₂²×C₄	Dic₂₅₄	1	4	yes	—
83	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
84	C₂×C₈	—	2	12	no	—
85	C₂×C₈	—	127	4	no	—
86	C₂×C₈	—	254	12	no	—
87	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
88	C₄²	—	127	12	no	—
89	C₄²	—	127	4	no	—
90	C₂³	Dic₁₂₇⋊C₄	1	4	yes	—
91	C₂³	C₈×D₁₂₇	1	4	yes	—
92	C₂³	#14523231	1	4	yes	—
93	C₂³	C₂×C₄×D₁₂₇	1	3	yes	—
94	C₂×C₄	—	127	24	no	—
95	C₂×C₄	—	127	4	no	—
96	C₂×C₄	—	127	4	no	—
97	C₂×C₄	—	254	12	no	—
98	C₂×C₄	Dic₁₂₇⋊C₄	1	12	yes	—
99	C₂×C₄	C₂×C₄×D₁₂₇	1	3	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	6	yes	—
101	C₂×C₄	C₂×Dic₂₅₄	1	6	yes	—
102	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	254	4	no	—
105	C₂²	Dic₁₂₇⋊C₈	1	16	yes	direct product factor
106	C₂²	C₂×Dic₁₂₇⋊C₄	1	6	yes	—
107	C₂²	C₂×C₈×D₁₂₇	1	6	yes	—
108	C₂²	C₂×#14523231	1	6	yes	—
109	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
110	C₄	—	127	8	no	—
111	C₄	—	254	4	no	—
112	C₄	C₂×Dic₁₂₇⋊C₄	1	6	yes	—
113	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
114	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
115	C₂	C₂×Dic₁₂₇⋊C₈	1	12	yes	direct product factor
116	C₂	C₂²×Dic₁₂₇⋊C₄	1	1	yes	characteristic
117	C₂	C₂²×C₈×D₁₂₇	1	1	yes	characteristic
118	C₂	C₂²×#14523231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal, direct product factor

C₂³×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂³×Dic₁₂₇

C₄

yes

characteristic

C₂³×Dic₁₂₇

C₄

yes

characteristic

C₂²×C₁₂₇⋊C₈

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₄

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

direct product factor

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

C₈

yes

complemented

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂⁴×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₈

yes

complemented

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₄⋊C₄

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

—

C₂³×C₁₂₇

M₁₆

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊C₈

yes

complemented

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

—

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂²×C₈

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

complemented

C₂×C₁₂₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂³×C₈

yes

characteristic

C₂×C₁₂₇

C₂²×M₁₆

yes

characteristic

C₂²×C₄⋊C₈

—

127

maximal, Sylow

C₁₂₇

C₂²×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂³×C₈

—

127

—

C₂³×C₈

D₁₂₇

yes

characteristic

C₂²×C₄²

—

127

—

C₂×C₄⋊C₈

—

127

—

C₂³×C₄

—

127

—

C₂³×C₄

—

127

—

C₂³×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

127

—

C₂²×C₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₄⋊C₈

—

127

—

C₂⁴

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

Dic₂₅₄

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

—

127

—

C₂³

Dic₁₂₇⋊C₄

yes

—

C₂³

C₈×D₁₂₇

yes

—

C₂³

#14523231

yes

—

C₂³

C₂×C₄×D₁₂₇

yes

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

101

C₂×C₄

C₂×Dic₂₅₄

yes

—

102

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

103

C₈

—

104

C₈

—

254

—

105

C₂²

Dic₁₂₇⋊C₈

yes

direct product factor

106

C₂²

C₂×Dic₁₂₇⋊C₄

yes

—

107

C₂²

C₂×C₈×D₁₂₇

yes

—

108

C₂²

C₂×#14523231

yes

—

109

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

110

C₄

—

127

—

111

C₄

—

254

—

112

C₄

C₂×Dic₁₂₇⋊C₄

yes

—

113

C₄

C₂²×C₂²⋊D₁₂₇

yes

characteristic

114

C₄

C₂²×Dic₂₅₄

yes

characteristic

115

C₂

C₂×Dic₁₂₇⋊C₈

yes

direct product factor

116

C₂

C₂²×Dic₁₂₇⋊C₄

yes

characteristic

117

C₂

C₂²×C₈×D₁₂₇

yes

characteristic

118

C₂

C₂²×#14523231

yes

characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂³×C₄

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂³×C₈×C₁₂₇

C₂

socle, prime radical

C₂⁴×C₁₂₇

C₂×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₂³×C₈

D₁₂₇

hypocenter

C₁₂₇

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. C₂×Dic₁₂₇⋊C₈	C₂	direct (N × H)
11. Dic₁₂₇⋊C₈	C₂²	direct (N × H)
16. C₂²×Dic₁₂₇	C₈	semidirect
31. C₂×Dic₁₂₇	C₂×C₈	semidirect
45. C₂²×C₁₂₇	C₄⋊C₈	semidirect
54. Dic₁₂₇	C₂²×C₈	semidirect
55. C₂×C₁₂₇	C₂×C₄⋊C₈	semidirect
60. C₁₂₇	C₂²×C₄⋊C₈	semidirect
105. C₂²	Dic₁₂₇⋊C₈	direct (N × H)
115. C₂	C₂×Dic₁₂₇⋊C₈	direct (N × H)

H ≅ G/N

type

3. C₂×Dic₁₂₇⋊C₈

C₂

direct (N × H)

11. Dic₁₂₇⋊C₈

C₂²

direct (N × H)

16. C₂²×Dic₁₂₇

C₈

semidirect

31. C₂×Dic₁₂₇

C₂×C₈

semidirect

45. C₂²×C₁₂₇

C₄⋊C₈

semidirect

54. Dic₁₂₇

C₂²×C₈

semidirect

55. C₂×C₁₂₇

C₂×C₄⋊C₈

semidirect

60. C₁₂₇

C₂²×C₄⋊C₈

semidirect

105. C₂²

Dic₁₂₇⋊C₈

direct (N × H)

115. C₂

C₂×Dic₁₂₇⋊C₈

direct (N × H)

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

subgroup	index
C₂³×C₁₂₇⋊C₈	2
C₂²×C₄×Dic₁₂₇	2
C₂×Dic₁₂₇⋊C₈	2
C₂³×C₈×C₁₂₇	2
C₂²×C₄⋊C₈	127

subgroup

index

C₂³×C₁₂₇⋊C₈

C₂²×C₄×Dic₁₂₇

C₂×Dic₁₂₇⋊C₈

C₂³×C₈×C₁₂₇

C₂²×C₄⋊C₈

127

Subgroups — C₂²×Dic₁₂₇⋊C₈ #30647280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×Dic127⋊C8 #30647280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₁₂₇⋊C₈ #30647280