Group #30649238 — subgroups

Subgroups — C₂×#22550281 #30649238

Order 16256.

424 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 213 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×#16861209	C₂	1	2	yes	maximal
2	#22550281	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861143	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	2	no	—
6	#16861209	—	2	4	no	—
7	#16861209	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
10	#16861143	C₂²	1	16	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
14	C₂×Dic₂₅₄	—	2	8	no	—
15	C₂×Dic₂₅₄	—	4	8	no	—
16	C₂²×Dic₁₂₇	—	2	4	no	—
17	C₄⋊Dic₁₂₇	—	2	4	no	—
18	C₄⋊Dic₁₂₇	—	2	8	no	—
19	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
20	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	Q₈	1	8	yes	complemented
24	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
25	C₄²×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	—	4	16	no	—
28	C₂×Dic₁₂₇	—	2	8	no	—
29	C₂×Dic₁₂₇	—	4	8	no	—
30	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	D₈	1	4	yes	—
32	C₂×C₄×C₁₂₇	Q₁₆	1	4	yes	—
33	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
37	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂×Q₈	1	8	yes	complemented
39	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₂₇	C₂×D₈	1	2	yes	—
41	C₂²×C₁₂₇	C₂×Q₁₆	1	2	yes	—
42	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₂₇	C₂×D₈	1	4	yes	—
45	C₄×C₁₂₇	C₂×Q₁₆	1	4	yes	—
46	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
47	Dic₁₂₇	—	4	8	no	—
48	C₂×C₁₂₇	#500	1	4	yes	complemented
49	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
51	C₂×C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic
52	C₂×#500	—	127	1	no	maximal, Sylow
53	C₁₂₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊Q₈	—	127	2	no	—
55	#500	—	127	8	no	—
56	C₂×#107	—	127	4	no	—
57	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
58	C₂²×Q₈	—	254	2	no	—
59	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
60	C₄⋊Q₈	—	127	8	no	—
61	C₄⋊Q₈	—	254	4	no	—
62	C₂×C₄⋊C₄	—	127	4	no	—
63	C₂×C₄⋊C₄	—	254	2	no	—
64	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
65	#107	—	127	16	no	—
66	C₄×C₈	C₂×D₁₂₇	1	4	yes	—
67	C₂×Q₈	—	508	8	no	—
68	C₂×Q₈	—	254	8	no	—
69	C₂²×C₄	—	254	4	no	—
70	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
71	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₂×C₈	Dic₂₅₄	1	8	yes	complemented
74	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
75	C₄⋊C₄	—	127	8	no	—
76	C₄⋊C₄	—	254	4	no	—
77	C₄⋊C₄	—	254	8	no	—
78	C₄²	D₅₀₈	1	2	yes	—
79	C₄²	C₂²×D₁₂₇	1	2	yes	—
80	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
81	Q₈	—	508	16	no	—
82	C₂×C₄	—	508	8	no	—
83	C₂×C₄	—	254	8	no	—
84	C₂×C₄	D₁₀₁₆	1	4	yes	—
85	C₂×C₄	Dic₅₀₈	1	4	yes	—
86	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
87	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
88	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
89	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
90	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
91	C₈	C₂×Dic₂₅₄	1	8	yes	complemented
92	C₂²	#16861209	1	2	yes	—
93	C₂²	C₂×D₁₀₁₆	1	2	yes	—
94	C₂²	C₂×Dic₅₀₈	1	2	yes	—
95	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
96	C₄	—	508	8	no	—
97	C₄	#16861209	1	2	yes	—
98	C₄	C₂×D₁₀₁₆	1	4	yes	—
99	C₄	C₂×Dic₅₀₈	1	4	yes	—
100	C₄	C₂²×Dic₂₅₄	1	2	yes	—
101	C₂	#22550281	1	4	yes	direct product factor
102	C₂	C₂×#16861209	1	1	yes	characteristic
103	C₂	C₂²×D₁₀₁₆	1	1	yes	characteristic
104	C₂	C₂²×Dic₅₀₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal

#22550281

C₂

yes

maximal, direct product factor

C₂×#16861143

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

#16861209

—

#16861209

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861143

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

complemented

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

complemented

C₂²×C₁₂₇

C₄⋊Q₈

yes

—

C₂²×C₁₂₇

C₂×D₈

yes

—

C₂²×C₁₂₇

C₂×Q₁₆

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

C₂×D₈

yes

—

C₄×C₁₂₇

C₂×Q₁₆

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#500

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂²×D₈

yes

characteristic

C₂×C₁₂₇

C₂²×Q₁₆

yes

characteristic

C₂×#500

—

127

maximal, Sylow

C₁₂₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

#500

—

127

—

C₂×#107

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₂²×Q₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

complemented

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

D₅₀₈

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

D₁₀₁₆

yes

—

C₂×C₄

Dic₅₀₈

yes

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

—

C₂×C₄

C₂×D₅₀₈

yes

—

C₂×C₄

C₂×Dic₂₅₄

yes

—

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

C₈

C₂×Dic₂₅₄

yes

complemented

C₂²

#16861209

yes

—

C₂²

C₂×D₁₀₁₆

yes

—

C₂²

C₂×Dic₅₀₈

yes

—

C₂²

C₂²×D₅₀₈

yes

characteristic

C₄

—

508

—

C₄

#16861209

yes

—

C₄

C₂×D₁₀₁₆

yes

—

C₄

C₂×Dic₅₀₈

yes

—

100

C₄

C₂²×Dic₂₅₄

yes

—

101

C₂

#22550281

yes

direct product factor

102

C₂

C₂×#16861209

yes

characteristic

103

C₂

C₂²×D₁₀₁₆

yes

characteristic

104

C₂

C₂²×Dic₅₀₈

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#500
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

C₂×#500

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22550281	C₂	direct (N × H)
23. C₂×C₈×C₁₂₇	Q₈	semidirect
38. C₈×C₁₂₇	C₂×Q₈	semidirect
48. C₂×C₁₂₇	#500	semidirect
53. C₁₂₇	C₂×#500	semidirect
73. C₂×C₈	Dic₂₅₄	semidirect
91. C₈	C₂×Dic₂₅₄	semidirect
101. C₂	#22550281	direct (N × H)

H ≅ G/N

type

2. #22550281

C₂

direct (N × H)

23. C₂×C₈×C₁₂₇

Q₈

semidirect

38. C₈×C₁₂₇

C₂×Q₈

semidirect

48. C₂×C₁₂₇

#500

semidirect

53. C₁₂₇

C₂×#500

semidirect

73. C₂×C₈

Dic₂₅₄

semidirect

91. C₈

C₂×Dic₂₅₄

semidirect

101. C₂

#22550281

direct (N × H)

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

subgroup	index
C₂×#16861209	2
#22550281	2
C₂×#16861143	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#500	127

subgroup

index

C₂×C₄×C₈×C₁₂₇

127

Subgroups — C₂×#22550281 #30649238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550281 #30649238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550281 #30649238