Group #30634658 — subgroups

Subgroups — C₁₂₇×#1706 #30634658

Order 16256.

224 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 88 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	#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#492×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#486×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
7	C₂×D₈×C₁₂₇	—	2	2	no	—
8	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	#127×C₁₂₇	C₂²	1	2	yes	—
10	#107×C₁₂₇	—	2	2	no	—
11	#102×C₁₂₇	—	2	4	no	—
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₂₇	—	2	2	no	—
16	C₂×D₄×C₁₂₇	—	2	2	no	—
17	C₂×D₄×C₁₂₇	—	2	2	no	—
18	C₂×D₄×C₁₂₇	—	2	2	no	—
19	SD₁₆×C₁₂₇	—	4	8	no	—
20	D₈×C₁₂₇	—	4	4	no	—
21	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
22	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₂₇	—	2	2	no	—
25	C₄⋊C₄×C₁₂₇	—	2	2	no	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂³×C₁₂₇	—	4	2	no	—
28	Q₈×C₁₂₇	—	4	4	no	—
29	D₄×C₁₂₇	—	4	4	no	—
30	D₄×C₁₂₇	—	4	4	no	—
31	D₄×C₁₂₇	—	4	4	no	—
32	C₂×C₄×C₁₂₇	—	4	2	no	—
33	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₂₇	—	2	2	no	—
37	C₈×C₁₂₇	D₈	1	4	yes	complemented
38	C₈×C₁₂₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₂₇	—	8	2	no	—
40	C₂²×C₁₂₇	—	8	2	no	—
41	C₂²×C₁₂₇	—	8	2	no	—
42	C₂²×C₁₂₇	#127	1	1	yes	characteristic
43	C₄×C₁₂₇	—	8	2	no	—
44	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
45	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₂₇	#135	1	2	yes	—
47	C₂×C₁₂₇	—	16	2	no	—
48	C₂×C₁₂₇	#492	1	1	yes	characteristic
49	C₂×C₁₂₇	#493	1	1	yes	characteristic
50	C₂×C₁₂₇	#495	1	1	yes	characteristic
51	#1706	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₁₂₇	#1706	1	1	yes	characteristic, direct product factor, Sylow
53	#500	C₂×C₁₂₇	1	1	yes	characteristic, complemented
54	#493	C₂×C₁₂₇	1	1	yes	characteristic
55	#492	C₂×C₁₂₇	1	2	yes	complemented
56	#486	C₂×C₁₂₇	1	2	yes	complemented
57	C₈²	C₂×C₁₂₇	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	2	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
61	#127	C₂²×C₁₂₇	1	2	yes	—
62	#107	—	2	2	no	—
63	#102	—	2	4	no	—
64	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
66	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	4	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
74	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
75	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	4	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	D₈×C₁₂₇	1	4	yes	complemented
90	C₈	SD₁₆×C₁₂₇	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#127×C₁₂₇	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	C₂×D₈×C₁₂₇	1	2	yes	—
97	C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
98	C₄	#135×C₁₂₇	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#492×C₁₂₇	1	1	yes	characteristic
101	C₂	#493×C₁₂₇	1	1	yes	characteristic
102	C₂	#495×C₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#493×C₁₂₇

C₂

yes

maximal, characteristic

#492×C₁₂₇

C₂

yes

maximal, complemented

#486×C₁₂₇

C₂

yes

maximal, complemented

C₈²×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₂₇

—

C₂×D₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

#127×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

—

#102×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

SD₁₆×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

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

D₈

yes

complemented

C₈×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#127

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₈

yes

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

#135

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1706

yes

characteristic, direct product factor, Sylow

#500

C₂×C₁₂₇

yes

characteristic, complemented

#493

C₂×C₁₂₇

yes

characteristic

#492

C₂×C₁₂₇

yes

complemented

#486

C₂×C₁₂₇

yes

complemented

C₈²

C₂×C₁₂₇

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₁₂₇

yes

—

#127

C₂²×C₁₂₇

yes

—

#107

—

#102

—

C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₁₂₇

yes

—

C₂×C₈

D₄×C₁₂₇

yes

—

C₂×C₈

D₄×C₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂³×C₁₂₇

yes

characteristic

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

C₈

—

C₈

D₈×C₁₂₇

yes

complemented

C₈

SD₁₆×C₁₂₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#127×C₁₂₇

yes

characteristic

C₄

—

C₄

C₂×D₈×C₁₂₇

yes

—

C₄

C₂×SD₁₆×C₁₂₇

yes

—

C₄

#135×C₁₂₇

yes

—

C₂

—

100

C₂

#492×C₁₂₇

yes

characteristic

101

C₂

#493×C₁₂₇

yes

characteristic

102

C₂

#495×C₁₂₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₁₂₇
prime radical	#493×C₁₂₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₂₇

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₁₂₇

prime radical

#493×C₁₂₇

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ G

N	H ≅ G/N	type
1. #500×C₁₂₇	C₂	semidirect
3. #492×C₁₂₇	C₂	semidirect
4. #486×C₁₂₇	C₂	semidirect
5. C₈²×C₁₂₇	C₂	semidirect
37. C₈×C₁₂₇	D₈	semidirect
38. C₈×C₁₂₇	SD₁₆	semidirect
51. #1706	C₁₂₇	direct (N × H)
52. C₁₂₇	#1706	direct (N × H)
53. #500	C₂×C₁₂₇	semidirect
55. #492	C₂×C₁₂₇	semidirect
56. #486	C₂×C₁₂₇	semidirect
57. C₈²	C₂×C₁₂₇	semidirect
89. C₈	D₈×C₁₂₇	semidirect
90. C₈	SD₁₆×C₁₂₇	semidirect

H ≅ G/N

type

1. #500×C₁₂₇

C₂

semidirect

3. #492×C₁₂₇

C₂

semidirect

4. #486×C₁₂₇

C₂

semidirect

5. C₈²×C₁₂₇

C₂

semidirect

37. C₈×C₁₂₇

D₈

semidirect

38. C₈×C₁₂₇

SD₁₆

semidirect

51. #1706

C₁₂₇

direct (N × H)

52. C₁₂₇

#1706

direct (N × H)

53. #500

C₂×C₁₂₇

semidirect

55. #492

C₂×C₁₂₇

semidirect

56. #486

C₂×C₁₂₇

semidirect

57. C₈²

C₂×C₁₂₇

semidirect

89. C₈

D₈×C₁₂₇

semidirect

90. C₈

SD₁₆×C₁₂₇

semidirect

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

subgroup	index
#500×C₁₂₇	2
#493×C₁₂₇	2
#492×C₁₂₇	2
#486×C₁₂₇	2
C₈²×C₁₂₇	2
#1706	127

subgroup

index

#500×C₁₂₇

#493×C₁₂₇

#492×C₁₂₇

#486×C₁₂₇

C₈²×C₁₂₇

#1706

127

Subgroups — C₁₂₇×#1706 #30634658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1706 #30634658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1706 #30634658