Group #30635815 — subgroups

Subgroups — C₁₂₇×#2863 #30635815

Order 16256.

408 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 216 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	#391×C₁₂₇	C₂	1	3	yes	maximal
2	#389×C₁₂₇	C₂	1	3	yes	maximal
3	#387×C₁₂₇	C₂	1	6	yes	maximal
4	#383×C₁₂₇	C₂	1	2	yes	maximal
5	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	6	yes	—
8	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#95×C₁₂₇	C₂²	1	6	yes	—
13	#95×C₁₂₇	C₂²	1	6	yes	—
14	#95×C₁₂₇	C₂²	1	6	yes	—
15	C₂×Q₈×C₁₂₇	—	2	6	no	—
16	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	—	2	12	no	—
22	C₄⋊C₄×C₁₂₇	—	2	6	no	—
23	C₄²×C₁₂₇	—	2	6	no	—
24	C₄²×C₁₂₇	—	2	6	no	—
25	C₄²×C₁₂₇	Q₈	1	4	yes	complemented
26	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
27	Q₈×C₁₂₇	—	4	4	no	—
28	C₂×C₄×C₁₂₇	—	2	3	no	—
29	C₂×C₄×C₁₂₇	—	2	3	no	—
30	C₂×C₄×C₁₂₇	—	2	3	no	—
31	C₂×C₄×C₁₂₇	—	2	3	no	—
32	C₂×C₄×C₁₂₇	—	2	6	no	—
33	C₂×C₄×C₁₂₇	—	2	12	no	—
34	C₂×C₄×C₁₂₇	—	2	6	no	—
35	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
36	C₂×C₄×C₁₂₇	#41	1	6	yes	—
37	C₂×C₄×C₁₂₇	#41	1	6	yes	—
38	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₂₇	C₂×#41	1	3	yes	—
40	C₂²×C₁₂₇	C₂×#41	1	3	yes	—
41	C₄×C₁₂₇	—	2	6	no	—
42	C₄×C₁₂₇	—	2	6	no	—
43	C₄×C₁₂₇	—	4	2	no	—
44	C₄×C₁₂₇	—	4	6	no	—
45	C₄×C₁₂₇	#126	1	4	yes	—
46	C₂×C₁₂₇	C₂×#126	1	1	yes	characteristic
47	C₂×C₁₂₇	#529	1	3	yes	—
48	C₂×C₁₂₇	#533	1	3	yes	—
49	#2863	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₁₂₇	#2863	1	1	yes	characteristic, direct product factor, Sylow
51	#391	C₂×C₁₂₇	1	3	yes	—
52	#389	C₂×C₁₂₇	1	3	yes	—
53	#387	C₂×C₁₂₇	1	6	yes	—
54	#383	C₂×C₁₂₇	1	2	yes	—
55	C₄³	C₂×C₁₂₇	1	1	yes	characteristic
56	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	6	yes	—
58	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	3	yes	—
59	C₂×C₄²	C₂²×C₁₂₇	1	3	yes	—
60	C₂×C₄²	C₂²×C₁₂₇	1	3	yes	—
61	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
62	#95	C₂²×C₁₂₇	1	6	yes	—
63	#95	C₂²×C₁₂₇	1	6	yes	—
64	#95	C₂²×C₁₂₇	1	6	yes	—
65	C₂×Q₈	—	2	6	no	—
66	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
67	C₂²×C₄	C₂³×C₁₂₇	1	6	yes	—
68	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
69	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
70	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
71	C₄⋊C₄	—	2	12	no	—
72	C₄⋊C₄	—	2	6	no	—
73	C₄²	—	2	6	no	—
74	C₄²	—	2	6	no	—
75	C₄²	Q₈×C₁₂₇	1	4	yes	complemented
76	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
77	Q₈	—	4	4	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	2	6	no	—
83	C₂×C₄	—	2	12	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
86	C₂×C₄	#41×C₁₂₇	1	6	yes	—
87	C₂×C₄	#41×C₁₂₇	1	6	yes	—
88	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
89	C₂²	C₂×#41×C₁₂₇	1	3	yes	—
90	C₂²	C₂×#41×C₁₂₇	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	2	6	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	6	no	—
95	C₄	#126×C₁₂₇	1	4	yes	—
96	C₂	C₂×#126×C₁₂₇	1	1	yes	characteristic
97	C₂	#529×C₁₂₇	1	3	yes	—
98	C₂	#533×C₁₂₇	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#391×C₁₂₇

C₂

yes

maximal

#389×C₁₂₇

C₂

yes

maximal

#387×C₁₂₇

C₂

yes

maximal

#383×C₁₂₇

C₂

yes

maximal

C₄³×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

C₂²

yes

—

#95×C₁₂₇

C₂²

yes

—

#95×C₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

Q₈

yes

complemented

C₂³×C₁₂₇

C₂⁴

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×#41

yes

—

C₂²×C₁₂₇

C₂×#41

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#126

yes

—

C₂×C₁₂₇

C₂×#126

yes

characteristic

C₂×C₁₂₇

#529

yes

—

C₂×C₁₂₇

#533

yes

—

#2863

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#2863

yes

characteristic, direct product factor, Sylow

#391

C₂×C₁₂₇

yes

—

#389

C₂×C₁₂₇

yes

—

#387

C₂×C₁₂₇

yes

—

#383

C₂×C₁₂₇

yes

—

C₄³

C₂×C₁₂₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₂₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

characteristic

#95

C₂²×C₁₂₇

yes

—

#95

C₂²×C₁₂₇

yes

—

#95

C₂²×C₁₂₇

yes

—

C₂×Q₈

—

C₂²×C₄

C₂³×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

Q₈×C₁₂₇

yes

complemented

C₂³

C₂⁴×C₁₂₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Q₈×C₁₂₇

yes

—

C₂×C₄

#41×C₁₂₇

yes

—

C₂×C₄

#41×C₁₂₇

yes

—

C₂²

C₂²×Q₈×C₁₂₇

yes

characteristic

C₂²

C₂×#41×C₁₂₇

yes

—

C₂²

C₂×#41×C₁₂₇

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#126×C₁₂₇

yes

—

C₂

C₂×#126×C₁₂₇

yes

characteristic

C₂

#529×C₁₂₇

yes

—

C₂

#533×C₁₂₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₂₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₂₇

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₂₇ ◁ G

N	H ≅ G/N	type
25. C₄²×C₁₂₇	Q₈	semidirect
49. #2863	C₁₂₇	direct (N × H)
50. C₁₂₇	#2863	direct (N × H)
75. C₄²	Q₈×C₁₂₇	semidirect

H ≅ G/N

type

25. C₄²×C₁₂₇

Q₈

semidirect

49. #2863

C₁₂₇

direct (N × H)

50. C₁₂₇

#2863

direct (N × H)

75. C₄²

Q₈×C₁₂₇

semidirect

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

subgroup	index
#391×C₁₂₇	2
#389×C₁₂₇	2
#387×C₁₂₇	2
#383×C₁₂₇	2
C₄³×C₁₂₇	2
#2863	127

subgroup

index

#391×C₁₂₇

#389×C₁₂₇

#387×C₁₂₇

#383×C₁₂₇

C₄³×C₁₂₇

#2863

127

Subgroups — C₁₂₇×#2863 #30635815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#2863 #30635815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2863 #30635815