Group #30647112 — subgroups

Subgroups — #30647112

Order 16256.

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

normal subgroups only

This group has 408 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	#22549782	C₂	1	3	yes	maximal
2	#22549774	C₂	1	3	yes	maximal
3	#22549760	C₂	1	6	yes	maximal
4	#22549744	C₂	1	2	yes	maximal
5	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
8	C₂×Dic₁₂₇⋊C₄	C₂²	1	6	yes	—
9	C₁₂₇⋊#95	C₂²	1	6	yes	—
10	C₁₂₇⋊#95	C₂²	1	6	yes	—
11	C₁₂₇⋊#95	C₂²	1	6	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	6	no	—
16	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
17	C₂²×Dic₁₂₇	C₂³	1	6	yes	—
18	C₄⋊Dic₁₂₇	—	2	6	no	—
19	Dic₁₂₇⋊C₄	—	2	12	no	—
20	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
21	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄²×C₁₂₇	—	2	6	no	—
24	C₄²×C₁₂₇	—	2	6	no	—
25	C₄²×C₁₂₇	Q₈	1	4	yes	complemented
26	Dic₂₅₄	—	4	4	no	—
27	C₂×Dic₁₂₇	—	2	12	no	—
28	C₂×Dic₁₂₇	—	2	6	no	—
29	C₂×Dic₁₂₇	—	2	6	no	—
30	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	—	2	3	no	—
32	C₂×C₄×C₁₂₇	—	2	3	no	—
33	C₂×C₄×C₁₂₇	—	2	3	no	—
34	C₂×C₄×C₁₂₇	—	2	3	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	#126	1	4	yes	—
44	Dic₁₂₇	—	4	2	no	—
45	Dic₁₂₇	—	4	6	no	—
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	—	127	1	no	maximal, Sylow
50	C₁₂₇	#2863	1	1	yes	characteristic, complemented, Sylow
51	#391	—	127	3	no	—
52	#389	—	127	3	no	—
53	#387	—	127	6	no	—
54	#383	—	127	2	no	—
55	C₄³	D₁₂₇	1	1	yes	characteristic
56	C₂²×Q₈	—	127	1	no	—
57	C₂×C₄⋊C₄	—	127	6	no	—
58	C₂×C₄⋊C₄	—	127	3	no	—
59	C₂×C₄²	C₂×D₁₂₇	1	3	yes	—
60	C₂×C₄²	C₂×D₁₂₇	1	3	yes	—
61	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
62	#95	—	127	6	no	—
63	#95	—	127	6	no	—
64	#95	—	127	6	no	—
65	C₂×Q₈	—	254	6	no	—
66	C₂²×C₄	—	127	2	no	—
67	C₂²×C₄	—	127	6	no	—
68	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
69	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
70	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
71	C₄⋊C₄	—	254	12	no	—
72	C₄⋊C₄	—	254	6	no	—
73	C₄²	—	2	6	no	—
74	C₄²	—	2	6	no	—
75	C₄²	Dic₂₅₄	1	4	yes	complemented
76	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
77	Q₈	—	508	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	—	254	6	no	—
83	C₂×C₄	—	254	12	no	—
84	C₂×C₄	—	254	6	no	—
85	C₂×C₄	C₂×Dic₂₅₄	1	6	yes	—
86	C₂×C₄	#14523248	1	6	yes	—
87	C₂×C₄	#14523248	1	6	yes	—
88	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
89	C₂²	C₂×#14523248	1	3	yes	—
90	C₂²	C₂×#14523248	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	2	6	no	—
93	C₄	—	508	2	no	—
94	C₄	—	508	6	no	—
95	C₄	#16861203	1	4	yes	—
96	C₂	C₂×#16861203	1	1	yes	characteristic
97	C₂	#22550410	1	3	yes	—
98	C₂	#22550433	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549782

C₂

yes

maximal

#22549774

C₂

yes

maximal

#22549760

C₂

yes

maximal

#22549744

C₂

yes

maximal

C₄³×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₁₂₇⋊#95

C₂²

yes

—

C₁₂₇⋊#95

C₂²

yes

—

C₁₂₇⋊#95

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

Q₈

yes

complemented

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂⁴

yes

characteristic

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

#126

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂×#126

yes

characteristic

C₂×C₁₂₇

#529

yes

—

C₂×C₁₂₇

#533

yes

—

#2863

—

127

maximal, Sylow

C₁₂₇

#2863

yes

characteristic, complemented, Sylow

#391

—

127

—

#389

—

127

—

#387

—

127

—

#383

—

127

—

C₄³

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#95

—

127

—

#95

—

127

—

#95

—

127

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

C₄²

—

C₄²

Dic₂₅₄

yes

complemented

C₂³

C₂³×D₁₂₇

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×Dic₂₅₄

yes

—

C₂×C₄

#14523248

yes

—

C₂×C₄

#14523248

yes

—

C₂²

C₂²×Dic₂₅₄

yes

characteristic

C₂²

C₂×#14523248

yes

—

C₂²

C₂×#14523248

yes

—

C₄

—

C₄

—

C₄

—

508

—

C₄

—

508

—

C₄

#16861203

yes

—

C₂

C₂×#16861203

yes

characteristic

C₂

#22550410

yes

—

C₂

#22550433

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator, socle, prime radical, cosocle	C₂³×C₁₂₇	C₂⁴
Fitting	C₄³×C₁₂₇	C₂
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#2863
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₂₇

commutator, socle, prime radical, cosocle

C₂³×C₁₂₇

C₂⁴

Fitting

C₄³×C₁₂₇

C₂

hypercenter

C₄³

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
25. C₄²×C₁₂₇	Q₈	semidirect
50. C₁₂₇	#2863	semidirect
75. C₄²	Dic₂₅₄	semidirect

H ≅ G/N

type

25. C₄²×C₁₂₇

Q₈

semidirect

50. C₁₂₇

#2863

semidirect

75. C₄²

Dic₂₅₄

semidirect

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

subgroup	index
#22549782	2
#22549774	2
#22549760	2
#22549744	2
C₄³×C₁₂₇	2
#2863	127

subgroup

index

C₄³×C₁₂₇

127

Subgroups — #30647112

Conjugacy classes of subgroups

Complements

Maximal subgroups