Group #30640326 — subgroups

Subgroups — #30640326

Order 16256.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861149	C₂	1	1	yes	maximal, characteristic
2	#22549831	C₂	1	1	yes	maximal, characteristic
3	#22549651	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₄	1	1	yes	characteristic
7	#16861168	C₄	1	1	yes	characteristic
8	#16861149	C₄	1	4	yes	complemented
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
10	#16861119	C₄	1	2	yes	—
11	C₈×Dic₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₂₇	C₄	1	4	yes	complemented
14	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
16	#14523232	C₈	1	4	yes	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊Dic₁₂₇	C₈	1	2	yes	—
19	C₂²⋊Dic₁₂₇	C₈	1	2	yes	—
20	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
21	C₁₂₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₂₇	C₂×C₈	1	4	yes	—
30	C₁₂₇⋊C₈	C₂×C₈	1	4	yes	—
31	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₂₇	C₄²	1	2	yes	—
37	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	C₄×C₈	1	2	yes	—
41	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₂₇	C₄×C₈	1	2	yes	—
43	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₂₇	—	2	4	no	—
46	C₂×C₁₂₇	—	2	2	no	—
47	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₂₇	#504	1	2	yes	—
49	#2108	—	127	1	no	maximal, Sylow
50	C₁₂₇	#2108	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	127	1	no	—
52	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
53	#405	—	127	1	no	—
54	#346	—	127	2	no	—
55	C₄×C₁₆	—	127	2	no	—
56	C₂×M₁₆	—	127	1	no	—
57	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
58	#117	—	127	1	no	—
59	M₃₂	—	127	4	no	—
60	C₂×C₁₆	—	127	2	no	—
61	C₂×C₁₆	Dic₁₂₇	1	4	yes	complemented
62	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
64	#97	—	127	2	no	—
65	C₄×C₈	—	127	2	no	—
66	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
67	M₁₆	—	127	4	no	—
68	C₂×C₈	—	127	2	no	—
69	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
70	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
71	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
72	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	—	127	2	no	—
74	C₂²⋊C₄	—	127	2	no	—
75	C₄²	—	127	2	no	—
76	C₁₆	—	127	4	no	—
77	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
78	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
79	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	—	127	4	no	—
81	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
82	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
83	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
85	C₈	—	127	4	no	—
86	C₈	C₄×Dic₁₂₇	1	2	yes	—
87	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
88	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₁₂₇	1	2	yes	—
91	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
92	C₄	—	254	4	no	—
93	C₄	C₈×Dic₁₂₇	1	2	yes	—
94	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
95	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₁₂₇	1	1	yes	characteristic
98	C₂	#22550292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861149

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal, characteristic

#22549651

C₂

yes

maximal, complemented

C₁₆×Dic₁₂₇

C₂

yes

maximal, complemented

C₂²×C₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₄

yes

characteristic

#16861168

C₄

yes

characteristic

#16861149

C₄

yes

complemented

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

#16861119

C₄

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₄

yes

complemented

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#14523232

C₈

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

C₈

yes

—

C₂²⋊Dic₁₂₇

C₈

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₁₂₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

C₂×C₄

yes

—

C₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₁₂₇⋊C₈

C₂×C₈

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₄²

yes

—

C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×C₈

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₄×C₁₂₇

C₄×C₈

yes

—

C₄×C₁₂₇

C₂×C₄²

yes

characteristic

C₄×C₁₂₇

C₂²×C₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₂₇

#504

yes

—

#2108

—

127

maximal, Sylow

C₁₂₇

#2108

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

C₂²×C₁₆

D₁₂₇

yes

characteristic

#405

—

127

—

#346

—

127

—

C₄×C₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

M₃₂

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

Dic₁₂₇

yes

complemented

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#97

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₁₆

—

127

—

C₁₆

C₂×Dic₁₂₇

yes

—

C₁₆

C₂×Dic₁₂₇

yes

—

C₂³

C₈×D₁₂₇

yes

characteristic

C₂×C₄

—

127

—

C₂×C₄

C₄×Dic₁₂₇

yes

—

C₂×C₄

C₈×D₁₂₇

yes

—

C₂×C₄

C₈×D₁₂₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

C₄×Dic₁₂₇

yes

—

C₈

C₂×C₄×D₁₂₇

yes

characteristic

C₈

C₂²×Dic₁₂₇

yes

characteristic

C₂²

—

C₂²

C₈×Dic₁₂₇

yes

—

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

C₄

—

254

—

C₄

C₈×Dic₁₂₇

yes

—

C₄

C₂×C₄×Dic₁₂₇

yes

characteristic

C₄

C₂×C₈×D₁₂₇

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₁₂₇

yes

characteristic

C₂

#22550292

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₈
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2108
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

C₂²×C₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×C₈

prime radical

C₂³×C₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂²×C₁₆

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #22549651	C₂	semidirect
4. C₁₆×Dic₁₂₇	C₂	semidirect
8. #16861149	C₄	semidirect
13. C₂×C₁₆×C₁₂₇	C₄	semidirect
50. C₁₂₇	#2108	semidirect
61. C₂×C₁₆	Dic₁₂₇	semidirect

H ≅ G/N

type

3. #22549651

C₂

semidirect

4. C₁₆×Dic₁₂₇

C₂

semidirect

8. #16861149

C₄

semidirect

13. C₂×C₁₆×C₁₂₇

C₄

semidirect

50. C₁₂₇

#2108

semidirect

61. C₂×C₁₆

Dic₁₂₇

semidirect

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

subgroup	index
C₂×#16861149	2
#22549831	2
#22549651	2
C₁₆×Dic₁₂₇	2
C₂²×C₁₆×C₁₂₇	2
#2108	127

subgroup

index

C₂×#16861149

#22549831

#22549651

C₁₆×Dic₁₂₇

C₂²×C₁₆×C₁₂₇

#2108

127

Subgroups — #30640326

Conjugacy classes of subgroups

Complements

Maximal subgroups