Group #30647158 — subgroups

Subgroups — #30647158

Order 16256.

556 conjugacy classes of subgroups; only the 373 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 373 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×#16861216	C₂	1	2	yes	maximal
2	#22549832	C₂	1	6	yes	maximal, complemented
3	C₄×#14523232	C₂	1	3	yes	maximal, complemented
4	C₂×#16861120	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861216	C₄	1	16	yes	complemented
7	C₂×#14523232	C₂²	1	6	yes	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	6	yes	—
9	#16861120	C₂²	1	1	yes	characteristic
10	#16861120	C₂²	1	3	yes	—
11	#16861120	C₂²	1	6	yes	—
12	C₄×C₁₂₇⋊C₈	C₂²	1	6	yes	—
13	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₂₇	C₄	1	2	yes	—
15	C₄×D₄×C₁₂₇	C₄	1	6	yes	—
16	#117×C₁₂₇	C₂²	1	3	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
18	#14523232	C₂×C₄	1	24	yes	complemented
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	24	yes	complemented
20	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
21	C₂×C₁₂₇⋊C₈	C₂³	1	6	yes	—
22	#41×C₁₂₇	C₂×C₄	1	8	yes	—
23	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₁₂₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	6	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
27	C₄⋊C₄×C₁₂₇	C₂×C₄	1	6	yes	—
28	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	6	yes	—
29	C₄²×C₁₂₇	C₂×C₄	1	6	yes	—
30	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₄²×C₁₂₇	C₂³	1	3	yes	—
32	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
33	C₁₂₇⋊C₈	C₂²×C₄	1	12	yes	—
34	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
35	Q₈×C₁₂₇	C₄²	1	4	yes	—
36	D₄×C₁₂₇	C₄²	1	12	yes	—
37	C₂×C₄×C₁₂₇	—	4	6	no	—
38	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
43	C₂²×C₁₂₇	—	2	6	no	—
44	C₂²×C₁₂₇	C₂×C₄²	1	6	yes	—
45	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	2	no	—
47	C₄×C₁₂₇	—	2	6	no	—
48	C₄×C₁₂₇	C₂×C₄²	1	6	yes	—
49	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
50	C₂×C₁₂₇	—	2	6	no	—
51	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
52	C₂×C₁₂₇	#568	1	2	yes	—
53	#2875	—	127	1	no	maximal, Sylow
54	C₁₂₇	#2875	1	1	yes	characteristic, complemented, Sylow
55	C₂×#131	—	127	2	no	—
56	C₄×#41	D₁₂₇	1	1	yes	characteristic
57	#405	—	127	6	no	—
58	C₄×M₁₆	—	127	3	no	—
59	C₂×#97	—	127	3	no	—
60	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
61	#131	—	127	16	no	—
62	C₂×M₁₆	—	127	6	no	—
63	C₂²×C₈	—	127	6	no	—
64	C₄×Q₈	Dic₁₂₇	1	2	yes	—
65	C₄×D₄	Dic₁₂₇	1	6	yes	—
66	#117	C₂×D₁₂₇	1	3	yes	—
67	C₂×C₄²	C₂×D₁₂₇	1	3	yes	—
68	#97	—	127	1	no	—
69	#97	—	127	3	no	—
70	#97	—	127	6	no	—
71	C₄×C₈	—	127	6	no	—
72	#41	C₄×D₁₂₇	1	8	yes	—
73	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
74	C₂×D₄	C₂×Dic₁₂₇	1	3	yes	—
75	C₂²×C₄	C₂×Dic₁₂₇	1	6	yes	—
76	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
77	M₁₆	—	127	24	no	—
78	C₂×C₈	—	127	24	no	—
79	C₂×C₈	—	127	6	no	—
80	C₂×C₈	—	127	2	no	—
81	C₄⋊C₄	C₂×Dic₁₂₇	1	6	yes	—
82	C₂²⋊C₄	C₂×Dic₁₂₇	1	6	yes	—
83	C₄²	C₂×Dic₁₂₇	1	6	yes	—
84	C₄²	C₂²×D₁₂₇	1	3	yes	—
85	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₂³	C₂²×Dic₁₂₇	1	3	yes	—
87	Q₈	C₄×Dic₁₂₇	1	4	yes	—
88	D₄	C₄×Dic₁₂₇	1	12	yes	—
89	C₂×C₄	—	4	6	no	—
90	C₂×C₄	C₂×C₄×D₁₂₇	1	12	yes	—
91	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
93	C₂×C₄	C₂²×Dic₁₂₇	1	3	yes	—
94	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
95	C₈	—	127	4	no	—
96	C₈	—	127	12	no	—
97	C₂²	—	2	6	no	—
98	C₂²	C₂×C₄×Dic₁₂₇	1	6	yes	—
99	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	6	no	—
102	C₄	C₂×C₄×Dic₁₂₇	1	6	yes	—
103	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
104	C₂	—	2	6	no	—
105	C₂	C₂²×C₄×Dic₁₂₇	1	1	yes	characteristic
106	C₂	#22550643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861216

C₂

yes

maximal

#22549832

C₂

yes

maximal, complemented

C₄×#14523232

C₂

yes

maximal, complemented

C₂×#16861120

C₂

yes

maximal, complemented

C₄×#41×C₁₂₇

C₂

yes

maximal, characteristic

#16861216

C₄

yes

complemented

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861120

C₂²

yes

characteristic

#16861120

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂×#41×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

C₄

yes

—

C₄×D₄×C₁₂₇

C₄

yes

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

#14523232

C₂×C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

#41×C₁₂₇

C₂×C₄

yes

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

C₂²×C₄

yes

—

C₁₂₇⋊C₈

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

—

Q₈×C₁₂₇

C₄²

yes

—

D₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄²

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄²

yes

—

C₄×C₁₂₇

C₂³×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×C₄²

yes

characteristic

C₂×C₁₂₇

#568

yes

—

#2875

—

127

maximal, Sylow

C₁₂₇

#2875

yes

characteristic, complemented, Sylow

C₂×#131

—

127

—

C₄×#41

D₁₂₇

yes

characteristic

#405

—

127

—

C₄×M₁₆

—

127

—

C₂×#97

—

127

—

C₂×#41

C₂×D₁₂₇

yes

characteristic

#131

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₄×Q₈

Dic₁₂₇

yes

—

C₄×D₄

Dic₁₂₇

yes

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

—

#97

—

127

—

#97

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

#41

C₄×D₁₂₇

yes

—

C₂×Q₈

C₂×Dic₁₂₇

yes

characteristic

C₂×D₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

C₂²×Dic₁₂₇

yes

—

Q₈

C₄×Dic₁₂₇

yes

—

D₄

C₄×Dic₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

127

—

C₂²

—

C₂²

C₂×C₄×Dic₁₂₇

yes

—

C₂²

C₂³×Dic₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

C₂×C₄×Dic₁₂₇

yes

—

103

C₄

C₂²×C₄×D₁₂₇

yes

—

104

C₂

—

105

C₂

C₂²×C₄×Dic₁₂₇

yes

characteristic

106

C₂

#22550643

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Fitting	C₄×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂×#41×C₁₂₇	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#2875
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator

C₂×C₁₂₇

C₂²×C₄²

Fitting

C₄×#41×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂³×C₄

prime radical

C₂×#41×C₁₂₇

C₂²

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₄×#41

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
2. #22549832	C₂	semidirect
3. C₄×#14523232	C₂	semidirect
4. C₂×#16861120	C₂	semidirect
6. #16861216	C₄	semidirect
18. #14523232	C₂×C₄	semidirect
19. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
54. C₁₂₇	#2875	semidirect

H ≅ G/N

type

2. #22549832

C₂

semidirect

3. C₄×#14523232

C₂

semidirect

4. C₂×#16861120

C₂

semidirect

6. #16861216

C₄

semidirect

18. #14523232

C₂×C₄

semidirect

19. C₂×C₁₂₇⋊C₈

C₂×C₄

semidirect

54. C₁₂₇

#2875

semidirect

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

subgroup	index
C₂×#16861216	2
#22549832	2
C₄×#14523232	2
C₂×#16861120	2
C₄×#41×C₁₂₇	2
#2875	127

subgroup

index

C₄×#41×C₁₂₇

127

Subgroups — #30647158

Conjugacy classes of subgroups

Complements

Maximal subgroups