Group #30749605 — subgroups

Subgroups — #30749605

Order 17152.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#22560366	C₂	1	2	yes	maximal
2	#22560337	C₂	1	4	yes	maximal
3	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523975	—	2	4	no	—
5	#16863266	C₂²	1	4	yes	—
6	#16863159	—	2	8	no	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₈²×C₆₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₆₇	—	2	8	no	—
11	C₂×C₄⋊Dic₆₇	—	2	4	no	—
12	#14523975	—	4	8	no	—
13	C₂²×C₈×C₆₇	D₄	1	4	yes	—
14	C₂²×C₈×C₆₇	D₄	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₇	Q₈	1	4	yes	—
19	C₄×C₈×C₆₇	Q₈	1	2	yes	—
20	C₂²×Dic₆₇	—	4	4	no	—
21	C₄⋊Dic₆₇	—	4	8	no	—
22	C₄⋊Dic₆₇	—	4	16	no	—
23	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
25	C₂×C₈×C₆₇	—	2	8	no	—
26	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
27	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₆₇	SD₁₆	1	8	yes	—
29	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
32	C₂³×C₆₇	#127	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₆₇	#135	1	8	yes	—
39	C₂×Dic₆₇	—	8	4	no	—
40	C₂×Dic₆₇	—	8	8	no	—
41	C₈×C₆₇	—	2	8	no	—
42	C₈×C₆₇	#106	1	8	yes	—
43	C₂²×C₆₇	#384	1	1	yes	characteristic
44	C₂²×C₆₇	#492	1	2	yes	—
45	C₂²×C₆₇	#495	1	4	yes	—
46	C₄×C₆₇	C₂×#106	1	2	yes	—
47	C₄×C₆₇	#427	1	4	yes	—
48	C₄×C₆₇	#498	1	2	yes	—
49	C₄×C₆₇	#499	1	4	yes	—
50	Dic₆₇	—	16	4	no	—
51	#8815	—	67	1	no	maximal, Sylow
52	C₂×C₆₇	#1708	1	4	yes	—
53	C₂×C₆₇	#1844	1	1	yes	characteristic
54	C₂×C₆₇	#1846	1	2	yes	—
55	#1844	—	67	2	no	—
56	#1834	—	67	4	no	—
57	C₂×C₈²	D₆₇	1	1	yes	characteristic
58	C₆₇	#8815	1	1	yes	characteristic, complemented, Sylow
59	C₂×#106	—	134	4	no	—
60	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
61	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
62	#384	—	67	4	no	—
63	#340	—	134	8	no	—
64	C₈²	Dic₆₇	1	4	yes	complemented
65	C₂²×C₈	D₂₆₈	1	4	yes	—
66	C₂²×C₈	D₂₆₈	1	2	yes	—
67	C₂×C₄⋊C₄	—	134	8	no	—
68	C₂×C₄⋊C₄	—	134	4	no	—
69	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
70	#106	—	268	8	no	—
71	C₄×C₈	Dic₁₃₄	1	4	yes	—
72	C₄×C₈	Dic₁₃₄	1	2	yes	—
73	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
74	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
75	C₂²×C₄	—	268	4	no	—
76	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
81	C₂×C₈	#11761617	1	8	yes	—
82	C₄⋊C₄	—	268	8	no	—
83	C₄⋊C₄	—	268	16	no	—
84	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
86	C₄²	C₂×Dic₁₃₄	1	2	yes	—
87	C₂³	#14524040	1	1	yes	characteristic
88	C₂×C₄	—	536	4	no	—
89	C₂×C₄	—	536	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
92	C₂×C₄	#14524043	1	1	yes	characteristic
93	C₂×C₄	#14524043	1	2	yes	—
94	C₂×C₄	C₂×#11761617	1	4	yes	—
95	C₂×C₄	#14524065	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#14523975	1	8	yes	—
98	C₂²	#16863266	1	1	yes	characteristic
99	C₂²	#16863769	1	2	yes	—
100	C₂²	#16863780	1	4	yes	—
101	C₄	—	1072	4	no	—
102	C₄	C₂×#14523975	1	2	yes	—
103	C₄	#16863434	1	4	yes	—
104	C₄	#16863794	1	2	yes	—
105	C₄	#16863798	1	4	yes	—
106	C₂	#22559810	1	4	yes	—
107	C₂	#22560366	1	1	yes	characteristic
108	C₂	#22560373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560366

C₂

yes

maximal

#22560337

C₂

yes

maximal

C₂×C₈²×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523975

—

#16863266

C₂²

yes

—

#16863159

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₈²×C₆₇

C₄

yes

complemented

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523975

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#106

yes

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#492

yes

—

C₂²×C₆₇

#495

yes

—

C₄×C₆₇

C₂×#106

yes

—

C₄×C₆₇

#427

yes

—

C₄×C₆₇

#498

yes

—

C₄×C₆₇

#499

yes

—

Dic₆₇

—

#8815

—

maximal, Sylow

C₂×C₆₇

#1708

yes

—

C₂×C₆₇

#1844

yes

characteristic

C₂×C₆₇

#1846

yes

—

#1844

—

#1834

—

C₂×C₈²

D₆₇

yes

characteristic

C₆₇

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

—

#340

—

134

—

C₈²

Dic₆₇

yes

complemented

C₂²×C₈

D₂₆₈

yes

—

C₂²×C₈

D₂₆₈

yes

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#106

—

268

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

C₂²×C₄

C₂×D₂₆₈

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

#11761617

yes

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

C₂²×Dic₆₇

yes

characteristic

C₄²

C₂×Dic₁₃₄

yes

characteristic

C₄²

C₂×Dic₁₃₄

yes

—

C₂³

#14524040

yes

characteristic

C₂×C₄

—

536

—

C₂×C₄

—

536

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

C₂×C₄

#14524043

yes

characteristic

C₂×C₄

#14524043

yes

—

C₂×C₄

C₂×#11761617

yes

—

C₂×C₄

#14524065

yes

—

C₈

—

C₈

#14523975

yes

—

C₂²

#16863266

yes

characteristic

C₂²

#16863769

yes

—

100

C₂²

#16863780

yes

—

101

C₄

—

1072

—

102

C₄

C₂×#14523975

yes

—

103

C₄

#16863434

yes

—

104

C₄

#16863794

yes

—

105

C₄

#16863798

yes

—

106

C₂

#22559810

yes

—

107

C₂

#22560366

yes

characteristic

108

C₂

#22560373

yes

—

	subgroup	G/H
center	C₂³	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#8815
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14524040

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×C₈²×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

#127

cosocle

C₂×C₄²×C₆₇

C₂³

hypercenter

C₂×C₈²

D₆₇

hypocenter

C₆₇

#8815

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₆₇	C₄	semidirect
58. C₆₇	#8815	semidirect
64. C₈²	Dic₆₇	semidirect

H ≅ G/N

type

9. C₈²×C₆₇

C₄

semidirect

58. C₆₇

#8815

semidirect

64. C₈²

Dic₆₇

semidirect

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

subgroup	index
#22560366	2
#22560337	2
C₂×C₈²×C₆₇	2
#8815	67

subgroup

index

#22560366

#22560337

C₂×C₈²×C₆₇

#8815

Subgroups — #30749605

Conjugacy classes of subgroups

Complements

Maximal subgroups