Group #30770536 — subgroups

Subgroups — #30770536

Order 17152.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22562434	C₂	1	4	yes	maximal, complemented
2	#22559824	C₂	1	1	yes	maximal, characteristic
3	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863773	—	2	2	no	—
6	#16863773	C₂²	1	2	yes	—
7	#16863196	—	2	8	no	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₆₇	C₂²	1	2	yes	—
10	#487×C₆₇	—	2	2	no	—
11	C₈²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×D₅₃₆	—	2	4	no	—
13	C₂×D₅₃₆	—	4	4	no	—
14	#14524040	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
16	C₄⋊Q₈×C₆₇	—	2	2	no	—
17	#107×C₆₇	—	2	4	no	—
18	#103×C₆₇	—	4	4	no	—
19	C₄×C₈×C₆₇	D₄	1	2	yes	—
20	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×D₂₆₈	—	4	2	no	—
22	C₂×D₂₆₈	—	4	4	no	—
23	D₅₃₆	—	4	8	no	—
24	D₅₃₆	—	8	8	no	—
25	C₆₇⋊C₁₆	—	2	4	no	—
26	C₂×Q₈×C₆₇	—	4	2	no	—
27	C₂×C₈×C₆₇	D₈	1	4	yes	—
28	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
29	C₄⋊C₄×C₆₇	—	4	2	no	—
30	C₄⋊C₄×C₆₇	—	4	2	no	—
31	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₆₇	—	8	2	no	—
33	Q₈×C₆₇	—	8	4	no	—
34	C₂×C₄×C₆₇	—	8	2	no	—
35	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
37	D₂₆₈	—	8	4	no	—
38	D₂₆₈	—	8	8	no	—
39	C₈×C₆₇	—	2	4	no	—
40	C₈×C₆₇	#135	1	4	yes	—
41	C₂²×C₆₇	#493	1	1	yes	characteristic
42	C₂×D₆₇	—	16	2	no	—
43	C₂×D₆₇	—	16	4	no	—
44	C₄×C₆₇	—	16	2	no	—
45	C₄×C₆₇	#495	1	2	yes	—
46	C₄×C₆₇	#509	1	4	yes	—
47	#12513	—	67	1	no	maximal, Sylow
48	C₂×C₆₇	#1709	1	1	yes	characteristic
49	C₂×C₆₇	#2250	1	2	yes	—
50	D₆₇	—	32	2	no	—
51	#2240	—	67	4	no	—
52	#1712	—	67	1	no	—
53	#1710	D₆₇	1	1	yes	characteristic, complemented
54	#1311	—	67	1	no	—
55	C₆₇	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	C₂×D₆₇	1	2	yes	—
57	#493	—	67	2	no	—
58	#493	—	134	2	no	—
59	#487	—	2	2	no	—
60	#357	—	134	8	no	—
61	C₄×C₁₆	—	67	2	no	—
62	C₈²	C₂×D₆₇	1	1	yes	characteristic
63	C₂×D₈	—	268	4	no	—
64	C₂×D₈	—	134	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	134	2	no	—
67	C₂×C₁₆	—	67	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
71	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	268	2	no	—
74	C₂×D₄	—	268	4	no	—
75	D₈	—	268	8	no	—
76	D₈	—	536	8	no	—
77	C₂×C₈	#11761613	1	4	yes	—
78	C₂×C₈	D₄×D₆₇	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
82	C₁₆	—	134	4	no	—
83	C₂³	—	536	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	536	4	no	—
86	D₄	—	536	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14524039	1	1	yes	characteristic
89	C₂×C₄	C₂×#11761613	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14524066	1	4	yes	—
92	C₂²	—	1072	2	no	—
93	C₂²	—	1072	4	no	—
94	C₂²	#16863772	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#16863781	1	2	yes	—
97	C₄	#16863831	1	4	yes	—
98	C₂	—	2144	2	no	—
99	C₂	#22559816	1	1	yes	characteristic
100	C₂	#22562476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562434

C₂

yes

maximal, complemented

#22559824

C₂

yes

maximal, characteristic

#22558126

C₂

yes

maximal, characteristic, complemented

C₆₇×#1710

C₂

yes

maximal, characteristic, complemented

#16863773

—

#16863773

C₂²

yes

—

#16863196

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#500×C₆₇

C₂²

yes

—

#487×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₄⋊Q₈×C₆₇

—

#107×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

#135

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#495

yes

—

C₄×C₆₇

#509

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

#2240

—

#1712

—

#1710

D₆₇

yes

characteristic, complemented

#1311

—

C₆₇

#12513

yes

characteristic, complemented, Sylow

#500

C₂×D₆₇

yes

—

#493

—

#493

—

134

—

#487

—

#357

—

134

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×D₈

—

268

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

#127

—

134

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

D₈

—

268

—

D₈

—

536

—

C₂×C₈

#11761613

yes

—

C₂×C₈

D₄×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

—

536

—

Q₈

—

D₄

—

536

—

D₄

—

536

—

C₂×C₄

—

C₂×C₄

#14524039

yes

characteristic

C₂×C₄

C₂×#11761613

yes

—

C₈

—

C₈

#14524066

yes

—

C₂²

—

1072

—

C₂²

—

1072

—

C₂²

#16863772

yes

characteristic

C₄

—

C₄

#16863781

yes

—

C₄

#16863831

yes

—

C₂

—

2144

—

C₂

#22559816

yes

characteristic

100

C₂

#22562476

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1710	C₂
socle	C₂²×C₆₇	#493
prime radical	#22559824	C₂
hypercenter	#1710	D₆₇
hypocenter	C₆₇	#12513
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1710

C₂

C₂²×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #22562434	C₂	semidirect
3. #22558126	C₂	semidirect
4. C₆₇×#1710	C₂	semidirect
53. #1710	D₆₇	semidirect
55. C₆₇	#12513	semidirect

H ≅ G/N

type

1. #22562434

C₂

semidirect

3. #22558126

C₂

semidirect

4. C₆₇×#1710

C₂

semidirect

53. #1710

D₆₇

semidirect

55. C₆₇

#12513

semidirect

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

subgroup	index
#22562434	2
#22559824	2
#22558126	2
C₆₇×#1710	2
#12513	67

subgroup

index

C₆₇×#1710

Subgroups — #30770536

Conjugacy classes of subgroups

Complements

Maximal subgroups