Group #30770548 — subgroups

Subgroups — #30770548

Order 17152.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22562464	C₂	1	4	yes	maximal, complemented
2	#22559827	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1712	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523987	—	2	8	no	—
6	#16863801	—	2	2	no	—
7	#16863776	C₂²	1	2	yes	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#493×C₆₇	—	2	2	no	—
10	#493×C₆₇	C₂²	1	2	yes	—
11	C₈²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₆₈	—	2	4	no	—
13	#14524043	—	2	2	no	—
14	#14523987	—	4	16	no	—
15	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
16	#14523977	—	4	4	no	—
17	C₂×D₈×C₆₇	—	2	4	no	—
18	C₂×D₈×C₆₇	—	4	4	no	—
19	#127×C₆₇	—	2	2	no	—
20	C₄×C₈×C₆₇	D₄	1	2	yes	—
21	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₃₄	—	4	2	no	—
23	Dic₂₆₈	—	4	8	no	—
24	C₄⋊Dic₆₇	—	4	4	no	—
25	C₆₇⋊C₁₆	D₈	1	8	yes	complemented
26	C₂×D₄×C₆₇	—	4	2	no	—
27	C₂×D₄×C₆₇	—	4	4	no	—
28	D₈×C₆₇	—	4	8	no	—
29	D₈×C₆₇	—	8	8	no	—
30	C₂×C₈×C₆₇	D₈	1	4	yes	—
31	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
32	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂³×C₆₇	—	8	2	no	—
34	D₄×C₆₇	—	8	4	no	—
35	D₄×C₆₇	—	8	8	no	—
36	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
38	C₂×Dic₆₇	—	8	2	no	—
39	Dic₁₃₄	—	8	4	no	—
40	C₈×C₆₇	SD₃₂	1	8	yes	complemented
41	C₈×C₆₇	C₂×D₈	1	4	yes	—
42	C₂²×C₆₇	—	16	2	no	—
43	C₂²×C₆₇	—	16	4	no	—
44	C₂²×C₆₇	#493	1	1	yes	characteristic
45	C₄×C₆₇	#493	1	2	yes	—
46	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
47	Dic₆₇	—	16	2	no	—
48	#12517	—	67	1	no	maximal, Sylow
49	C₂×C₆₇	—	32	2	no	—
50	C₂×C₆₇	#1712	1	1	yes	characteristic
51	C₂×C₆₇	#2248	1	2	yes	—
52	#2248	—	67	4	no	—
53	#1713	—	67	1	no	—
54	#1712	D₆₇	1	1	yes	characteristic
55	C₈×C₁₆	—	67	1	no	—
56	C₆₇	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	134	8	no	—
58	#500	—	134	2	no	—
59	#494	—	67	2	no	—
60	#493	—	2	2	no	—
61	#493	C₂×D₆₇	1	2	yes	—
62	C₄×C₁₆	—	67	2	no	—
63	C₈²	C₂×D₆₇	1	1	yes	characteristic
64	C₂×Q₁₆	—	134	4	no	—
65	C₂×D₈	—	2	4	no	—
66	C₂×D₈	—	4	4	no	—
67	C₄⋊Q₈	—	134	2	no	—
68	#127	—	2	2	no	—
69	SD₃₂	—	268	16	no	—
70	C₂×C₁₆	—	67	4	no	—
71	#107	—	268	4	no	—
72	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
73	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
74	C₂×Q₈	—	268	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	—	4	4	no	—
77	Q₁₆	—	268	8	no	—
78	D₈	—	4	8	no	—
79	D₈	—	8	8	no	—
80	C₂×C₈	#11761613	1	4	yes	—
81	C₂×C₈	D₄×D₆₇	1	2	yes	—
82	C₄⋊C₄	—	268	4	no	—
83	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
84	C₁₆	—	67	8	no	—
85	C₂³	—	8	2	no	—
86	Q₈	—	536	4	no	—
87	D₄	—	8	4	no	—
88	D₄	—	8	8	no	—
89	C₂×C₄	—	536	2	no	—
90	C₂×C₄	#14524039	1	1	yes	characteristic
91	C₂×C₄	C₂×#11761613	1	2	yes	—
92	C₈	#14523987	1	8	yes	complemented
93	C₈	D₈×D₆₇	1	4	yes	—
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#16863772	1	1	yes	characteristic
97	C₄	—	1072	2	no	—
98	C₄	#16863771	1	2	yes	—
99	C₄	C₂×#14523987	1	4	yes	—
100	C₂	—	32	2	no	—
101	C₂	#22559823	1	1	yes	characteristic
102	C₂	#22562464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562464

C₂

yes

maximal, complemented

#22559827

C₂

yes

maximal, characteristic, complemented

C₈×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₆₇×#1712

C₂

yes

maximal, characteristic

C₂×#14523987

—

#16863801

—

#16863776

C₂²

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#493×C₆₇

—

#493×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

#14524043

—

#14523987

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

#127×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

D₈

yes

complemented

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

SD₃₂

yes

complemented

C₈×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

#493

yes

—

C₄×C₆₇

C₂×SD₃₂

yes

—

Dic₆₇

—

#12517

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1712

yes

characteristic

C₂×C₆₇

#2248

yes

—

#2248

—

#1713

—

#1712

D₆₇

yes

characteristic

C₈×C₁₆

—

C₆₇

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

134

—

#500

—

134

—

#494

—

#493

—

#493

C₂×D₆₇

yes

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

134

—

#127

—

SD₃₂

—

268

—

C₂×C₁₆

—

#107

—

268

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

268

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

268

—

D₈

—

D₈

—

C₂×C₈

#11761613

yes

—

C₂×C₈

D₄×D₆₇

yes

—

C₄⋊C₄

—

268

—

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

536

—

D₄

—

D₄

—

C₂×C₄

—

536

—

C₂×C₄

#14524039

yes

characteristic

C₂×C₄

C₂×#11761613

yes

—

C₈

#14523987

yes

complemented

C₈

D₈×D₆₇

yes

—

C₂²

—

C₂²

—

C₂²

#16863772

yes

characteristic

C₄

—

1072

—

C₄

#16863771

yes

—

C₄

C₂×#14523987

yes

—

100

C₂

—

101

C₂

#22559823

yes

characteristic

102

C₂

#22562464

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#1712	C₂
socle	C₂²×C₆₇	#493
hypercenter	#1712	D₆₇
hypocenter	C₆₇	#12517
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting, prime radical

C₆₇×#1712

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22562464	C₂	semidirect
2. #22559827	C₂	semidirect
3. C₈×C₆₇⋊C₁₆	C₂	semidirect
25. C₆₇⋊C₁₆	D₈	semidirect
40. C₈×C₆₇	SD₃₂	semidirect
56. C₆₇	#12517	semidirect
92. C₈	#14523987	semidirect

H ≅ G/N

type

1. #22562464

C₂

semidirect

2. #22559827

C₂

semidirect

3. C₈×C₆₇⋊C₁₆

C₂

semidirect

25. C₆₇⋊C₁₆

D₈

semidirect

40. C₈×C₆₇

SD₃₂

semidirect

56. C₆₇

#12517

semidirect

92. C₈

#14523987

semidirect

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

subgroup	index
#22562464	2
#22559827	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1712	2
#12517	67

subgroup

index

#22562464

#22559827

C₈×C₆₇⋊C₁₆

C₆₇×#1712

#12517

Subgroups — #30770548

Conjugacy classes of subgroups

Complements

Maximal subgroups