Group #30770569 — subgroups

Subgroups — #30770569

Order 17152.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22562489	C₂	1	2	yes	maximal
2	#22562482	C₂	1	2	yes	maximal
3	#22559819	C₂	1	1	yes	maximal, characteristic
4	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1710	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863747	—	2	2	no	—
8	#16863223	—	2	4	no	—
9	#16863221	—	2	4	no	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#500×C₆₇	C₂²	1	2	yes	—
13	#487×C₆₇	—	2	2	no	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	#14524043	—	2	2	no	—
16	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
17	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
18	#14523977	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523968	—	4	4	no	—
21	C₄⋊Q₈×C₆₇	—	2	2	no	—
22	#107×C₆₇	—	2	2	no	—
23	#107×C₆₇	—	2	2	no	—
24	#103×C₆₇	—	4	4	no	—
25	C₄×C₈×C₆₇	D₄	1	2	yes	—
26	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₃₄	—	4	2	no	—
28	C₄⋊Dic₆₇	—	4	2	no	—
29	C₄⋊Dic₆₇	—	4	2	no	—
30	C₆₇⋊C₁₆	Q₁₆	1	4	yes	—
31	C₆₇⋊C₁₆	Q₁₆	1	4	yes	—
32	C₂×Q₈×C₆₇	—	4	2	no	—
33	C₂×C₈×C₆₇	D₈	1	4	yes	—
34	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	—	4	2	no	—
37	C₄⋊C₄×C₆₇	—	4	2	no	—
38	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₆₇	—	8	4	no	—
40	C₂×C₄×C₆₇	—	8	2	no	—
41	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
43	C₂×Dic₆₇	—	8	2	no	—
44	Dic₁₃₄	—	8	4	no	—
45	C₈×C₆₇	—	2	4	no	—
46	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
47	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
48	C₂²×C₆₇	#493	1	1	yes	characteristic
49	C₄×C₆₇	—	16	2	no	—
50	C₄×C₆₇	#494	1	1	yes	characteristic
51	C₄×C₆₇	#494	1	1	yes	characteristic
52	C₄×C₆₇	#508	1	4	yes	—
53	Dic₆₇	—	16	2	no	—
54	#12522	—	67	1	no	maximal, Sylow
55	C₂×C₆₇	#1713	1	1	yes	characteristic
56	C₂×C₆₇	#2249	1	2	yes	—
57	#2254	—	67	2	no	—
58	#2252	—	67	2	no	—
59	#1710	—	67	1	no	—
60	#1710	D₆₇	1	1	yes	characteristic
61	C₈×C₁₆	—	67	1	no	—
62	C₆₇	#12522	1	1	yes	characteristic, complemented, Sylow
63	#500	—	67	2	no	—
64	#500	C₂×D₆₇	1	2	yes	—
65	#487	—	2	2	no	—
66	#487	—	134	2	no	—
67	#367	—	134	4	no	—
68	#366	—	134	4	no	—
69	C₄×C₁₆	—	67	1	no	—
70	C₄×C₁₆	—	67	1	no	—
71	C₈²	C₂×D₆₇	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	134	2	no	—
74	C₂×C₁₆	—	67	2	no	—
75	C₂×C₁₆	—	67	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	134	2	no	—
79	#107	—	134	2	no	—
80	#103	—	4	4	no	—
81	#103	—	268	4	no	—
82	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
83	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	268	2	no	—
86	C₂×C₈	#11761613	1	4	yes	—
87	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
88	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	268	2	no	—
92	C₄⋊C₄	—	268	2	no	—
93	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
94	C₁₆	—	67	4	no	—
95	C₁₆	—	67	4	no	—
96	Q₈	—	8	4	no	—
97	Q₈	—	536	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	536	2	no	—
100	C₂×C₄	#14524039	1	1	yes	characteristic
101	C₂×C₄	C₂×#11761613	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₆₇	1	2	yes	—
104	C₈	Q₁₆×D₆₇	1	2	yes	—
105	C₂²	#16863772	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	1072	2	no	—
108	C₄	#16863774	1	1	yes	characteristic
109	C₄	#16863774	1	1	yes	characteristic
110	C₄	#16863826	1	4	yes	—
111	C₂	#22559825	1	1	yes	characteristic
112	C₂	#22562468	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562489

C₂

yes

maximal

#22562482

C₂

yes

maximal

#22559819

C₂

yes

maximal, characteristic

C₈×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic

C₆₇×#1710

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

#16863747

—

#16863223

—

#16863221

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#487×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523977

—

#14523968

—

C₄⋊Q₈×C₆₇

—

#107×C₆₇

—

#107×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

Q₁₆

yes

—

C₆₇⋊C₁₆

Q₁₆

yes

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₈×C₁₆

—

C₆₇

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₆₇

yes

—

#487

—

#487

—

134

—

#367

—

134

—

#366

—

134

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

134

—

#107

—

134

—

#103

—

#103

—

268

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×C₈

#11761613

yes

—

C₂×C₈

D₄×D₆₇

yes

characteristic

C₂×C₈

D₄×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

536

—

C₂×C₄

—

C₂×C₄

—

536

—

100

C₂×C₄

#14524039

yes

characteristic

101

C₂×C₄

C₂×#11761613

yes

—

102

C₈

—

103

C₈

Q₁₆×D₆₇

yes

—

104

C₈

Q₁₆×D₆₇

yes

—

105

C₂²

#16863772

yes

characteristic

106

C₄

—

107

C₄

—

1072

—

108

C₄

#16863774

yes

characteristic

109

C₄

#16863774

yes

characteristic

110

C₄

#16863826

yes

—

111

C₂

#22559825

yes

characteristic

112

C₂

#22562468

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1710	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1710	D₆₇
hypocenter	C₆₇	#12522
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

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
62. C₆₇	#12522	semidirect

H ≅ G/N

type

62. C₆₇

#12522

semidirect

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

subgroup	index
#22562489	2
#22562482	2
#22559819	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1710	2
#12522	67

subgroup

index

#22562489

#22562482

#22559819

C₈×C₆₇⋊C₁₆

C₆₇×#1710

#12522

Subgroups — #30770569

Conjugacy classes of subgroups

Complements

Maximal subgroups