Group #30770571 — subgroups

Subgroups — #30770571

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	#22562491	C₂	1	1	yes	maximal, characteristic
2	#22562482	C₂	1	2	yes	maximal
3	#22559818	C₂	1	2	yes	maximal
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863748	—	2	2	no	—
8	#16863746	—	2	2	no	—
9	#16863224	—	2	2	no	—
10	#16863221	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#367×C₆₇	—	2	2	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#14524043	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
18	#14523977	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523967	—	4	4	no	—
21	#14523966	—	4	4	no	—
22	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊Q₈×C₆₇	—	2	2	no	—
24	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
25	#107×C₆₇	—	2	2	no	—
26	#107×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₃₄	—	4	2	no	—
29	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
30	C₄⋊Dic₆₇	—	4	2	no	—
31	C₄⋊Dic₆₇	—	4	2	no	—
32	C₆₇⋊C₁₆	Q₁₆	1	4	yes	—
33	C₂×Q₈×C₆₇	—	4	2	no	—
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	C₁₆×C₆₇	Q₁₆	1	4	yes	—
40	Q₈×C₆₇	—	8	4	no	—
41	C₂×C₄×C₆₇	—	8	2	no	—
42	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
44	C₂×Dic₆₇	—	8	2	no	—
45	Dic₁₃₄	—	8	4	no	—
46	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
47	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
48	C₆₇⋊C₈	—	2	4	no	—
49	C₂²×C₆₇	#493	1	1	yes	characteristic
50	C₄×C₆₇	—	16	2	no	—
51	C₄×C₆₇	#494	1	1	yes	characteristic
52	C₄×C₆₇	#494	1	1	yes	characteristic
53	C₄×C₆₇	#508	1	4	yes	—
54	Dic₆₇	—	16	2	no	—
55	#12522	—	67	1	no	maximal, Sylow
56	C₂×C₆₇	#1713	1	1	yes	characteristic
57	C₂×C₆₇	#2249	1	2	yes	—
58	#2254	—	67	1	no	—
59	#2254	D₆₇	1	1	yes	characteristic
60	#2252	—	67	2	no	—
61	#1710	—	67	2	no	—
62	C₈×C₁₆	—	67	1	no	—
63	C₆₇	#12522	1	1	yes	characteristic, complemented, Sylow
64	#500	—	67	2	no	—
65	#500	C₂×D₆₇	1	2	yes	—
66	#487	—	134	2	no	—
67	#487	—	134	2	no	—
68	#367	—	2	2	no	—
69	#367	—	134	2	no	—
70	#366	—	134	4	no	—
71	C₄×C₁₆	—	67	1	no	—
72	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
73	C₈²	—	67	1	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	134	2	no	—
76	C₂×C₁₆	—	67	2	no	—
77	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	134	2	no	—
81	#107	—	134	2	no	—
82	#103	—	268	4	no	—
83	#103	—	268	4	no	—
84	C₄×C₈	—	67	2	no	—
85	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	268	2	no	—
88	C₂×C₈	—	67	4	no	—
89	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
90	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	268	2	no	—
94	C₄⋊C₄	—	268	2	no	—
95	C₄²	D₄×D₆₇	1	1	yes	characteristic
96	C₁₆	—	67	4	no	—
97	C₁₆	#11761618	1	4	yes	—
98	Q₈	—	8	4	no	—
99	Q₈	—	536	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	536	2	no	—
102	C₂×C₄	#14524039	1	1	yes	characteristic
103	C₂×C₄	D₈×D₆₇	1	2	yes	—
104	C₈	—	134	4	no	—
105	C₈	Q₁₆×D₆₇	1	2	yes	—
106	C₈	C₂×#11761618	1	2	yes	—
107	C₂²	#16863771	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	1072	2	no	—
110	C₄	#16863774	1	1	yes	characteristic
111	C₄	#16863775	1	1	yes	characteristic
112	C₄	#16863827	1	4	yes	—
113	C₂	#22559826	1	1	yes	characteristic
114	C₂	#22562469	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562491

C₂

yes

maximal, characteristic

#22562482

C₂

yes

maximal

#22559818

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2254

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#367×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

Q₁₆

yes

—

C₂×Q₈×C₆₇

—

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

C₁₆×C₆₇

Q₁₆

yes

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2254

—

#2254

D₆₇

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

C₆₇

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₆₇

yes

—

#487

—

134

—

#487

—

134

—

#367

—

#367

—

134

—

#366

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

#107

—

#107

—

134

—

#107

—

134

—

#103

—

268

—

#103

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×C₈

—

C₂×C₈

D₄×D₆₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

D₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

#11761618

yes

—

Q₈

—

Q₈

—

536

—

100

C₂×C₄

—

101

C₂×C₄

—

536

—

102

C₂×C₄

#14524039

yes

characteristic

103

C₂×C₄

D₈×D₆₇

yes

—

104

C₈

—

134

—

105

C₈

Q₁₆×D₆₇

yes

—

106

C₈

C₂×#11761618

yes

—

107

C₂²

#16863771

yes

characteristic

108

C₄

—

109

C₄

—

1072

—

110

C₄

#16863774

yes

characteristic

111

C₄

#16863775

yes

characteristic

112

C₄

#16863827

yes

—

113

C₂

#22559826

yes

characteristic

114

C₂

#22562469

yes

—

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2254	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#2254	D₆₇
hypocenter	C₆₇	#12522
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2254

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

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

H ≅ G/N

type

63. C₆₇

#12522

semidirect

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

subgroup	index
#22562491	2
#22562482	2
#22559818	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2254	2
#12522	67

subgroup

index

#22562491

#22562482

#22559818

C₁₆×C₆₇⋊C₈

C₆₇×#2254

#12522

Subgroups — #30770571

Conjugacy classes of subgroups

Complements

Maximal subgroups