Group #30770506 — subgroups

Subgroups — #30770506

Order 17152.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562462	C₂	1	1	yes	maximal, characteristic
3	#22562437	C₂	1	2	yes	maximal
4	#22559806	C₂	1	2	yes	maximal
5	C₆₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	—	2	2	no	—
7	#16863801	C₂²	1	2	yes	—
8	#16863794	—	2	2	no	—
9	#16863776	C₂²	1	2	yes	—
10	#16863747	—	2	2	no	—
11	#16863224	—	2	2	no	—
12	#16863201	—	2	4	no	—
13	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₈²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₆₈	—	2	2	no	—
17	C₂×Dic₂₆₈	—	2	2	no	—
18	#14524043	—	2	2	no	—
19	#14524043	—	2	2	no	—
20	Dic₅₃₆	—	4	4	no	—
21	#14523977	—	2	2	no	—
22	#14523977	—	2	2	no	—
23	#14523975	—	4	4	no	—
24	#14523968	—	4	4	no	—
25	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
27	C₄×C₈×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	2	no	—
30	C₂×Dic₁₃₄	—	4	2	no	—
31	Dic₂₆₈	—	4	4	no	—
32	Dic₂₆₈	—	4	4	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	D₈	1	4	yes	—
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₆₇	—	2	2	no	—
42	C₁₆×C₆₇	SD₁₆	1	4	yes	—
43	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
45	C₂×Dic₆₇	—	8	2	no	—
46	C₂×Dic₆₇	—	8	2	no	—
47	Dic₁₃₄	—	8	2	no	—
48	Dic₁₃₄	—	8	2	no	—
49	Dic₁₃₄	—	8	4	no	—
50	C₈×C₆₇	—	2	4	no	—
51	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
52	C₈×C₆₇	#135	1	2	yes	—
53	C₂²×C₆₇	#493	1	1	yes	characteristic
54	C₄×C₆₇	#492	1	1	yes	characteristic
55	C₄×C₆₇	#495	1	1	yes	characteristic
56	C₄×C₆₇	#510	1	4	yes	—
57	Dic₆₇	—	16	2	no	—
58	Dic₆₇	—	16	2	no	—
59	#12507	—	67	1	no	maximal, Sylow
60	C₂×C₆₇	#1706	1	1	yes	characteristic
61	C₂×C₆₇	#2251	1	2	yes	—
62	#2254	—	67	1	no	—
63	#2247	—	67	1	no	—
64	#2241	—	67	2	no	—
65	#1707	—	67	2	no	—
66	#1311	D₆₇	1	1	yes	characteristic
67	C₆₇	#12507	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₃₂	—	134	2	no	—
69	#500	—	67	2	no	—
70	#498	—	134	2	no	—
71	#494	—	67	2	no	—
72	#487	—	134	2	no	—
73	#367	—	134	2	no	—
74	#358	—	134	4	no	—
75	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
76	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
77	C₈²	C₂×D₆₇	1	1	yes	characteristic
78	C₂×Q₁₆	—	134	2	no	—
79	C₂×Q₁₆	—	134	2	no	—
80	C₄⋊Q₈	—	134	2	no	—
81	C₄⋊Q₈	—	134	2	no	—
82	Q₃₂	—	268	4	no	—
83	C₂×C₁₆	D₂₆₈	1	2	yes	—
84	C₂×C₁₆	D₂₆₈	1	2	yes	—
85	#107	—	134	2	no	—
86	#107	—	134	2	no	—
87	#106	—	268	4	no	—
88	#103	—	268	4	no	—
89	C₄×C₈	D₂₆₈	1	2	yes	—
90	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
91	C₂×Q₈	—	268	2	no	—
92	C₂×Q₈	—	268	2	no	—
93	Q₁₆	—	268	4	no	—
94	Q₁₆	—	268	4	no	—
95	C₂×C₈	D₅₃₆	1	4	yes	—
96	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
97	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
98	C₄⋊C₄	—	268	2	no	—
99	C₄⋊C₄	—	268	2	no	—
100	C₄⋊C₄	—	268	2	no	—
101	C₄⋊C₄	—	268	2	no	—
102	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11761617	1	4	yes	—
105	Q₈	—	536	2	no	—
106	Q₈	—	536	2	no	—
107	Q₈	—	536	4	no	—
108	C₂×C₄	—	536	2	no	—
109	C₂×C₄	—	536	2	no	—
110	C₂×C₄	#14524040	1	1	yes	characteristic
111	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#11761617	1	2	yes	—
114	C₈	#14524065	1	2	yes	—
115	C₂²	#16863773	1	1	yes	characteristic
116	C₄	—	1072	2	no	—
117	C₄	—	1072	2	no	—
118	C₄	#16863769	1	1	yes	characteristic
119	C₄	#16863780	1	1	yes	characteristic
120	C₄	#16863839	1	4	yes	—
121	C₂	#22559801	1	1	yes	characteristic
122	C₂	#22562480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562491

C₂

yes

maximal, characteristic

#22562462

C₂

yes

maximal, characteristic

#22562437

C₂

yes

maximal

#22559806

C₂

yes

maximal

C₆₇×#1311

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

#16863801

C₂²

yes

—

#16863794

—

#16863776

C₂²

yes

—

#16863747

—

#16863224

—

#16863201

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

#14524043

—

Dic₅₃₆

—

—

—

—

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

#135

yes

—

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#12507

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

#500

—

#498

—

134

—

#494

—

#487

—

134

—

#367

—

134

—

#358

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

Q₃₂

—

268

—

C₂×C₁₆

D₂₆₈

yes

—

C₂×C₁₆

D₂₆₈

yes

—

#107

—

134

—

#107

—

134

—

#106

—

268

—

#103

—

268

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

268

—

C₂×Q₈

—

268

—

Q₁₆

—

268

—

Q₁₆

—

268

—

C₂×C₈

D₅₃₆

yes

—

C₂×C₈

C₂×D₂₆₈

yes

characteristic

C₂×C₈

C₂×D₂₆₈

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

100

C₄⋊C₄

—

268

—

101

C₄⋊C₄

—

268

—

102

C₄²

C₂×D₂₆₈

yes

characteristic

103

C₁₆

—

104

C₁₆

#11761617

yes

—

105

Q₈

—

536

—

106

Q₈

—

536

—

107

Q₈

—

536

—

108

C₂×C₄

—

536

—

109

C₂×C₄

—

536

—

110

C₂×C₄

#14524040

yes

characteristic

111

C₂×C₄

C₂×D₅₃₆

yes

—

112

C₈

—

113

C₈

C₂×#11761617

yes

—

114

C₈

#14524065

yes

—

115

C₂²

#16863773

yes

characteristic

116

C₄

—

1072

—

117

C₄

—

1072

—

118

C₄

#16863769

yes

characteristic

119

C₄

#16863780

yes

characteristic

120

C₄

#16863839

yes

—

121

C₂

#22559801

yes

characteristic

122

C₂

#22562480

yes

—

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1311	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1311	D₆₇
hypocenter	C₆₇	#12507
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1311

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
67. C₆₇	#12507	semidirect

H ≅ G/N

type

67. C₆₇

#12507

semidirect

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

subgroup	index
#22562491	2
#22562462	2
#22562437	2
#22559806	2
C₆₇×#1311	2
#12507	67

subgroup

index

C₆₇×#1311

Subgroups — #30770506

Conjugacy classes of subgroups

Complements

Maximal subgroups