Group #30770524 — subgroups

Subgroups — #30770524

Order 17152.

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

normal subgroups only

This group has 260 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	#22562436	C₂	1	2	yes	maximal, complemented
2	#22562432	C₂	1	2	yes	maximal
3	#22559819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863747	—	2	2	no	—
8	#16863200	—	2	4	no	—
9	#16863197	—	2	4	no	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₆₇	C₂²	1	1	yes	characteristic
12	#493×C₆₇	C₂²	1	1	yes	characteristic
13	#486×C₆₇	—	2	2	no	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	#14524043	—	2	2	no	—
16	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
17	#14523977	—	2	2	no	—
18	#14523977	—	2	2	no	—
19	#14523968	—	4	4	no	—
20	C₂×Q₁₆×C₆₇	—	2	2	no	—
21	C₂×D₈×C₆₇	—	2	2	no	—
22	C₄⋊Q₈×C₆₇	—	2	1	no	—
23	#127×C₆₇	—	2	1	no	—
24	#102×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	—	2	4	no	—
31	C₂×Q₈×C₆₇	—	4	1	no	—
32	C₂×D₄×C₆₇	—	4	1	no	—
33	C₂×D₄×C₆₇	—	4	2	no	—
34	Q₁₆×C₆₇	—	4	4	no	—
35	D₈×C₆₇	—	4	4	no	—
36	C₂×C₈×C₆₇	D₈	1	4	yes	—
37	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₆₇	—	4	2	no	—
39	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂³×C₆₇	—	8	1	no	—
41	Q₈×C₆₇	—	8	2	no	—
42	D₄×C₆₇	—	8	2	no	—
43	D₄×C₆₇	—	8	4	no	—
44	C₂×C₄×C₆₇	—	8	1	no	—
45	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
47	C₂×Dic₆₇	—	8	2	no	—
48	Dic₁₃₄	—	8	4	no	—
49	C₈×C₆₇	—	2	4	no	—
50	C₈×C₆₇	#135	1	4	yes	—
51	C₂²×C₆₇	—	16	1	no	—
52	C₂²×C₆₇	—	16	2	no	—
53	C₂²×C₆₇	#493	1	1	yes	characteristic
54	C₄×C₆₇	—	16	1	no	—
55	C₄×C₆₇	#495	1	2	yes	—
56	C₄×C₆₇	#508	1	4	yes	—
57	Dic₆₇	—	16	2	no	—
58	#12511	—	67	1	no	maximal, Sylow
59	C₂×C₆₇	—	32	1	no	—
60	C₂×C₆₇	#1709	1	1	yes	characteristic
61	C₂×C₆₇	#2249	1	2	yes	—
62	#2241	—	67	2	no	—
63	#2240	—	67	2	no	—
64	#1710	—	67	1	no	—
65	#1709	D₆₇	1	1	yes	characteristic
66	C₈×C₁₆	—	67	1	no	—
67	C₆₇	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	—	67	2	no	—
69	#494	C₂×D₆₇	1	1	yes	characteristic
70	#493	C₂×D₆₇	1	1	yes	characteristic
71	#487	—	134	2	no	—
72	#486	—	2	2	no	—
73	#358	—	134	4	no	—
74	#357	—	134	4	no	—
75	C₄×C₁₆	—	67	2	no	—
76	C₈²	C₂×D₆₇	1	1	yes	characteristic
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	134	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	67	4	no	—
83	#107	—	134	2	no	—
84	#107	—	134	2	no	—
85	#103	—	268	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
88	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	268	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	#11761613	1	4	yes	—
96	C₂×C₈	D₄×D₆₇	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	268	2	no	—
99	C₄⋊C₄	—	268	2	no	—
100	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
101	C₁₆	—	134	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	536	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	536	2	no	—
109	C₂×C₄	#14524039	1	1	yes	characteristic
110	C₂×C₄	C₂×#11761613	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14524062	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#16863772	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	1072	2	no	—
118	C₄	#16863779	1	2	yes	—
119	C₄	#16863826	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#22559814	1	1	yes	characteristic
122	C₂	#22562468	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562436

C₂

yes

maximal, complemented

#22562432

C₂

yes

maximal

#22559819

C₂

yes

maximal, characteristic, complemented

C₈×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₆₇×#1709

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

#16863747

—

#16863200

—

#16863197

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#494×C₆₇

C₂²

yes

characteristic

#493×C₆₇

C₂²

yes

characteristic

#486×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523977

—

#14523968

—

C₂×Q₁₆×C₆₇

—

C₂×D₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

#127×C₆₇

—

#102×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

Q₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#135

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#495

yes

—

C₄×C₆₇

#508

yes

—

Dic₆₇

—

#12511

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1709

yes

characteristic

C₂×C₆₇

#2249

yes

—

—

—

—

D₆₇

yes

characteristic

C₈×C₁₆

—

C₆₇

#12511

yes

characteristic, complemented, Sylow

#500

—

#494

C₂×D₆₇

yes

characteristic

#493

C₂×D₆₇

yes

characteristic

#487

—

134

—

#486

—

#358

—

134

—

#357

—

134

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

#127

—

C₂×C₁₆

—

#107

—

134

—

#107

—

134

—

#103

—

268

—

#102

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#11761613

yes

—

C₂×C₈

D₄×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

100

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

101

C₁₆

—

134

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

536

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

536

—

109

C₂×C₄

#14524039

yes

characteristic

110

C₂×C₄

C₂×#11761613

yes

—

111

C₈

—

112

C₈

#14524062

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16863772

yes

characteristic

116

C₄

—

117

C₄

—

1072

—

118

C₄

#16863779

yes

—

119

C₄

#16863826

yes

—

120

C₂

—

121

C₂

#22559814

yes

characteristic

122

C₂

#22562468

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1709	C₂
socle	C₂²×C₆₇	#493
prime radical	#493×C₆₇	C₂²
hypercenter	#1709	D₆₇
hypocenter	C₆₇	#12511
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1709

C₂

socle

C₂²×C₆₇

#493

prime radical

#493×C₆₇

C₂²

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #22562436	C₂	semidirect
3. #22559819	C₂	semidirect
4. C₈×C₆₇⋊C₁₆	C₂	semidirect
67. C₆₇	#12511	semidirect

H ≅ G/N

type

1. #22562436

C₂

semidirect

3. #22559819

C₂

semidirect

4. C₈×C₆₇⋊C₁₆

C₂

semidirect

67. C₆₇

#12511

semidirect

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

subgroup	index
#22562436	2
#22562432	2
#22559819	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1709	2
#12511	67

subgroup

index

#22562436

#22562432

#22559819

C₈×C₆₇⋊C₁₆

C₆₇×#1709

#12511

Subgroups — #30770524

Conjugacy classes of subgroups

Complements

Maximal subgroups