Group #30749489 — subgroups

Subgroups — #30749489

Order 17152.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22558957	C₂	1	2	yes	maximal
2	#22558054	C₂	1	2	yes	maximal
3	#22558038	C₂	1	2	yes	maximal
4	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	4	yes	—
6	#16863149	—	2	2	no	—
7	#16863147	—	2	2	no	—
8	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	#376×C₆₇	C₂²	1	1	yes	characteristic
11	#336×C₆₇	C₄	1	4	yes	—
12	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
14	C₄×C₆₇⋊C₈	—	2	4	no	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
19	#97×C₆₇	—	2	2	no	—
20	C₄×C₈×C₆₇	D₄	1	2	yes	—
21	C₄×C₈×C₆₇	Q₈	1	2	yes	—
22	#95×C₆₇	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	4	4	no	—
24	C₂²×C₄×C₆₇	—	2	2	no	—
25	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	#41	1	2	yes	—
27	C₂×C₈×C₆₇	—	2	4	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	4	no	—
30	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
31	C₄²×C₆₇	—	2	4	no	—
32	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₆₇	#41	1	2	yes	—
35	C₂³×C₆₇	#117	1	1	yes	characteristic
36	C₂×C₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	—	4	4	no	—
38	C₂×C₄×C₆₇	—	4	2	no	—
39	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#124	1	2	yes	—
42	C₂×C₄×C₆₇	#125	1	2	yes	—
43	C₂×C₄×C₆₇	#131	1	4	yes	—
44	C₈×C₆₇	—	2	4	no	—
45	C₈×C₆₇	—	4	2	no	—
46	C₆₇⋊C₈	—	8	4	no	—
47	C₂²×C₆₇	—	2	2	no	—
48	C₂²×C₆₇	#382	1	1	yes	characteristic
49	C₂²×C₆₇	#433	1	2	yes	—
50	C₄×C₆₇	—	2	2	no	—
51	C₄×C₆₇	—	2	2	no	—
52	C₄×C₆₇	—	8	2	no	—
53	C₄×C₆₇	#424	1	2	yes	—
54	C₄×C₆₇	#433	1	2	yes	—
55	#8783	—	67	1	no	maximal, Sylow
56	C₂×C₆₇	—	2	2	no	—
57	C₂×C₆₇	#1832	1	1	yes	characteristic
58	C₂×C₆₇	#1833	1	2	yes	—
59	#1746	D₆₇	1	1	yes	characteristic
60	#1564	—	67	2	no	—
61	#1281	—	67	2	no	—
62	#1274	—	67	2	no	—
63	C₆₇	#8783	1	1	yes	characteristic, complemented, Sylow
64	#431	—	67	4	no	—
65	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
67	#376	C₂×D₆₇	1	1	yes	characteristic
68	#336	Dic₆₇	1	4	yes	—
69	#335	—	134	2	no	—
70	#334	—	134	2	no	—
71	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
72	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
73	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
74	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
75	C₄⋊C₈	—	134	4	no	—
76	C₂²⋊C₈	—	134	4	no	—
77	#97	—	2	2	no	—
78	C₄×C₈	—	134	4	no	—
79	C₄×C₈	Dic₁₃₄	1	2	yes	—
80	C₄×C₈	D₂₆₈	1	2	yes	—
81	#95	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
84	C₂²×C₄	#11761628	1	2	yes	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	268	4	no	—
89	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
90	C₄²	—	2	4	no	—
91	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
92	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
93	C₄²	#11761628	1	2	yes	—
94	C₂³	#14524002	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
99	C₂×C₄	#14524002	1	1	yes	characteristic
100	C₂×C₄	#14524031	1	2	yes	—
101	C₂×C₄	#14524035	1	2	yes	—
102	C₂×C₄	#14524050	1	4	yes	—
103	C₈	—	2	4	no	—
104	C₈	—	4	2	no	—
105	C₈	—	536	4	no	—
106	C₂²	—	2	2	no	—
107	C₂²	#16863262	1	1	yes	characteristic
108	C₂²	#16863457	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	8	2	no	—
112	C₄	#16863423	1	2	yes	—
113	C₄	#16863457	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	#22560329	1	1	yes	characteristic
116	C₂	#22560333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558957

C₂

yes

maximal

#22558054

C₂

yes

maximal

#22558038

C₂

yes

maximal

C₆₇×#1746

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863149

—

#16863147

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#97×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#125

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#382

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

#433

yes

—

#8783

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1832

yes

characteristic

C₂×C₆₇

#1833

yes

—

#1746

D₆₇

yes

—

—

—

C₆₇

#8783

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

C₂×D₆₇

yes

characteristic

#336

Dic₆₇

yes

—

#335

—

134

—

#334

—

134

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

134

—

#97

—

C₄×C₈

—

134

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

#11761628

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₄²

—

C₄²

C₂×D₂₆₈

yes

characteristic

C₄²

C₂×Dic₁₃₄

yes

characteristic

C₄²

#11761628

yes

—

C₂³

#14524002

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

#14524002

yes

characteristic

100

C₂×C₄

#14524031

yes

—

101

C₂×C₄

#14524035

yes

—

102

C₂×C₄

#14524050

yes

—

103

C₈

—

104

C₈

—

105

C₈

—

536

—

106

C₂²

—

107

C₂²

#16863262

yes

characteristic

108

C₂²

#16863457

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#16863423

yes

—

113

C₄

#16863457

yes

—

114

C₂

—

115

C₂

#22560329

yes

characteristic

116

C₂

#22560333

yes

—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1746	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1746	D₆₇
hypocenter	C₆₇	#8783
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂×C₄

#14524002

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1746

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

N	H ≅ G/N	type
63. C₆₇	#8783	semidirect

H ≅ G/N

type

63. C₆₇

#8783

semidirect

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

subgroup	index
#22558957	2
#22558054	2
#22558038	2
C₆₇×#1746	2
#8783	67

subgroup

index

C₆₇×#1746

Subgroups — #30749489

Conjugacy classes of subgroups

Complements

Maximal subgroups