Group #30742940 — subgroups

Subgroups — #30742940

Order 17152.

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

normal subgroups only

This group has 200 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	#22558968	C₂	1	2	yes	maximal
2	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
3	#16863420	—	2	2	no	—
4	#16863149	—	2	2	no	—
5	#16863149	C₄	1	4	yes	—
6	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₆₇	—	2	2	no	—
9	C₂×#11761612	—	4	2	no	—
10	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
15	M₃₂×C₆₇	—	4	2	no	—
16	C₂×C₁₆×C₆₇	—	2	2	no	—
17	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₇	D₄	1	2	yes	—
19	#11761612	—	8	4	no	—
20	C₂×C₆₇⋊C₈	—	4	4	no	—
21	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
23	C₂×C₈×C₆₇	—	2	1	no	—
24	C₂×C₈×C₆₇	—	2	1	no	—
25	C₂×C₈×C₆₇	—	4	1	no	—
26	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₆₇	D₈	1	1	yes	characteristic
31	C₄²×C₆₇	Q₁₆	1	1	yes	characteristic
32	C₁₆×C₆₇	—	4	2	no	—
33	C₂³×C₆₇	#106	1	1	yes	characteristic
34	C₂×C₄×C₆₇	—	2	1	no	—
35	C₂×C₄×C₆₇	—	2	1	no	—
36	C₂×C₄×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
39	C₂×C₄×C₆₇	#102	1	2	yes	—
40	C₂×C₄×C₆₇	#103	1	2	yes	—
41	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
42	C₈×C₆₇	—	2	2	no	—
43	C₈×C₆₇	—	2	1	no	—
44	C₈×C₆₇	#100	1	1	yes	characteristic
45	C₈×C₆₇	#101	1	1	yes	characteristic
46	C₆₇⋊C₈	—	8	4	no	—
47	C₂²×C₆₇	—	2	1	no	—
48	C₂²×C₆₇	—	2	2	no	—
49	C₂²×C₆₇	#340	1	1	yes	characteristic
50	C₄×C₆₇	—	2	1	no	—
51	C₄×C₆₇	—	2	2	no	—
52	C₄×C₆₇	#338	1	1	yes	characteristic
53	C₄×C₆₇	#343	1	1	yes	characteristic
54	C₄×C₆₇	#361	1	2	yes	—
55	C₄×C₆₇	#362	1	2	yes	—
56	#7245	—	67	1	no	maximal, Sylow
57	C₂×C₆₇	—	4	1	no	—
58	C₂×C₆₇	#1292	1	1	yes	characteristic
59	C₂×C₆₇	#1387	1	2	yes	—
60	#2150	D₆₇	1	1	yes	characteristic
61	#1567	—	67	2	no	—
62	C₆₇	#7245	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₃₂	Dic₆₇	1	2	yes	—
64	#423	—	134	2	no	—
65	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
66	C₄⋊C₁₆	—	2	2	no	—
67	#335	—	67	4	no	—
68	#335	—	134	2	no	—
69	C₂×M₁₆	—	268	2	no	—
70	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
72	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₄⋊C₈	—	268	2	no	—
76	C₄⋊C₈	—	134	4	no	—
77	C₄×C₈	C₄×D₆₇	1	2	yes	—
78	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
79	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
80	C₂²×C₄	#11761617	1	2	yes	—
81	M₁₆	—	536	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	268	4	no	—
86	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
88	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
89	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
90	C₄²	D₅₃₆	1	1	yes	characteristic
91	C₄²	Dic₂₆₈	1	1	yes	characteristic
92	C₁₆	—	4	2	no	—
93	C₂³	#14523975	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
99	C₂×C₄	#14523965	1	2	yes	—
100	C₂×C₄	#14523968	1	2	yes	—
101	C₂×C₄	#14523977	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	536	4	no	—
105	C₈	#14523960	1	1	yes	characteristic
106	C₈	#14523962	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#16863159	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	C₆₇⋊#338	1	1	yes	characteristic
113	C₄	#16863165	1	1	yes	characteristic
114	C₄	#16863210	1	2	yes	—
115	C₄	#16863213	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	#22558084	1	1	yes	characteristic
118	C₂	#22558318	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558968

C₂

yes

maximal

C₆₇×#2150

C₂

yes

maximal, characteristic

#16863420

—

#16863149

—

#16863149

C₄

yes

—

C₂×M₃₂×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

Q₁₆

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

#106

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#95

yes

characteristic

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

#107

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#340

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#362

yes

—

#7245

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1292

yes

characteristic

C₂×C₆₇

#1387

yes

—

#2150

D₆₇

yes

characteristic

#1567

—

C₆₇

#7245

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₆₇

yes

—

#423

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

134

—

C₂×M₁₆

—

268

—

C₂²×C₈

Dic₁₃₄

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₄²

D₂₆₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

268

—

C₄⋊C₈

—

134

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂²×C₄

#11761617

yes

—

M₁₆

—

536

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄×Dic₆₇

yes

characteristic

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₈

Dic₆₇⋊C₄

yes

—

C₄²

(C₂×D₆₇)⋊C₄

yes

—

C₄²

D₅₃₆

yes

characteristic

C₄²

Dic₂₆₈

yes

characteristic

C₁₆

—

C₂³

#14523975

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₆₇⋊#95

yes

characteristic

C₂×C₄

#14523965

yes

—

100

C₂×C₄

#14523968

yes

—

101

C₂×C₄

#14523977

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

536

—

105

C₈

#14523960

yes

characteristic

106

C₈

#14523962

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

#16863159

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₆₇⋊#338

yes

characteristic

113

C₄

#16863165

yes

characteristic

114

C₄

#16863210

yes

—

115

C₄

#16863213

yes

—

116

C₂

—

117

C₂

#22558084

yes

characteristic

118

C₂

#22558318

yes

—

	subgroup	G/H
center	C₂²	#16863159
commutator	C₂×C₈×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#106
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#7245
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

#16863159

commutator

C₂×C₈×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₆₇×#2150

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
62. C₆₇	#7245	semidirect

H ≅ G/N

type

62. C₆₇

#7245

semidirect

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

subgroup	index
#22558968	2
C₆₇×#2150	2
#7245	67

subgroup

index

#22558968

C₆₇×#2150

#7245

Subgroups — #30742940

Conjugacy classes of subgroups

Complements

Maximal subgroups