Group #30749587 — subgroups

Subgroups — #30749587

Order 17152.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#22560361	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	4	yes	maximal
3	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	#16863149	—	2	8	no	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
8	C₄³×C₆₇	C₂²	1	1	yes	characteristic
9	#336×C₆₇	C₄	1	4	yes	—
10	C₂²×C₆₇⋊C₈	—	2	4	no	—
11	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
12	C₄⋊(C₆₇⋊C₈)	—	4	8	no	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₆₇	D₄	1	2	yes	—
16	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₇	D₄	1	4	yes	—
19	C₄×C₈×C₆₇	Q₈	1	4	yes	—
20	C₂×C₆₇⋊C₈	—	4	4	no	—
21	C₂×C₆₇⋊C₈	—	4	8	no	—
22	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
23	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₆₇	—	2	8	no	—
27	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
28	C₄²×C₆₇	—	2	4	no	—
29	C₄²×C₆₇	—	4	2	no	—
30	C₄²×C₆₇	D₈	1	4	yes	—
31	C₄²×C₆₇	Q₁₆	1	4	yes	—
32	C₄²×C₆₇	C₂×D₄	1	2	yes	—
33	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	—	2	4	no	—
36	C₂×C₄×C₆₇	—	2	4	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	#107	1	8	yes	—
39	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#131	1	4	yes	—
44	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
46	C₈×C₆₇	—	2	8	no	—
47	C₆₇⋊C₈	—	8	4	no	—
48	C₂²×C₆₇	#384	1	1	yes	characteristic
49	C₂²×C₆₇	#424	1	2	yes	—
50	C₂²×C₆₇	C₂×#107	1	2	yes	—
51	C₂²×C₆₇	#430	1	2	yes	—
52	C₄×C₆₇	—	2	8	no	—
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	#424	1	4	yes	—
55	C₄×C₆₇	#493	1	1	yes	characteristic
56	C₄×C₆₇	#494	1	1	yes	characteristic
57	C₄×C₆₇	#500	1	2	yes	—
58	#8810	—	67	1	no	maximal, Sylow
59	C₂×C₆₇	#1571	1	4	yes	—
60	C₂×C₆₇	#1843	1	1	yes	characteristic
61	C₂×C₆₇	#1845	1	1	yes	characteristic
62	C₂×C₆₇	#1849	1	1	yes	characteristic
63	#1842	—	67	2	no	—
64	#1744	D₆₇	1	1	yes	characteristic
65	C₂×#335	—	67	4	no	—
66	C₆₇	#8810	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	67	4	no	—
68	C₂×C₄⋊C₈	—	134	4	no	—
69	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
70	C₄³	C₂×D₆₇	1	1	yes	characteristic
71	#336	Dic₆₇	1	4	yes	—
72	#335	—	134	8	no	—
73	C₂²×C₈	—	134	4	no	—
74	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
75	C₂×C₄²	Dic₁₃₄	1	2	yes	—
76	C₂×C₄²	D₂₆₈	1	2	yes	—
77	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
78	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
79	C₄⋊C₈	—	268	8	no	—
80	C₄⋊C₈	—	134	8	no	—
81	C₄×C₈	Dic₁₃₄	1	4	yes	—
82	C₄×C₈	D₂₆₈	1	4	yes	—
83	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
84	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	268	4	no	—
89	C₂×C₈	—	268	8	no	—
90	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₅₃₆	1	4	yes	—
94	C₄²	Dic₂₆₈	1	4	yes	—
95	C₄²	C₂×D₂₆₈	1	2	yes	—
96	C₄²	C₂×Dic₁₃₄	1	2	yes	—
97	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	#14523977	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
103	C₂×C₄	#14524040	1	1	yes	characteristic
104	C₂×C₄	#14524043	1	2	yes	—
105	C₂×C₄	#14524043	1	1	yes	characteristic
106	C₂×C₄	#14524050	1	4	yes	—
107	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	536	4	no	—
111	C₂²	#16863266	1	1	yes	characteristic
112	C₂²	#16863423	1	2	yes	—
113	C₂²	C₂×#14523977	1	2	yes	—
114	C₂²	#16863445	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	#16863423	1	4	yes	—
118	C₄	#16863773	1	1	yes	characteristic
119	C₄	#16863776	1	1	yes	characteristic
120	C₄	#16863801	1	2	yes	—
121	C₂	#22558984	1	4	yes	—
122	C₂	#22560363	1	1	yes	characteristic
123	C₂	#22560369	1	1	yes	characteristic
124	C₂	#22560384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560361

C₂

yes

maximal

C₂×#16863149

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863149

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₈

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#424

yes

—

C₂²×C₆₇

C₂×#107

yes

—

C₂²×C₆₇

#430

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#8810

—

maximal, Sylow

C₂×C₆₇

#1571

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₂×#335

—

C₆₇

#8810

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

C₂×D₆₇

yes

characteristic

#336

Dic₆₇

yes

—

#335

—

134

—

C₂²×C₈

—

134

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

Dic₁₃₄

yes

—

C₂×C₄²

D₂₆₈

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

268

—

C₄⋊C₈

—

134

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

C₂²×C₄

C₄⋊Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₄²

—

C₄²

—

C₄²

D₅₃₆

yes

—

C₄²

Dic₂₆₈

yes

—

C₄²

C₂×D₂₆₈

yes

—

C₄²

C₂×Dic₁₃₄

yes

—

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#14523977

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

103

C₂×C₄

#14524040

yes

characteristic

104

C₂×C₄

#14524043

yes

—

105

C₂×C₄

#14524043

yes

characteristic

106

C₂×C₄

#14524050

yes

—

107

C₂×C₄

C₂×D₅₃₆

yes

—

108

C₂×C₄

C₂×Dic₂₆₈

yes

—

109

C₈

—

110

C₈

—

536

—

111

C₂²

#16863266

yes

characteristic

112

C₂²

#16863423

yes

—

113

C₂²

C₂×#14523977

yes

—

114

C₂²

#16863445

yes

—

115

C₄

—

116

C₄

—

117

C₄

#16863423

yes

—

118

C₄

#16863773

yes

characteristic

119

C₄

#16863776

yes

characteristic

120

C₄

#16863801

yes

—

121

C₂

#22558984

yes

—

122

C₂

#22560363

yes

characteristic

123

C₂

#22560369

yes

characteristic

124

C₂

#22560384

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8810
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1744

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
66. C₆₇	#8810	semidirect

H ≅ G/N

type

66. C₆₇

#8810

semidirect

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

subgroup	index
#22560361	2
C₂×#16863149	2
C₆₇×#1744	2
#8810	67

subgroup

index

#22560361

C₂×#16863149

C₆₇×#1744

#8810

Subgroups — #30749587

Conjugacy classes of subgroups

Complements

Maximal subgroups