Group #30749529 — subgroups

Subgroups — #30749529

Order 17152.

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

normal subgroups only

This group has 270 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	#22560381	C₂	1	1	yes	maximal, characteristic
2	#22560340	C₂	1	1	yes	maximal, characteristic
3	#22558068	C₂	1	4	yes	maximal
4	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	2	no	—
6	#16863420	C₂²	1	2	yes	—
7	#16863262	C₂²	1	2	yes	—
8	#16863159	—	2	2	no	—
9	#431×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#322×C₆₇	C₄	1	4	yes	complemented
12	C₂×#11761612	—	2	2	no	—
13	C₂×C₄⋊Dic₆₇	—	2	2	no	—
14	#14523979	—	4	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₆₇⋊#95	—	2	4	no	—
17	C₂²×C₈×C₆₇	D₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₆₇	—	2	2	no	—
20	C₂²⋊C₈×C₆₇	—	2	2	no	—
21	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₇	Q₈	1	2	yes	—
24	C₂²×Dic₆₇	—	4	2	no	—
25	#11761612	—	4	4	no	—
26	C₂×C₆₇⋊C₈	—	4	2	no	—
27	C₄⋊Dic₆₇	—	4	2	no	—
28	C₄⋊Dic₆₇	—	4	2	no	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	#41	1	2	yes	—
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
35	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₆₇	#41	1	2	yes	—
38	C₂³×C₆₇	#124	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	4	2	no	—
40	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#117	1	2	yes	—
42	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#125	1	2	yes	—
44	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₆₇	#137	1	2	yes	—
46	C₂×Dic₆₇	—	8	4	no	—
47	C₂×Dic₆₇	—	8	2	no	—
48	C₈×C₆₇	—	2	4	no	—
49	C₈×C₆₇	—	4	4	no	—
50	C₆₇⋊C₈	—	8	2	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	#382	1	1	yes	characteristic
53	C₂²×C₆₇	#489	1	2	yes	—
54	C₄×C₆₇	—	2	2	no	—
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	#428	1	2	yes	—
57	C₄×C₆₇	#490	1	1	yes	characteristic
58	C₄×C₆₇	#491	1	1	yes	characteristic
59	Dic₆₇	—	16	2	no	—
60	#8794	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	#1833	1	2	yes	—
63	C₂×C₆₇	#1836	1	1	yes	characteristic
64	#1848	—	67	1	no	—
65	#1835	—	67	1	no	—
66	#1452	D₆₇	1	1	yes	characteristic
67	#1286	—	67	4	no	—
68	C₆₇	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	C₂×D₆₇	1	2	yes	—
70	C₂×#108	—	134	2	no	—
71	#423	—	67	2	no	—
72	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	#382	—	67	2	no	—
74	#340	—	134	2	no	—
75	#322	Dic₆₇	1	4	yes	complemented
76	C₂×M₁₆	—	134	2	no	—
77	C₂²×C₈	D₂₆₈	1	2	yes	—
78	C₂×C₄⋊C₄	—	134	2	no	—
79	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
80	#108	—	268	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	134	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	Dic₁₃₄	1	2	yes	—
85	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
86	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
87	#95	—	134	4	no	—
88	C₂²×C₄	—	268	2	no	—
89	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
90	C₂²×C₄	#11761628	1	2	yes	—
91	M₁₆	—	268	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	268	2	no	—
96	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
97	C₄⋊C₄	—	268	2	no	—
98	C₄⋊C₄	—	268	2	no	—
99	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
101	C₄²	#11761628	1	2	yes	—
102	C₂³	#14524031	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	536	4	no	—
105	C₂×C₄	—	536	2	no	—
106	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
107	C₂×C₄	#14524002	1	2	yes	—
108	C₂×C₄	#14524031	1	1	yes	characteristic
109	C₂×C₄	#14524035	1	2	yes	—
110	C₂×C₄	#14524071	1	2	yes	—
111	C₂×C₄	#14524076	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	536	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#16863262	1	1	yes	characteristic
117	C₂²	#16863756	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	1072	2	no	—
121	C₄	#16863438	1	2	yes	—
122	C₄	#16863761	1	1	yes	characteristic
123	C₄	#16863765	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#22560333	1	2	yes	—
126	C₂	#22560343	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal, characteristic

#22560340

C₂

yes

maximal, characteristic

#22558068

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

#16863262

C₂²

yes

—

#16863159

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

complemented

C₂×#11761612

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#125

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#382

yes

characteristic

C₂²×C₆₇

#489

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#428

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#8794

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1833

yes

—

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1286

—

C₆₇

#8794

yes

characteristic, complemented, Sylow

#431

C₂×D₆₇

yes

—

C₂×#108

—

134

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#382

—

#340

—

134

—

#322

Dic₆₇

yes

complemented

C₂×M₁₆

—

134

—

C₂²×C₈

D₂₆₈

yes

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#108

—

268

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

#95

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

C₂²×C₄

#11761628

yes

—

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

C₂²×Dic₆₇

yes

characteristic

100

C₄²

C₂×Dic₁₃₄

yes

characteristic

101

C₄²

#11761628

yes

—

102

C₂³

#14524031

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

536

—

105

C₂×C₄

—

536

—

106

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

107

C₂×C₄

#14524002

yes

—

108

C₂×C₄

#14524031

yes

characteristic

109

C₂×C₄

#14524035

yes

—

110

C₂×C₄

#14524071

yes

—

111

C₂×C₄

#14524076

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

536

—

115

C₂²

—

116

C₂²

#16863262

yes

characteristic

117

C₂²

#16863756

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

1072

—

121

C₄

#16863438

yes

—

122

C₄

#16863761

yes

characteristic

123

C₄

#16863765

yes

characteristic

124

C₂

—

125

C₂

#22560333

yes

—

126

C₂

#22560343

yes

characteristic

	subgroup	G/H
center	C₂²	#16863262
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#8794
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16863262

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
11. #322×C₆₇	C₄	semidirect
68. C₆₇	#8794	semidirect
75. #322	Dic₆₇	semidirect

H ≅ G/N

type

11. #322×C₆₇

C₄

semidirect

68. C₆₇

#8794

semidirect

75. #322

Dic₆₇

semidirect

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

subgroup	index
#22560381	2
#22560340	2
#22558068	2
C₆₇×#1452	2
#8794	67

subgroup

index

C₆₇×#1452

Subgroups — #30749529

Conjugacy classes of subgroups

Complements

Maximal subgroups