Group #30749632 — subgroups

Subgroups — #30749632

Order 17152.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	2	yes	maximal
2	#22558965	C₂	1	4	yes	maximal
3	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523979	—	2	4	no	—
5	#16863420	C₂²	1	4	yes	—
6	#16863147	—	2	8	no	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
9	C₈²×C₆₇	C₄	1	1	yes	characteristic
10	C₈²×C₆₇	C₄	1	2	yes	—
11	C₈²×C₆₇	C₄	1	1	yes	characteristic
12	C₂×#11761612	—	2	4	no	—
13	#14523979	—	4	8	no	—
14	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
15	C₂²×C₈×C₆₇	D₄	1	2	yes	—
16	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	D₄	1	4	yes	—
23	C₄×C₈×C₆₇	Q₈	1	2	yes	—
24	#11761612	—	4	8	no	—
25	C₂×C₆₇⋊C₈	—	4	4	no	—
26	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
28	C₂×C₈×C₆₇	—	2	8	no	—
29	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₆₇	SD₁₆	1	8	yes	—
34	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₆₇	C₂×D₄	1	2	yes	—
36	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	4	no	—
39	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
44	C₈×C₆₇	—	2	4	no	—
45	C₈×C₆₇	#106	1	4	yes	—
46	C₈×C₆₇	#106	1	4	yes	—
47	C₈×C₆₇	#108	1	8	yes	complemented
48	C₆₇⋊C₈	—	8	4	no	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	#384	1	1	yes	characteristic
51	C₂²×C₆₇	#498	1	2	yes	—
52	C₄×C₆₇	—	2	2	no	—
53	C₄×C₆₇	C₂×#106	1	2	yes	—
54	C₄×C₆₇	C₂×#108	1	4	yes	—
55	C₄×C₆₇	#492	1	2	yes	—
56	#8822	—	67	1	no	maximal, Sylow
57	C₂×C₆₇	—	2	2	no	—
58	C₂×C₆₇	#1844	1	1	yes	characteristic
59	C₂×C₆₇	#1848	1	2	yes	—
60	#1848	—	67	2	no	—
61	#1566	—	67	4	no	—
62	C₂×C₈²	D₆₇	1	1	yes	characteristic
63	C₆₇	#8822	1	1	yes	characteristic, complemented, Sylow
64	C₂×#108	—	134	4	no	—
65	#423	—	67	4	no	—
66	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
68	#334	—	134	8	no	—
69	C₈²	Dic₆₇	1	1	yes	characteristic
70	C₈²	Dic₆₇	1	2	yes	—
71	C₈²	Dic₆₇	1	1	yes	characteristic
72	C₂×M₁₆	—	134	4	no	—
73	C₂²×C₈	Dic₁₃₄	1	4	yes	—
74	C₂²×C₈	D₂₆₈	1	2	yes	—
75	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
76	#108	—	268	8	no	—
77	C₄⋊C₈	—	134	8	no	—
78	C₄×C₈	Dic₁₃₄	1	2	yes	—
79	C₄×C₈	D₂₆₈	1	4	yes	—
80	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
81	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
84	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
86	M₁₆	—	268	8	no	—
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	268	4	no	—
89	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
90	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
91	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
92	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
93	C₂×C₈	#11761617	1	8	yes	—
94	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
95	C₄²	C₂×D₂₆₈	1	2	yes	—
96	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
97	C₂³	#14524043	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
101	C₂×C₄	#14524040	1	1	yes	characteristic
102	C₂×C₄	#14524043	1	2	yes	—
103	C₂×C₄	C₂×#11761617	1	4	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	536	4	no	—
106	C₈	#14523975	1	4	yes	—
107	C₈	#14523975	1	4	yes	—
108	C₈	#14523979	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	#16863266	1	1	yes	characteristic
111	C₂²	#16863794	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	C₂×#14523975	1	2	yes	—
114	C₄	C₂×#14523979	1	4	yes	—
115	C₄	#16863769	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	#22560366	1	1	yes	characteristic
118	C₂	#22560381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal

#22558965

C₂

yes

maximal

C₂×C₈²×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

#16863147

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₄

yes

characteristic

C₈²×C₆₇

C₄

yes

—

C₈²×C₆₇

C₄

yes

characteristic

C₂×#11761612

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#108

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#498

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#106

yes

—

C₄×C₆₇

C₂×#108

yes

—

C₄×C₆₇

#492

yes

—

#8822

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1844

yes

characteristic

C₂×C₆₇

#1848

yes

—

#1848

—

#1566

—

C₂×C₈²

D₆₇

yes

characteristic

C₆₇

#8822

yes

characteristic, complemented, Sylow

C₂×#108

—

134

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

—

#334

—

134

—

C₈²

Dic₆₇

yes

characteristic

C₈²

Dic₆₇

yes

—

C₈²

Dic₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₂²×C₈

Dic₁₃₄

yes

—

C₂²×C₈

D₂₆₈

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#108

—

268

—

C₄⋊C₈

—

134

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₄

yes

—

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

#11761617

yes

—

C₄²

C₂²×Dic₆₇

yes

characteristic

C₄²

C₂×D₂₆₈

yes

—

C₄²

C₂×Dic₁₃₄

yes

characteristic

C₂³

#14524043

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

100

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

101

C₂×C₄

#14524040

yes

characteristic

102

C₂×C₄

#14524043

yes

—

103

C₂×C₄

C₂×#11761617

yes

—

104

C₈

—

105

C₈

—

536

—

106

C₈

#14523975

yes

—

107

C₈

#14523975

yes

—

108

C₈

#14523979

yes

complemented

109

C₂²

—

110

C₂²

#16863266

yes

characteristic

111

C₂²

#16863794

yes

—

112

C₄

—

113

C₄

C₂×#14523975

yes

—

114

C₄

C₂×#14523979

yes

—

115

C₄

#16863769

yes

—

116

C₂

—

117

C₂

#22560366

yes

characteristic

118

C₂

#22560381

yes

—

	subgroup	G/H
center	C₂×C₄	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#8822
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14524040

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×C₈²×C₆₇

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

hypercenter

C₂×C₈²

D₆₇

hypocenter

C₆₇

#8822

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
47. C₈×C₆₇	#108	semidirect
63. C₆₇	#8822	semidirect
108. C₈	#14523979	semidirect

H ≅ G/N

type

47. C₈×C₆₇

#108

semidirect

63. C₆₇

#8822

semidirect

108. C₈

#14523979

semidirect

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

subgroup	index
#22560381	2
#22558965	2
C₂×C₈²×C₆₇	2
#8822	67

subgroup

index

#22560381

#22558965

C₂×C₈²×C₆₇

#8822

Subgroups — #30749632

Conjugacy classes of subgroups

Complements

Maximal subgroups