Group #30749635 — subgroups

Subgroups — #30749635

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	#22558968	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	#16863149	—	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	D₈	1	4	yes	—
34	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
35	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₆₇	C₂×D₄	1	2	yes	—
37	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	4	no	—
40	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
46	C₈×C₆₇	—	2	4	no	—
47	C₈×C₆₇	#107	1	4	yes	—
48	C₈×C₆₇	#107	1	4	yes	—
49	C₈×C₆₇	#108	1	8	yes	complemented
50	C₆₇⋊C₈	—	8	4	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	#384	1	1	yes	characteristic
53	C₂²×C₆₇	#500	1	2	yes	—
54	C₄×C₆₇	—	2	2	no	—
55	C₄×C₆₇	C₂×#107	1	2	yes	—
56	C₄×C₆₇	C₂×#108	1	4	yes	—
57	C₄×C₆₇	#493	1	1	yes	characteristic
58	C₄×C₆₇	#494	1	1	yes	characteristic
59	#8823	—	67	1	no	maximal, Sylow
60	C₂×C₆₇	—	2	2	no	—
61	C₂×C₆₇	#1845	1	1	yes	characteristic
62	C₂×C₆₇	#1848	1	2	yes	—
63	#1848	—	67	2	no	—
64	#1567	—	67	4	no	—
65	C₂×C₈²	D₆₇	1	1	yes	characteristic
66	C₆₇	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	134	4	no	—
68	#423	—	67	4	no	—
69	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
71	#335	—	134	8	no	—
72	C₈²	Dic₆₇	1	1	yes	characteristic
73	C₈²	Dic₆₇	1	2	yes	—
74	C₈²	Dic₆₇	1	1	yes	characteristic
75	C₂×M₁₆	—	134	4	no	—
76	C₂²×C₈	Dic₁₃₄	1	4	yes	—
77	C₂²×C₈	D₂₆₈	1	2	yes	—
78	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
79	#108	—	268	8	no	—
80	C₄⋊C₈	—	134	8	no	—
81	C₄×C₈	Dic₁₃₄	1	2	yes	—
82	C₄×C₈	D₂₆₈	1	4	yes	—
83	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
84	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
87	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
89	M₁₆	—	268	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	268	4	no	—
92	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
96	C₂×C₈	D₅₃₆	1	4	yes	—
97	C₂×C₈	Dic₂₆₈	1	4	yes	—
98	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
99	C₄²	C₂×D₂₆₈	1	2	yes	—
100	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
101	C₂³	#14524043	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
105	C₂×C₄	#14524040	1	1	yes	characteristic
106	C₂×C₄	#14524043	1	2	yes	—
107	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	536	4	no	—
111	C₈	#14523977	1	4	yes	—
112	C₈	#14523977	1	4	yes	—
113	C₈	#14523979	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#16863266	1	1	yes	characteristic
116	C₂²	#16863801	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#14523977	1	2	yes	—
119	C₄	C₂×#14523979	1	4	yes	—
120	C₄	#16863773	1	1	yes	characteristic
121	C₄	#16863776	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#22560369	1	1	yes	characteristic
124	C₂	#22560381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal

#22558968

C₂

yes

maximal

C₂×C₈²×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

#16863149

—

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

D₈

yes

—

C₂×C₈×C₆₇

Q₁₆

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×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#107

yes

—

C₈×C₆₇

#107

yes

—

C₈×C₆₇

#108

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#500

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#107

yes

—

C₄×C₆₇

C₂×#108

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1845

yes

characteristic

C₂×C₆₇

#1848

yes

—

#1848

—

#1567

—

C₂×C₈²

D₆₇

yes

characteristic

C₆₇

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

134

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

—

#335

—

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

D₅₃₆

yes

—

C₂×C₈

Dic₂₆₈

yes

—

C₄²

C₂²×Dic₆₇

yes

characteristic

C₄²

C₂×D₂₆₈

yes

—

100

C₄²

C₂×Dic₁₃₄

yes

characteristic

101

C₂³

#14524043

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

105

C₂×C₄

#14524040

yes

characteristic

106

C₂×C₄

#14524043

yes

—

107

C₂×C₄

C₂×D₅₃₆

yes

—

108

C₂×C₄

C₂×Dic₂₆₈

yes

—

109

C₈

—

110

C₈

—

536

—

111

C₈

#14523977

yes

—

112

C₈

#14523977

yes

—

113

C₈

#14523979

yes

complemented

114

C₂²

—

115

C₂²

#16863266

yes

characteristic

116

C₂²

#16863801

yes

—

117

C₄

—

118

C₄

C₂×#14523977

yes

—

119

C₄

C₂×#14523979

yes

—

120

C₄

#16863773

yes

characteristic

121

C₄

#16863776

yes

characteristic

122

C₂

—

123

C₂

#22560369

yes

characteristic

124

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	#8823
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

#8823

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₆₇	#108	semidirect
66. C₆₇	#8823	semidirect
113. C₈	#14523979	semidirect

H ≅ G/N

type

49. C₈×C₆₇

#108

semidirect

66. C₆₇

#8823

semidirect

113. C₈

#14523979

semidirect

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

subgroup	index
#22560381	2
#22558968	2
C₂×C₈²×C₆₇	2
#8823	67

subgroup

index

#22560381

#22558968

C₂×C₈²×C₆₇

#8823

Subgroups — #30749635

Conjugacy classes of subgroups

Complements

Maximal subgroups