Group #30749640 — subgroups

Subgroups — #30749640

Order 17152.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22558964	C₂	1	4	yes	maximal
3	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	2	no	—
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863146	—	2	8	no	—
9	#16863117	C₄	1	4	yes	—
10	C₂×#108×C₆₇	—	2	2	no	—
11	#423×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#11761612	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
15	#14523979	—	4	4	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
19	C₂×M₁₆×C₆₇	—	2	2	no	—
20	C₂²×C₈×C₆₇	D₄	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
22	#108×C₆₇	—	4	4	no	—
23	C₄⋊C₈×C₆₇	—	2	4	no	—
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	Q₈	1	2	yes	—
26	#11761612	—	4	4	no	—
27	C₂×C₆₇⋊C₈	—	4	4	no	—
28	C₂×C₆₇⋊C₈	—	4	2	no	—
29	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
30	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
32	M₁₆×C₆₇	—	4	4	no	—
33	C₂×C₈×C₆₇	—	4	2	no	—
34	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
36	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
37	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	C₂×D₄	1	2	yes	—
39	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
48	C₈×C₆₇	—	8	2	no	—
49	C₈×C₆₇	#106	1	8	yes	—
50	C₆₇⋊C₈	—	2	8	no	—
51	C₆₇⋊C₈	—	8	2	no	—
52	C₂²×C₆₇	—	2	2	no	—
53	C₂²×C₆₇	#384	1	1	yes	characteristic
54	C₂²×C₆₇	#498	1	2	yes	—
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	C₂×#106	1	2	yes	—
57	C₄×C₆₇	#430	1	4	yes	—
58	C₄×C₆₇	#492	1	1	yes	characteristic
59	C₄×C₆₇	#492	1	1	yes	characteristic
60	#8825	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	#1844	1	1	yes	characteristic
63	C₂×C₆₇	#1849	1	2	yes	—
64	#1848	—	67	1	no	—
65	#1848	D₆₇	1	1	yes	characteristic
66	#1566	—	67	4	no	—
67	C₂×#322	—	67	1	no	—
68	C₆₇	#8825	1	1	yes	characteristic, complemented, Sylow
69	C₂×#108	—	2	2	no	—
70	C₂×#108	—	134	2	no	—
71	#423	—	67	2	no	—
72	#423	C₂×D₆₇	1	2	yes	—
73	C₂×C₄×C₈	—	67	2	no	—
74	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
75	#334	—	134	8	no	—
76	#322	—	67	4	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	134	2	no	—
79	C₂²×C₈	—	67	4	no	—
80	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
81	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	268	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	134	4	no	—
86	C₄×C₈	—	67	4	no	—
87	C₄×C₈	—	67	4	no	—
88	C₄×C₈	Dic₁₃₄	1	2	yes	—
89	C₄×C₈	C₄×D₆₇	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
91	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
92	M₁₆	—	4	4	no	—
93	M₁₆	—	268	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	268	2	no	—
96	C₂×C₈	—	268	4	no	—
97	C₂×C₈	—	67	8	no	—
98	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
99	C₂×C₈	#11761615	1	4	yes	—
100	C₂×C₈	#11761616	1	4	yes	—
101	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
102	C₄²	D₄×D₆₇	1	2	yes	—
103	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
104	C₂³	#14524041	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#14524039	1	1	yes	characteristic
109	C₂×C₄	#14524042	1	2	yes	—
110	C₂×C₄	C₂×#11761615	1	2	yes	—
111	C₂×C₄	C₂×#11761616	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	536	2	no	—
114	C₈	—	134	8	no	—
115	C₈	#14523974	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#16863265	1	1	yes	characteristic
118	C₂²	#16863792	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#14523974	1	2	yes	—
121	C₄	#16863443	1	4	yes	—
122	C₄	#16863768	1	1	yes	characteristic
123	C₄	#16863770	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#22560365	1	1	yes	characteristic
126	C₂	#22560382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal, characteristic

#22558964

C₂

yes

maximal

C₂×#16863117

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal, characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863146

—

#16863117

C₄

yes

—

C₂×#108×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

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

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#106

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#498

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#106

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1844

yes

characteristic

C₂×C₆₇

#1849

yes

—

#1848

—

#1848

D₆₇

yes

characteristic

#1566

—

C₂×#322

—

C₆₇

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

134

—

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#334

—

134

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#108

—

#108

—

268

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

C₄×D₆₇

yes

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₂²×C₄

Q₈×D₆₇

yes

—

M₁₆

—

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

C₂×C₈

Dic₆₇⋊C₄

yes

—

C₂×C₈

#11761615

yes

—

100

C₂×C₈

#11761616

yes

—

101

C₄²

C₂×C₄×D₆₇

yes

characteristic

102

C₄²

D₄×D₆₇

yes

—

103

C₄²

C₂×Dic₁₃₄

yes

characteristic

104

C₂³

#14524041

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

107

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

108

C₂×C₄

#14524039

yes

characteristic

109

C₂×C₄

#14524042

yes

—

110

C₂×C₄

C₂×#11761615

yes

—

111

C₂×C₄

C₂×#11761616

yes

—

112

C₈

—

113

C₈

—

536

—

114

C₈

—

134

—

115

C₈

#14523974

yes

—

116

C₂²

—

117

C₂²

#16863265

yes

characteristic

118

C₂²

#16863792

yes

—

119

C₄

—

120

C₄

C₂×#14523974

yes

—

121

C₄

#16863443

yes

—

122

C₄

#16863768

yes

characteristic

123

C₄

#16863770

yes

characteristic

124

C₂

—

125

C₂

#22560365

yes

characteristic

126

C₂

#22560382

yes

—

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#8825
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1848

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
68. C₆₇	#8825	semidirect

H ≅ G/N

type

68. C₆₇

#8825

semidirect

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

subgroup	index
#22560381	2
#22558964	2
C₂×#16863117	2
C₆₇×#1848	2
#8825	67

subgroup

index

C₆₇×#1848

Subgroups — #30749640

Conjugacy classes of subgroups

Complements

Maximal subgroups