Finite Group Databasebeta

← #13056415

Subgroups — #13056415

Order 1344.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621421C211yesmaximal, characteristic
2#10621375C211yesmaximal, characteristic, complemented
3C3×C7×#102C211yesmaximal, characteristic, complemented
4#10620671C211yesmaximal, characteristic, complemented
5#10620659C211yesmaximal, characteristic, complemented
6#10620656C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89551—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66636—21no—
15#66631—21no—
16C4⋊Dic21C2211yescharacteristic
17C4⋊Dic21C2211yescharacteristic
18C4×Dic21—21no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6501—31no—
21#6455—31no—
22C7×#102D311yescharacteristic, complemented
23#6406—31no—
24(C2×D7)⋊C8—31no—
25#6391—31no—
26#6381—31no—
27#4779—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×D21D411yescharacteristic
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34#4268—42no—
35C2×Dic21—21no—
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C4×D21—21no—
39C4×D21—21no—
40C3×C7×C8—21no—
41C3⋊(C7⋊C8)—21no—
42C2×C7×D4C2×D311yescharacteristic
43C2×C22⋊D7—61no—
44C2×C4×D7—31no—
45C2×C7×C8C2×D311yescharacteristic
46C7×C4⋊C4C2×D311yescharacteristic
47C22⋊Dic7—61no—
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50C4⋊Dic7—31no—
51C4×Dic7—61no—
52C2×C7⋊C8—31no—
53#938—71no—
54#892—71no—
55C3×#102D711yescharacteristic, complemented
56#832—71no—
57(C2×D3)⋊C8—71no—
58#817—71no—
59#807—71no—
60C22×C3×C7—41no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7C2×D411yescharacteristic
64C2×D21—21no—
65C2×D21—21no—
66C2×D21—21no—
67C3×C4×C7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—21no—
71Dic21—41no—
72Dic21—41no—
73#482—211noSylow
74C23×C7—21no—
75C22×D7—31no—
76C7×D4—21no—
77C7×D4—21no—
78C2×C4×C7—21no—
79C2×C4×C7C22×D311yescharacteristic
80C22⋊D7—122no—
81C2×Dic7—31no—
82C2×Dic7—61no—
83C2×Dic7—61no—
84C4×D7—61no—
85C4×D7—61no—
86C7×C8—21no—
87C7⋊C8—61no—
88C2×C3×D4C2×D711yescharacteristic
89C2×#67—141no—
90C2×C4×D3—71no—
91C2×C3×C8C2×D711yescharacteristic
92C3×C4⋊C4C2×D711yescharacteristic
93#217—141no—
94(C2×D3)⋊C4—141no—
95C4⋊Dic3—71no—
96C4⋊Dic3—71no—
97C4×Dic3—141no—
98C2×C3⋊C8—71no—
99C2×C3×C7—81no—
100C2×C3×C7#12311yescharacteristic
101C2×C3×C7#13511yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103D21—41no—
104#121—211no—
105#117—211no—
106#107—211no—
107#106—211no—
108#102—211no—
109#102D2111yescharacteristic, complemented
110C22⋊C8—211no—
111C22×C7—41no—
112C22×C7—41no—
113C22×C7—41no—
114C22×C7D3×D411yescharacteristic
115C2×D7—61no—
116C2×D7—61no—
117C2×D7—61no—
118C4×C7—41no—
119C4×C7#23511yescharacteristic
120C4×C7#23711yescharacteristic
121Dic7—61no—
122Dic7—121no—
123Dic7—121no—
124C23×C3—21no—
125C22×D3—71no—
126C3×D4—21no—
127C3×D4—21no—
128C2×C3×C4—21no—
129C2×C3×C4C22×D711yescharacteristic
130#67—282no—
131C2×Dic3—71no—
132C2×Dic3—141no—
133C2×Dic3—141no—
134C4×D3—141no—
135C4×D3—141no—
136C3×C8—21no—
137C3⋊C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—421no—
140C2×D4C2×D2111yescharacteristic
141C22×C4—211no—
142C2×C8—211no—
143C2×C8C2×D2111yescharacteristic
144C4⋊C4—211no—
145C4⋊C4—211no—
146C4⋊C4C2×D2111yescharacteristic
147C22⋊C4—421no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7—81no—
151C2×C7#88311yescharacteristic
152C2×C7#91111yescharacteristic
153C2×C7#91611yescharacteristic
154D7—121no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3D4×D711yescharacteristic
159C2×D3—141no—
160C2×D3—141no—
161C2×D3—141no—
162C3×C4—41no—
163C3×C4#115711yescharacteristic
164C3×C4#115911yescharacteristic
165Dic3—141no—
166Dic3—281no—
167Dic3—281no—
168C23—21no—
169C23—211no—
170D4—21no—
171D4—21no—
172D4—842no—
173C2×C4—21no—
174C2×C4—211no—
175C2×C4—421no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4—421no—
179C2×C4C22×D2111yescharacteristic
180C8—21no—
181C8—421no—
182C7#477911yescharacteristic, complemented, Sylow
183C2×C3—81no—
184C2×C3#644611yescharacteristic
185C2×C3#647411yescharacteristic
186C2×C3#647911yescharacteristic
187D3—281no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—421no—
192C22—421no—
193C22—421no—
194C22D4×D2111yescharacteristic
195C4—41no—
196C4—421no—
197C4—841no—
198C4—841no—
199C4#6672811yescharacteristic
200C4#6673011yescharacteristic
201C3#8955111yescharacteristic, complemented, Sylow
202C2—81no—
203C2—841no—
204C2#1062136611yescharacteristic
205C2#1062139411yescharacteristic
206C2#1062139911yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radical#10621421C2
hypercenter#102D21
hypocenterC3×C7#482
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #10621375C2semidirect
3. C3×C7×#102C2semidirect
4. #10620671C2semidirect
5. #10620659C2semidirect
6. #10620656C2semidirect
7. #10620646C2semidirect
22. C7×#102D3semidirect
55. C3×#102D7semidirect
109. #102D21semidirect
138. C3×C7#482semidirect
182. C7#4779semidirect
201. C3#89551semidirect

Maximal subgroups

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

subgroupindex
#106214212
#106213752
C3×C7×#1022
#106206712
#106206592
#106206562
#106206462
#895513
#47797