Finite Group Databasebeta

← #13054407

Subgroups — #13054407

Order 1344.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#66730C211yesmaximal, characteristic, complemented
2C2×C7×#213C211yesmaximal, characteristic, complemented
3#10621028C211yesmaximal, characteristic, complemented
4C2×#66567C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89930—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C22×Dic21—21no—
12#66730—22no—
13#66730—22no—
14C2×C4×D21C2211yescharacteristic
15C2×Dic42C2211yescharacteristic
16C2×C7×D12C2211yescharacteristic
17C2×#4247—21no—
18C7×#213—22no—
19C2×C7×C3⋊C8C2211yescharacteristic
20C3×C4⋊Dic7C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22(C2×D3)⋊Dic7—21no—
23#66567—22no—
24C2×#1159—31no—
25C2×C7×D8—31no—
26#6501—31no—
27C2×#1142—31no—
28#6406D311yescharacteristic, complemented
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#4781—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C22×D3×C7—21no—
35C3×C7×D4—21no—
36C3×C7×D4D412yescomplemented
37C2×C3×C4×C7C2311yescharacteristic
38#4268—24no—
39C2×Dic21—24no—
40C2×Dic21—21no—
41C2×Dic21D411yescharacteristic
42C4×D21—21no—
43C4×D21—21no—
44Dic42—21no—
45Dic42D412yescomplemented
46C7×D12—21no—
47C7×D12—21no—
48C2×C3×Dic7—21no—
49#4247—42no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×D4—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—61no—
55C2×C22⋊D7—61no—
56C22×Dic7—61no—
57#1159—62no—
58#1159—62no—
59C2×C4×D7—31no—
60C2×Dic14—31no—
61C7×D8—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1142—62no—
65C4⋊Dic7C2×D311yescharacteristic
66C2×C7⋊C8C2×D311yescharacteristic
67C2×#237—71no—
68C2×#213D711yescharacteristic, complemented
69C2×#204—71no—
70#895—71no—
71C3×#102—71no—
72(C2×D3)⋊C8—71no—
73#810—71no—
74C22×C3×C7—22no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—21no—
79C2×D21—21no—
80C2×D21—21no—
81C2×D3×C7—41no—
82C2×D3×C7—41no—
83C2×D3×C7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—22no—
87Dic21—21no—
88C3×Dic7—41no—
89#449—211noSylow
90C23×C7—21no—
91C23×C7—61no—
92C22×D7—31no—
93C7×D4—21no—
94C7×D4—61no—
95C7×D4—61no—
96C7×D4D1212yescomplemented
97C2×C4×C7C22×D311yescharacteristic
98C22⋊D7—64no—
99C22⋊D7—122no—
100C2×Dic7—21no—
101C2×Dic7—31no—
102C2×Dic7—64no—
103C2×Dic7—61no—
104C4×D7—61no—
105C4×D7—61no—
106Dic14—32no—
107Dic14—61no—
108C7×C8—61no—
109C7⋊C8—21no—
110C2×C3×D4C2×D711yescharacteristic
111C2×#67—141no—
112C22×Dic3—141no—
113#237—142no—
114#237—142no—
115C2×D12—141no—
116C2×D12C2×D711yescharacteristic
117C2×C4×D3—71no—
118C2×Dic6—71no—
119C2×C3×C8—71no—
120C3×C4⋊C4—71no—
121#213—22no—
122(C2×D3)⋊C4—141no—
123C2×C3⋊C8C2×D711yescharacteristic
124#204—142no—
125C2×C3×C7—42no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—41no—
130D3×C7—81no—
131C2×#41—211no—
132C2×SD16—211no—
133C2×D8—31no—
134#121—211no—
135#103—211no—
136#102—71no—
137C22⋊C8—211no—
138C22×C7—22no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7—121no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7D3×D411yescharacteristic
145C2×D7—61no—
146C2×D7—61no—
147C2×D7—61no—
148C4×C7C2×D1211yescharacteristic
149C4×C7D3×D411yescharacteristic
150Dic7—41no—
151Dic7—62no—
152Dic7—61no—
153C23×C3—21no—
154C22×D3—21no—
155C22×D3—71no—
156C3×D4—21no—
157C3×D4C22⋊D712yescomplemented
158C2×C3×C4—141no—
159C2×C3×C4C22×D711yescharacteristic
160#67—144no—
161C2×Dic3—71no—
162C2×Dic3—144no—
163C2×Dic3—141no—
164D12—21no—
165D12—21no—
166D12—282no—
167C4×D3—141no—
168C4×D3—141no—
169Dic6—72no—
170Dic6—141no—
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—211no—
177C2×D4—31no—
178C2×D4—421no—
179C2×D4—421no—
180C2×D4D3×D711yescharacteristic
181C22×C4—211no—
182C22×C4—421no—
183SD16—422no—
184D8—62no—
185C2×C8—31no—
186C2×C8—71no—
187C4⋊C4—71no—
188C22⋊C4—421no—
189C2×C7—42no—
190C2×C7—241no—
191C2×C7#88211yescharacteristic
192C2×C7#91111yescharacteristic
193C2×C7#91611yescharacteristic
194D7—121no—
195C22×C3—22no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3D4×D711yescharacteristic
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—141no—
203C2×D3—141no—
204C2×D3—141no—
205C3×C4—281no—
206C3×C4D4×D711yescharacteristic
207C3×C4C2×C22⋊D711yescharacteristic
208Dic3—142no—
209Dic3—141no—
210C23—21no—
211C23—61no—
212C23—211no—
213Q8—212no—
214Q8—421no—
215D4—21no—
216D4—61no—
217D4—61no—
218D4—424no—
219D4—842no—
220D4#424712yescomplemented
221C2×C4—141no—
222C2×C4—211no—
223C2×C4—424no—
224C2×C4—421no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—141no—
230C7#478111yescharacteristic, complemented, Sylow
231C2×C3—42no—
232C2×C3#647411yescharacteristic
233C2×C3#647511yescharacteristic
234C2×C3#650011yescharacteristic
235D3—81no—
236D3—281no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—421no—
244C22—421no—
245C22—421no—
246C22#6669411yescharacteristic
247C4—281no—
248C4—421no—
249C4—422no—
250C4C2×#424711yescharacteristic
251C4#6669411yescharacteristic
252C3#8993011yescharacteristic, complemented, Sylow
253C2—42no—
254C2—241no—
255C2—841no—
256C2#1062104411yescharacteristic
257C2#1062106111yescharacteristic
258C2#1062113511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66730C2semidirect
2. C2×C7×#213C2semidirect
3. #10621028C2semidirect
4. C2×#66567C2semidirect
5. C3×#6406C2semidirect
6. #10620509C2semidirect
7. #10620492C2semidirect
28. #6406D3semidirect
36. C3×C7×D4D4semidirect
45. Dic42D4semidirect
68. C2×#213D7semidirect
96. C7×D4D12semidirect
157. C3×D4C22⋊D7semidirect
173. C3×C7#449semidirect
220. D4#4247semidirect
230. C7#4781semidirect
252. C3#89930semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×C7×#2132
#106210282
C2×#665672
C3×#64062
#106205092
#106204922
#899303
#47817