Finite Group Databasebeta

← #13054555

Subgroups — #13054555

Order 1344.

324 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66670C211yesmaximal, characteristic, complemented
2#10621421C211yesmaximal, characteristic, complemented
3C2×C7×#214C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10620508C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89944—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C7×Dic6C2211yescharacteristic
13C2×C3×D28C2211yescharacteristic
14C2×#4246—21no—
15C2×Dic3×D7—21no—
16#66670—22no—
17#66670—22no—
18#66636—21no—
19C4⋊Dic21C2211yescharacteristic
20C7×#214—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×#1141—22no—
23C2×C3×C7⋊C8C2211yescharacteristic
24C2×#1161—31no—
25C2×C7×SD16—31no—
26#6501—31no—
27C2×#1141D311yescharacteristic, complemented
28#6409—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5158—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C22×C3×D7—21no—
35C3×C7×D4—21no—
36C3×C7×D4—21no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—42no—
39C2×Dic21—21no—
40C2×Dic21D411yescharacteristic
41C4×D21—21no—
42C4×D21—21no—
43C2×C7×Dic3—21no—
44C7×Dic6—21no—
45C7×Dic6D412yescomplemented
46C3×D28—21no—
47C3×D28D412yescomplemented
48#4246—24no—
49Dic3×D7—24no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—61no—
55#1161—62no—
56#1161—62no—
57C2×D28—61no—
58C2×D28C2×D311yescharacteristic
59C2×C4×D7—31no—
60C2×C4×D7—61no—
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1141—22no—
65C4⋊Dic7—31no—
66C2×C7⋊C8C2×D311yescharacteristic
67C2×#237—71no—
68C2×C3×D8—71no—
69#938—71no—
70C2×#214D711yescharacteristic, complemented
71#832—71no—
72(C2×D3)⋊C8—71no—
73#816—71no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—21no—
79C2×D21—21no—
80C2×D21—21no—
81C2×C3×D7—22no—
82C2×C3×D7—41no—
83C2×C3×D7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—21no—
87Dic21—41no—
88C7×Dic3—22no—
89#449—211noSylow
90C23×C7—21no—
91C22×D7—21no—
92C22×D7—31no—
93C7×Q8—32no—
94C7×Q8—61no—
95C7×D4—21no—
96C7×D4—21no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—122no—
100C2×Dic7—31no—
101C2×Dic7—61no—
102D28—21no—
103D28—64no—
104D28#6712yescomplemented
105C4×D7—64no—
106C4×D7—61no—
107C4×D7—61no—
108C7×C8—61no—
109C7⋊C8—21no—
110C2×C3×D4—71no—
111C2×C3×D4C2×D711yescharacteristic
112C2×#67—141no—
113C2×#67—141no—
114C22×Dic3—141no—
115#237—142no—
116#237—142no—
117C2×C4×D3—71no—
118C2×Dic6C2×D711yescharacteristic
119C3×D8—142no—
120C2×C3×C8—71no—
121#217—141no—
122#214—22no—
123C4⋊Dic3—71no—
124C2×C3⋊C8C2×D711yescharacteristic
125C2×C3×C7—81no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—41no—
130C3×D7—42no—
131C2×#41—211no—
132C2×SD16—31no—
133C2×D8—71no—
134#121—211no—
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7D3×D411yescharacteristic
142C2×D7—22no—
143C2×D7—41no—
144C2×D7—41no—
145C2×D7—61no—
146C2×D7—61no—
147C2×D7—61no—
148C4×C7—62no—
149C4×C7D3×D411yescharacteristic
150C4×C7C2×#6711yescharacteristic
151Dic7—61no—
152Dic7—121no—
153C23×C3—21no—
154C23×C3—141no—
155C22×D3—71no—
156C3×D4—21no—
157C3×D4—21no—
158C3×D4—72no—
159C3×D4—141no—
160C2×C3×C4C22×D711yescharacteristic
161#67—144no—
162#67—282no—
163C2×Dic3—21no—
164C2×Dic3—71no—
165C2×Dic3—144no—
166C2×Dic3—141no—
167C4×D3—141no—
168C4×D3—141no—
169Dic6—21no—
170Dic6C22⋊D712yescomplemented
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—31no—
177C2×D4—71no—
178C2×D4—421no—
179C2×D4—421no—
180C2×D4D3×D711yescharacteristic
181C22×C4—211no—
182C22×C4—421no—
183SD16—62no—
184D8—142no—
185C2×C8—31no—
186C2×C8—71no—
187C4⋊C4—211no—
188C22⋊C4—421no—
189C2×C7—81no—
190C2×C7#91111yescharacteristic
191C2×C7#91211yescharacteristic
192C2×C7#93711yescharacteristic
193D7—42no—
194D7—121no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—142no—
199C22×C3—281no—
200C22×C3—281no—
201C22×C3D4×D711yescharacteristic
202C2×D3—141no—
203C2×D3—141no—
204C2×D3—141no—
205C3×C4D4×D711yescharacteristic
206C3×C4C2×C22⋊D711yescharacteristic
207Dic3—22no—
208Dic3—141no—
209Dic3—281no—
210C23—21no—
211C23—141no—
212C23—211no—
213Q8—32no—
214Q8—61no—
215D4—21no—
216D4—21no—
217D4—72no—
218D4—141no—
219D4—424no—
220D4—842no—
221C2×C4—61no—
222C2×C4—211no—
223C2×C4—424no—
224C2×C4—421no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—141no—
230C7#515811yescharacteristic, complemented, Sylow
231C2×C3—81no—
232C2×C3—282no—
233C2×C3#647711yescharacteristic
234C2×C3#647911yescharacteristic
235C2×C3#650011yescharacteristic
236D3—281no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—142no—
241C22—281no—
242C22—281no—
243C22—421no—
244C22—421no—
245C22—421no—
246C22#6669411yescharacteristic
247C4—62no—
248C4—421no—
249C4—841no—
250C4D7×#6711yescharacteristic
251C4D3×C22⋊D711yescharacteristic
252C3#8994411yescharacteristic, complemented, Sylow
253C2—81no—
254C2—282no—
255C2—841no—
256C2#1062105911yescharacteristic
257C2#1062106011yescharacteristic
258C2#1062113911yescharacteristic
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×#66670C2semidirect
2. #10621421C2semidirect
3. C2×C7×#214C2semidirect
4. C2×C3×#1141C2semidirect
5. #10620508C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
27. C2×#1141D3semidirect
45. C7×Dic6D4semidirect
47. C3×D28D4semidirect
70. C2×#214D7semidirect
104. D28#67semidirect
170. Dic6C22⋊D7semidirect
173. C3×C7#449semidirect
230. C7#5158semidirect
252. C3#89944semidirect

Maximal subgroups

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

subgroupindex
C2×#666702
#106214212
C2×C7×#2142
C2×C3×#11412
#106205082
#106205002
#106204922
#899443
#51587