Finite Group Databasebeta

← #13059839

Subgroups — #13059839

Order 1344.

424 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#4268C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic, complemented
3C3×#6489C211yesmaximal, characteristic, complemented
4#10621092C212yesmaximal, complemented
5#10621088C212yesmaximal
6#10621032C211yesmaximal, characteristic, complemented
7#10620949C212yesmaximal, complemented
8#10620945C211yesmaximal, characteristic, complemented
9#10620909C211yesmaximal, characteristic, complemented
10#10620906C211yesmaximal, characteristic, complemented
11#10620885C212yesmaximal, complemented
12#90332—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C2×C4×D21C2211yescharacteristic
16C2×C7×Dic6C2211yescharacteristic
17C2×#4248C2212yes—
18#66636C2211yescharacteristic
19#66631C2211yescharacteristic
20#66629C2211yescharacteristic
21C4×Dic21C2211yescharacteristic
22C7×#217C2212yes—
23C7×C4⋊Dic3C2211yescharacteristic
24C7×Dic3⋊C4C2212yes—
25C3×C22⋊Dic7C2212yes—
26C3×C4⋊Dic7C2211yescharacteristic
27C3×C4⋊Dic7C2211yescharacteristic
28C3×C4×Dic7C2212yes—
29Dic3⋊Dic7C2212yes—
30#66577C2212yes—
31Dic7⋊Dic3C2212yes—
32#66575C2212yes—
33#66575C2212yes—
34Dic3×Dic7C2212yes—
35Dic3×Dic7C2212yes—
36C7×C22⋊Q8—31no—
37#6509—31no—
38Q8×Dic7—31no—
39C4×C22⋊D7—31no—
40#6489D311yescharacteristic, complemented
41#6461—32no—
42#6455—31no—
43#6451—31no—
44#6450—32no—
45#6448—32no—
46#6442—32no—
47#5533—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C22×D21C2311yescharacteristic
50C2×C3×C4×C7—22no—
51C2×C3×C4×C7C2311yescharacteristic
52C2×C3×C4×C7C2311yescharacteristic
53#4268—22no—
54C2×Dic21C2311yescharacteristic
55C2×Dic21C2311yescharacteristic
56C2×Dic21C2311yescharacteristic
57C4×D21—22no—
58C2×C7×Dic3C2312yes—
59C2×C7×Dic3C2312yes—
60C7×Dic6—22no—
61C2×C3×Dic7C2312yes—
62C2×C3×Dic7C2312yes—
63#4248—24no—
64C2×C7×Q8—31no—
65C22×C4×C7C2×D311yescharacteristic
66C2×C22⋊D7—31no—
67C2×C4×D7—31no—
68C2×Dic14—32no—
69C7×C4⋊C4—32no—
70C7×C4⋊C4—31no—
71C7×C22⋊C4—32no—
72C22⋊Dic7—31no—
73C22⋊Dic7C2×D312yes—
74(C2×D7)⋊C4—31no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—32no—
77C4⋊Dic7—32no—
78C4⋊Dic7C2×D311yescharacteristic
79C4⋊Dic7C2×D311yescharacteristic
80Dic7⋊C4—31no—
81Dic7⋊C4—32no—
82Dic7⋊C4—32no—
83C4×Dic7—31no—
84C4×Dic7—32no—
85C4×Dic7—32no—
86C4×Dic7C2×D312yes—
87C3×#117—71no—
88C4×#67—71no—
89#924D711yescharacteristic, complemented
90#897—71no—
91#892—71no—
92#888—71no—
93#887—71no—
94#885—72no—
95#879—72no—
96#876—72no—
97C4×Dic6—72no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C22×C3×C7—21no—
101C22×C3×C7C2411yescharacteristic
102C2×D21—21no—
103C2×D21—21no—
104C2×D21—21no—
105C3×C4×C7—21no—
106C3×C4×C7#4112yes—
107Dic21—21no—
108Dic21—21no—
109Dic21#4112yes—
110C3×Dic7—22no—
111C3×Dic7—22no—
112C7×Dic3—22no—
113C7×Dic3—22no—
114#556—211noSylow
115C23×C7C22×D311yescharacteristic
116C22×D7—31no—
117C7×Q8—62no—
118C2×C4×C7—22no—
119C2×C4×C7—32no—
120C2×C4×C7—32no—
121C2×C4×C7C22×D311yescharacteristic
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—62no—
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—31no—
127C2×Dic7C22×D312yes—
128C2×Dic7C22×D312yes—
129C4×D7—62no—
130Dic14—64no—
131C22×C3×C4C2×D711yescharacteristic
132C2×#67—71no—
133C2×C4×D3—71no—
134C2×Dic6—72no—
135C2×Dic6C2×D711yescharacteristic
136C3×C4⋊C4—71no—
137C3×C4⋊C4—71no—
138C3×C22⋊C4—72no—
139C3×C42—72no—
140#217—71no—
141#217C2×D712yes—
142(C2×D3)⋊C4—71no—
143(C2×D3)⋊C4—72no—
144(C2×D3)⋊C4—72no—
145C4⋊Dic3—72no—
146C4⋊Dic3C2×D711yescharacteristic
147Dic3⋊C4—71no—
148Dic3⋊C4—72no—
149Dic3⋊C4—72no—
150Dic3⋊C4C2×D712yes—
151C4×Dic3—71no—
152C4×Dic3—72no—
153C4×Dic3—72no—
154C2×C3×C7—41no—
155C2×C3×C7C2×#4111yescharacteristic
156C2×C3×C7C2×#4111yescharacteristic
157C2×C3×C72−1+411yescharacteristic
158D21—41no—
159C4⋊Q8—211no—
160#126—212no—
161#126—212no—
162#124—212no—
163C22⋊Q8—31no—
164C22⋊Q8—211no—
165C4×Q8—212no—
166C4×Q8—211no—
167C4×D4—211no—
168#117—71no—
169#117—211no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7C23×D311yescharacteristic
174C2×D7—61no—
175C2×D7—61no—
176C2×D7—61no—
177C4×C7—21no—
178C4×C7—62no—
179C4×C7—62no—
180C4×C7#23512yes—
181Dic7—22no—
182Dic7—22no—
183Dic7—32no—
184Dic7—61no—
185Dic7—61no—
186C23×C3C22×D711yescharacteristic
187C22×D3—71no—
188C2×C3×C4—22no—
189C2×C3×C4—72no—
190C2×C3×C4—72no—
191C2×C3×C4C22×D711yescharacteristic
192C2×C3×C4C22×D711yescharacteristic
193#67—142no—
194C2×Dic3—71no—
195C2×Dic3—71no—
196C2×Dic3—71no—
197C2×Dic3C22×D712yes—
198C2×Dic3C22×D712yes—
199C4×D3—142no—
200Dic6—22no—
201Dic6—144no—
202C3×C7#55611yescharacteristic, complemented, Hall
203C2×Q8—31no—
204C2×Q8—212no—
205C2×D4—211no—
206C22×C4—211no—
207C22×C4D3×D711yescharacteristic
208C4⋊C4—31no—
209C4⋊C4—32no—
210C4⋊C4—71no—
211C4⋊C4—71no—
212C4⋊C4—212no—
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C22⋊C4—32no—
217C22⋊C4—72no—
218C22⋊C4—212no—
219C22⋊C4—212no—
220C22⋊C4—211no—
221C22⋊C4—211no—
222C42—72no—
223C42—212no—
224C42—212no—
225C42—211no—
226C2×C7—41no—
227C2×C7C2×#23511yescharacteristic
228C2×C7D3×#4111yescharacteristic
229C2×C7#101011yescharacteristic
230D7—121no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3C23×D711yescharacteristic
235C2×D3—141no—
236C2×D3—141no—
237C2×D3—141no—
238C3×C4—21no—
239C3×C4—142no—
240C3×C4—142no—
241C3×C4#115912yes—
242Dic3—22no—
243Dic3—22no—
244Dic3—72no—
245Dic3—141no—
246Dic3—141no—
247C23—211no—
248C23C2×D3×D711yescharacteristic
249Q8—62no—
250Q8—424no—
251D4—422no—
252C2×C4—22no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—72no—
256C2×C4—72no—
257C2×C4—211no—
258C2×C4—211no—
259C2×C4—211no—
260C2×C4—422no—
261C2×C4C2×D3×D711yescharacteristic
262C2×C4C2×D3×D711yescharacteristic
263C7#553311yescharacteristic, complemented, Sylow
264C2×C3—41no—
265C2×C3C2×#115911yescharacteristic
266C2×C3#655111yescharacteristic
267C2×C3D7×#4111yescharacteristic
268D3—281no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—421no—
273C22—421no—
274C22—421no—
275C22C22×D3×D711yescharacteristic
276C4—21no—
277C4—62no—
278C4—62no—
279C4—142no—
280C4—142no—
281C4—212no—
282C4—421no—
283C4—421no—
284C4#6668612yes—
285C3#9033211yescharacteristic, complemented, Sylow
286C2—41no—
287C2—841no—
288C2#1062154711yescharacteristic
289C2#1062156311yescharacteristic
290C2C2×#6668611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×#4268C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#556
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C7×#924C2semidirect
3. C3×#6489C2semidirect
4. #10621092C2semidirect
6. #10621032C2semidirect
7. #10620949C2semidirect
8. #10620945C2semidirect
9. #10620909C2semidirect
10. #10620906C2semidirect
11. #10620885C2semidirect
40. #6489D3semidirect
89. #924D7semidirect
202. C3×C7#556semidirect
263. C7#5533semidirect
285. C3#90332semidirect

Maximal subgroups

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

subgroupindex
C4×#42682
C7×#9242
C3×#64892
#106210922
#106210882
#106210322
#106209492
#106209452
#106209092
#106209062
#106208852
#903323
#55337