Finite Group Databasebeta

← #13058642

Subgroups — #13058642

Order 1344.

468 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 136 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#10621373C211yesmaximal, characteristic, complemented
2C7×#897C211yesmaximal, characteristic, complemented
3C3×#6456C211yesmaximal, characteristic, complemented
4#10621019C212yesmaximal, complemented
5#10621018C211yesmaximal, characteristic
6#10621002C212yesmaximal, complemented
7C4×D3×Dic7C211yesmaximal, characteristic, complemented
8#10620930C211yesmaximal, characteristic, complemented
9#10620923C212yesmaximal, complemented
10#10620920C212yesmaximal, complemented
11Dic6×Dic7C211yesmaximal, characteristic, complemented
12#90315—31nomaximal, Hall
13C2×C4×D3×C7C2211yescharacteristic
14C2×C7×Dic6C2211yescharacteristic
15C2×C3×D28C2211yescharacteristic
16C2×C3×C4×D7C2212yes—
17C2×#4245C2212yes—
18C2×D3×Dic7C2211yescharacteristic
19C2×D3×Dic7C2211yescharacteristic
20C3×C7×C4⋊C4C2211yescharacteristic
21C4⋊Dic21C2211yescharacteristic
22C4⋊Dic21C2212yes—
23#66629C2212yes—
24C4×Dic21C2211yescharacteristic
25C7×(C2×D3)⋊C4C2212yes—
26C7×C4⋊Dic3C2212yes—
27C3×(C2×D7)⋊C4C2212yes—
28C3×C4×Dic7C2211yescharacteristic
29Dic3⋊Dic7C2212yes—
30(C2×D7)⋊Dic3C2212yes—
31(C2×D7)⋊Dic3C2212yes—
32(C2×D7)⋊Dic3C2212yes—
33Dic3×Dic7C2211yescharacteristic
34Dic3×Dic7C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C7×C22⋊Q8—31no—
37#6510—31no—
38Q8×Dic7—31no—
39#6493—31no—
40C2×C4×Dic7—31no—
41#6461—32no—
42#6456D311yescharacteristic, complemented
43#6455—32no—
44#6453—31no—
45#6449—32no—
46(C22⋊D7)⋊C4—32no—
47#5557—71nomaximal, Hall
48C22×D3×C7C2311yescharacteristic
49C22×C3×D7C2312yes—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2312yes—
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C4×D3×C7—22no—
58C7×Dic6—22no—
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2311yescharacteristic
61C3×D28—22no—
62C3×C4×D7—24no—
63#4245—24no—
64D3×Dic7—22no—
65D3×Dic7—22no—
66C2×C7×Q8—31no—
67C22×C4×C7—31no—
68C2×C22⋊D7—32no—
69C22×Dic7—31no—
70C22×Dic7—31no—
71C2×D28C2×D311yescharacteristic
72C2×C4×D7C2×D312yes—
73C7×C4⋊C4—32no—
74C7×C4⋊C4C2×D311yescharacteristic
75C7×C22⋊C4—32no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4C2×D312yes—
80C4⋊Dic7—32no—
81C4⋊Dic7—31no—
82C4⋊Dic7—32no—
83Dic7⋊C4—32no—
84C4×Dic7—31no—
85C4×Dic7—32no—
86C4×Dic7—31no—
87C4×Dic7—31no—
88C4×Dic7C2×D311yescharacteristic
89C3×C4×D4—71no—
90#938—71no—
91#936—71no—
92C4×#67—72no—
93#926—72no—
94#897D711yescharacteristic, complemented
95#890—71no—
96#883—72no—
97#879—72no—
98C42×D3—71no—
99C4×Dic6—71no—
100C22×C3×C7C2411yescharacteristic
101C2×D3×C7—21no—
102C2×D3×C7—21no—
103C2×D3×C7#4112yes—
104C2×C3×D7—22no—
105C2×C3×D7—22no—
106C2×C3×D7—22no—
107C3×C4×C7—22no—
108C3×C4×C7#4112yes—
109Dic21—21no—
110Dic21—21no—
111Dic21—22no—
112C3×Dic7—21no—
113C3×Dic7#4112yes—
114C7×Dic3—22no—
115C7×Dic3—21no—
116#529—211noSylow
117C23×C7—31no—
118C22×D7C22×D312yes—
119C7×Q8—62no—
120C2×C4×C7—31no—
121C2×C4×C7—32no—
122C2×C4×C7—62no—
123C2×C4×C7C22×D312yes—
124C2×C4×C7C22×D311yescharacteristic
125C22⋊D7—64no—
126C2×Dic7—31no—
127C2×Dic7—31no—
128C2×Dic7—32no—
129C2×Dic7—62no—
130C2×Dic7—62no—
131C2×Dic7C22×D311yescharacteristic
132C2×Dic7C22×D311yescharacteristic
133D28—22no—
134C4×D7—24no—
135C2×C3×D4—71no—
136C22×C3×C4—72no—
137C2×#67—72no—
138C2×C4×D3—71no—
139C2×C4×D3—71no—
140C2×C4×D3C2×D711yescharacteristic
141C2×Dic6C2×D711yescharacteristic
142C3×C4⋊C4C2×D711yescharacteristic
143C3×C22⋊C4—72no—
144C3×C42—71no—
145#217—72no—
146#217—72no—
147#217—72no—
148(C2×D3)⋊C4C2×D712yes—
149C4⋊Dic3—71no—
150C4⋊Dic3—72no—
151C4⋊Dic3C2×D712yes—
152Dic3⋊C4—72no—
153Dic3⋊C4—72no—
154C4×Dic3—71no—
155C4×Dic3—72no—
156C4×Dic3—71no—
157C4×Dic3—71no—
158C2×C3×C7C2×#4111yescharacteristic
159C2×C3×C7C2×#4111yescharacteristic
160C2×C3×C7C2×#4111yescharacteristic
161C3×D7—42no—
162D3×C7—22no—
163#126—212no—
164#125—211no—
165#124—211no—
166#123—212no—
167C22⋊Q8—31no—
168#121—211no—
169C4×Q8—211no—
170C4×D4—71no—
171C4×D4—212no—
172#117—212no—
173C2×C42—211no—
174C22×C7—32no—
175C22×C7—61no—
176C22×C7—61no—
177C22×C7C23×D311yescharacteristic
178C2×D7—22no—
179C2×D7—22no—
180C2×D7—22no—
181C4×C7—22no—
182C4×C7—61no—
183C4×C7—62no—
184C4×C7#23512yes—
185Dic7—21no—
186Dic7—61no—
187Dic7—61no—
188Dic7—62no—
189Dic7#23712yes—
190C23×C3—72no—
191C22×D3C22×D711yescharacteristic
192C3×D4—142no—
193C2×C3×C4—71no—
194C2×C3×C4—71no—
195C2×C3×C4—144no—
196C2×C3×C4C22×D712yes—
197C2×C3×C4C22×D711yescharacteristic
198#67—144no—
199C2×Dic3—71no—
200C2×Dic3—71no—
201C2×Dic3—72no—
202C2×Dic3C22×D711yescharacteristic
203C2×Dic3C22×D712yes—
204C4×D3—22no—
205C4×D3—142no—
206C4×D3—142no—
207Dic6—22no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—31no—
210C2×D4—71no—
211C2×D4—212no—
212C22×C4—31no—
213C22×C4—72no—
214C22×C4—211no—
215C22×C4—211no—
216C4⋊C4—32no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—212no—
220C4⋊C4—211no—
221C4⋊C4D3×D711yescharacteristic
222C22⋊C4—32no—
223C22⋊C4—72no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C22⋊C4—212no—
227C42—71no—
228C42—211no—
229C42—211no—
230C42—212no—
231C42—211no—
232C2×C7—62no—
233C2×C7C2×#23511yescharacteristic
234C2×C7C2×#23711yescharacteristic
235C2×C7D3×#4111yescharacteristic
236D7—42no—
237C22×C3—142no—
238C22×C3—142no—
239C22×C3—142no—
240C22×C3C23×D711yescharacteristic
241C2×D3—21no—
242C2×D3—21no—
243C2×D3#116112yes—
244C3×C4—22no—
245C3×C4—72no—
246C3×C4—141no—
247C3×C4#115912yes—
248Dic3—22no—
249Dic3—21no—
250Dic3—141no—
251Dic3—142no—
252Dic3—141no—
253C23—31no—
254C23—72no—
255Q8—62no—
256D4—142no—
257D4—424no—
258C2×C4—31no—
259C2×C4—32no—
260C2×C4—62no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—144no—
264C2×C4—211no—
265C2×C4—211no—
266C2×C4—212no—
267C2×C4—422no—
268C2×C4—422no—
269C2×C4C2×D3×D712yes—
270C2×C4C2×D3×D711yescharacteristic
271C7#555711yescharacteristic, complemented, Sylow
272C2×C3—282no—
273C2×C3C2×#115911yescharacteristic
274C2×C3C2×#116111yescharacteristic
275C2×C3D7×#4111yescharacteristic
276D3—22no—
277C22—32no—
278C22—61no—
279C22—61no—
280C22—142no—
281C22—142no—
282C22—142no—
283C22C22×D3×D711yescharacteristic
284C4—22no—
285C4—61no—
286C4—62no—
287C4—72no—
288C4—141no—
289C4—421no—
290C4—421no—
291C4—422no—
292C4#6667612yes—
293C3#9031511yescharacteristic, complemented, Sylow
294C2—62no—
295C2—282no—
296C2C2×#6667611yescharacteristic
297C2D7×#23711yescharacteristic
298C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radical#10621018C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#529
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621373C2semidirect
2. C7×#897C2semidirect
3. C3×#6456C2semidirect
4. #10621019C2semidirect
6. #10621002C2semidirect
7. C4×D3×Dic7C2semidirect
8. #10620930C2semidirect
9. #10620923C2semidirect
10. #10620920C2semidirect
11. Dic6×Dic7C2semidirect
42. #6456D3semidirect
94. #897D7semidirect
208. C3×C7#529semidirect
271. C7#5557semidirect
293. C3#90315semidirect

Maximal subgroups

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

subgroupindex
#106213732
C7×#8972
C3×#64562
#106210192
#106210182
#106210022
C4×D3×Dic72
#106209302
#106209232
#106209202
Dic6×Dic72
#903153
#55577