Finite Group Databasebeta

← #13060133

Subgroups — #13060133

Order 1344.

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

normal subgroups only

This group has 692 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
1C2×D4×D21C211yesmaximal, characteristic, complemented
2C22×#4246C211yesmaximal, characteristic, complemented
3C7×#935C211yesmaximal, characteristic, complemented
4C3×#6500C211yesmaximal, characteristic, complemented
5#10621134C212yesmaximal, complemented
6#10621126C211yesmaximal, characteristic, complemented
7#10621123C211yesmaximal, characteristic, complemented
8#10621104C211yesmaximal, characteristic, complemented
9#10621095C212yesmaximal, complemented
10#10621021C212yesmaximal, complemented
11#10620916C212yesmaximal, complemented
12#90344—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C23×D21C2211yescharacteristic, complemented
15C23×C3×D7C2211yescharacteristic, complemented
16C2×C3×C7×D4C2211yescharacteristic
17C2×#4268C2211yescharacteristic
18C2×#4268C2211yescharacteristic, complemented
19D4×D21—24no—
20C2×D84C2211yescharacteristic, complemented
21C2×C4×D21C2211yescharacteristic, complemented
22C22×C7×Dic3C2211yescharacteristic, complemented
23C2×C3×C22⋊D7C2212yescomplemented
24C2×#4246—24no—
25C2×#4246C2212yes—
26C2×#4246C2212yescomplemented
27C2×#4246C2212yescomplemented
28C2×D21⋊C4C2212yescomplemented
29C7×#217C2212yescomplemented
30C7×#217C2211yescharacteristic, complemented
31C7×Dic3⋊C4C2212yescomplemented
32C3×C22⋊Dic7C2211yescharacteristic, complemented
33C3×(C2×D7)⋊C4C2212yescomplemented
34Dic7⋊Dic3C2212yescomplemented
35#66575C2212yescomplemented
36#66575C2212yescomplemented
37(C2×D7)⋊Dic3C2212yescomplemented
38C2×D4×D7—31no—
39C22×D28—31no—
40C7×#123—31no—
41#6500D311yescharacteristic, complemented
42#6492—31no—
43#6458—32no—
44#6457—32no—
45#6447—32no—
46#6445—31no—
47#6445—32no—
48D7×C22⋊C4—31no—
49#5589—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C23×C3×C7C2311yescharacteristic
52C22×D21—22no—
53C22×D21—42no—
54C22×D21—42no—
55C22×D21D414yescomplemented
56C22×D21C2311yescharacteristic
57C22×D21C2311yescharacteristic
58C22×D21C2311yescharacteristic
59C22×C3×D7—22no—
60C22×C3×D7—42no—
61C22×C3×D7C2312yes—
62C3×C7×D4—22no—
63C2×C3×C4×C7C2311yescharacteristic
64#4268—22no—
65#4268—22no—
66C2×Dic21C2311yescharacteristic
67D84—22no—
68C4×D21—22no—
69C2×C7×Dic3—22no—
70C2×C7×Dic3C2312yes—
71C2×C7×Dic3C2312yes—
72C3×C22⋊D7—42no—
73C2×C3×Dic7C2312yes—
74#4246—24no—
75#4246—24no—
76#4246—42no—
77D21⋊C4—42no—
78C23×D7—31no—
79C23×D7—31no—
80C23×D7C2×D311yescharacteristic, complemented
81C2×C7×D4C2×D311yescharacteristic
82C22×C4×C7—31no—
83C2×C22⋊D7—31no—
84C2×C22⋊D7—31no—
85C2×C22⋊D7C2×D312yescomplemented
86D4×D7—64no—
87C2×D28—32no—
88C2×D28—32no—
89C2×D28—31no—
90C2×D28—32no—
91C2×D28—64no—
92C2×C4×D7—31no—
93C2×C4×D7—32no—
94C7×C4⋊C4—32no—
95C7×C22⋊C4—31no—
96C7×C22⋊C4—32no—
97C22⋊Dic7C2×D311yescharacteristic, complemented
98(C2×D7)⋊C4—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101(C2×D7)⋊C4C2×D312yescomplemented
102Dic7⋊C4—32no—
103C22×#67—71no—
104C2×D3×D4—71no—
105C3×#120—71no—
106#953—71no—
107#938—72no—
108#937—72no—
109#935D711yescharacteristic, complemented
110#886—71no—
111#884—72no—
112#883—72no—
113D3×C22⋊C4—71no—
114C22×C3×C7—22no—
115C22×C3×C7—21no—
116C22×C3×C7—22no—
117C22×C3×C7C2×D412yes—
118C22×C3×C7C2411yescharacteristic
119C2×D21—22no—
120C2×D21—22no—
121C2×D21—21no—
122C2×D21—22no—
123C2×D21—44no—
124C2×D21—44no—
125C2×D21C2×D412yes—
126C2×D21C2×D412yes—
127C2×C3×D7—22no—
128C2×C3×D7—22no—
129C2×C3×D7—22no—
130C2×C3×D7—44no—
131C3×C4×C7—21no—
132Dic21—21no—
133C3×Dic7—22no—
134C7×Dic3—22no—
135C7×Dic3—22no—
136#534—211noSylow
137C23×C7C22×D311yescharacteristic
138C23×C7C22×D311yescharacteristic
139C22×D7—22no—
140C22×D7—31no—
141C22×D7—31no—
142C22×D7—34no—
143C22×D7—31no—
144C22×D7—42no—
145C22×D7—62no—
146C22×D7—122no—
147C22×D7—122no—
148C22×D7C22×D312yes—
149C7×D4—22no—
150C2×C4×C7—32no—
151C2×C4×C7—32no—
152C2×C4×C7—62no—
153C2×C4×C7C22×D311yescharacteristic
154C22⋊D7—42no—
155C22⋊D7—62no—
156C22⋊D7—62no—
157C2×Dic7—31no—
158C2×Dic7C22×D312yes—
159D28—62no—
160D28—64no—
161D28—64no—
162D28—122no—
163C4×D7—62no—
164C4×D7—122no—
165C24×C3—71no—
166C23×D3—71no—
167C23×D3—71no—
168C2×C3×D4—72no—
169C2×C3×D4C2×D711yescharacteristic
170C2×#67—71no—
171C2×#67—71no—
172C2×#67—72no—
173C2×#67—72no—
174C2×#67—72no—
175C2×#67—144no—
176C22×Dic3C2×D711yescharacteristic, complemented
177D3×D4—144no—
178C2×D12—71no—
179C2×C4×D3—71no—
180C2×C4×D3—72no—
181C3×C22⋊C4—71no—
182C3×C22⋊C4—72no—
183#217—72no—
184#217C2×D711yescharacteristic, complemented
185#217C2×D712yescomplemented
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4—72no—
188C4⋊Dic3—72no—
189Dic3⋊C4C2×D712yescomplemented
190C2×C3×C7—22no—
191C2×C3×C7—41no—
192C2×C3×C7C22×D411yescharacteristic
193C2×C3×C7H(5,𝔽2)12yes—
194D21—22no—
195D21—22no—
196D21—41no—
197C3×D7—42no—
198C22×D4—211no—
199C22×D4—211no—
200#123—31no—
201#123—212no—
202#123—211no—
203#121—212no—
204#121—212no—
205#120—71no—
206#120—212no—
207#120—211no—
208C2×C22⋊C4—211no—
209C22×C7—22no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7D3×D412yes—
213C22×C7C23×D311yescharacteristic
214C2×D7—22no—
215C2×D7—22no—
216C2×D7—22no—
217C2×D7—32no—
218C2×D7—32no—
219C2×D7—44no—
220C2×D7—62no—
221C2×D7—62no—
222C2×D7—61no—
223C2×D7—62no—
224C2×D7—124no—
225C2×D7—124no—
226C4×C7—21no—
227C4×C7—62no—
228C4×C7—62no—
229Dic7—22no—
230Dic7—61no—
231C23×C3—72no—
232C23×C3—142no—
233C23×C3—282no—
234C23×C3C22×D711yescharacteristic
235C23×C3C22×D711yescharacteristic
236C22×D3—71no—
237C22×D3—71no—
238C22×D3—74no—
239C22×D3—71no—
240C22×D3—142no—
241C22×D3—282no—
242C22×D3—282no—
243C3×D4—22no—
244C3×D4—282no—
245C2×C3×C4—72no—
246C2×C3×C4C22×D711yescharacteristic
247#67—144no—
248#67—144no—
249#67—142no—
250#67—142no—
251#67—282no—
252C2×Dic3—22no—
253C2×Dic3—71no—
254C2×Dic3C22×D712yes—
255C2×Dic3C22×D712yes—
256D12—142no—
257C4×D3—142no—
258C4×D3—282no—
259C3×C7#53411yescharacteristic, complemented, Hall
260C24—71no—
261C24—211no—
262C24—211no—
263C2×D4—72no—
264C2×D4—212no—
265C2×D4—212no—
266C2×D4—212no—
267C2×D4—211no—
268C2×D4—211no—
269C2×D4—211no—
270C2×D4—424no—
271C2×D4—424no—
272C2×D4D3×D711yescharacteristic
273C22×C4—31no—
274C22×C4—212no—
275C22×C4—211no—
276C4⋊C4—32no—
277C4⋊C4—212no—
278C22⋊C4—31no—
279C22⋊C4—32no—
280C22⋊C4—71no—
281C22⋊C4—72no—
282C22⋊C4—212no—
283C22⋊C4—212no—
284C22⋊C4—212no—
285C2×C7—22no—
286C2×C7—41no—
287C2×C7C2×D3×D411yescharacteristic
288C2×C7#100412yes—
289D7—42no—
290D7—62no—
291D7—62no—
292D7—121no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—142no—
297C22×C3—142no—
298C22×C3—142no—
299C22×C3—284no—
300C22×C3D4×D712yes—
301C22×C3C23×D711yescharacteristic
302C2×D3—72no—
303C2×D3—72no—
304C2×D3—142no—
305C2×D3—142no—
306C2×D3—141no—
307C2×D3—142no—
308C2×D3—284no—
309C2×D3—284no—
310C3×C4—21no—
311C3×C4—142no—
312Dic3—22no—
313Dic3—22no—
314Dic3—141no—
315C23—72no—
316C23—142no—
317C23—211no—
318C23—211no—
319C23—214no—
320C23—211no—
321C23—282no—
322C23—422no—
323C23—842no—
324C23—842no—
325C23C2×D3×D711yescharacteristic
326C23C2×D3×D711yescharacteristic
327D4—22no—
328D4—282no—
329D4—422no—
330D4—424no—
331D4—424no—
332D4—422no—
333D4—422no—
334D4—842no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—62no—
338C2×C4—72no—
339C2×C4—211no—
340C2×C4—422no—
341C2×C4—842no—
342C2×C4C2×D3×D711yescharacteristic
343C7#558911yescharacteristic, complemented, Sylow
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—282no—
347C2×C3C2×D4×D711yescharacteristic
348C2×C3#655312yes—
349D3—142no—
350D3—142no—
351D3—281no—
352C22—22no—
353C22—22no—
354C22—21no—
355C22—142no—
356C22—142no—
357C22—142no—
358C22—212no—
359C22—212no—
360C22—284no—
361C22—422no—
362C22—422no—
363C22—422no—
364C22—421no—
365C22—844no—
366C22—844no—
367C22#6669412yes—
368C22C22×D3×D711yescharacteristic
369C4—21no—
370C4—62no—
371C4—62no—
372C4—142no—
373C4—421no—
374C3#9034411yescharacteristic, complemented, Sylow
375C2—22no—
376C2—41no—
377C2—282no—
378C2—422no—
379C2—422no—
380C2—841no—
381C2#1062157312yes—
382C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#534
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×D21C2semidirect
2. C22×#4246C2semidirect
3. C7×#935C2semidirect
4. C3×#6500C2semidirect
5. #10621134C2semidirect
6. #10621126C2semidirect
7. #10621123C2semidirect
8. #10621104C2semidirect
9. #10621095C2semidirect
10. #10621021C2semidirect
11. #10620916C2semidirect
14. C23×D21C22semidirect
15. C23×C3×D7C22semidirect
18. C2×#4268C22semidirect
20. C2×D84C22semidirect
21. C2×C4×D21C22semidirect
22. C22×C7×Dic3C22semidirect
23. C2×C3×C22⋊D7C22semidirect
26. C2×#4246C22semidirect
27. C2×#4246C22semidirect
28. C2×D21⋊C4C22semidirect
29. C7×#217C22semidirect
30. C7×#217C22semidirect
31. C7×Dic3⋊C4C22semidirect
32. C3×C22⋊Dic7C22semidirect
33. C3×(C2×D7)⋊C4C22semidirect
34. Dic7⋊Dic3C22semidirect
35. #66575C22semidirect
36. #66575C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
41. #6500D3semidirect
55. C22×D21D4semidirect
80. C23×D7C2×D3semidirect
85. C2×C22⋊D7C2×D3semidirect
97. C22⋊Dic7C2×D3semidirect
101. (C2×D7)⋊C4C2×D3semidirect
109. #935D7semidirect
176. C22×Dic3C2×D7semidirect
184. #217C2×D7semidirect
185. #217C2×D7semidirect
189. Dic3⋊C4C2×D7semidirect
259. C3×C7#534semidirect
343. C7#5589semidirect
374. C3#90344semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C22×#42462
C7×#9352
C3×#65002
#106211342
#106211262
#106211232
#106211042
#106210952
#106210212
#106209162
#903443
#55897