Finite Group Databasebeta

← D7×#3805 #72245762

Subgroups — D7×#3805 #72245762

Order 2016.

710 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 148 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#11759637C211yesmaximal, characteristic, complemented
2#11759616C212yesmaximal, complemented
3C7×#3805C211yesmaximal, characteristic, complemented
4C2×D7×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×D7×#67C212yesmaximal, complemented
6#11759444C211yesmaximal, characteristic, complemented
7#11759443C211yesmaximal, characteristic, complemented
8#11759442C212yesmaximal, complemented
9D7×#627C211yesmaximal, characteristic, complemented
10D7×#625C211yesmaximal, characteristic, complemented
11D3×Dic3×D7C212yesmaximal, complemented
12D7×#237—32nomaximal
13C22×C32×D7C2211yescharacteristic
14C2×C3×D3×D7C2212yes—
15C2×C3⋊Dic21C2211yescharacteristic, complemented
16C2×C7×C3⋊Dic3C2211yescharacteristic, complemented
17C3×#4268C2212yescomplemented
18C3×C7×#67C2212yescomplemented
19#92696C2211yescharacteristic, complemented
20#92695C2211yescharacteristic, complemented
21D21⋊Dic3C2212yescomplemented
22#92691C2212yescomplemented
23#92688C2212yescomplemented
24D3×Dic21C2212yescomplemented
25Dic3×D21C2212yescomplemented
26C7×#627C2211yescharacteristic, complemented
27C7×#625C2211yescharacteristic, complemented
28D3×C7×Dic3C2212yescomplemented
29D7×C3⋊Dic3C2211yescharacteristic
30D7×C3⋊Dic3C2211yescharacteristic
31D7×C3⋊Dic3C2211yescharacteristic, complemented
32D7×C3⋊Dic3C2211yescharacteristic, complemented
33C3×#4246C2212yescomplemented
34C3×#4245C2212yescomplemented
35C3×Dic3×D7C2212yes—
36#66730—32no—
37C7×#237—32no—
38C3×D4×D7—32no—
39D7×#67—32no—
40D7×#67D312yescomplemented
41#66684—32no—
42#66684—32no—
43#66683—32no—
44#66683—32no—
45C2×Dic3×D7—32no—
46C2×Dic3×D7—32no—
47C2×Dic3×D7—61no—
48C4×D3×D7—32no—
49#66670—32no—
50#66669—32no—
51D7×Dic6—32no—
52C2×#3805—71nomaximal, Hall
53C22×C32×C7C2311yescharacteristic
54C2×C3×D21C2312yes—
55C2×C3×D3×C7C2312yes—
56C2×C32×D7—22no—
57C2×C32×D7C2311yescharacteristic
58C2×C32×D7C2311yescharacteristic
59C3×D3×D7—24no—
60C3⋊Dic21C2311yescharacteristic
61C3⋊Dic21C2311yescharacteristic
62C7×C3⋊Dic3C2311yescharacteristic
63C7×C3⋊Dic3C2311yescharacteristic
64C3×Dic21C2312yes—
65C3×C7×Dic3C2312yes—
66D7×#41—91noHall
67C22×C3×D7—21no—
68C22×C3×D7—32no—
69C22×C3×D7C2×D312yes—
70C2×D3×D7C2×D312yes—
71C3×C7×D4—32no—
72#4268—32no—
73#4268C2×D312yescomplemented
74C2×Dic21—32no—
75C2×Dic21—32no—
76C2×Dic21—61no—
77C4×D21—32no—
78Dic42—32no—
79C7×#67—32no—
80C7×#67C2×D312yescomplemented
81C2×C7×Dic3—32no—
82C2×C7×Dic3—32no—
83C2×C7×Dic3—61no—
84C4×D3×C7—32no—
85C7×Dic6—32no—
86C3×C22⋊D7—32no—
87C3×C22⋊D7—32no—
88C3×D28—32no—
89C3×C4×D7—32no—
90#4248—32no—
91#4248—32no—
92#4246—32no—
93#4246C2×D312yescomplemented
94#4245—32no—
95#4245C2×D312yescomplemented
96D21⋊C4—32no—
97D3×Dic7—32no—
98Dic3×D7—32no—
99Dic3×D7—32no—
100Dic3×D7—32no—
101Dic3×D7—32no—
102Dic3×D7—32no—
103Dic3×D7—32no—
104Dic3×D7—61no—
105Dic3×D7—61no—
106Dic3×D7—61no—
107Dic3×D7—61no—
108Dic3×D7C2×D312yes—
109C22×C3⋊Dic3—71no—
110C2×C3×#67—72no—
111C2×#627—71no—
112C2×#625—71no—
113#3805—71no—
114#3805—72no—
115#3805—71no—
116#3805—71no—
117#3805—72no—
118#3805D711yescharacteristic, direct product factor
119C2×D3×Dic3—72no—
120C2×C32×C7—21no—
121C2×C32×C7C2411yescharacteristic
122C3×D21—22no—
123C3×D3×C7—22no—
124C32×D7—21no—
125C32×D7#4112yescomplemented
126C7×#41—91no—
127#1161—91no—
128Q8×D7—91no—
129#1159—91no—
130#1159—92no—
131D4×D7—32no—
132D4×D7—91no—
133#1157—92no—
134#1157—91no—
135C2×C4×D7—92no—
136C2×C4×D7—91no—
137C2×#237—212no—
138C22×C3×C7—21no—
139C22×C3×C7—32no—
140C22×C3×C7C22×D312yes—
141C2×D21C22×D312yes—
142C2×D3×C7C22×D312yes—
143C2×C3×D7—22no—
144C2×C3×D7—22no—
145C2×C3×D7—24no—
146C2×C3×D7—21no—
147C2×C3×D7—21no—
148C2×C3×D7—32no—
149C2×C3×D7—64no—
150C2×C3×D7C22×D312yes—
151C2×C3×D7C22×D312yes—
152D3×D7—24no—
153C3×C4×C7—32no—
154Dic21—32no—
155Dic21—32no—
156Dic21—61no—
157Dic21—61no—
158Dic21C22×D312yes—
159C3×Dic7—32no—
160C7×Dic3—32no—
161C7×Dic3—32no—
162C7×Dic3—61no—
163C7×Dic3—61no—
164C7×Dic3C22×D312yes—
165C23×C32—71no—
166C22×C3×D3—72no—
167C2×C3⋊Dic3—71no—
168C2×C3⋊Dic3—71no—
169C2×C3⋊Dic3—71no—
170C2×C3⋊Dic3—71no—
171C2×C3⋊Dic3—71no—
172C2×C3⋊Dic3C2×D711yescharacteristic, complemented
173C3×#67—72no—
174C3×#67—72no—
175C3×#67—72no—
176C3×#67C2×D712yescomplemented
177C2×C3×Dic3—72no—
178#627—71no—
179#627—72no—
180#627C2×D711yescharacteristic, complemented
181#625—72no—
182#625—71no—
183#625C2×D711yescharacteristic, complemented
184D3×Dic3—72no—
185D3×Dic3—72no—
186D3×Dic3—72no—
187D3×Dic3C2×D712yescomplemented
188C32×C7C2×#4111yescharacteristic, complemented, Hall
189C22×D7—32no—
190C22×D7D3211yescharacteristic
191C7×Q8—91no—
192C7×D4—32no—
193C7×D4—91no—
194C2×C4×C7—92no—
195C2×C4×C7—91no—
196C22⋊D7—32no—
197C22⋊D7—32no—
198C22⋊D7—92no—
199C2×Dic7—92no—
200C2×Dic7—91no—
201D28—32no—
202D28—91no—
203C4×D7—32no—
204C4×D7—92no—
205C4×D7—92no—
206C4×D7—91no—
207C4×D7—91no—
208C4×D7—91no—
209C4×D7—91no—
210Dic14—91no—
211Dic14—92no—
212C2×C3×D4—212no—
213C2×#67—72no—
214C2×#67—212no—
215C22×Dic3—212no—
216C22×Dic3—212no—
217C22×Dic3—421no—
218#237—32no—
219#237—212no—
220#237—212no—
221#237—212no—
222#237—212no—
223#237—212no—
224#237—212no—
225#237—212no—
226C2×C4×D3—212no—
227C2×Dic6—212no—
228C2×C3×C7—21no—
229C2×C3×C7—21no—
230C2×C3×C7—22no—
231C2×C3×C7—21no—
232C2×C3×C7—62no—
233C2×C3×C7C23×D312yes—
234D21—22no—
235C3×D7—21no—
236C3×D7—21no—
237C3×D7—22no—
238C3×D7—22no—
239C3×D7—62no—
240C3×D7#23714yescomplemented
241D3×C7—22no—
242C22×C32—71no—
243C22×C32—71no—
244C22×C32—142no—
245C22×C32C22×D711yescharacteristic
246C2×C3×D3—72no—
247C2×C3×D3—144no—
248C2×C3×D3C22×D712yes—
249C3⋊Dic3—71no—
250C3⋊Dic3—71no—
251C3⋊Dic3C22×D711yescharacteristic
252C3⋊Dic3C22×D711yescharacteristic
253C3×Dic3—72no—
254C3×Dic3C22×D712yes—
255C2×#41—631noSylow
256C22×C7—32no—
257C22×C7C2×D3211yescharacteristic
258C2×D7—22no—
259C2×D7—32no—
260C2×D7—64no—
261C2×D7C2×D3211yescharacteristic
262C2×D7C2×D3211yescharacteristic
263C4×C7—32no—
264C4×C7—91no—
265C4×C7—91no—
266Dic7—32no—
267Dic7—91no—
268Dic7—91no—
269C23×C3—72no—
270C23×C3—141no—
271C23×C3—212no—
272C22×D3—72no—
273C3×D4—32no—
274C3×D4—212no—
275C3×D4—212no—
276C3×D4—212no—
277C2×C3×C4—212no—
278#67—32no—
279#67—72no—
280#67—72no—
281#67—72no—
282#67—212no—
283#67—212no—
284#67—212no—
285#67D3×D712yescomplemented
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—61no—
289C2×Dic3—72no—
290C2×Dic3—212no—
291C2×Dic3—212no—
292C2×Dic3—212no—
293C2×Dic3—212no—
294C2×Dic3—212no—
295C2×Dic3—212no—
296C2×Dic3—212no—
297C2×Dic3—212no—
298C2×Dic3—421no—
299C2×Dic3—421no—
300C2×Dic3—421no—
301C2×Dic3—421no—
302C2×Dic3—421no—
303C4×D3—32no—
304C4×D3—212no—
305C4×D3—212no—
306C4×D3—212no—
307Dic6—32no—
308Dic6—212no—
309Dic6—212no—
310Dic6—212no—
311C3×C7—21no—
312C3×C7C2×#23712yescomplemented
313C2×C32—21no—
314C2×C32—72no—
315C2×C32—141no—
316C2×C32C23×D711yescharacteristic
317C3×D3—22no—
318C3×D3—142no—
319#41—91no—
320#41—632no—
321#41—631no—
322#41—632no—
323#41—631no—
324#41—631no—
325C2×Q8—631no—
326C2×D4—212no—
327C2×D4—631no—
328C22×C4—632no—
329C22×C4—631no—
330C2×C7—21no—
331C2×C7—62no—
332C2×C7C22×D3211yescharacteristic
333D7—21no—
334D7—62no—
335D7#380512yesdirect product factor
336C22×C3—21no—
337C22×C3—32no—
338C22×C3—72no—
339C22×C3—72no—
340C22×C3—141no—
341C22×C3—141no—
342C22×C3—144no—
343C22×C3—142no—
344C22×C3—142no—
345C22×C3—212no—
346C22×C3—424no—
347C22×C3C2×D3×D712yes—
348C2×D3—72no—
349C2×D3—144no—
350C2×D3C2×D3×D712yes—
351C3×C4—32no—
352C3×C4—212no—
353Dic3—32no—
354Dic3—32no—
355Dic3—61no—
356Dic3—61no—
357Dic3—72no—
358Dic3—212no—
359Dic3—212no—
360Dic3—421no—
361Dic3—421no—
362Dic3C2×D3×D712yes—
363C32D7×#4111yescharacteristic, complemented, Sylow
364C23—71no—
365C23—212no—
366Q8—91no—
367Q8—632no—
368Q8—631no—
369D4—32no—
370D4—91no—
371D4—212no—
372D4—212no—
373D4—212no—
374D4—631no—
375D4—632no—
376C2×C4—92no—
377C2×C4—91no—
378C2×C4—212no—
379C2×C4—632no—
380C2×C4—632no—
381C2×C4—631no—
382C2×C4—631no—
383C2×C4—631no—
384C2×C4—631no—
385C2×C4—632no—
386C2×C4—631no—
387C7C2×#380511yescharacteristic, complemented, Sylow
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—22no—
391C2×C3—21no—
392C2×C3—62no—
393C2×C3—74no—
394C2×C3—142no—
395C2×C3—141no—
396C2×C3—141no—
397C2×C3—142no—
398C2×C3—422no—
399C2×C3C22×D3×D712yes—
400D3—22no—
401D3—142no—
402C22—32no—
403C22—71no—
404C22—71no—
405C22—142no—
406C22—212no—
407C22—424no—
408C22D32×D711yescharacteristic
409C4—32no—
410C4—91no—
411C4—91no—
412C4—212no—
413C4—631no—
414C4—631no—
415C3—21no—
416C3D7×#23712yescomplemented
417C2—21no—
418C2—62no—
419C2—72no—
420C2—141no—
421C2—422no—
422C2C2×D32×D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D7
commutator, socle, cosocleC2×C32×C7C24
FittingC22×C32×C7C23
hypercenterC22D32×D7
hypocenterC32×C7C2×#41
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11759637C2semidirect
2. #11759616C2semidirect
3. C7×#3805C2semidirect
4. C2×D7×C3⋊Dic3C2semidirect
5. C3×D7×#67C2semidirect
6. #11759444C2semidirect
7. #11759443C2semidirect
8. #11759442C2semidirect
9. D7×#627C2semidirect
10. D7×#625C2semidirect
11. D3×Dic3×D7C2semidirect
15. C2×C3⋊Dic21C22semidirect
16. C2×C7×C3⋊Dic3C22semidirect
17. C3×#4268C22semidirect
18. C3×C7×#67C22semidirect
19. #92696C22semidirect
20. #92695C22semidirect
21. D21⋊Dic3C22semidirect
22. #92691C22semidirect
23. #92688C22semidirect
24. D3×Dic21C22semidirect
25. Dic3×D21C22semidirect
26. C7×#627C22semidirect
27. C7×#625C22semidirect
28. D3×C7×Dic3C22semidirect
31. D7×C3⋊Dic3C22semidirect
32. D7×C3⋊Dic3C22semidirect
33. C3×#4246C22semidirect
34. C3×#4245C22semidirect
40. D7×#67D3semidirect
73. #4268C2×D3semidirect
80. C7×#67C2×D3semidirect
93. #4246C2×D3semidirect
95. #4245C2×D3semidirect
118. #3805D7direct (N × H)
125. C32×D7#41semidirect
172. C2×C3⋊Dic3C2×D7semidirect
176. C3×#67C2×D7semidirect
180. #627C2×D7semidirect
183. #625C2×D7semidirect
187. D3×Dic3C2×D7semidirect
188. C32×C7C2×#41semidirect
240. C3×D7#237semidirect
285. #67D3×D7semidirect
312. C3×C7C2×#237semidirect
335. D7#3805direct (N × H)
363. C32D7×#41semidirect
387. C7C2×#3805semidirect
416. C3D7×#237semidirect

Maximal subgroups

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

subgroupindex
#117596372
#117596162
C7×#38052
C2×D7×C3⋊Dic32
C3×D7×#672
#117594442
#117594432
#117594422
D7×#6272
D7×#6252
D3×Dic3×D72
D7×#2373
C2×#38057