Finite Group Databasebeta

← #13367652

Subgroups — #13367652

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740672C211yesmaximal, characteristic, complemented
2C2×#89858C211yesmaximal, characteristic
3#11731766C211yesmaximal, characteristic
4C7×#1956C211yesmaximal, characteristic, complemented
5#11728035C212yesmaximal, complemented
6#11728030C212yesmaximal
7#11727994C212yesmaximal
8#11727993C212yesmaximal, complemented
9#11727527C211yesmaximal, characteristic
10#11727505C211yesmaximal, characteristic, complemented
11#11727329C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×Q8×Dic7C2212yes—
14C22×#1144C2212yes—
15C22×#1143C2212yes—
16C2×#6456C2211yescharacteristic
17C2×#6450C2211yescharacteristic
18C2×#6438C2211yescharacteristic
19C2×C7×#103C2212yes—
20C2×C7×#103C2212yes—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#384C2211yescharacteristic
23#90002C2212yes—
24C2×#6395C2212yes—
25C2×#6390C2212yes—
26#89858C418yescomplemented
27#89762—22no—
28#89761—22no—
29#89760C2211yescharacteristic
30#89753—22no—
31#89753—22no—
32#89746C2211yescharacteristic
33#89708C2211yescharacteristic
34#89704C2211yescharacteristic
35C2×#6377C2211yescharacteristic
36#89345C2212yes—
37#89345C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#22278—71nomaximal, Sylow
41C22×C7×Q8C2312yes—
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C22×Dic14C2311yescharacteristic
45C22×C7×C8C2312yes—
46C7×C4⋊Q8—22no—
47C7×C4⋊Q8C2×C414yescomplemented
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4C2311yescharacteristic
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C42×C7C2311yescharacteristic
53Q8×Dic7—42no—
54Q8×Dic7—42no—
55C2×#1144—42no—
56C2×#1144—42no—
57C2×#1144C2×C418yescomplemented
58C2×#1143—42no—
59C2×#1143—42no—
60C2×#1143C2×C418yes—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—22no—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7C2312yes—
66C2×Dic7⋊C4—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7—22no—
69C2×C4×Dic7C2311yescharacteristic
70C2×C4×Dic7C2311yescharacteristic
71C22×C7⋊C8C2312yes—
72#6456—42no—
73#6450—42no—
74#6438—22no—
75#6438C2×C414yes—
76C7×#103—42no—
77C7×#103—42no—
78C4×C7×C8—22no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95D412yes—
84C7⋊#95D412yes—
85#6395—42no—
86#6390—42no—
87#6377C2×C414yescomplemented
88C2×#497—71no—
89#2327—71no—
90#2326—71no—
91#1958—71no—
92#1956D711yescharacteristic, complemented
93#1880—72no—
94#1878—72no—
95#1878—72no—
96#1877—72no—
97#1836—71no—
98#1746—71no—
99C23×D7—21no—
100C2×C7×Q8—22no—
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8C22×C414yes—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—22no—
113C22×Dic7C2×D411yescharacteristic
114C22×Dic7C2×D411yescharacteristic
115C22×Dic7C2×D412yes—
116C2×D28—22no—
117C2×D28—22no—
118C2×D28—21no—
119C2×D28C22×C412yes—
120C2×C4×D7—42no—
121C2×C4×D7—82no—
122C2×Dic14—22no—
123C2×Dic14—22no—
124C2×Dic14—21no—
125C2×Dic14C22×C412yes—
126C2×C7×C8—22no—
127C2×C7×C8—22no—
128C2×C7×C8—22no—
129C7×C4⋊C4—28no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—82no—
133C42×C7C22×C412yes—
134C42×C7#4112yes—
135#1144—48no—
136#1143—48no—
137(C2×D7)⋊C4—28no—
138(C2×D7)⋊C4—42no—
139(C2×D7)⋊C4—82no—
140C4⋊Dic7—22no—
141C4⋊Dic7—22no—
142C4⋊Dic7—44no—
143Dic7⋊C4—42no—
144C4×Dic7—22no—
145C4×Dic7—22no—
146C4×Dic7—42no—
147C4×Dic7—44no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—22no—
150C2×C7⋊C8C22×C414yes—
151C22×Q16—72no—
152C22×SD16—72no—
153C2×C4⋊Q8C2×D711yescharacteristic
154C2×#124—71no—
155C2×C4×Q8—72no—
156C2×C4×Q8—71no—
157C2×C4×D4—71no—
158#497—78no—
159C2×#103—72no—
160C2×#103C2×D712yes—
161C2×#103C2×D712yes—
162C2×#102—72no—
163C2×#97—71no—
164C2×C4×C8C2×D711yescharacteristic
165#391—72no—
166#391—71no—
167#390—71no—
168#388—142no—
169#387—142no—
170#387—142no—
171#385—142no—
172#384C2×D711yescharacteristic
173#382—71no—
174#376—71no—
175#340—72no—
176#340—72no—
177#336—72no—
178#336—72no—
179C23×C7C22×D411yescharacteristic
180C22×D7—22no—
181C22×D7—41no—
182C22×D7—41no—
183C22×D7—42no—
184C22×D7—42no—
185C7×Q8—24no—
186C7×Q8—24no—
187C7×Q8—42no—
188C7×Q8—42no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—24no—
192C2×C4×C7—42no—
193C2×C4×C7—42no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7C4×D414yes—
199C2×C4×C7C23×C411yescharacteristic
200C2×C4×C7C2×#4111yescharacteristic
201C2×C4×C7H(5,𝔽2)12yes—
202C2×C4×C7H(5,𝔽2)12yes—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—21no—
206C2×Dic7—21no—
207C2×Dic7—22no—
208C2×Dic7—22no—
209C2×Dic7—22no—
210C2×Dic7—22no—
211C2×Dic7—41no—
212C2×Dic7—41no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216D28—24no—
217D28—42no—
218C4×D7—84no—
219Dic14—24no—
220Dic14—42no—
221C7×C8—42no—
222C7⋊C8—24no—
223C22×Q8—71no—
224C22×Q8C22×D712yes—
225C22×D4—71no—
226C23×C4—141no—
227C2×Q16—78no—
228C2×Q16—282no—
229C2×Q16—282no—
230C2×SD16—78no—
231C2×SD16—282no—
232C2×SD16—282no—
233C22×C8—72no—
234C22×C8C22×D712yes—
235C4⋊Q8—22no—
236C4⋊Q8C4×D714yescomplemented
237#124—74no—
238#124—142no—
239C4×Q8—282no—
240C4×Q8—282no—
241C4×Q8—282no—
242C4×D4—282no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—72no—
246C2×C4⋊C4—141no—
247C2×C4⋊C4—142no—
248C2×C4⋊C4C22×D711yescharacteristic
249C2×C4⋊C4C22×D711yescharacteristic
250C2×C22⋊C4—142no—
251C2×C22⋊C4—141no—
252C2×C22⋊C4—142no—
253C2×C42—71no—
254C2×C42—71no—
255C2×C42—142no—
256C2×C42—141no—
257C2×C42C22×D711yescharacteristic
258#103—42no—
259#103—42no—
260#103—282no—
261#102—282no—
262#97—74no—
263C4×C8—22no—
264#95—72no—
265#95—72no—
266#95—142no—
267#95—142no—
268#95—142no—
269#95—142no—
270C22×C7C2×C4×D411yescharacteristic
271C22×C7#53511yescharacteristic
272C22×C7#53511yescharacteristic
273C22×C7#57712yes—
274C22×C7#57812yes—
275C2×D7—42no—
276C2×D7—44no—
277C2×D7—81no—
278C2×D7—81no—
279C2×D7—82no—
280C4×C7—24no—
281C4×C7—44no—
282C4×C7—81no—
283C4×C7—81no—
284C4×C7#51812yes—
285C4×C7#54012yes—
286Dic7—42no—
287Dic7—41no—
288Dic7—41no—
289Dic7—42no—
290Dic7—82no—
291C24—141no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—72no—
298C2×Q8—141no—
299C2×Q8—142no—
300C2×Q8—142no—
301C2×Q8C2×C4×D714yes—
302C2×D4—72no—
303C2×D4—141no—
304C2×D4—142no—
305C2×D4—142no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—71no—
310C22×C4—71no—
311C22×C4—72no—
312C22×C4—141no—
313C22×C4—142no—
314C22×C4—282no—
315C22×C4—562no—
316C22×C4D4×D712yes—
317C22×C4C23×D711yescharacteristic
318Q16—288no—
319SD16—288no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—74no—
324C2×C8—142no—
325C2×C8—142no—
326C4⋊C4—28no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—82no—
330C4⋊C4—142no—
331C4⋊C4—142no—
332C4⋊C4—282no—
333C4⋊C4—284no—
334C22⋊C4—148no—
335C22⋊C4—282no—
336C22⋊C4—562no—
337C42—142no—
338C42—142no—
339C42—284no—
340C42—282no—
341C42C2×C4×D712yes—
342C42#115912yes—
343C2×C7#232811yescharacteristic
344C2×C7#294712yes—
345C2×C7#322211yescharacteristic
346C2×C7#322311yescharacteristic
347C2×C7#322512yes—
348D7—82no—
349C23—142no—
350C23—281no—
351C23—282no—
352C23—282no—
353C23—281no—
354C23C2×D4×D711yescharacteristic
355Q8—24no—
356Q8—24no—
357Q8—42no—
358Q8—42no—
359Q8—144no—
360Q8—282no—
361D4—144no—
362D4—282no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—142no—
375C2×C4—142no—
376C2×C4—142no—
377C2×C4—142no—
378C2×C4—142no—
379C2×C4—142no—
380C2×C4—281no—
381C2×C4—281no—
382C2×C4—282no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4—564no—
386C2×C4(C22⋊D7)⋊C414yes—
387C2×C4C22×C4×D711yescharacteristic
388C2×C4C2×#115911yescharacteristic
389C2×C4#654812yes—
390C2×C4#655312yes—
391C8—42no—
392C8—144no—
393C7#2227811yescharacteristic, complemented, Sylow
394C22—284no—
395C22—282no—
396C22—561no—
397C22—561no—
398C22—562no—
399C22C2×(C22⋊D7)⋊C411yescharacteristic
400C22#9036511yescharacteristic
401C22#9040611yescharacteristic
402C22#9045312yes—
403C22#9045712yes—
404C4—24no—
405C4—44no—
406C4—81no—
407C4—81no—
408C4—281no—
409C4—281no—
410C4—282no—
411C4—282no—
412C4—562no—
413C4#9023612yes—
414C4#9031012yes—
415C2—562no—
416C2#1173374311yescharacteristic
417C2#1173621912yes—
418C2#1173905112yes—
419C2#1173921411yescharacteristic
420C2#1173924911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1956C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1956D7
hypocenterC7#22278
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. #11740672C2semidirect
4. C7×#1956C2semidirect
5. #11728035C2semidirect
8. #11727993C2semidirect
10. #11727505C2semidirect
11. #11727329C2semidirect
26. #89858C4semidirect
47. C7×C4⋊Q8C2×C4semidirect
57. C2×#1144C2×C4semidirect
87. #6377C2×C4semidirect
92. #1956D7semidirect
236. C4⋊Q8C4×D7semidirect
393. C7#22278semidirect

Maximal subgroups

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

subgroupindex
#117406722
C2×#898582
#117317662
C7×#19562
#117280352
#117280302
#117279942
#117279932
#117275272
#117275052
#117273292
#222787