Finite Group Databasebeta

← #13367656

Subgroups — #13367656

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
1C2×C7×#497C211yesmaximal, characteristic
2#11740672C211yesmaximal, characteristic, complemented
3#11740398C211yesmaximal, characteristic
4#11729819C212yesmaximal, complemented
5#11729819C212yesmaximal, complemented
6#11729802C212yesmaximal
7#11729801C212yesmaximal
8#11729539C211yesmaximal, characteristic, complemented
9#11729479C211yesmaximal, characteristic
10#11727569C211yesmaximal, characteristic, complemented
11#11727560C211yesmaximal, characteristic, complemented
12C22×C7×Q16C2212yes—
13C22×C7×SD16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#124C2211yescharacteristic
16C2×Q8×Dic7C2212yes—
17C2×Q8×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C7×#497C418yescomplemented
20C2×C7×#97C2211yescharacteristic
21#90002C2212yes—
22#90002C2211yescharacteristic
23C2×#6409C2212yes—
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26#89993C2211yescharacteristic
27C2×#6406C2212yes—
28#89753—22no—
29#89753—22no—
30#89723—22no—
31#89722—22no—
32#89708C2211yescharacteristic
33#89707C2211yescharacteristic
34#89704C2211yescharacteristic
35C2×C4×C7⋊C8C2211yescharacteristic
36#89346C2212yes—
37#89346C2212yes—
38#89345C2212yes—
39#89345C2212yes—
40#22278—71nomaximal, Sylow
41C22×C7×Q8C2311yescharacteristic
42C22×C7×Q8C2312yes—
43C22×C7×D4C2311yescharacteristic
44C23×Dic7—21no—
45C2×C7×Q16—42no—
46C2×C7×Q16—42no—
47C2×C7×Q16C2×C418yescomplemented
48C2×C7×SD16—42no—
49C2×C7×SD16—42no—
50C2×C7×SD16C2×C418yes—
51C22×C7×C8C2312yes—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8C2×C414yescomplemented
54C7×#124—22no—
55C7×#124C2×C414yes—
56C2×C7×C4⋊C4—22no—
57C2×C7×C22⋊C4—22no—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—21no—
60C2×C22⋊Dic7—22no—
61Q8×Dic7—42no—
62Q8×Dic7—42no—
63Q8×Dic7—42no—
64D4×Dic7—42no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2311yescharacteristic
70C2×C4⋊Dic7C2312yes—
71C2×C4×Dic7—21no—
72C2×C4×Dic7—22no—
73C2×C4×Dic7C2311yescharacteristic
74C2×C4×Dic7C2311yescharacteristic
75C22×C7⋊C8C2312yes—
76C7×#97C2×C414yescomplemented
77#6409—42no—
78#6409—42no—
79#6409—42no—
80#6406—42no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95D412yes—
86C7⋊#95D412yes—
87C4×C7⋊C8—22no—
88C2×#497D711yescharacteristic
89#2327—71no—
90#2326—71no—
91#1958—71no—
92#1956—71no—
93#1880—72no—
94#1878—72no—
95#1878—72no—
96#1877—72no—
97#1836—71no—
98#1746—71no—
99C24×C7—21no—
100C2×C7×Q8—21no—
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8—22no—
106C2×C7×Q8—22no—
107C2×C7×Q8—22no—
108C2×C7×Q8C22×C412yes—
109C2×C7×Q8C22×C414yes—
110C2×C7×D4—21no—
111C2×C7×D4—22no—
112C2×C7×D4—22no—
113C2×C7×D4C22×C412yes—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2411yescharacteristic
118C22×Dic7—22no—
119C22×Dic7—21no—
120C22×Dic7—21no—
121C22×Dic7—42no—
122C22×Dic7—82no—
123C22×Dic7C2×D411yescharacteristic
124C22×Dic7C2×D411yescharacteristic
125C22×Dic7C2×D412yes—
126C7×Q16—48no—
127C7×SD16—48no—
128C2×C7×C8—22no—
129C2×C7×C8—22no—
130C2×C7×C8C22×C414yes—
131C7×C4⋊C4—28no—
132C7×C22⋊C4—28no—
133C42×C7C22×C412yes—
134C42×C7#4112yes—
135C22⋊Dic7—42no—
136C22⋊Dic7—82no—
137C4⋊Dic7—22no—
138C4⋊Dic7—22no—
139C4⋊Dic7—22no—
140C4⋊Dic7—22no—
141C4⋊Dic7—44no—
142C4⋊Dic7—42no—
143C4⋊Dic7—82no—
144C4×Dic7—22no—
145C4×Dic7—22no—
146C4×Dic7—44no—
147C4×Dic7—42no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—22no—
150C2×C7⋊C8—22no—
151C22×Q16C2×D712yes—
152C22×SD16C2×D712yes—
153C2×C4⋊Q8C2×D711yescharacteristic
154C2×#124C2×D711yescharacteristic
155C2×C4×Q8—71no—
156C2×C4×Q8—72no—
157C2×C4×D4—71no—
158#497Dic718yescomplemented
159C2×#103—72no—
160C2×#103—72no—
161C2×#103—72no—
162C2×#102—72no—
163C2×#97C2×D711yescharacteristic
164C2×C4×C8—71no—
165#391—71no—
166#391—72no—
167#390—71no—
168#388—142no—
169#387—142no—
170#387—142no—
171#385—142no—
172#384—71no—
173#382—71no—
174#376—71no—
175#340—72no—
176#340—72no—
177#336—72no—
178#336—72no—
179C23×C7—22no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7—42no—
183C23×C7—42no—
184C23×C7C22×D411yescharacteristic
185C7×Q8—24no—
186C7×Q8—24no—
187C7×Q8—24no—
188C7×Q8—42no—
189C7×Q8—42no—
190C7×Q8—42no—
191C7×D4—24no—
192C7×D4—42no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—24no—
197C2×C4×C7—41no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—41no—
201C2×C4×C7C4×D414yes—
202C2×C4×C7C23×C411yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×C4×C7H(5,𝔽2)12yes—
205C2×C4×C7H(5,𝔽2)12yes—
206C2×Dic7—22no—
207C2×Dic7—21no—
208C2×Dic7—21no—
209C2×Dic7—22no—
210C2×Dic7—22no—
211C2×Dic7—22no—
212C2×Dic7—22no—
213C2×Dic7—41no—
214C2×Dic7—42no—
215C2×Dic7—41no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220C2×Dic7—84no—
221C7×C8—24no—
222C7⋊C8—42no—
223C22×Q8C22×D711yescharacteristic
224C22×Q8C22×D712yes—
225C22×D4C22×D711yescharacteristic
226C23×C4—141no—
227C2×Q16—42no—
228C2×Q16—42no—
229C2×Q16C2×Dic718yescomplemented
230C2×SD16—42no—
231C2×SD16—42no—
232C2×SD16C2×Dic718yes—
233C22×C8—72no—
234C22×C8C22×D712yes—
235C4⋊Q8—22no—
236C4⋊Q8C2×Dic714yescomplemented
237#124—22no—
238#124C2×Dic714yes—
239C4×Q8—282no—
240C4×Q8—282no—
241C4×Q8—282no—
242C4×D4—282no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—71no—
245C2×C4⋊C4—72no—
246C2×C4⋊C4—71no—
247C2×C4⋊C4—141no—
248C2×C4⋊C4—142no—
249C2×C4⋊C4—142no—
250C2×C22⋊C4—22no—
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—282no—
259#103—282no—
260#103—282no—
261#102—282no—
262#97C2×Dic714yescomplemented
263C4×C8—142no—
264#95—72no—
265#95—72no—
266#95—142no—
267#95—142no—
268#95—142no—
269#95—142no—
270C22×C7—42no—
271C22×C7—44no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—82no—
275C22×C7C2×C4×D411yescharacteristic
276C22×C7#53511yescharacteristic
277C22×C7#53511yescharacteristic
278C22×C7#57712yes—
279C22×C7#57812yes—
280C4×C7—24no—
281C4×C7—42no—
282C4×C7—44no—
283C4×C7#51812yes—
284C4×C7#54012yes—
285Dic7—41no—
286Dic7—41no—
287Dic7—42no—
288Dic7—81no—
289Dic7—81no—
290Dic7—82no—
291C24—21no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C22×Dic712yes—
301C2×Q8C22×Dic714yes—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4C22×Dic712yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—71no—
309C22×C4—71no—
310C22×C4—72no—
311C22×C4—142no—
312C22×C4—141no—
313C22×C4—141no—
314C22×C4—282no—
315C22×C4—562no—
316C22×C4D4×D712yes—
317C22×C4C23×D711yescharacteristic
318Q16—48no—
319SD16—48no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—142no—
323C2×C8—142no—
324C2×C8—142no—
325C2×C8C22×Dic714yes—
326C4⋊C4—28no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—142no—
330C4⋊C4—142no—
331C4⋊C4—284no—
332C4⋊C4—282no—
333C4⋊C4—562no—
334C22⋊C4—28no—
335C22⋊C4—282no—
336C22⋊C4—562no—
337C42—142no—
338C42—142no—
339C42—282no—
340C42—284no—
341C42#115912yes—
342C42C22×Dic712yes—
343C2×C7—82no—
344C2×C7#232811yescharacteristic
345C2×C7#294712yes—
346C2×C7#322211yescharacteristic
347C2×C7#322311yescharacteristic
348C2×C7#322512yes—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23C2×D4×D711yescharacteristic
355Q8—24no—
356Q8—24no—
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361D4—24no—
362D4—42no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—141no—
372C2×C4—142no—
373C2×C4—141no—
374C2×C4—142no—
375C2×C4—142no—
376C2×C4—142no—
377C2×C4—142no—
378C2×C4—281no—
379C2×C4—282no—
380C2×C4—281no—
381C2×C4—282no—
382C2×C4—282no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4—564no—
386C2×C4D4×Dic714yes—
387C2×C4C2×#115911yescharacteristic
388C2×C4#654812yes—
389C2×C4#654812yes—
390C2×C4C23×Dic711yescharacteristic
391C8—24no—
392C8—282no—
393C7#2227811yescharacteristic, complemented, Sylow
394C22—42no—
395C22—44no—
396C22—81no—
397C22—81no—
398C22—82no—
399C22#9040611yescharacteristic
400C22#9040611yescharacteristic
401C22#9046912yes—
402C22#9047012yes—
403C22C2×D4×Dic711yescharacteristic
404C4—24no—
405C4—42no—
406C4—44no—
407C4—281no—
408C4—281no—
409C4—282no—
410C4—561no—
411C4—561no—
412C4—562no—
413C4#9041012yes—
414C4#9048912yes—
415C2—82no—
416C2#1173932311yescharacteristic
417C2#1173932411yescharacteristic
418C2#1173932512yes—
419C2#1174061711yescharacteristic
420C2#1174141012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#497C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#497D7
hypocenterC7#22278
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. #11740672C2semidirect
4. #11729819C2semidirect
5. #11729819C2semidirect
8. #11729539C2semidirect
10. #11727569C2semidirect
11. #11727560C2semidirect
19. C7×#497C4semidirect
47. C2×C7×Q16C2×C4semidirect
53. C7×C4⋊Q8C2×C4semidirect
76. C7×#97C2×C4semidirect
158. #497Dic7semidirect
229. C2×Q16C2×Dic7semidirect
236. C4⋊Q8C2×Dic7semidirect
262. #97C2×Dic7semidirect
393. C7#22278semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4972
#117406722
#117403982
#117298192
#117298022
#117298012
#117295392
#117294792
#117275692
#117275602
#222787