Finite Group Databasebeta

← #13337735

Subgroups — #13337735

Order 1792.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3160C211yesmaximal, characteristic, complemented
2#11740799C212yesmaximal
3C8×#1157C211yesmaximal, characteristic
4#11729586C212yesmaximal, complemented
5#11729132C211yesmaximal, characteristic, complemented
6#11729131C211yesmaximal, characteristic, complemented
7#11729126C211yesmaximal, characteristic, complemented
8#11729125C211yesmaximal, characteristic, complemented
9#11729102C212yesmaximal, complemented
10#11729098C212yesmaximal, complemented
11#11727486C211yesmaximal, characteristic, complemented
12C7×#533C2212yes—
13C4×#1157C2211yescharacteristic
14C7×#500C2211yescharacteristic
15C7×#500C2211yescharacteristic
16C7×#499—21no—
17C7×#499—21no—
18C7×#498C2212yes—
19C7×#427—21no—
20C7×#427—21no—
21C2×C4×C7×C8C2211yescharacteristic
22C8×C22⋊D7—21no—
23C8×C22⋊D7—21no—
24#89863C2212yes—
25#89862C2212yes—
26#89856C2212yes—
27#89855C2212yes—
28#89819—22no—
29#89817—22no—
30#89802—22no—
31#89798—22no—
32#89698C2212yes—
33#89697C2211yescharacteristic
34C8×D28C2211yescharacteristic
35C8×D28C2211yescharacteristic
36#89457—21no—
37C4×C8×D7—21no—
38C8×Dic14C2211yescharacteristic
39C8×Dic14C2211yescharacteristic
40#89342C2212yes—
41#89342C2212yes—
42#89341C2212yes—
43#89341C2212yes—
44#89253C2211yescharacteristic
45#89253C2211yescharacteristic
46#89252C2212yes—
47#19273—71nomaximal, Sylow
48C2×#1157D411yescharacteristic
49C22×C7×C8D411yescharacteristic
50C22×C7×C8D411yescharacteristic
51C7×C4⋊Q8C2312yes—
52C7×C4⋊Q8C2312yes—
53C7×#125—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—22no—
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C7×#117—22no—
59C2×C42×C7C2311yescharacteristic
60C2×#1144—22no—
61C2×#1143—22no—
62C4×C22⋊D7—21no—
63C4×C22⋊D7—21no—
64#6489D411yescharacteristic
65C2×C8×D7—21no—
66C2×C8×D7—21no—
67C4×D28D411yescharacteristic
68C4×D28C2311yescharacteristic
69#6435—21no—
70C42×D7—21no—
71C4×Dic14D411yescharacteristic
72C4×Dic14C2311yescharacteristic
73C7×#107—21no—
74C7×#107—21no—
75C7×#107—21no—
76C7×#107—21no—
77C7×#106—22no—
78C7×#106—22no—
79C4×C7×C8D411yescharacteristic
80C4×C7×C8D411yescharacteristic
81C4×C7×C8C2311yescharacteristic
82C4×C7×C8C2311yescharacteristic
83#6409—22no—
84#6409—22no—
85C22⋊(C7⋊C8)—21no—
86C22⋊(C7⋊C8)—22no—
87(C2×D7)⋊C8—21no—
88(C2×D7)⋊C8—21no—
89Dic7⋊C8—21no—
90Dic7⋊C8—21no—
91C8×Dic7—21no—
92C8×Dic7—21no—
93#6384—22no—
94#6383—22no—
95#6382—22no—
96#6381—22no—
97C4⋊(C7⋊C8)C2312yes—
98C4⋊(C7⋊C8)C2312yes—
99C4⋊(C7⋊C8)C2311yescharacteristic
100C4⋊(C7⋊C8)C2311yescharacteristic
101#6377—22no—
102C4×C7⋊C8—21no—
103#3160D711yescharacteristic, complemented
104#3116—72no—
105C8×#41—71no—
106#1676—71no—
107#1675—71no—
108#1674—71no—
109#1673—71no—
110#1672—72no—
111#1671—72no—
112#1568—71no—
113#1554—72no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7#4111yescharacteristic
119C2×C22⋊D7—21no—
120#1157—22no—
121#1157—22no—
122C2×D28C2×D411yescharacteristic
123C2×C4×D7—21no—
124C2×C4×D7—21no—
125C2×C4×D7—21no—
126C2×Dic14C2×D411yescharacteristic
127C2×C7×C8—22no—
128C2×C7×C8—22no—
129C2×C7×C8C2×D411yescharacteristic
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8C2×D411yescharacteristic
132C2×C7×C8C2×D411yescharacteristic
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—42no—
138C7×C4⋊C4—42no—
139C7×C22⋊C4—42no—
140C42×C7—42no—
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7#4111yescharacteristic
144C42×C7C2411yescharacteristic
145C22⋊Dic7—21no—
146#1144—44no—
147#1143—44no—
148(C2×D7)⋊C4—21no—
149(C2×D7)⋊C4—21no—
150C4⋊Dic7C2×D411yescharacteristic
151C4⋊Dic7C2×D411yescharacteristic
152Dic7⋊C4—21no—
153Dic7⋊C4—21no—
154C4×Dic7—21no—
155C4×Dic7—21no—
156C4×Dic7—21no—
157C2×C7⋊C8—22no—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—21no—
160C2×C7⋊C8—21no—
161C8×D7—42no—
162C8×D7—42no—
163#533C2×D712yes—
164C4×#41—71no—
165#500C2×D711yescharacteristic
166#500C2×D711yescharacteristic
167#499—21no—
168#499—21no—
169#498C2×D712yes—
170#485—142no—
171#477—72no—
172#476—72no—
173#462—72no—
174#461—72no—
175#452—142no—
176C8×Q8—71no—
177C8×Q8—71no—
178#441—142no—
179#440—142no—
180C8×D4—71no—
181C8×D4—71no—
182C8×D4—141no—
183C8×D4—141no—
184#432—72no—
185#431—71no—
186#431—141no—
187#427—21no—
188#427—21no—
189C2×C4×C8—141no—
190C2×C4×C8C2×D711yescharacteristic
191#335—71no—
192#335—71no—
193#334—72no—
194#333—72no—
195#333—72no—
196#332—72no—
197#332—72no—
198C23×C7#12111yescharacteristic
199C22×D7—21no—
200C7×Q8—42no—
201C7×Q8—42no—
202C7×Q8—82no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—41no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7#12111yescharacteristic
211C2×C4×C7#12111yescharacteristic
212C2×C4×C7#12111yescharacteristic
213C2×C4×C7C22×D411yescharacteristic
214C2×C4×C7C22×D411yescharacteristic
215C2×C4×C7C2×#4111yescharacteristic
216C22⋊D7—24no—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7—21no—
220D28—21no—
221D28#13512yes—
222C4×D7—24no—
223C4×D7—42no—
224C4×D7—41no—
225C4×D7—41no—
226Dic14—21no—
227Dic14#13512yes—
228C7×C8—21no—
229C7×C8—21no—
230C7×C8#13712yes—
231C7×C8#13712yes—
232C7⋊C8—42no—
233C7⋊C8—42no—
234C7⋊C8—41no—
235C7⋊C8—41no—
236C2×#41—71no—
237C2×Q16—142no—
238C2×SD16—142no—
239C22×C8—141no—
240C22×C8—141no—
241C22×C8C22⋊D711yescharacteristic
242C22×C8C22⋊D711yescharacteristic
243C4⋊Q8C22×D712yes—
244C4⋊Q8C22×D712yes—
245#125—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—71no—
251C4×Q8—71no—
252C4×D4—71no—
253C4×D4—71no—
254C4×D4—141no—
255C4×D4—141no—
256#117—22no—
257#117—71no—
258#117—141no—
259C2×C42—141no—
260C2×C42C22×D711yescharacteristic
261#107—21no—
262#107—21no—
263#107—21no—
264#107—21no—
265#107—142no—
266#106—22no—
267#106—22no—
268#106—142no—
269C4⋊C8—71no—
270C4⋊C8—71no—
271C4⋊C8—72no—
272C4⋊C8—72no—
273C4⋊C8—141no—
274C4⋊C8—141no—
275#103—142no—
276#103—142no—
277#103—142no—
278#102—142no—
279C22⋊C8—142no—
280C22⋊C8—141no—
281C22⋊C8—141no—
282C22⋊C8—141no—
283#97—142no—
284C4×C8—141no—
285C4×C8—141no—
286C4×C8—141no—
287C4×C8C22⋊D711yescharacteristic
288C4×C8C22⋊D711yescharacteristic
289C4×C8C22×D711yescharacteristic
290C4×C8C22×D711yescharacteristic
291C22×C7—21no—
292C22×C7—21no—
293C22×C7—21no—
294C22×C7C2×#12111yescharacteristic
295C2×D7—22no—
296C2×D7—41no—
297C2×D7—41no—
298C4×C7—21no—
299C4×C7—21no—
300C4×C7—21no—
301C4×C7—21no—
302C4×C7—82no—
303C4×C7—82no—
304C4×C7C2×#13511yescharacteristic
305C4×C7C2×#13511yescharacteristic
306C4×C7C2×#13711yescharacteristic
307C4×C7C2×#13711yescharacteristic
308Dic7—22no—
309Dic7—41no—
310Dic7—41no—
311#41—142no—
312#41—142no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—71no—
316C2×D4—71no—
317C2×D4—141no—
318C22×C4—141no—
319C22×C4—141no—
320C22×C4—141no—
321C22×C4D4×D711yescharacteristic
322C22×C4#115911yescharacteristic
323C22×C4C2×C22⋊D711yescharacteristic
324Q16—284no—
325SD16—284no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—142no—
329C2×C8—142no—
330C2×C8—141no—
331C2×C8—141no—
332C2×C8—282no—
333C2×C8—282no—
334C2×C8C2×C22⋊D711yescharacteristic
335C2×C8C2×C22⋊D711yescharacteristic
336C2×C8C2×C22⋊D711yescharacteristic
337C2×C8C2×C22⋊D711yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—71no—
345C4⋊C4—71no—
346C4⋊C4—141no—
347C4⋊C4—141no—
348C22⋊C4—42no—
349C22⋊C4—141no—
350C22⋊C4—141no—
351C22⋊C4—141no—
352C42—42no—
353C42—141no—
354C42—141no—
355C42—141no—
356C42D4×D711yescharacteristic
357C42#115911yescharacteristic
358C42C2×C22⋊D711yescharacteristic
359C42C23×D711yescharacteristic
360C2×C7—41no—
361C2×C7#303911yescharacteristic
362C2×C7#305011yescharacteristic
363C2×C7#306711yescharacteristic
364D7—42no—
365C23—141no—
366C23#650111yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—72no—
370Q8—82no—
371Q8—141no—
372D4—72no—
373D4—144no—
374D4—141no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—144no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—282no—
387C2×C4—281no—
388C2×C4—281no—
389C2×C4#650111yescharacteristic
390C2×C4#650111yescharacteristic
391C2×C4#650111yescharacteristic
392C2×C4C2×D4×D711yescharacteristic
393C2×C4C2×#115911yescharacteristic
394C2×C4C22×C22⋊D711yescharacteristic
395C8—21no—
396C8—21no—
397C8—282no—
398C8—282no—
399C8—281no—
400C8—281no—
401C8#650512yes—
402C8#650512yes—
403C7#1927311yescharacteristic, complemented, Sylow
404C22—21no—
405C22—21no—
406C22—21no—
407C22—142no—
408C22—281no—
409C22—281no—
410C22C2×#650111yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—82no—
417C4—142no—
418C4—281no—
419C4—281no—
420C4D7×#13511yescharacteristic
421C4D7×#13511yescharacteristic
422C4C2×#650511yescharacteristic
423C4C2×#650511yescharacteristic
424C2—41no—
425C2—282no—
426C2#1174065411yescharacteristic
427C2#1174135911yescharacteristic
428C2#1174137311yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3160C2
socleC22×C7C2×#121
prime radicalC2×#1157D4
hypercenter#3160D7
hypocenterC7#19273
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
1. C7×#3160C2semidirect
4. #11729586C2semidirect
5. #11729132C2semidirect
6. #11729131C2semidirect
7. #11729126C2semidirect
8. #11729125C2semidirect
9. #11729102C2semidirect
10. #11729098C2semidirect
11. #11727486C2semidirect
103. #3160D7semidirect
403. C7#19273semidirect

Maximal subgroups

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

subgroupindex
C7×#31602
#117407992
C8×#11572
#117295862
#117291322
#117291312
#117291262
#117291252
#117291022
#117290982
#117274862
#192737