Finite Group Databasebeta

← #13324938

Subgroups — #13324938

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737405C211yesmaximal, characteristic, complemented
2#11737350C211yesmaximal, characteristic
3#11737192C211yesmaximal, characteristic
4C7×#1584C211yesmaximal, characteristic, complemented
5#11728704C211yesmaximal, characteristic, complemented
6C8×#1144C211yesmaximal, characteristic, complemented
7#11728656C211yesmaximal, characteristic, complemented
8#11726328C211yesmaximal, characteristic, complemented
9#11726319C211yesmaximal, characteristic, complemented
10#11725997C211yesmaximal, characteristic, complemented
11#11725984C211yesmaximal, characteristic
12#11725977C211yesmaximal, characteristic, complemented
13#11725975C211yesmaximal, characteristic, complemented
14#11725839C211yesmaximal, characteristic, complemented
15#11725827C211yesmaximal, characteristic, complemented
16#90269C2211yescharacteristic
17C4×#1161C2211yescharacteristic
18C7×#446C2211yescharacteristic
19C7×C8×Q8C2211yescharacteristic
20C4×C7×Q16C2211yescharacteristic
21#89959C411yescharacteristic
22#89954C411yescharacteristic
23C4×#1144C2211yescharacteristic
24C4×#1143C2211yescharacteristic
25Q8×C7⋊C8C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89662C411yescharacteristic
28#89660C411yescharacteristic
29#89611—21no—
30#89610—21no—
31#89610C2211yescharacteristic
32#89607—21no—
33#89607C2211yescharacteristic
34D7×C4⋊C8C2211yescharacteristic
35#89603—21no—
36#89603C2211yescharacteristic
37#89590C411yescharacteristic
38#89587C411yescharacteristic
39#89582C411yescharacteristic
40#89563C411yescharacteristic
41#89480—21no—
42#89473—21no—
43C4×#1132C2211yescharacteristic
44#89460—21no—
45#89460C2211yescharacteristic
46C4×#1131—21no—
47C4×#1131C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×Q8⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56Q8⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89275C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89252C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89250C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#18013—71nomaximal, Sylow
68C2×#1161C2×C411yescharacteristic
69C2×C7×Q16C2×C411yescharacteristic
70C4×C7×Q8C2311yescharacteristic
71C4×C7×Q8C2311yescharacteristic
72#6509C2×C411yescharacteristic
73Q8×Dic7C2×C411yescharacteristic
74#6507C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77C2×#1132C2×C411yescharacteristic
78C2×#1131—21no—
79C2×#1131—21no—
80C2×#1131D411yescharacteristic
81C2×#1131D411yescharacteristic
82C2×C8×D7—21no—
83#6460C2×C411yescharacteristic
84#6459—21no—
85#6456—21no—
86#6456C2×C411yescharacteristic
87#6455—21no—
88#6455C2×C411yescharacteristic
89#6452—21no—
90#6451C2×C411yescharacteristic
91C4×D28—21no—
92C4×D28C2311yescharacteristic
93#6435—21no—
94C42×D7—21no—
95C42×D7C2311yescharacteristic
96C4×Dic14—21no—
97C4×Dic14C2311yescharacteristic
98C7×#107C2×C411yescharacteristic
99C7×C4⋊C8—21no—
100C7×C4⋊C8—21no—
101C7×C4⋊C8C2311yescharacteristic
102C7×C4⋊C8C2311yescharacteristic
103C7×#103C2×C411yescharacteristic
104C7×#103C2×C411yescharacteristic
105C7×#97—21no—
106C4×C7×C8—21no—
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110#6409C2×C411yescharacteristic
111#6409C2×C411yescharacteristic
112#6395C2×C411yescharacteristic
113(C2×D7)⋊C8—21no—
114(C2×D7)⋊C8—21no—
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8C2×C411yescharacteristic
118(C2×D7)⋊C8C2×C411yescharacteristic
119#6391C2×C411yescharacteristic
120#6390C2×C411yescharacteristic
121#6389—21no—
122#6389—21no—
123#6389D411yescharacteristic
124#6389D411yescharacteristic
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129Dic7⋊C8C2×C411yescharacteristic
130Dic7⋊C8C2×C411yescharacteristic
131C8×Dic7—21no—
132#6384C2×C411yescharacteristic
133#6383C2×C411yescharacteristic
134#6382C2×C411yescharacteristic
135#6381C2×C411yescharacteristic
136C4⋊(C7⋊C8)—21no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)C2311yescharacteristic
139C4⋊(C7⋊C8)C2311yescharacteristic
140C4×C7⋊C8—21no—
141C4×C7⋊C8—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3107—71no—
146#2971—71no—
147#2962—71no—
148#1592—71no—
149#1590—71no—
150#1588—71no—
151#1587—71no—
152#1584D711yescharacteristic, complemented
153#1583—71no—
154#1579—71no—
155C8×Q16—71no—
156#1573—71no—
157#1497—71no—
158#1477—71no—
159#1453—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×Q8C22×C411yescharacteristic
162#1161—22no—
163#1161—41no—
164C2×D28—21no—
165C2×D28C22×C411yescharacteristic
166C2×C4×D7—21no—
167C2×C4×D7—21no—
168C2×C4×D7—21no—
169C2×C4×D7C22×C411yescharacteristic
170C2×C4×D7C2×D411yescharacteristic
171C2×C4×D7#4111yescharacteristic
172C2×Dic14—21no—
173C2×Dic14C22×C411yescharacteristic
174C7×Q16—22no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C411yescharacteristic
179C2×C7×C8C22×C411yescharacteristic
180C2×C7×C8C2×D411yescharacteristic
181C2×C7×C8C2×D411yescharacteristic
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4C22×C411yescharacteristic
185C7×C4⋊C4C22×C411yescharacteristic
186C42×C7—21no—
187C42×C7—21no—
188C42×C7C2411yescharacteristic
189#1144—22no—
190#1143—41no—
191(C2×D7)⋊C4—21no—
192(C2×D7)⋊C4—21no—
193(C2×D7)⋊C4—21no—
194(C2×D7)⋊C4—21no—
195(C2×D7)⋊C4—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7C22×C411yescharacteristic
199C4⋊Dic7C22×C411yescharacteristic
200Dic7⋊C4—21no—
201Dic7⋊C4—21no—
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7C22×C411yescharacteristic
209C4×Dic7C2×D411yescharacteristic
210C4×Dic7#4111yescharacteristic
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C411yescharacteristic
215C2×C7⋊C8C22×C411yescharacteristic
216C2×C7⋊C8C2×D411yescharacteristic
217C2×C7⋊C8C2×D411yescharacteristic
218#1132—41no—
219#1131—22no—
220#1131—41no—
221#1131—42no—
222#1131—42no—
223C8×D7—42no—
224#533—71no—
225C4×#41—71no—
226#485—71no—
227#485—71no—
228#475—71no—
229#474—71no—
230#462—71no—
231#461—71no—
232#452—71no—
233#452—71no—
234#446—71no—
235#446—141no—
236#446—141no—
237#446C2×D711yescharacteristic
238C8×Q8—71no—
239C8×Q8—71no—
240C8×Q8—71no—
241C8×Q8C2×D711yescharacteristic
242C4×Q16—71no—
243C4×Q16C2×D711yescharacteristic
244C4×SD16—71no—
245C4×SD16—71no—
246#436—71no—
247#436—71no—
248#436—141no—
249#436—141no—
250#432—71no—
251#432—141no—
252#431—71no—
253#431—71no—
254#424—141no—
255C2×C4⋊C8—71no—
256#405—141no—
257C4×M16—71no—
258C4×M16—141no—
259#335—71no—
260#335C2×D711yescharacteristic
261#334—71no—
262#334—71no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8C2×D711yescharacteristic
268Q8⋊C8C2×D711yescharacteristic
269D4⋊C8—71no—
270D4⋊C8—71no—
271#322—71no—
272#322—71no—
273#322C2×D711yescharacteristic
274C82—71no—
275C22×D7—21no—
276C22×D7C4×D411yescharacteristic
277C7×Q8—21no—
278C7×Q8—21no—
279C7×Q8—21no—
280C7×Q8#13112yes—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7C4×D411yescharacteristic
292C2×Dic7C4×D411yescharacteristic
293C2×Dic7C4×D411yescharacteristic
294D28—21no—
295D28—21no—
296D28—42no—
297C4×D7—21no—
298C4×D7—21no—
299C4×D7—21no—
300C4×D7—21no—
301C4×D7—41no—
302C4×D7—42no—
303C4×D7—41no—
304C4×D7—41no—
305C4×D7—81no—
306Dic14—21no—
307Dic14—21no—
308Dic14—81no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—41no—
313C7×C8—41no—
314C7×C8—41no—
315C7×C8#13712yes—
316C7⋊C8—21no—
317C7⋊C8—21no—
318C7⋊C8—21no—
319C7⋊C8—21no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C2×#41—71no—
324C2×Q16—71no—
325C2×Q16C4×D711yescharacteristic
326C2×SD16—71no—
327C2×SD16—71no—
328C2×M16—71no—
329C2×M16—71no—
330C2×M16—141no—
331C2×M16—141no—
332C22×C8—141no—
333C4⋊Q8—71no—
334C4⋊Q8—71no—
335#125—141no—
336C22⋊Q8—71no—
337C22⋊Q8—71no—
338C22⋊Q8—141no—
339C4×Q8—71no—
340C4×Q8—71no—
341C4×Q8—141no—
342C4×Q8C22×D711yescharacteristic
343C4×Q8C22×D711yescharacteristic
344C4×D4—71no—
345C4×D4—71no—
346C4×D4—141no—
347C4×D4—141no—
348#117—71no—
349#117—141no—
350#117—141no—
351C2×C42—71no—
352C2×C42—141no—
353#107—71no—
354#107C4×D711yescharacteristic
355#106—71no—
356#106—71no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—71no—
360C4⋊C8—71no—
361C4⋊C8—71no—
362C4⋊C8—71no—
363C4⋊C8—141no—
364C4⋊C8—141no—
365C4⋊C8—141no—
366C4⋊C8—141no—
367C4⋊C8—141no—
368C4⋊C8—141no—
369C4⋊C8C22×D711yescharacteristic
370C4⋊C8C22×D711yescharacteristic
371#103—71no—
372#103—71no—
373#103—71no—
374#103—71no—
375#103C4×D711yescharacteristic
376#103C4×D711yescharacteristic
377#102—71no—
378#102—71no—
379C22⋊C8—71no—
380C22⋊C8—71no—
381C22⋊C8—141no—
382C22⋊C8—141no—
383C22⋊C8—141no—
384C22⋊C8—141no—
385#97—21no—
386#97—71no—
387#97—71no—
388#97—141no—
389#97—141no—
390C4×C8—21no—
391C4×C8—71no—
392C4×C8—71no—
393C4×C8—71no—
394C4×C8—141no—
395C4×C8—141no—
396C4×C8—141no—
397C4×C8C22×D711yescharacteristic
398C4×C8C22×D711yescharacteristic
399C4×C8C22×D711yescharacteristic
400C22×C7C2×C4×D411yescharacteristic
401C2×D7—21no—
402C2×D7—21no—
403C2×D7—21no—
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—41no—
407C4×C7—22no—
408C4×C7—21no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7C2×#13111yescharacteristic
413C4×C7#56811yescharacteristic
414C4×C7C2×#13711yescharacteristic
415C4×C7#57511yescharacteristic
416Dic7—21no—
417Dic7—21no—
418Dic7—21no—
419Dic7—41no—
420Dic7—41no—
421Dic7—41no—
422#41—142no—
423#41—281no—
424C2×Q8—71no—
425C2×Q8—141no—
426C2×Q8C2×C4×D711yescharacteristic
427C2×Q8C2×C4×D711yescharacteristic
428C2×D4—71no—
429C2×D4—141no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436Q16—22no—
437Q16—142no—
438SD16—281no—
439SD16—281no—
440M16—142no—
441M16—281no—
442M16—282no—
443M16—282no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—71no—
448C2×C8—71no—
449C2×C8—71no—
450C2×C8—71no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—282no—
455C2×C8C2×C4×D711yescharacteristic
456C2×C8C2×C4×D711yescharacteristic
457C2×C8D4×D711yescharacteristic
458C2×C8D4×D711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—71no—
462C4⋊C4—71no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4C2×C4×D711yescharacteristic
471C4⋊C4C2×C4×D711yescharacteristic
472C22⋊C4—141no—
473C22⋊C4—141no—
474C22⋊C4—141no—
475C22⋊C4—141no—
476C22⋊C4—141no—
477C42—21no—
478C42—21no—
479C42—71no—
480C42—71no—
481C42—71no—
482C42—141no—
483C42—141no—
484C42—141no—
485C42C23×D711yescharacteristic
486C2×C7#293311yescharacteristic
487C2×C7C4×#13711yescharacteristic
488C2×C7#295711yescharacteristic
489D7—41no—
490D7—81no—
491C23—71no—
492C23—141no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—141no—
497Q8—141no—
498Q8—561no—
499Q8#647012yes—
500D4—141no—
501D4—141no—
502D4—282no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—71no—
508C2×C4—71no—
509C2×C4—71no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—281no—
518C2×C4—282no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—561no—
522C2×C4C22×C4×D711yescharacteristic
523C2×C4C2×D4×D711yescharacteristic
524C2×C4D7×#4111yescharacteristic
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—141no—
532C8—141no—
533C8—141no—
534C8—141no—
535C8—281no—
536C8—281no—
537C8—281no—
538C8#648112yes—
539C7#1801311yescharacteristic, complemented, Sylow
540C22—141no—
541C22—141no—
542C22—141no—
543C22—281no—
544C22—281no—
545C22—281no—
546C22C4×D4×D711yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—141no—
553C4—141no—
554C4—141no—
555C4—281no—
556C4—281no—
557C4—281no—
558C4C2×#647011yescharacteristic
559C4#9044111yescharacteristic
560C4C2×#648111yescharacteristic
561C4#9045911yescharacteristic
562C2—281no—
563C2—561no—
564C2#1173692611yescharacteristic
565C2C4×#648111yescharacteristic
566C2#1173734411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1584C2
socleC22×C7C2×C4×D4
prime radicalC2×#1161C2×C4
cosocleC42×C7C24
hypercenter#1584D7
hypocenterC7#18013
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
1. #11737405C2semidirect
4. C7×#1584C2semidirect
5. #11728704C2semidirect
6. C8×#1144C2semidirect
7. #11728656C2semidirect
8. #11726328C2semidirect
9. #11726319C2semidirect
10. #11725997C2semidirect
12. #11725977C2semidirect
13. #11725975C2semidirect
14. #11725839C2semidirect
15. #11725827C2semidirect
152. #1584D7semidirect
539. C7#18013semidirect

Maximal subgroups

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

subgroupindex
#117374052
#117373502
#117371922
C7×#15842
#117287042
C8×#11442
#117286562
#117263282
#117263192
#117259972
#117259842
#117259772
#117259752
#117258392
#117258272
#180137