Finite Group Databasebeta

← #13325056

Subgroups — #13325056

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737366C211yesmaximal, characteristic, complemented
2#11737354C211yesmaximal, characteristic
3#11737168C211yesmaximal, characteristic
4C7×#1579C211yesmaximal, characteristic, complemented
5#11728688C211yesmaximal, characteristic, complemented
6#11728687C211yesmaximal, characteristic, complemented
7#11728657C211yesmaximal, characteristic, complemented
8#11726288C211yesmaximal, characteristic, complemented
9#11726273C211yesmaximal, characteristic, complemented
10#11725995C211yesmaximal, characteristic
11D7×Q8⋊C8C211yesmaximal, characteristic, complemented
12#11725981C211yesmaximal, characteristic, complemented
13#11725968C211yesmaximal, characteristic, complemented
14#11725836C211yesmaximal, characteristic, complemented
15#11725825C211yesmaximal, characteristic, complemented
16#90271C2211yescharacteristic
17C4×Q8×D7C2211yescharacteristic
18C7×C8×Q8C2211yescharacteristic
19C7×C8×Q8C2211yescharacteristic
20C4×C7×Q16C2211yescharacteristic
21#89958C411yescharacteristic
22#89957C411yescharacteristic
23C4×#1144C2211yescharacteristic
24C4×#1143C2211yescharacteristic
25#89796C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89654C411yescharacteristic
28#89649C411yescharacteristic
29#89611—21no—
30#89608—21no—
31#89607—21no—
32#89607C2211yescharacteristic
33#89606C2211yescharacteristic
34D7×C4⋊C8C2211yescharacteristic
35#89603—21no—
36#89603C2211yescharacteristic
37#89578C411yescharacteristic
38#89577C411yescharacteristic
39#89571C411yescharacteristic
40#89570C411yescharacteristic
41C4×#1132C2211yescharacteristic
42#89462—21no—
43#89460C2211yescharacteristic
44C8×D28—21no—
45#89457—21no—
46#89457C2211yescharacteristic
47#89457C2211yescharacteristic
48C4×C8×D7C2211yescharacteristic
49#89451—21no—
50#89451C2211yescharacteristic
51C7×#334C2211yescharacteristic
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#89253C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66#89249C2211yescharacteristic
67#18023—71nomaximal, Sylow
68C2×Q8×D7C2×C411yescharacteristic
69C2×C7×Q16C2×C411yescharacteristic
70C4×C7×Q8C2311yescharacteristic
71C4×C7×Q8C2311yescharacteristic
72#6510C2×C411yescharacteristic
73#6509C2×C411yescharacteristic
74Q8×Dic7C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77C2×#1132C2×C411yescharacteristic
78C2×#1131—21no—
79C2×C8×D7—21no—
80C2×C8×D7—21no—
81C2×C8×D7—21no—
82#6461—21no—
83#6458C2×C411yescharacteristic
84#6457—21no—
85D7×C4⋊C4—21no—
86D7×C4⋊C4C2×C411yescharacteristic
87#6453C2×C411yescharacteristic
88#6450—21no—
89#6450C2×C411yescharacteristic
90C4×D28—21no—
91C4×D28C2311yescharacteristic
92#6435—21no—
93C42×D7—21no—
94C42×D7C2311yescharacteristic
95C4×Dic14—21no—
96C4×Dic14C2311yescharacteristic
97C7×#107C2×C411yescharacteristic
98C7×C4⋊C8—21no—
99C7×C4⋊C8—21no—
100C7×C4⋊C8C2311yescharacteristic
101C7×C4⋊C8C2311yescharacteristic
102C7×#103C2×C411yescharacteristic
103C7×#103C2×C411yescharacteristic
104C4×C7×C8—21no—
105C4×C7×C8—21no—
106C4×C7×C8C2311yescharacteristic
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109#6409C2×C411yescharacteristic
110#6409C2×C411yescharacteristic
111#6395C2×C411yescharacteristic
112(C2×D7)⋊C8—21no—
113(C2×D7)⋊C8—21no—
114(C2×D7)⋊C8—21no—
115(C2×D7)⋊C8C2×C411yescharacteristic
116(C2×D7)⋊C8C2×C411yescharacteristic
117(C2×D7)⋊C8D411yescharacteristic
118(C2×D7)⋊C8D411yescharacteristic
119#6391C2×C411yescharacteristic
120#6390C2×C411yescharacteristic
121#6389—21no—
122#6389—21no—
123#6389—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8C2×C411yescharacteristic
128Dic7⋊C8C2×C411yescharacteristic
129Dic7⋊C8D411yescharacteristic
130Dic7⋊C8D411yescharacteristic
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
140#6377—21no—
141C4×C7⋊C8—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3108—71no—
146#2994—71no—
147#2984—71no—
148#1593—71no—
149#1592—71no—
150#1584—71no—
151#1584—71no—
152#1583—71no—
153#1582—71no—
154#1579D711yescharacteristic, complemented
155SD16⋊C8—71no—
156#1573—71no—
157#1494—71no—
158C2×Q8⋊C8—71no—
159#1452—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×Q8C22×C411yescharacteristic
162Q8×D7—22no—
163Q8×D7—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—41no—
190#1143—22no—
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—81no—
220C8×D7—42no—
221C8×D7—81no—
222C8×D7—81no—
223#529—71no—
224C2×C4×Q8—71no—
225#483—71no—
226#480—71no—
227#479—71no—
228#479—71no—
229#464—71no—
230#464—71no—
231#451—71no—
232#448—71no—
233#446—71no—
234#446—71no—
235#446—71no—
236#446—141no—
237#446—141no—
238C8×Q8—71no—
239C8×Q8C2×D711yescharacteristic
240C8×Q8C2×D711yescharacteristic
241C4×Q16—71no—
242C4×Q16C2×D711yescharacteristic
243C4×SD16—71no—
244C4×SD16—71no—
245#436—71no—
246#435—141no—
247C8×D4—71no—
248C8×D4—141no—
249#433—141no—
250#432—71no—
251#432—141no—
252#431—71no—
253#431—71no—
254#431—141no—
255#424—141no—
256C2×C4⋊C8—71no—
257C2×C4×C8—71no—
258#335—71no—
259#335—71no—
260#334—71no—
261#334C2×D711yescharacteristic
262Q8⋊C8—71no—
263Q8⋊C8—71no—
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8C2×D711yescharacteristic
267Q8⋊C8C2×D711yescharacteristic
268D4⋊C8—71no—
269D4⋊C8—71no—
270#322—71no—
271#322—71no—
272#322—71no—
273#322C2×D711yescharacteristic
274C22×D7—21no—
275C22×D7C4×D411yescharacteristic
276C7×Q8—21no—
277C7×Q8—21no—
278C7×Q8—21no—
279C7×Q8#13112yes—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7C4×D411yescharacteristic
291C2×Dic7C4×D411yescharacteristic
292C2×Dic7C4×D411yescharacteristic
293D28—21no—
294D28—21no—
295D28—42no—
296C4×D7—22no—
297C4×D7—21no—
298C4×D7—21no—
299C4×D7—21no—
300C4×D7—21no—
301C4×D7—42no—
302C4×D7—42no—
303C4×D7—41no—
304C4×D7—41no—
305Dic14—21no—
306Dic14—21no—
307Dic14—42no—
308C7×C8—21no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—41no—
313C7×C8—41no—
314C7×C8—41no—
315C7⋊C8—21no—
316C7⋊C8—21no—
317C7⋊C8—21no—
318C7⋊C8—21no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C22×Q8—71no—
323C2×Q16—71no—
324C2×Q16C4×D711yescharacteristic
325C2×SD16—71no—
326C2×SD16—71no—
327C2×M16—141no—
328C22×C8—141no—
329C22×C8—141no—
330C22×C8—141no—
331#126—141no—
332#125—71no—
333#124—71no—
334#123—141no—
335C22⋊Q8—71no—
336#121—71no—
337C4×Q8—71no—
338C4×Q8—71no—
339C4×Q8—71no—
340C4×Q8—141no—
341C4×Q8—141no—
342C4×Q8C22×D711yescharacteristic
343C4×Q8C22×D711yescharacteristic
344C4×D4—71no—
345C4×D4—141no—
346#117—141no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—141no—
349C2×C42—71no—
350C2×C42—141no—
351#107—71no—
352#107C4×D711yescharacteristic
353#106—71no—
354#106—71no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—71no—
358C4⋊C8—71no—
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⋊C8C22×D711yescharacteristic
369C4⋊C8C22×D711yescharacteristic
370#103—71no—
371#103—71no—
372#103—71no—
373#103—71no—
374#103C4×D711yescharacteristic
375#103C4×D711yescharacteristic
376#102—71no—
377#102—71no—
378C22⋊C8—71no—
379C22⋊C8—71no—
380C22⋊C8—71no—
381C22⋊C8—71no—
382C22⋊C8—141no—
383C22⋊C8—141no—
384C22⋊C8—141no—
385#97—141no—
386#97—141no—
387#97—141no—
388#97—141no—
389C4×C8—21no—
390C4×C8—21no—
391C4×C8—71no—
392C4×C8—71no—
393C4×C8—71no—
394C4×C8—141no—
395C4×C8—141no—
396C4×C8C22×D711yescharacteristic
397C4×C8C22×D711yescharacteristic
398C4×C8C22×D711yescharacteristic
399C22×C7C2×C4×D411yescharacteristic
400C2×D7—21no—
401C2×D7—21no—
402C2×D7—41no—
403C2×D7—41no—
404C2×D7—41no—
405C2×D7#44412yes—
406C4×C7—22no—
407C4×C7—21no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7C2×#13111yescharacteristic
412C4×C7#56811yescharacteristic
413C4×C7#57711yescharacteristic
414C4×C7#57811yescharacteristic
415Dic7—21no—
416Dic7—21no—
417Dic7—21no—
418Dic7—41no—
419Dic7—41no—
420Dic7—41no—
421C2×Q8—71no—
422C2×Q8—141no—
423C2×Q8—142no—
424C2×Q8—281no—
425C2×Q8C2×C4×D711yescharacteristic
426C2×Q8C2×C4×D711yescharacteristic
427C2×D4—71no—
428C2×D4—141no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435Q16—22no—
436Q16—281no—
437SD16—142no—
438SD16—281no—
439M16—561no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—71no—
444C2×C8—71no—
445C2×C8—71no—
446C2×C8—71no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—282no—
451C2×C8—561no—
452C2×C8—561no—
453C2×C8C2×C4×D711yescharacteristic
454C2×C8C2×C4×D711yescharacteristic
455C2×C8D4×D711yescharacteristic
456C2×C8D4×D711yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—71no—
460C4⋊C4—71no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4C2×C4×D711yescharacteristic
469C4⋊C4C2×C4×D711yescharacteristic
470C22⋊C4—141no—
471C22⋊C4—141no—
472C22⋊C4—141no—
473C22⋊C4—141no—
474C22⋊C4—141no—
475C42—21no—
476C42—21no—
477C42—71no—
478C42—71no—
479C42—71no—
480C42—141no—
481C42—141no—
482C42—141no—
483C42C23×D711yescharacteristic
484C2×C7#293311yescharacteristic
485C2×C7#294711yescharacteristic
486C2×C7C2×#44411yescharacteristic
487D7—22no—
488D7—81no—
489C23—71no—
490C23—141no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—141no—
495Q8—141no—
496Q8—282no—
497Q8#646412yes—
498D4—141no—
499D4—141no—
500D4—282no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—71no—
506C2×C4—71no—
507C2×C4—71no—
508C2×C4—142no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4C22×C4×D711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4D7×#4111yescharacteristic
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—141no—
531C8—141no—
532C8—141no—
533C8—141no—
534C8—281no—
535C8—281no—
536C8—281no—
537C7#1802311yescharacteristic, complemented, Sylow
538C22—72no—
539C22—141no—
540C22—141no—
541C22—281no—
542C22—281no—
543C22—281no—
544C22C4×D4×D711yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—141no—
551C4—141no—
552C4—141no—
553C4—281no—
554C4—281no—
555C4—281no—
556C4C2×#646411yescharacteristic
557C4#9044111yescharacteristic
558C4#9046911yescharacteristic
559C4#9047011yescharacteristic
560C2—142no—
561C2—561no—
562C2#1173686811yescharacteristic
563C2#1173727511yescharacteristic
564C2D7×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1579C2
socleC22×C7C2×C4×D4
prime radical#6458C2×C4
cosocleC42×C7C24
hypercenter#1579D7
hypocenterC7#18023
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
1. #11737366C2semidirect
4. C7×#1579C2semidirect
5. #11728688C2semidirect
6. #11728687C2semidirect
7. #11728657C2semidirect
8. #11726288C2semidirect
9. #11726273C2semidirect
11. D7×Q8⋊C8C2semidirect
12. #11725981C2semidirect
13. #11725968C2semidirect
14. #11725836C2semidirect
15. #11725825C2semidirect
154. #1579D7semidirect
537. C7#18023semidirect

Maximal subgroups

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

subgroupindex
#117373662
#117373542
#117371682
C7×#15792
#117286882
#117286872
#117286572
#117262882
#117262732
#117259952
D7×Q8⋊C82
#117259812
#117259682
#117258362
#117258252
#180237