Finite Group Databasebeta

← #13324597

Subgroups — #13324597

Order 1792.

578 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737403C211yesmaximal, characteristic, complemented
2#11737348C211yesmaximal, characteristic
3#11737171C211yesmaximal, characteristic
4C7×#1584C211yesmaximal, characteristic, complemented
5#11728707C211yesmaximal, characteristic, complemented
6#11728688C211yesmaximal, characteristic, complemented
7#11728658C211yesmaximal, characteristic, complemented
8#11726325C211yesmaximal, characteristic, complemented
9#11726316C211yesmaximal, characteristic, complemented
10#11725997C211yesmaximal, characteristic, complemented
11#11725996C211yesmaximal, characteristic
12#11725977C211yesmaximal, characteristic, complemented
13#11725976C211yesmaximal, characteristic, complemented
14#11725833C211yesmaximal, characteristic, complemented
15#11725823C211yesmaximal, characteristic, complemented
16#90271C2211yescharacteristic
17#90269C2211yescharacteristic
18C7×#446C2211yescharacteristic
19C7×C8×Q8C2211yescharacteristic
20C4×C7×Q16C2211yescharacteristic
21#89962C411yescharacteristic
22#89961C411yescharacteristic
23C4×#1144C2211yescharacteristic
24C4×#1143C2211yescharacteristic
25#89796C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89658C411yescharacteristic
28#89648C411yescharacteristic
29#89612—21no—
30#89611—21no—
31#89610C2211yescharacteristic
32#89609C2211yescharacteristic
33#89608—21no—
34#89607C2211yescharacteristic
35#89602—21no—
36#89602C2211yescharacteristic
37#89588C411yescharacteristic
38#89584C411yescharacteristic
39#89569C411yescharacteristic
40#89567C411yescharacteristic
41#89480—21no—
42#89473—21no—
43C4×#1132C2211yescharacteristic
44#89461—21no—
45C4×#1131C2211yescharacteristic
46C8×D28—21no—
47C8×D28C2211yescharacteristic
48#89457C2211yescharacteristic
49C4×C8×D7C2211yescharacteristic
50#89451C2211yescharacteristic
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
66#89249C2211yescharacteristic
67#17986—71nomaximal, Sylow
68C2×C7×Q16C2×C411yescharacteristic
69C4×C7×Q8C2311yescharacteristic
70C4×C7×Q8C2311yescharacteristic
71#6510C2×C411yescharacteristic
72#6509C2×C411yescharacteristic
73#6509C2×C411yescharacteristic
74#6507C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77C2×#1132C2×C411yescharacteristic
78C2×#1131—21no—
79C2×#1131—21no—
80C2×#1131C2×C411yescharacteristic
81C2×#1131C2×C411yescharacteristic
82C2×C8×D7—21no—
83C2×C8×D7D411yescharacteristic
84C2×C8×D7D411yescharacteristic
85#6461—21no—
86#6460C2×C411yescharacteristic
87#6459—21no—
88#6458C2×C411yescharacteristic
89#6457—21no—
90#6453C2×C411yescharacteristic
91#6452—21no—
92#6451C2×C411yescharacteristic
93C4×D28—21no—
94C4×D28C2311yescharacteristic
95#6435—21no—
96#6435—21no—
97C42×D7C2311yescharacteristic
98C4×Dic14—21no—
99C4×Dic14C2311yescharacteristic
100C7×#107C2×C411yescharacteristic
101C7×C4⋊C8—21no—
102C7×C4⋊C8—21no—
103C7×C4⋊C8C2311yescharacteristic
104C7×C4⋊C8C2311yescharacteristic
105C7×#103C2×C411yescharacteristic
106C7×#103C2×C411yescharacteristic
107C7×#97—21no—
108C4×C7×C8—21no—
109C4×C7×C8C2311yescharacteristic
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112#6409C2×C411yescharacteristic
113#6409C2×C411yescharacteristic
114#6395C2×C411yescharacteristic
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8—21no—
118(C2×D7)⋊C8—21no—
119#6391C2×C411yescharacteristic
120#6390C2×C411yescharacteristic
121#6389—21no—
122#6389—21no—
123#6389C2×C411yescharacteristic
124#6389C2×C411yescharacteristic
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129C8×Dic7—21no—
130C8×Dic7D411yescharacteristic
131C8×Dic7D411yescharacteristic
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#3172—71no—
146#2966—71no—
147#2960—71no—
148#1588—71no—
149#1588—71no—
150#1587—71no—
151#1587—71no—
152#1584—71no—
153#1584D711yescharacteristic, complemented
154#1582—71no—
155#1582—71no—
156#1569—71no—
157#1497—71no—
158#1495—71no—
159#1451—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×Q8C22×C411yescharacteristic
162C2×D28—21no—
163C2×D28C22×C411yescharacteristic
164C2×C4×D7—21no—
165C2×C4×D7C22×C411yescharacteristic
166C2×C4×D7C2×D411yescharacteristic
167C2×C4×D7#4111yescharacteristic
168C2×Dic14—21no—
169C2×Dic14C22×C411yescharacteristic
170C7×Q16—41no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8C22×C411yescharacteristic
175C2×C7×C8C22×C411yescharacteristic
176C2×C7×C8C2×D411yescharacteristic
177C2×C7×C8C2×D411yescharacteristic
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4C22×C411yescharacteristic
181C7×C4⋊C4C22×C411yescharacteristic
182C42×C7—21no—
183C42×C7—21no—
184C42×C7C2411yescharacteristic
185#1144—41no—
186#1143—41no—
187(C2×D7)⋊C4—21no—
188(C2×D7)⋊C4—21no—
189(C2×D7)⋊C4—21no—
190(C2×D7)⋊C4—21no—
191(C2×D7)⋊C4—21no—
192(C2×D7)⋊C4—21no—
193(C2×D7)⋊C4—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7C22×C411yescharacteristic
197C4⋊Dic7C22×C411yescharacteristic
198Dic7⋊C4—21no—
199Dic7⋊C4—21no—
200Dic7⋊C4—21no—
201Dic7⋊C4—21no—
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205C4×Dic7—21no—
206C4×Dic7C22×C411yescharacteristic
207C4×Dic7C2×D411yescharacteristic
208C4×Dic7#4111yescharacteristic
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8C22×C411yescharacteristic
213C2×C7⋊C8C22×C411yescharacteristic
214C2×C7⋊C8C2×D411yescharacteristic
215C2×C7⋊C8C2×D411yescharacteristic
216#1132—41no—
217#1131—41no—
218#1131—41no—
219#1131—42no—
220#1131—42no—
221C8×D7—22no—
222C8×D7—41no—
223C8×D7—42no—
224#533—71no—
225#529—71no—
226#501—71no—
227#497—71no—
228#491—71no—
229#489—71no—
230#470—71no—
231#470—71no—
232#468—71no—
233#468—71no—
234#446—71no—
235#446—71no—
236#446—141no—
237#446C2×D711yescharacteristic
238C8×Q8—71no—
239C8×Q8—71no—
240C8×Q8—141no—
241C8×Q8C2×D711yescharacteristic
242C4×Q16—71no—
243C4×Q16C2×D711yescharacteristic
244C4×SD16—71no—
245C4×SD16—71no—
246#436—71no—
247#435—141no—
248C8×D4—71no—
249C8×D4—141no—
250#433—141no—
251#432—71no—
252#431—71no—
253#424—141no—
254#423—71no—
255#405—141no—
256#405—141no—
257C4×M16—71no—
258C2×C4×C8—71no—
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#322—71no—
274#322C2×D711yescharacteristic
275C22×D7—21no—
276C22×D7C4×D411yescharacteristic
277C7×Q8—21no—
278C7×Q8—21no—
279C7×Q8—21no—
280C7×Q8—21no—
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—41no—
303C4×D7—41no—
304Dic14—21no—
305Dic14—21no—
306Dic14—42no—
307C7×C8—21no—
308C7×C8—21no—
309C7×C8—21no—
310C7×C8—41no—
311C7×C8—41no—
312C7×C8—41no—
313C7×C8#13712yes—
314C7⋊C8—21no—
315C7⋊C8—21no—
316C7⋊C8—21no—
317C7⋊C8—21no—
318C7⋊C8—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C2×Q16—71no—
322C2×Q16C4×D711yescharacteristic
323C2×SD16—71no—
324C2×SD16—71no—
325C2×M16—71no—
326C2×M16—71no—
327C2×M16—141no—
328C2×M16—141no—
329C22×C8—71no—
330C22×C8—71no—
331C22×C8—141no—
332C4⋊Q8—71no—
333C4⋊Q8—71no—
334#126—141no—
335#125—71no—
336#125—141no—
337#124—71no—
338#123—141no—
339C22⋊Q8—71no—
340C22⋊Q8—71no—
341C22⋊Q8—71no—
342C22⋊Q8—141no—
343#121—71no—
344C4×Q8—71no—
345C4×Q8—141no—
346C4×Q8C22×D711yescharacteristic
347C4×Q8C22×D711yescharacteristic
348C4×D4—71no—
349C4×D4—141no—
350#117—141no—
351#117—141no—
352C2×C42—71no—
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—141no—
362C4⋊C8—141no—
363C4⋊C8—141no—
364C4⋊C8—141no—
365C4⋊C8—141no—
366C4⋊C8—141no—
367C4⋊C8C22×D711yescharacteristic
368C4⋊C8C22×D711yescharacteristic
369#103—71no—
370#103—71no—
371#103—71no—
372#103—71no—
373#103C4×D711yescharacteristic
374#103C4×D711yescharacteristic
375#102—71no—
376#102—71no—
377C22⋊C8—141no—
378C22⋊C8—141no—
379C22⋊C8—141no—
380C22⋊C8—141no—
381#97—21no—
382#97—71no—
383#97—71no—
384#97—141no—
385#97—141no—
386#97—141no—
387C4×C8—21no—
388C4×C8—71no—
389C4×C8—71no—
390C4×C8—71no—
391C4×C8—71no—
392C4×C8—71no—
393C4×C8—141no—
394C4×C8—141no—
395C4×C8C22×D711yescharacteristic
396C4×C8C22×D711yescharacteristic
397C4×C8C22×D711yescharacteristic
398C22×C7C2×C4×D411yescharacteristic
399C2×D7—21no—
400C2×D7—21no—
401C2×D7—21no—
402C2×D7—41no—
403C2×D7—41no—
404C2×D7—41no—
405C4×C7—21no—
406C4×C7—41no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7#56811yescharacteristic
411C4×C7#56811yescharacteristic
412C4×C7C2×#13711yescharacteristic
413C4×C7#57511yescharacteristic
414Dic7—21no—
415Dic7—21no—
416Dic7—21no—
417Dic7—41no—
418Dic7—41no—
419Dic7—41no—
420C2×Q8—71no—
421C2×Q8—141no—
422C2×Q8C2×C4×D711yescharacteristic
423C2×Q8C2×C4×D711yescharacteristic
424C2×D4—71no—
425C2×D4—141no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—141no—
430Q16—41no—
431Q16—281no—
432SD16—281no—
433SD16—281no—
434M16—281no—
435M16—281no—
436M16—282no—
437M16—282no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—71no—
442C2×C8—71no—
443C2×C8—71no—
444C2×C8—71no—
445C2×C8—142no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—281no—
450C2×C8—282no—
451C2×C8C2×C4×D711yescharacteristic
452C2×C8C2×C4×D711yescharacteristic
453C2×C8D4×D711yescharacteristic
454C2×C8D4×D711yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—71no—
458C4⋊C4—71no—
459C4⋊C4—141no—
460C4⋊C4—141no—
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—
475C22⋊C4—141no—
476C22⋊C4—141no—
477C42—21no—
478C42—21no—
479C42—71no—
480C42—71no—
481C42—71no—
482C42—141no—
483C42C23×D711yescharacteristic
484C2×C7#293211yescharacteristic
485C2×C7C4×#13711yescharacteristic
486C2×C7#295511yescharacteristic
487D7—41no—
488D7—81no—
489C23—71no—
490C23—141no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—141no—
496Q8—141no—
497Q8—282no—
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—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4C22×C4×D711yescharacteristic
519C2×C4C2×D4×D711yescharacteristic
520C2×C4D7×#4111yescharacteristic
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—141no—
528C8—141no—
529C8—141no—
530C8—141no—
531C8—281no—
532C8—281no—
533C8—281no—
534C8#648112yes—
535C7#1798611yescharacteristic, complemented, Sylow
536C22—141no—
537C22—141no—
538C22—141no—
539C22—281no—
540C22—281no—
541C22—281no—
542C22C4×D4×D711yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—141no—
549C4—141no—
550C4—141no—
551C4—281no—
552C4—281no—
553C4—281no—
554C4#9043611yescharacteristic
555C4#9044111yescharacteristic
556C4C2×#648111yescharacteristic
557C4#9045911yescharacteristic
558C2—281no—
559C2—561no—
560C2#1173692511yescharacteristic
561C2C4×#648111yescharacteristic
562C2#1173734211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1584C2
socleC22×C7C2×C4×D4
prime radical#6458C2×C4
cosocleC42×C7C24
hypercenter#1584D7
hypocenterC7#17986
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. #11737403C2semidirect
4. C7×#1584C2semidirect
5. #11728707C2semidirect
6. #11728688C2semidirect
7. #11728658C2semidirect
8. #11726325C2semidirect
9. #11726316C2semidirect
10. #11725997C2semidirect
12. #11725977C2semidirect
13. #11725976C2semidirect
14. #11725833C2semidirect
15. #11725823C2semidirect
153. #1584D7semidirect
535. C7#17986semidirect

Maximal subgroups

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

subgroupindex
#117374032
#117373482
#117371712
C7×#15842
#117287072
#117286882
#117286582
#117263252
#117263162
#117259972
#117259962
#117259772
#117259762
#117258332
#117258232
#179867