Finite Group Databasebeta

← #13324664

Subgroups — #13324664

Order 1792.

576 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737392C211yesmaximal, characteristic, complemented
2#11737353C211yesmaximal, characteristic, complemented
3#11737160C211yesmaximal, characteristic
4C7×#1584C211yesmaximal, characteristic, complemented
5#11728708C211yesmaximal, characteristic, complemented
6#11728690C211yesmaximal, characteristic, complemented
7#11728657C211yesmaximal, characteristic, complemented
8#11726323C211yesmaximal, characteristic, complemented
9#11726313C211yesmaximal, characteristic, complemented
10#11725996C211yesmaximal, characteristic, complemented
11#11725996C211yesmaximal, characteristic, complemented
12#11725977C211yesmaximal, characteristic, complemented
13#11725974C211yesmaximal, characteristic, complemented
14#11725832C211yesmaximal, characteristic, complemented
15#11725825C211yesmaximal, characteristic, complemented
16#90271C2211yescharacteristic
17#90271C2211yescharacteristic
18C7×#446C2211yescharacteristic
19C7×C8×Q8C2211yescharacteristic
20C4×C7×Q16C2211yescharacteristic
21#89963C411yescharacteristic
22#89960C411yescharacteristic
23C4×#1143C2211yescharacteristic
24C4×#1143C2211yescharacteristic
25#89796C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89655C411yescharacteristic
28#89650C411yescharacteristic
29#89611—21no—
30#89611—21no—
31#89610—21no—
32#89610C2211yescharacteristic
33#89609C2211yescharacteristic
34#89609C2211yescharacteristic
35#89608—21no—
36#89606C2211yescharacteristic
37#89579C411yescharacteristic
38#89579C411yescharacteristic
39#89572C411yescharacteristic
40#89572C411yescharacteristic
41#89481—21no—
42#89478—21no—
43C4×D56C2211yescharacteristic
44#89462—21no—
45#89460—21no—
46#89460C2211yescharacteristic
47C8×D28C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50C4×C8×D7C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×Q8⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56Q8⋊(C7⋊C8)C2211yescharacteristic
57#89278C2211yescharacteristic
58#89278C2211yescharacteristic
59#89274C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89254C2211yescharacteristic
63#89253C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66#89249C2211yescharacteristic
67#17992—71nomaximal, Sylow
68C2×C7×Q16C2×C411yescharacteristic
69C4×C7×Q8C2311yescharacteristic
70C4×C7×Q8C2311yescharacteristic
71#6510C2×C411yescharacteristic
72#6510C2×C411yescharacteristic
73#6509C2×C411yescharacteristic
74#6509C2×C411yescharacteristic
75C2×#1143C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77C2×D56C2×C411yescharacteristic
78C2×#1131—21no—
79C2×#1131—21no—
80C2×#1131—21no—
81C2×#1131—21no—
82C2×C8×D7—21no—
83C2×C8×D7—21no—
84C2×C8×D7C2×C411yescharacteristic
85C2×C8×D7C2×C411yescharacteristic
86#6461—21no—
87#6461—21no—
88#6458C2×C411yescharacteristic
89#6458C2×C411yescharacteristic
90#6457—21no—
91#6457—21no—
92#6453C2×C411yescharacteristic
93#6453C2×C411yescharacteristic
94C4×D28—21no—
95C4×D28—21no—
96C4×D28C2311yescharacteristic
97C4×D28C2311yescharacteristic
98#6435—21no—
99#6435—21no—
100C42×D7C2311yescharacteristic
101C7×#107C2×C411yescharacteristic
102C7×C4⋊C8—21no—
103C7×C4⋊C8—21no—
104C7×C4⋊C8C2311yescharacteristic
105C7×C4⋊C8C2311yescharacteristic
106C7×#103C2×C411yescharacteristic
107C7×#103C2×C411yescharacteristic
108C7×#97—21no—
109C4×C7×C8—21no—
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112C4×C7×C8C2311yescharacteristic
113#6409C2×C411yescharacteristic
114#6409C2×C411yescharacteristic
115#6395C2×C411yescharacteristic
116#6395C2×C411yescharacteristic
117(C2×D7)⋊C8—21no—
118(C2×D7)⋊C8—21no—
119(C2×D7)⋊C8—21no—
120(C2×D7)⋊C8—21no—
121(C2×D7)⋊C8—21no—
122(C2×D7)⋊C8D411yescharacteristic
123(C2×D7)⋊C8D411yescharacteristic
124#6392C2×C411yescharacteristic
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8D411yescharacteristic
129Dic7⋊C8D411yescharacteristic
130C8×Dic7C2×C411yescharacteristic
131C8×Dic7C2×C411yescharacteristic
132#6383C2×C411yescharacteristic
133#6383C2×C411yescharacteristic
134#6382C2×C411yescharacteristic
135#6382C2×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#3168—71no—
146#2993—71no—
147#2993—71no—
148#1588—71no—
149#1588—71no—
150#1585—71no—
151#1585—71no—
152#1584—71no—
153#1584D711yescharacteristic, complemented
154#1581—71no—
155#1581—71no—
156#1567—71no—
157#1495—71no—
158#1495—71no—
159#1452—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×Q8C22×C411yescharacteristic
162C2×D28—21no—
163C2×D28—21no—
164C2×D28C22×C411yescharacteristic
165C2×D28C22×C411yescharacteristic
166C2×C4×D7—21no—
167C2×C4×D7—21no—
168C2×C4×D7C22×C411yescharacteristic
169C2×C4×D7C2×D411yescharacteristic
170C2×C4×D7#4111yescharacteristic
171C7×Q16—41no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8C22×C411yescharacteristic
176C2×C7×C8C22×C411yescharacteristic
177C2×C7×C8C2×D411yescharacteristic
178C2×C7×C8C2×D411yescharacteristic
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C22×C411yescharacteristic
182C7×C4⋊C4C22×C411yescharacteristic
183C42×C7—21no—
184C42×C7—21no—
185C42×C7C2411yescharacteristic
186#1143—41no—
187#1143—41no—
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—
194(C2×D7)⋊C4—21no—
195(C2×D7)⋊C4—21no—
196(C2×D7)⋊C4—21no—
197(C2×D7)⋊C4—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7C22×C411yescharacteristic
201C4⋊Dic7C22×C411yescharacteristic
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—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
216D56—41no—
217#1131—81no—
218#1131—81no—
219#1131—81no—
220#1131—81no—
221C8×D7—41no—
222C8×D7—41no—
223C8×D7—81no—
224C8×D7—81no—
225#529—71no—
226#529—71no—
227#499—71no—
228#495—71no—
229#488—71no—
230#488—71no—
231#467—71no—
232#466—71no—
233#465—71no—
234#465—71no—
235#446—71no—
236#446C2×D711yescharacteristic
237C8×Q8—71no—
238C8×Q8C2×D711yescharacteristic
239C4×Q16C2×D711yescharacteristic
240C4×SD16—71no—
241C4×SD16—71no—
242C4×D8—71no—
243#436—71no—
244#436—71no—
245#436—141no—
246#436—141no—
247#435—141no—
248#435—141no—
249C8×D4—71no—
250C8×D4—71no—
251#433—141no—
252#433—141no—
253#431—71no—
254#431—71no—
255#424—141no—
256#424—141no—
257#423—71no—
258#423—71no—
259C2×C4×C8—71no—
260#335—71no—
261#335—71no—
262#335—71no—
263#335C2×D711yescharacteristic
264Q8⋊C8—71no—
265Q8⋊C8—71no—
266Q8⋊C8C2×D711yescharacteristic
267Q8⋊C8C2×D711yescharacteristic
268D4⋊C8—71no—
269D4⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8—71no—
272#322—71no—
273#322—71no—
274#322—71no—
275#322C2×D711yescharacteristic
276C22×D7—21no—
277C22×D7—21no—
278C22×D7C4×D411yescharacteristic
279C7×Q8—21no—
280C7×Q8—21no—
281C7×Q8—21no—
282C7×Q8—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7C23×C411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7C4×D411yescharacteristic
293C2×Dic7C4×D411yescharacteristic
294C2×Dic7C4×D411yescharacteristic
295D28—21no—
296D28—21no—
297D28—21no—
298D28—21no—
299D28—81no—
300D28—81no—
301C4×D7—21no—
302C4×D7—21no—
303C4×D7—21no—
304C4×D7—21no—
305C4×D7—41no—
306C4×D7—41no—
307C4×D7—41no—
308C4×D7—41no—
309C4×D7—41no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—21no—
313C7×C8—21no—
314C7×C8—41no—
315C7×C8—41no—
316C7×C8—41no—
317C7⋊C8—21no—
318C7⋊C8—21no—
319C7⋊C8—21no—
320C7⋊C8—21no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C7⋊C8—41no—
324C2×Q16C4×D711yescharacteristic
325C2×SD16—71no—
326C2×SD16—71no—
327C2×D8—71no—
328C2×M16—141no—
329C2×M16—141no—
330C2×M16—141no—
331C2×M16—141no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8—141no—
335C22×C8—141no—
336#126—141no—
337#126—141no—
338#125—71no—
339#125—71no—
340#124—71no—
341#124—71no—
342#123—141no—
343#123—141no—
344C22⋊Q8—71no—
345C22⋊Q8—71no—
346#121—71no—
347#121—71no—
348C4×Q8C22×D711yescharacteristic
349C4×Q8C22×D711yescharacteristic
350C4×D4—71no—
351C4×D4—71no—
352C4×D4—141no—
353C4×D4—141no—
354#117—141no—
355#117—141no—
356C2×C42—71no—
357#107—71no—
358#107C4×D711yescharacteristic
359#106—71no—
360#106—71no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—71no—
364C4⋊C8—71no—
365C4⋊C8—71no—
366C4⋊C8—71no—
367C4⋊C8—141no—
368C4⋊C8—141no—
369C4⋊C8—141no—
370C4⋊C8—141no—
371C4⋊C8—141no—
372C4⋊C8C22×D711yescharacteristic
373C4⋊C8C22×D711yescharacteristic
374#103—71no—
375#103—71no—
376#103C4×D711yescharacteristic
377#103C4×D711yescharacteristic
378#102—71no—
379#102—71no—
380#102—71no—
381#102—71no—
382C22⋊C8—71no—
383C22⋊C8—71no—
384C22⋊C8—141no—
385C22⋊C8—141no—
386C22⋊C8—141no—
387C22⋊C8—141no—
388C22⋊C8—141no—
389#97—21no—
390#97—141no—
391C4×C8—21no—
392C4×C8—71no—
393C4×C8—71no—
394C4×C8—71no—
395C4×C8—71no—
396C4×C8—71no—
397C4×C8—141no—
398C4×C8C22×D711yescharacteristic
399C4×C8C22×D711yescharacteristic
400C4×C8C22×D711yescharacteristic
401C22×C7C2×C4×D411yescharacteristic
402C2×D7—21no—
403C2×D7—21no—
404C2×D7—21no—
405C2×D7—41no—
406C2×D7—41no—
407C2×D7—41no—
408C2×D7—41no—
409C2×D7—41no—
410C2×D7—41no—
411C4×C7—21no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7#56811yescharacteristic
417C4×C7#56811yescharacteristic
418C4×C7#57611yescharacteristic
419C4×C7#57811yescharacteristic
420Dic7—21no—
421Dic7—21no—
422Dic7—21no—
423Dic7—41no—
424Dic7—41no—
425C2×Q8C2×C4×D711yescharacteristic
426C2×Q8C2×C4×D711yescharacteristic
427C2×D4—71no—
428C2×D4—71no—
429C2×D4—141no—
430C2×D4—141no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—141no—
435C22×C4—141no—
436Q16—41no—
437SD16—281no—
438SD16—281no—
439D8—281no—
440M16—561no—
441M16—561no—
442M16—561no—
443M16—561no—
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—281no—
455C2×C8—281no—
456C2×C8—561no—
457C2×C8—561no—
458C2×C8C2×C4×D711yescharacteristic
459C2×C8C2×C4×D711yescharacteristic
460C2×C8D4×D711yescharacteristic
461C2×C8D4×D711yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—71no—
465C4⋊C4—71no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4C2×C4×D711yescharacteristic
473C4⋊C4C2×C4×D711yescharacteristic
474C22⋊C4—141no—
475C22⋊C4—141no—
476C22⋊C4—141no—
477C22⋊C4—141no—
478C22⋊C4—141no—
479C22⋊C4—141no—
480C22⋊C4—141no—
481C22⋊C4—141no—
482C22⋊C4—141no—
483C22⋊C4—141no—
484C42—21no—
485C42—21no—
486C42—71no—
487C42—71no—
488C42—71no—
489C42C23×D711yescharacteristic
490C2×C7#293211yescharacteristic
491C2×C7#295111yescharacteristic
492C2×C7#295711yescharacteristic
493D7—41no—
494D7—81no—
495D7—81no—
496C23—71no—
497C23—141no—
498C23—141no—
499Q8—21no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503D4—141no—
504D4—141no—
505D4—141no—
506D4—141no—
507D4—561no—
508D4—561no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—71no—
514C2×C4—71no—
515C2×C4—71no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4C22×C4×D711yescharacteristic
528C2×C4C2×D4×D711yescharacteristic
529C2×C4D7×#4111yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—141no—
538C8—141no—
539C8—141no—
540C8—141no—
541C8—281no—
542C8—281no—
543C8—281no—
544C7#1799211yescharacteristic, complemented, Sylow
545C22—141no—
546C22—141no—
547C22—141no—
548C22—281no—
549C22—281no—
550C22—281no—
551C22—281no—
552C22—281no—
553C22—281no—
554C22C4×D4×D711yescharacteristic
555C4—21no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—141no—
561C4—141no—
562C4—141no—
563C4—281no—
564C4—281no—
565C4#9043611yescharacteristic
566C4#9044111yescharacteristic
567C4#9045711yescharacteristic
568C4#9046011yescharacteristic
569C2—281no—
570C2—561no—
571C2—561no—
572C2#1173692511yescharacteristic
573C2#1173716611yescharacteristic
574C2#1173734411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1584C2
socleC22×C7C2×C4×D4
prime radical#89655C4
cosocleC42×C7C24
hypercenter#1584D7
hypocenterC7#17992
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. #11737392C2semidirect
2. #11737353C2semidirect
4. C7×#1584C2semidirect
5. #11728708C2semidirect
6. #11728690C2semidirect
7. #11728657C2semidirect
8. #11726323C2semidirect
9. #11726313C2semidirect
10. #11725996C2semidirect
11. #11725996C2semidirect
12. #11725977C2semidirect
13. #11725974C2semidirect
14. #11725832C2semidirect
15. #11725825C2semidirect
153. #1584D7semidirect
544. C7#17992semidirect

Maximal subgroups

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

subgroupindex
#117373922
#117373532
#117371602
C7×#15842
#117287082
#117286902
#117286572
#117263232
#117263132
#117259962
#117259772
#117259742
#117258322
#117258252
#179927