Finite Group Databasebeta

← #13325508

Subgroups — #13325508

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6475C211yesmaximal, characteristic
2#11736969C211yesmaximal, characteristic, complemented
3#11736851C211yesmaximal, characteristic, complemented
4C7×#1581C211yesmaximal, characteristic, complemented
5#11728633C211yesmaximal, characteristic, complemented
6#11728614C211yesmaximal, characteristic, complemented
7#11728593C211yesmaximal, characteristic, complemented
8#11726323C211yesmaximal, characteristic, complemented
9#11726314C211yesmaximal, characteristic, complemented
10#11725928C211yesmaximal, characteristic, complemented
11#11725928C211yesmaximal, characteristic, complemented
12#11725916C211yesmaximal, characteristic, complemented
13#11725904C211yesmaximal, characteristic, complemented
14#11725840C211yesmaximal, characteristic, complemented
15#11725824C211yesmaximal, characteristic, complemented
16C2×#6475C411yescharacteristic
17C4×#1159C2211yescharacteristic
18C4×#1159C2211yescharacteristic
19C4×C7×D8C2211yescharacteristic
20C7×#436C2211yescharacteristic
21C7×C8×D4C2211yescharacteristic
22D8×Dic7C411yescharacteristic
23#89900—21no—
24#89890—21no—
25#89883—21no—
26C2×Dic7⋊C8—21no—
27C4×#1142C2211yescharacteristic
28C4×#1142C2211yescharacteristic
29#89784C2211yescharacteristic
30D4×C7⋊C8C2211yescharacteristic
31#89652C411yescharacteristic
32#89645C411yescharacteristic
33#89609C2211yescharacteristic
34#89609C2211yescharacteristic
35#89603C2211yescharacteristic
36#89602C2211yescharacteristic
37#89544C411yescharacteristic
38#89544C411yescharacteristic
39#89530C411yescharacteristic
40#89530C411yescharacteristic
41#89502—21no—
42#89501—21no—
43#89491—21no—
44#89490—21no—
45C4×Dic28C2211yescharacteristic
46C4×#1131C2211yescharacteristic
47#89457C2211yescharacteristic
48C4×C8×D7C2211yescharacteristic
49#89451C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×D4⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55D4⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89279C2211yescharacteristic
59#89274C2211yescharacteristic
60#89274C2211yescharacteristic
61#89253C2211yescharacteristic
62#89251C2211yescharacteristic
63#89251C2211yescharacteristic
64#89249C2211yescharacteristic
65#89249C2211yescharacteristic
66#89249C2211yescharacteristic
67#18061—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×#1159C2×C411yescharacteristic
70C2×C7×D8C2×C411yescharacteristic
71C2×C7×M16—21no—
72C22×C7×C8—21no—
73C4×C7×D4C2311yescharacteristic
74C4×C7×D4C2311yescharacteristic
75D4×Dic7C2×C411yescharacteristic
76D4×Dic7C2×C411yescharacteristic
77C2×#1142C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C4×C22⋊D7—21no—
80C4×C22⋊D7—21no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×#1136—21no—
84C22×C7⋊C8—21no—
85#6475—24no—
86C2×Dic28C2×C411yescharacteristic
87C2×#1131—21no—
88C2×#1131—21no—
89C2×#1131—21no—
90C2×C8×D7C2×C411yescharacteristic
91C2×C8×D7C2×C411yescharacteristic
92#6455C2×C411yescharacteristic
93#6455C2×C411yescharacteristic
94#6450C2×C411yescharacteristic
95#6450C2×C411yescharacteristic
96(C22⋊D7)⋊C4—21no—
97(C22⋊D7)⋊C4—21no—
98#6440—21no—
99#6440—21no—
100C42×D7C2311yescharacteristic
101C4×Dic14C2311yescharacteristic
102C4×Dic14C2311yescharacteristic
103C7×#107C2×C411yescharacteristic
104C7×C4⋊C8C2311yescharacteristic
105C7×C4⋊C8C2311yescharacteristic
106C7×#102C2×C411yescharacteristic
107C7×#102C2×C411yescharacteristic
108C7×C22⋊C8—21no—
109C7×C22⋊C8—21no—
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112C4×C7×C8C2311yescharacteristic
113#6406C2×C411yescharacteristic
114#6406C2×C411yescharacteristic
115C22⋊(C7⋊C8)—21no—
116C22⋊(C7⋊C8)—21no—
117(C2×D7)⋊C8D411yescharacteristic
118(C2×D7)⋊C8D411yescharacteristic
119#6392C2×C411yescharacteristic
120#6390C2×C411yescharacteristic
121#6390C2×C411yescharacteristic
122#6389—21no—
123#6389—21no—
124#6389—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129Dic7⋊C8—21no—
130Dic7⋊C8—21no—
131Dic7⋊C8D411yescharacteristic
132Dic7⋊C8D411yescharacteristic
133C8×Dic7—21no—
134C8×Dic7—21no—
135C8×Dic7C2×C411yescharacteristic
136C8×Dic7C2×C411yescharacteristic
137#6384C2×C411yescharacteristic
138#6384C2×C411yescharacteristic
139#6382C2×C411yescharacteristic
140#6382C2×C411yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4⋊(C7⋊C8)C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145C4×C7⋊C8C2311yescharacteristic
146#2991—71no—
147#2990—71no—
148C4×#135—71no—
149#1592—71no—
150#1592—71no—
151#1584—71no—
152#1584—71no—
153#1581—71no—
154#1581D711yescharacteristic, complemented
155SD16⋊C8—71no—
156SD16⋊C8—71no—
157#1567—71no—
158#1489—71no—
159#1489—71no—
160#1451—71no—
161C2×C7×D4C22×C411yescharacteristic
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C2×C22⋊D7—21no—
166C2×C22⋊D7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171#1159—22no—
172#1159—22no—
173#1159—41no—
174#1159—41no—
175C2×C4×D7C22×C411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7#4111yescharacteristic
178C2×Dic14C22×C411yescharacteristic
179C2×Dic14C22×C411yescharacteristic
180C7×D8—22no—
181C7×M16—81no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—81no—
186C2×C7×C8C22×C411yescharacteristic
187C2×C7×C8C22×C411yescharacteristic
188C2×C7×C8C2×D411yescharacteristic
189C2×C7×C8C2×D411yescharacteristic
190C7×C4⋊C4C22×C411yescharacteristic
191C7×C4⋊C4C22×C411yescharacteristic
192C7×C22⋊C4—21no—
193C7×C22⋊C4—21no—
194C42×C7C2411yescharacteristic
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197#1142—22no—
198#1142M1614yescomplemented
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—21no—
201C4⋊Dic7C22×C411yescharacteristic
202C4⋊Dic7C22×C411yescharacteristic
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206Dic7⋊C4—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7C22×C411yescharacteristic
214C4×Dic7C2×D411yescharacteristic
215C4×Dic7#4111yescharacteristic
216#1136—81no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—81no—
221C2×C7⋊C8C22×C411yescharacteristic
222C2×C7⋊C8C22×C411yescharacteristic
223C2×C7⋊C8C2×D411yescharacteristic
224C2×C7⋊C8C2×D411yescharacteristic
225Dic28—22no—
226#1131—42no—
227#1131—81no—
228#1131—81no—
229C8×D7—22no—
230C8×D7—41no—
231C2×#135—71no—
232C4×#41—71no—
233C4×#41—71no—
234#446—71no—
235#446—71no—
236C8×Q8—71no—
237C8×Q8—71no—
238C4×Q16—71no—
239C4×Q16—71no—
240C4×SD16—71no—
241C4×SD16—71no—
242C4×SD16—71no—
243C4×SD16—71no—
244C4×D8—71no—
245C4×D8C2×D711yescharacteristic
246#436—71no—
247#436—141no—
248#436—141no—
249#436C2×D711yescharacteristic
250#435—141no—
251#435—141no—
252C8×D4—71no—
253C8×D4C2×D711yescharacteristic
254#432—141no—
255#431—71no—
256#431—141no—
257#427—71no—
258#423—71no—
259#423—71no—
260#423—141no—
261C2×C4⋊C8—141no—
262#416—71no—
263#416—71no—
264C4×M16—71no—
265C2×C4×C8—71no—
266#335—71no—
267#335—71no—
268#335—71no—
269#335C2×D711yescharacteristic
270Q8⋊C8—71no—
271Q8⋊C8—71no—
272Q8⋊C8—71no—
273Q8⋊C8—71no—
274D4⋊C8—71no—
275D4⋊C8—71no—
276D4⋊C8C2×D711yescharacteristic
277D4⋊C8C2×D711yescharacteristic
278#322—71no—
279#322—71no—
280#322—71no—
281#322C2×D711yescharacteristic
282C23×C7—21no—
283C23×C7—21no—
284C22×D7C4×D411yescharacteristic
285C7×D4—21no—
286C7×D4—21no—
287C7×D4—21no—
288C7×D4C2×M1612yes—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C22⋊D7—42no—
299C22⋊D7—42no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7C4×D411yescharacteristic
309C2×Dic7C4×D411yescharacteristic
310C2×Dic7C4×D411yescharacteristic
311C4×D7—21no—
312C4×D7—21no—
313C4×D7—21no—
314C4×D7—21no—
315C4×D7—22no—
316Dic14—21no—
317Dic14—21no—
318Dic14—21no—
319Dic14C2×M1612yes—
320C7×C8—21no—
321C7×C8—21no—
322C7×C8—21no—
323C7×C8—21no—
324C7×C8—41no—
325C7×C8—41no—
326C7×C8—41no—
327C7⋊C8—21no—
328C7⋊C8—21no—
329C7⋊C8—21no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C7⋊C8C2×M1612yes—
334C2×#41—71no—
335C2×#41—71no—
336#135—144no—
337C2×Q16—71no—
338C2×SD16—71no—
339C2×SD16—71no—
340C2×D8C4×D711yescharacteristic
341C2×M16—21no—
342C2×M16—141no—
343C2×M16—141no—
344C2×M16—141no—
345C2×M16—141no—
346C22×C8—21no—
347C22×C8—71no—
348C22×C8—71no—
349C22×C8—141no—
350C4×Q8—71no—
351C4×Q8—71no—
352C4×Q8—71no—
353C4×Q8—71no—
354C4×D4—71no—
355C4×D4—71no—
356C4×D4—141no—
357C4×D4—141no—
358C4×D4—141no—
359C4×D4—141no—
360C4×D4C22×D711yescharacteristic
361C4×D4C22×D711yescharacteristic
362#117—71no—
363#117—71no—
364#117—141no—
365#117—141no—
366C2×C42—71no—
367C2×C42—141no—
368C2×C42—141no—
369#107—71no—
370#107C4×D711yescharacteristic
371#106—71no—
372#106—71no—
373C4⋊C8—71no—
374C4⋊C8—71no—
375C4⋊C8—71no—
376C4⋊C8—71no—
377C4⋊C8—141no—
378C4⋊C8—141no—
379C4⋊C8—141no—
380C4⋊C8—141no—
381C4⋊C8—141no—
382C4⋊C8—141no—
383C4⋊C8C22×D711yescharacteristic
384C4⋊C8C22×D711yescharacteristic
385#103—71no—
386#103—71no—
387#103—71no—
388#103—71no—
389#102—71no—
390#102—71no—
391#102C4×D711yescharacteristic
392#102C4×D711yescharacteristic
393C22⋊C8—21no—
394C22⋊C8—21no—
395C22⋊C8—71no—
396C22⋊C8—71no—
397C22⋊C8—141no—
398C22⋊C8—141no—
399#97—141no—
400#97—141no—
401#97—141no—
402C4×C8—71no—
403C4×C8—71no—
404C4×C8—71no—
405C4×C8—71no—
406C4×C8—71no—
407C4×C8—141no—
408C4×C8—141no—
409C4×C8C22×D711yescharacteristic
410C4×C8C22×D711yescharacteristic
411C4×C8C22×D711yescharacteristic
412C22×C7—22no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7C2×C4×D411yescharacteristic
419C2×D7—21no—
420C2×D7—21no—
421C2×D7—21no—
422C4×C7—21no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7C22×M1611yescharacteristic
426C4×C7#56811yescharacteristic
427C4×C7#57611yescharacteristic
428C4×C7#57811yescharacteristic
429Dic7—22no—
430Dic7—21no—
431Dic7—21no—
432Dic7—41no—
433Dic7—41no—
434Dic7—41no—
435Dic7#44412yes—
436#41—142no—
437#41—142no—
438#41—281no—
439#41—281no—
440C2×Q8—71no—
441C2×Q8—71no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4C2×C4×D711yescharacteristic
445C2×D4C2×C4×D711yescharacteristic
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455Q16—142no—
456SD16—74no—
457SD16—142no—
458D8—22no—
459M16—81no—
460M16—282no—
461M16—561no—
462M16—561no—
463M16—561no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—71no—
468C2×C8—71no—
469C2×C8—71no—
470C2×C8—71no—
471C2×C8—81no—
472C2×C8—142no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—281no—
477C2×C8—561no—
478C2×C8C2×C4×D711yescharacteristic
479C2×C8C2×C4×D711yescharacteristic
480C2×C8D4×D711yescharacteristic
481C2×C8D4×D711yescharacteristic
482C4⋊C4—71no—
483C4⋊C4—71no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4C2×C4×D711yescharacteristic
489C4⋊C4C2×C4×D711yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—141no—
493C22⋊C4—141no—
494C22⋊C4—141no—
495C22⋊C4—141no—
496C42—71no—
497C42—71no—
498C42—71no—
499C42—141no—
500C42—141no—
501C42—141no—
502C42—141no—
503C42—141no—
504C42—141no—
505C42C23×D711yescharacteristic
506C2×C7—42no—
507C2×C7—81no—
508C2×C7D4×M1611yescharacteristic
509C2×C7#295111yescharacteristic
510C2×C7C2×#44411yescharacteristic
511D7—41no—
512C23—21no—
513C23—21no—
514C23—71no—
515Q8—72no—
516Q8—141no—
517Q8—141no—
518Q8—141no—
519D4—21no—
520D4—21no—
521D4—21no—
522D4—282no—
523D4—282no—
524D4D7×M1612yes—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—71no—
532C2×C4—71no—
533C2×C4—71no—
534C2×C4—142no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4C22×C4×D711yescharacteristic
548C2×C4C2×D4×D711yescharacteristic
549C2×C4D7×#4111yescharacteristic
550C8—21no—
551C8—21no—
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—72no—
558C8—141no—
559C8—141no—
560C8—141no—
561C8—281no—
562C8—281no—
563C8—281no—
564C7#1806111yescharacteristic, complemented, Sylow
565C22—22no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—141no—
572C22—141no—
573C22—141no—
574C22C4×D4×D711yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—72no—
579C4—141no—
580C4—142no—
581C4—141no—
582C4—281no—
583C4—281no—
584C4—281no—
585C4C2×D7×M1611yescharacteristic
586C4#9044111yescharacteristic
587C4#9044811yescharacteristic
588C4#9046211yescharacteristic
589C2—42no—
590C2—81no—
591C2—281no—
592C2#1173692811yescharacteristic
593C2#1173703411yescharacteristic
594C2D7×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1581C2
socleC22×C7C2×C4×D4
prime radicalC2×#6475C4
cosocleC42×C7C24
hypercenter#1581D7
hypocenterC7#18061
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #11736969C2semidirect
3. #11736851C2semidirect
4. C7×#1581C2semidirect
5. #11728633C2semidirect
6. #11728614C2semidirect
7. #11728593C2semidirect
8. #11726323C2semidirect
9. #11726314C2semidirect
10. #11725928C2semidirect
11. #11725928C2semidirect
12. #11725916C2semidirect
13. #11725904C2semidirect
14. #11725840C2semidirect
15. #11725824C2semidirect
154. #1581D7semidirect
198. #1142M16semidirect
564. C7#18061semidirect

Maximal subgroups

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

subgroupindex
C4×#64752
#117369692
#117368512
C7×#15812
#117286332
#117286142
#117285932
#117263232
#117263142
#117259282
#117259162
#117259042
#117258402
#117258242
#180617