Finite Group Databasebeta

← #13508599

Subgroups — #13508599

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736625C211yesmaximal, characteristic
2D7×#427C211yesmaximal, characteristic, complemented
3C7×#2158C211yesmaximal, characteristic, complemented
4#11729166C211yesmaximal, characteristic, complemented
5C2×(C2×D7)⋊C16C211yesmaximal, characteristic, complemented
6#11728405C211yesmaximal, characteristic, complemented
7#11728402C211yesmaximal, characteristic, complemented
8#11727106C211yesmaximal, characteristic, complemented
9#11727105C211yesmaximal, characteristic, complemented
10#11727104C211yesmaximal, characteristic, complemented
11#11727096C211yesmaximal, characteristic, complemented
12#11727086C211yesmaximal, characteristic, complemented
13#11727085C211yesmaximal, characteristic, complemented
14#11727084C211yesmaximal, characteristic, complemented
15#11727077C211yesmaximal, characteristic, complemented
16C2×#6467C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18#90215—21no—
19D7×#117—21no—
20C22×C7×C16C2211yescharacteristic
21C7×#427C2211yescharacteristic
22C7×#427C2211yescharacteristic
23#89878C2211yescharacteristic
24C22×C7⋊C16C2211yescharacteristic
25#89774—21no—
26#89770—21no—
27#89659C2211yescharacteristic
28#89659C2211yescharacteristic
29#89655C2211yescharacteristic
30#89655C2211yescharacteristic
31#89652C2211yescharacteristic
32#89652C2211yescharacteristic
33D7×#107C2211yescharacteristic
34D7×#107C2211yescharacteristic
35#89636—21no—
36#89636—21no—
37#89632—21no—
38D7×#106—21no—
39C7×#367C2211yescharacteristic, complemented
40C7×#367C2211yescharacteristic, complemented
41C7×#366C2211yescharacteristic, complemented
42C7×#366C2211yescharacteristic, complemented
43#89304C2211yescharacteristic, complemented
44#89304C2211yescharacteristic, complemented
45(C2×D7)⋊C16C2211yescharacteristic
46(C2×D7)⋊C16C2211yescharacteristic
47(C2×D7)⋊C16C2211yescharacteristic
48(C2×D7)⋊C16C2211yescharacteristic
49#89298C2211yescharacteristic, complemented
50#89298C2211yescharacteristic, complemented
51#89287C2211yescharacteristic, complemented
52#89287C2211yescharacteristic, complemented
53#89286C2211yescharacteristic, complemented
54#89286C2211yescharacteristic, complemented
55#89285C2211yescharacteristic, complemented
56#89285C2211yescharacteristic, complemented
57#89284C2211yescharacteristic, complemented
58#89284C2211yescharacteristic, complemented
59#33645—71nomaximal, Sylow
60C2×#1157—21no—
61C22×C4×D7D411yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C7×#117—21no—
64C7×#117—21no—
65#6489—21no—
66C22×C7⋊C8D411yescharacteristic
67C2×Dic28C2311yescharacteristic
68#6467—22no—
69#6467—22no—
70C2×D56C2311yescharacteristic
71C2×#1132—21no—
72C2×C8×D7—22no—
73C2×C8×D7—22no—
74C2×C8×D7D411yescharacteristic
75C2×C8×D7D411yescharacteristic
76C2×C8×D7C2311yescharacteristic
77C2×C8×D7C2311yescharacteristic
78#6460—21no—
79#6460—21no—
80#6458—21no—
81#6458—21no—
82#6455—21no—
83#6455—21no—
84D7×C4⋊C4—21no—
85D7×C4⋊C4—21no—
86#6449—41no—
87#6445—41no—
88D7×C22⋊C4—41no—
89#6440—41no—
90C4×D28—41no—
91C4×D28—41no—
92#6435—41no—
93C42×D7—41no—
94C2×C7×C16—22no—
95C2×C7×C16C2311yescharacteristic
96C2×C7×C16C2311yescharacteristic
97C7×#107C2311yescharacteristic
98C7×#107C2311yescharacteristic
99C7×#107C2311yescharacteristic
100C7×#107C2311yescharacteristic
101C7×#106—21no—
102C7×#106—21no—
103#6392C2311yescharacteristic
104#6392C2311yescharacteristic
105#6391—21no—
106C2×C7⋊C16—22no—
107C2×C7⋊C16C2311yescharacteristic
108C2×C7⋊C16C2311yescharacteristic
109#6384—21no—
110#6384—21no—
111#6383—21no—
112#6383—21no—
113#6382—21no—
114#6382—21no—
115#6381—21no—
116#6381—21no—
117#3050—71no—
118C2×#427—71no—
119#2237—71no—
120#2237—71no—
121#2236—71no—
122#2235—71no—
123#2234—71no—
124#2234—71no—
125#2233—71no—
126#2232—71no—
127#2158—71no—
128#2158D711yescharacteristic, complemented
129#2138—71no—
130#2138—71no—
131C2×C22⋊C16—71no—
132C23×D7D811yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C2×C22⋊D7—41no—
135C22×Dic7D811yescharacteristic
136#1157—42no—
137#1157—42no—
138C2×D28—21no—
139C2×D28—41no—
140C2×D28—41no—
141C2×C4×D7—22no—
142C2×C4×D7—22no—
143C2×C4×D7—41no—
144C2×C4×D7—41no—
145C2×C4×D7—41no—
146C2×C4×D7D811yescharacteristic
147C2×C4×D7D811yescharacteristic
148C2×C4×D7C2×D411yescharacteristic
149C2×C4×D7C2×D411yescharacteristic
150C2×Dic14—21no—
151C2×C7×C8C2×D411yescharacteristic
152C2×C7×C8#4112yes—
153C2×C7×C8#4112yes—
154C2×C7×C8C2411yescharacteristic
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—21no—
159C7×C22⋊C4—41no—
160C7×C22⋊C4—41no—
161C42×C7—41no—
162C42×C7—41no—
163C22⋊Dic7—41no—
164(C2×D7)⋊C4—41no—
165(C2×D7)⋊C4—41no—
166(C2×D7)⋊C4—41no—
167(C2×D7)⋊C4—41no—
168(C2×D7)⋊C4—41no—
169(C2×D7)⋊C4—41no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—41no—
173C4⋊Dic7—41no—
174Dic7⋊C4—41no—
175Dic7⋊C4—41no—
176C4×Dic7—41no—
177C4×Dic7—41no—
178C4×Dic7—41no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8C2×D411yescharacteristic
181C2×C7⋊C8C2×D411yescharacteristic
182Dic28—21no—
183Dic28—21no—
184D56—21no—
185D56—21no—
186#1132—42no—
187C8×D7—22no—
188C8×D7—22no—
189C8×D7—21no—
190C8×D7—21no—
191C8×D7—21no—
192C8×D7—21no—
193C7×C16—21no—
194C7×C16—21no—
195C7⋊C16—21no—
196C7⋊C16—21no—
197C2×#135—71no—
198C23×C8—71no—
199#525—141no—
200C2×#117—141no—
201C22×C16—71no—
202C22×C16C2×D711yescharacteristic
203#467—71no—
204#467—71no—
205#466—71no—
206#466—71no—
207#465—141no—
208#465—141no—
209#427—71no—
210#427—71no—
211#427—71no—
212#427—141no—
213#427—141no—
214#427C2×D711yescharacteristic
215#427C2×D711yescharacteristic
216C2×#107—71no—
217C2×#107—71no—
218C2×#106—141no—
219#416—141no—
220#367—71no—
221#367—71no—
222#367C2×D711yescharacteristic, complemented
223#367C2×D711yescharacteristic, complemented
224#366—71no—
225#366—71no—
226#366C2×D711yescharacteristic, complemented
227#366C2×D711yescharacteristic, complemented
228#358—71no—
229#358—71no—
230#358—71no—
231#358—71no—
232#357—71no—
233#357—71no—
234#357—71no—
235#357—71no—
236C22⋊C16—71no—
237C22⋊C16—71no—
238C22⋊C16—71no—
239C22⋊C16—71no—
240C23×C7C2×D811yescharacteristic
241C22×D7—22no—
242C22×D7—22no—
243C22×D7—41no—
244C22×D7—41no—
245C22×D7C2×D811yescharacteristic
246C22×D7C2×D811yescharacteristic
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7#12312yes—
252C2×C4×C7C2×D811yescharacteristic
253C2×C4×C7#13712yes—
254C2×C4×C7C22×D411yescharacteristic
255C22⋊D7—82no—
256C2×Dic7—22no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7C2×D811yescharacteristic
261C2×Dic7C2×D811yescharacteristic
262D28—41no—
263D28—41no—
264D28—82no—
265D28—82no—
266C4×D7—21no—
267C4×D7—21no—
268C4×D7—21no—
269C4×D7—21no—
270C4×D7—22no—
271C4×D7—22no—
272C4×D7—82no—
273C4×D7—82no—
274C4×D7—82no—
275Dic14—41no—
276Dic14—41no—
277C7×C8#12312yes—
278C7×C8C2×#4111yescharacteristic
279C7×C8C2×#4111yescharacteristic
280C7⋊C8—21no—
281C7⋊C8—21no—
282C2×#41—141no—
283C23×C4—71no—
284#135—142no—
285#135—142no—
286C2×Q16—71no—
287C2×SD16—141no—
288C2×D8—71no—
289C22×C8—71no—
290C22×C8—71no—
291C22×C8—71no—
292C22×C8—71no—
293C22×C8—71no—
294C22×C8—142no—
295C22×C8—142no—
296C22×C8C22×D711yescharacteristic
297#123—281no—
298C22⋊Q8—141no—
299C22⋊Q8—141no—
300#121—141no—
301#121—141no—
302#120—281no—
303C4×D4—281no—
304C4×D4—281no—
305#117—21no—
306#117—21no—
307#117—141no—
308#117—141no—
309#117—141no—
310#117—281no—
311#117—281no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C22⋊C4—281no—
315C2×C42—281no—
316C2×C16—22no—
317C2×C16—71no—
318C2×C16—71no—
319C2×C16—142no—
320C2×C16C22×D711yescharacteristic
321C2×C16C22×D711yescharacteristic
322#107—71no—
323#107—71no—
324#107—141no—
325#107—141no—
326#107C22×D711yescharacteristic
327#107C22×D711yescharacteristic
328#107C22×D711yescharacteristic
329#107C22×D711yescharacteristic
330#106—21no—
331#106—21no—
332#106—141no—
333#106—141no—
334#106—141no—
335#103—141no—
336#103—141no—
337#102—141no—
338#102—141no—
339C22×C7—21no—
340C22×C7—21no—
341C22×C7#48012yes—
342C22×C7C22×D811yescharacteristic
343C2×D7—21no—
344C2×D7—21no—
345C2×D7—21no—
346C2×D7—21no—
347C2×D7—21no—
348C2×D7—44no—
349C2×D7—81no—
350C2×D7—81no—
351C2×D7—81no—
352C2×D7#50812yes—
353C4×C7—81no—
354C4×C7—81no—
355C4×C7—81no—
356C4×C7—81no—
357C4×C7#48012yes—
358C4×C7C2×#12311yescharacteristic
359C4×C7C2×#13711yescharacteristic
360Dic7—21no—
361Dic7—21no—
362Dic7—81no—
363Dic7—81no—
364Dic7—81no—
365C24—71no—
366#41—282no—
367#41—282no—
368C2×Q8—141no—
369C2×D4—141no—
370C2×D4—281no—
371C2×D4—281no—
372C2×D4—281no—
373C22×C4—71no—
374C22×C4—71no—
375C22×C4—71no—
376C22×C4—71no—
377C22×C4—71no—
378C22×C4—142no—
379C22×C4—142no—
380C22×C4—281no—
381C22×C4—281no—
382C22×C4—281no—
383C22×C4D4×D711yescharacteristic
384Q16—141no—
385Q16—141no—
386SD16—282no—
387D8—141no—
388D8—141no—
389C2×C8—71no—
390C2×C8—71no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—142no—
394C2×C8—141no—
395C2×C8—141no—
396C2×C8—141no—
397C2×C8—141no—
398C2×C8#115712yes—
399C2×C8D4×D711yescharacteristic
400C2×C8#116112yes—
401C2×C8C23×D711yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—141no—
407C4⋊C4—141no—
408C4⋊C4—281no—
409C4⋊C4—281no—
410C4⋊C4—281no—
411C4⋊C4—281no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—281no—
415C22⋊C4—281no—
416C22⋊C4—281no—
417C22⋊C4—281no—
418C22⋊C4—281no—
419C22⋊C4—281no—
420C22⋊C4—281no—
421C42—41no—
422C42—41no—
423C42—281no—
424C42—281no—
425C42—281no—
426C16—21no—
427C16—21no—
428C16—141no—
429C16—141no—
430C2×C7—22no—
431C2×C7C2×#48011yescharacteristic
432C2×C7C2×#50811yescharacteristic
433C2×C7#341411yescharacteristic
434D7—22no—
435D7—41no—
436D7—161no—
437C23—71no—
438C23—71no—
439C23—142no—
440C23—142no—
441C23—281no—
442C23—281no—
443C23D7×D811yescharacteristic
444Q8—281no—
445Q8—281no—
446D4—281no—
447D4—281no—
448D4—562no—
449D4—562no—
450D4—562no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—71no—
456C2×C4—71no—
457C2×C4—142no—
458C2×C4—142no—
459C2×C4—142no—
460C2×C4—141no—
461C2×C4—141no—
462C2×C4—141no—
463C2×C4—141no—
464C2×C4—281no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—562no—
468C2×C4—562no—
469C2×C4—562no—
470C2×C4#645712yes—
471C2×C4D7×D811yescharacteristic
472C2×C4#648112yes—
473C2×C4C2×D4×D711yescharacteristic
474C8—141no—
475C8—141no—
476C8#645712yes—
477C8C2×#115711yescharacteristic
478C8C2×#116111yescharacteristic
479C7#3364511yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—72no—
483C22—141no—
484C22—141no—
485C22—141no—
486C22—141no—
487C22—141no—
488C22—284no—
489C22—561no—
490C22—561no—
491C22—561no—
492C22#8965412yes—
493C22C2×D7×D811yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—141no—
499C4—141no—
500C4—561no—
501C4—561no—
502C4—561no—
503C4#8965412yes—
504C4C2×#645711yescharacteristic
505C4C2×#648111yescharacteristic
506C2—22no—
507C2—142no—
508C2—281no—
509C2—1121no—
510C2C2×#8965411yescharacteristic
511C2D7×#50811yescharacteristic
512C2#1173947511yescharacteristic
subgroupG/H
centerC2×C4D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2158C2
socleC22×C7C22×D8
prime radical#11736625C2
hypercenter#2158D7
hypocenterC7#33645
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
2. D7×#427C2semidirect
3. C7×#2158C2semidirect
4. #11729166C2semidirect
5. C2×(C2×D7)⋊C16C2semidirect
6. #11728405C2semidirect
7. #11728402C2semidirect
8. #11727106C2semidirect
9. #11727105C2semidirect
10. #11727104C2semidirect
11. #11727096C2semidirect
12. #11727086C2semidirect
13. #11727085C2semidirect
14. #11727084C2semidirect
15. #11727077C2semidirect
39. C7×#367C22semidirect
40. C7×#367C22semidirect
41. C7×#366C22semidirect
42. C7×#366C22semidirect
43. #89304C22semidirect
44. #89304C22semidirect
49. #89298C22semidirect
50. #89298C22semidirect
51. #89287C22semidirect
52. #89287C22semidirect
53. #89286C22semidirect
54. #89286C22semidirect
55. #89285C22semidirect
56. #89285C22semidirect
57. #89284C22semidirect
58. #89284C22semidirect
128. #2158D7semidirect
222. #367C2×D7semidirect
223. #367C2×D7semidirect
226. #366C2×D7semidirect
227. #366C2×D7semidirect
479. C7#33645semidirect

Maximal subgroups

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

subgroupindex
#117366252
D7×#4272
C7×#21582
#117291662
C2×(C2×D7)⋊C162
#117284052
#117284022
#117271062
#117271052
#117271042
#117270962
#117270862
#117270852
#117270842
#117270772
#336457