Finite Group Databasebeta

← #13426914

Subgroups — #13426914

Order 1792.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2494C211yesmaximal, characteristic
2#11730913C211yesmaximal, characteristic
3#11727388C211yesmaximal, characteristic, complemented
4#11727385C211yesmaximal, characteristic, complemented
5#11727374C211yesmaximal, characteristic, complemented
6#11727357C211yesmaximal, characteristic, complemented
7#11727314C211yesmaximal, characteristic, complemented
8#11727294C211yesmaximal, characteristic, complemented
9#11727281C211yesmaximal, characteristic
10#11727280C211yesmaximal, characteristic
11#11727260C211yesmaximal, characteristic
12#11727257C211yesmaximal, characteristic
13#11727229C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727206C211yesmaximal, characteristic
16C2×#6489C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C7×#401—21no—
20C7×#397—21no—
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#388—21no—
24C7×#387—21no—
25C7×#382C2211yescharacteristic
26C7×#380C2211yescharacteristic
27C7×#380C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89424—21no—
48#89421—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27061—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C22×Dic14C2311yescharacteristic
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4D411yescharacteristic
71C2×C7×C22⋊C4D411yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77#6489—22no—
78#6489—41no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82#6487—22no—
83#6487—22no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C4×Dic14—22no—
95C4×Dic14—41no—
96C7×#95—21no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C22⋊(C7⋊C8)—22no—
110C22⋊(C7⋊C8)—41no—
111#6384—22no—
112#6384—22no—
113#6384—22no—
114#6384—41no—
115#6382—22no—
116#6382—41no—
117#6381—22no—
118#6381—22no—
119C4⋊(C7⋊C8)—22no—
120#6377—41no—
121#2494D711yescharacteristic
122#2362—71no—
123#2074—71no—
124#2066—71no—
125#2036—71no—
126#2035—71no—
127#1949—71no—
128#1943—71no—
129#1924—71no—
130#1921—71no—
131#1920—71no—
132#1863—71no—
133#1854—71no—
134#1825—71no—
135#1763—71no—
136C24×C7C2×D411yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×Dic14—21no—
158C2×Dic14—21no—
159C2×Dic14—21no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14#4112yes—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—81no—
173C7×C22⋊C4—41no—
174C7×C22⋊C4—41no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—81no—
182C22⋊Dic7—24no—
183C22⋊Dic7—42no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7#4112yes—
189C4⋊Dic7#4112yes—
190Dic7⋊C4—24no—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193C4×Dic7—42no—
194C4×Dic7—42no—
195C4×Dic7—42no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8#4112yes—
208C2×C22⋊Q8—71no—
209C2×C4×Q8—71no—
210C2×#117—71no—
211C2×#107—71no—
212C2×#107—71no—
213C2×#106—71no—
214C2×#106—71no—
215C2×C4⋊C8—71no—
216C2×#103—71no—
217C2×#103—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×C22⋊C8—71no—
221C2×C22⋊C8—71no—
222C2×#97—71no—
223#401—21no—
224#397—21no—
225#391—71no—
226#389—71no—
227#389C2×D711yescharacteristic
228#389C2×D711yescharacteristic
229#388—21no—
230#388—141no—
231#388—141no—
232#387—21no—
233#387—141no—
234#382C2×D711yescharacteristic
235#380C2×D711yescharacteristic
236#380C2×D711yescharacteristic
237C4×C4⋊C4—71no—
238C4×C4⋊C4—141no—
239C4×C4⋊C4C2×D711yescharacteristic
240C4×C22⋊C4C2×D711yescharacteristic
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336—71no—
250#336—71no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C22×D411yescharacteristic
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308Dic14—22no—
309Dic14—22no—
310Dic14—41no—
311Dic14—41no—
312C7⋊C8—22no—
313C7⋊C8—41no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C22×Q8—71no—
317C23×C4C22×D711yescharacteristic
318C22×C8—71no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8—71no—
322C22⋊Q8—142no—
323C22⋊Q8—142no—
324C4×Q8—142no—
325C4×Q8—281no—
326#117—142no—
327#117—281no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4C22⋊D711yescharacteristic
345C2×C22⋊C4C22⋊D711yescharacteristic
346C2×C42—21no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42C22×D711yescharacteristic
351C2×C42C22×D711yescharacteristic
352#107—142no—
353#107—142no—
354#106—142no—
355#106—281no—
356C4⋊C8—142no—
357#103—142no—
358#103—142no—
359#103—142no—
360#103—281no—
361C22⋊C8—142no—
362C22⋊C8—281no—
363#97—281no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95C22⋊D711yescharacteristic
376#95C22⋊D711yescharacteristic
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—41no—
383C22×C7—41no—
384C22×C7#52511yescharacteristic
385C22×C7#54611yescharacteristic
386C22×C7#54711yescharacteristic
387C22×C7#57511yescharacteristic
388C22×C7#57511yescharacteristic
389C22×C7#57711yescharacteristic
390C22×C7#57811yescharacteristic
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7#52911yescharacteristic
401C4×C7#53911yescharacteristic
402C4×C7#55311yescharacteristic
403C4×C7#55611yescharacteristic
404Dic7—42no—
405Dic7—42no—
406Dic7—42no—
407Dic7—81no—
408C24C2×C22⋊D711yescharacteristic
409C2×Q8—72no—
410C2×Q8—141no—
411C2×Q8—141no—
412C2×Q8—141no—
413C2×Q8—141no—
414C2×Q8—141no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—141no—
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—72no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—72no—
457C4⋊C4—72no—
458C4⋊C4—81no—
459C4⋊C4—144no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—42no—
469C22⋊C4—81no—
470C22⋊C4—144no—
471C22⋊C4—282no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—81no—
477C42—282no—
478C42—282no—
479C42—282no—
480C2×C7—81no—
481C2×C7#246611yescharacteristic
482C2×C7#311111yescharacteristic
483C2×C7#312111yescharacteristic
484C2×C7#330711yescharacteristic
485C2×C7#332711yescharacteristic
486C2×C7#334911yescharacteristic
487C2×C7#335011yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C22×C22⋊D711yescharacteristic
496Q8—142no—
497Q8—142no—
498Q8—281no—
499Q8—281no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4D7×#4111yescharacteristic
544C2×C4D7×#4111yescharacteristic
545C2×C4D7×#4111yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C2×C4#655311yescharacteristic
548C2×C4#655411yescharacteristic
549C8—142no—
550C8—281no—
551C8—281no—
552C8—281no—
553C7#2706111yescharacteristic, complemented, Sylow
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#9051311yescharacteristic
562C22#9051311yescharacteristic
563C22#9051511yescharacteristic
564C22#9051911yescharacteristic
565C22#9052011yescharacteristic
566C22#9052711yescharacteristic
567C22#9052911yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—282no—
578C4—282no—
579C4—282no—
580C4—561no—
581C4#9039311yescharacteristic
582C4#9039711yescharacteristic
583C4#9039811yescharacteristic
584C4#9040011yescharacteristic
585C2—81no—
586C2#1174055011yescharacteristic
587C2#1174058711yescharacteristic
588C2#1174091211yescharacteristic
589C2#1174105011yescharacteristic
590C2#1174105611yescharacteristic
591C2#1174106011yescharacteristic
592C2#1174110511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2494C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2494D7
hypocenterC7#27061
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
3. #11727388C2semidirect
4. #11727385C2semidirect
5. #11727374C2semidirect
6. #11727357C2semidirect
7. #11727314C2semidirect
8. #11727294C2semidirect
13. #11727229C2semidirect
14. #11727214C2semidirect
553. C7#27061semidirect

Maximal subgroups

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

subgroupindex
C7×#24942
#117309132
#117273882
#117273852
#117273742
#117273572
#117273142
#117272942
#117272812
#117272802
#117272602
#117272572
#117272292
#117272142
#117272062
#270617