Finite Group Databasebeta

← #13422468

Subgroups — #13422468

Order 1792.

712 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2507C211yesmaximal, characteristic, complemented
2#11731100C211yesmaximal, characteristic
3#11727386C211yesmaximal, characteristic, complemented
4#11727384C211yesmaximal, characteristic
5#11727382C211yesmaximal, characteristic
6#11727382C211yesmaximal, characteristic
7#11727380C211yesmaximal, characteristic
8#11727368C211yesmaximal, characteristic, complemented
9#11727355C211yesmaximal, characteristic, complemented
10#11727354C211yesmaximal, characteristic, complemented
11#11727342C211yesmaximal, characteristic, complemented
12#11727339C211yesmaximal, characteristic
13#11727236C211yesmaximal, characteristic, complemented
14C4×#6383C211yesmaximal, characteristic
15#11727219C211yesmaximal, characteristic, complemented
16C2×#6438C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C7×#398—21no—
20C7×#398—21no—
21C7×#389—21no—
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#387—21no—
27C7×#384C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30#89762—21no—
31#89761—21no—
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89711C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44#89701C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89441—21no—
48#89438—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26763—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2311yescharacteristic
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93#6438—22no—
94#6438—22no—
95C4×D28—22no—
96C4×D28—41no—
97C4×Dic14—22no—
98C4×Dic14—41no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95D411yescharacteristic
105C7×#95D411yescharacteristic
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109#6384—22no—
110#6384—22no—
111#6384—22no—
112#6384—41no—
113#6383—22no—
114#6383—22no—
115#6383—22no—
116#6383—41no—
117#6382—22no—
118#6381—22no—
119C4⋊(C7⋊C8)—41no—
120C4×C7⋊C8—41no—
121#2507D711yescharacteristic, complemented
122#2367—71no—
123#2071—71no—
124#2067—71no—
125#2042—71no—
126#2042—71no—
127#1933—71no—
128#1928—71no—
129#1919—71no—
130#1918—71no—
131#1916—71no—
132#1865—71no—
133#1864—71no—
134#1803—71no—
135C4×#102—71no—
136C23×D7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28#4112yes—
157C2×C4×D7—42no—
158C2×C4×D7—42no—
159C2×C4×D7—81no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14#4112yes—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—81no—
177C7×C4⋊C4—81no—
178C7×C4⋊C4—81no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4#4112yes—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—81no—
189(C2×D7)⋊C4—24no—
190(C2×D7)⋊C4—24no—
191(C2×D7)⋊C4—42no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—81no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7#4112yes—
198Dic7⋊C4—42no—
199C4×Dic7—42no—
200C4×Dic7—81no—
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—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×#124—71no—
214C2×C4×Q8—71no—
215C2×C4×D4—71no—
216C2×#107—71no—
217C2×#106—71no—
218C2×C4⋊C8—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#102—71no—
224C2×#102—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×C4×C8—71no—
228#398—21no—
229#398—21no—
230#391—71no—
231#390—71no—
232#389—21no—
233#389C2×D711yescharacteristic
234#389C2×D711yescharacteristic
235#389C2×D711yescharacteristic
236#389C2×D711yescharacteristic
237#388—141no—
238#388—141no—
239#387—21no—
240#385—141no—
241#384C2×D711yescharacteristic
242C4×C4⋊C4—71no—
243C4×C4⋊C4C2×D711yescharacteristic
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C22⋊C4—141no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#336—71no—
253#336—71no—
254#336—71no—
255#336—71no—
256C23×C7C22×D411yescharacteristic
257C22×D7—22no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C22×D7—41no—
263C22×D7—41no—
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—22no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
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×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7#13512yes—
293C2×C4×C7#13712yes—
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309D28—22no—
310D28—22no—
311D28—41no—
312D28—41no—
313C4×D7—84no—
314Dic14—22no—
315Dic14—22no—
316Dic14—41no—
317Dic14—41no—
318C7⋊C8—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C22×Q8—71no—
323C22×D4—71no—
324C23×C4—141no—
325C22×C8—71no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—71no—
329#124—142no—
330#124—142no—
331C4×Q8—142no—
332C4×Q8—281no—
333C4×D4—142no—
334C4×D4—281no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4C22⋊D711yescharacteristic
345C2×C4⋊C4C22⋊D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C42—21no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42C22×D711yescharacteristic
359C2×C42C22×D711yescharacteristic
360C2×C42C22×D711yescharacteristic
361#107—142no—
362#106—142no—
363C4⋊C8—281no—
364#103—142no—
365#103—142no—
366#103—142no—
367#103—281no—
368#102—142no—
369#102—142no—
370#102—142no—
371#102—281no—
372C4×C8—281no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95C22⋊D711yescharacteristic
382#95C22⋊D711yescharacteristic
383C22×C7#52511yescharacteristic
384C22×C7#54611yescharacteristic
385C22×C7#54811yescharacteristic
386C22×C7C2×#13511yescharacteristic
387C22×C7C2×#13711yescharacteristic
388C22×C7#57611yescharacteristic
389C22×C7#57711yescharacteristic
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—42no—
393C2×D7—81no—
394C2×D7—81no—
395C2×D7—81no—
396C2×D7—81no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—42no—
400C4×C7—41no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—41no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7—81no—
407C4×C7#52911yescharacteristic
408C4×C7#53911yescharacteristic
409C4×C7#55211yescharacteristic
410C4×C7#55311yescharacteristic
411Dic7—42no—
412Dic7—42no—
413Dic7—81no—
414C24—141no—
415C2×Q8—72no—
416C2×Q8—141no—
417C2×Q8—141no—
418C2×Q8—141no—
419C2×Q8—141no—
420C2×Q8—141no—
421C2×D4—72no—
422C2×D4—141no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—282no—
435C22×C4—282no—
436C22×C4—561no—
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C23×D711yescharacteristic
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—72no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—282no—
476C4⋊C4#116112yes—
477C22⋊C4—144no—
478C22⋊C4—144no—
479C22⋊C4—282no—
480C22⋊C4—282no—
481C22⋊C4—561no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—81no—
489C42—282no—
490C42—561no—
491C2×C7#246011yescharacteristic
492C2×C7#310711yescharacteristic
493C2×C7#311911yescharacteristic
494C2×C7#330111yescharacteristic
495C2×C7#331411yescharacteristic
496C2×C7#336111yescharacteristic
497C2×C7#336911yescharacteristic
498D7—82no—
499C23—142no—
500C23—281no—
501C23—281no—
502C23—281no—
503C23—281no—
504C23—281no—
505C23—281no—
506C23C22×C22⋊D711yescharacteristic
507Q8—142no—
508Q8—142no—
509Q8—281no—
510Q8—281no—
511D4—142no—
512D4—142no—
513D4—281no—
514D4—281no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—564no—
553C2×C4#650512yes—
554C2×C4#651312yes—
555C2×C4C2×#116111yescharacteristic
556C2×C4#655111yescharacteristic
557C2×C4D7×#4111yescharacteristic
558C2×C4D7×#4111yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4#655311yescharacteristic
561C8—281no—
562C8—281no—
563C8—281no—
564C8—281no—
565C7#2676311yescharacteristic, complemented, Sylow
566C22—282no—
567C22—282no—
568C22—282no—
569C22—561no—
570C22—561no—
571C22—561no—
572C22—561no—
573C22C2×#650511yescharacteristic
574C22#9050811yescharacteristic
575C22C2×#651311yescharacteristic
576C22#9051911yescharacteristic
577C22#9052011yescharacteristic
578C22#9052611yescharacteristic
579C22#9052711yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—282no—
591C4—282no—
592C4—561no—
593C4#9038211yescharacteristic
594C4#9038311yescharacteristic
595C4#9038411yescharacteristic
596C4#9038611yescharacteristic
597C2—562no—
598C2#1174050411yescharacteristic
599C2#1174079711yescharacteristic
600C2#1174087911yescharacteristic
601C2#1174106311yescharacteristic
602C2#1174108011yescharacteristic
603C2#1174108811yescharacteristic
604C2#1174114911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2507C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2507D7
hypocenterC7#26763
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2507

Complements

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

NH ≅ G/Ntype
1. C7×#2507C2semidirect
3. #11727386C2semidirect
8. #11727368C2semidirect
9. #11727355C2semidirect
10. #11727354C2semidirect
11. #11727342C2semidirect
13. #11727236C2semidirect
15. #11727219C2semidirect
121. #2507D7semidirect
565. C7#26763semidirect

Maximal subgroups

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

subgroupindex
C7×#25072
#117311002
#117273862
#117273842
#117273822
#117273802
#117273682
#117273552
#117273542
#117273422
#117273392
#117272362
C4×#63832
#117272192
#267637