Finite Group Databasebeta

← #13421806

Subgroups — #13421806

Order 1792.

680 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2370C211yesmaximal, characteristic, complemented
2#11731118C211yesmaximal, characteristic
3#11729498C211yesmaximal, characteristic, complemented
4#11729497C211yesmaximal, characteristic
5#11727385C211yesmaximal, characteristic, complemented
6#11727382C211yesmaximal, characteristic
7#11727376C211yesmaximal, characteristic, complemented
8#11727371C211yesmaximal, characteristic
9#11727242C211yesmaximal, characteristic
10#11727237C211yesmaximal, characteristic, complemented
11#11727229C211yesmaximal, characteristic, complemented
12#11727219C211yesmaximal, characteristic, complemented
13#11727216C211yesmaximal, characteristic
14#11727216C211yesmaximal, characteristic
15#11727215C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×#6438C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C7×#389—21no—
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#387—21no—
23C7×#387—21no—
24C4×C7×C4⋊C4—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29#89764—21no—
30#89762—21no—
31C2×#6384C2211yescharacteristic
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89711C2211yescharacteristic
42#89711C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89447—21no—
48#89441—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26719—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×Dic14C2311yescharacteristic
62C7×#125—22no—
63C7×#125—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6438—22no—
96#6438—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—22no—
113#6383—22no—
114#6383—22no—
115#6383—22no—
116#6383—41no—
117#6381—22no—
118#6381—41no—
119C4⋊(C7⋊C8)—41no—
120#6377—41no—
121#2448—71no—
122#2370D711yescharacteristic, complemented
123#2086—71no—
124#2042—71no—
125#2026—71no—
126#2025—71no—
127#1983—71no—
128#1936—71no—
129#1928—71no—
130#1925—71no—
131#1925—71no—
132#1921—71no—
133#1917—71no—
134#1863—71no—
135#1763—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—21no—
157C2×Dic14—21no—
158C2×Dic14—21no—
159C2×Dic14—21no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14#4112yes—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—41no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4#4112yes—
175C7×C4⋊C4#4112yes—
176C7×C4⋊C4#4112yes—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—42no—
184C42×C7—81no—
185C42×C7—81no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—42no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—42no—
190(C2×D7)⋊C4—81no—
191(C2×D7)⋊C4—81no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196Dic7⋊C4—42no—
197Dic7⋊C4—81no—
198C4×Dic7—42no—
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—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×#125C2×D711yescharacteristic
212C2×#124—71no—
213C2×C4×Q8—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×C4⋊C8—71no—
217C2×#103—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#102—71no—
225C2×#97—71no—
226#399—141no—
227#397—141no—
228#391—71no—
229#390—71no—
230#390—71no—
231#389—21no—
232#389—71no—
233#389C2×D711yescharacteristic
234#389C2×D711yescharacteristic
235#388—141no—
236#388—141no—
237#387—21no—
238#387—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2×D711yescharacteristic
241C4×C4⋊C4C2×D711yescharacteristic
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C4⋊C4C2×D711yescharacteristic
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336—71no—
251#336—71no—
252#336—71no—
253#336—71no—
254C23×C7C22×D411yescharacteristic
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
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—21no—
282C2×C4×C7—21no—
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×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
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—
307D28—41no—
308D28—41no—
309D28—41no—
310D28—41no—
311Dic14—22no—
312Dic14—22no—
313Dic14—41no—
314Dic14—41no—
315C7⋊C8—41no—
316C7⋊C8—41no—
317C7⋊C8—41no—
318C7⋊C8—41no—
319C22×Q8—71no—
320C22×D4—71no—
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—71no—
324C22×C8—71no—
325#125—22no—
326#125—22no—
327#124—142no—
328#124—281no—
329C4×Q8—142no—
330C4×Q8—281no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4C22⋊D711yescharacteristic
340C2×C4⋊C4C22⋊D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—141no—
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357C2×C42C22×D711yescharacteristic
358#107—142no—
359#107—281no—
360C4⋊C8—281no—
361#103—142no—
362#103—142no—
363#103—142no—
364#103—142no—
365#102—142no—
366#102—142no—
367#102—142no—
368#102—281no—
369#97—281no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95C22⋊D711yescharacteristic
379#95C22⋊D711yescharacteristic
380C22×C7#52511yescharacteristic
381C22×C7#54611yescharacteristic
382C22×C7#54811yescharacteristic
383C22×C7#57511yescharacteristic
384C22×C7#57511yescharacteristic
385C22×C7#57611yescharacteristic
386C22×C7#57811yescharacteristic
387C2×D7—81no—
388C2×D7—81no—
389C2×D7—81no—
390C2×D7—81no—
391C2×D7—81no—
392C2×D7—81no—
393C2×D7—81no—
394C4×C7—41no—
395C4×C7—42no—
396C4×C7—42no—
397C4×C7—41no—
398C4×C7—42no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—81no—
404C4×C7#52911yescharacteristic
405C4×C7#54211yescharacteristic
406C4×C7#55211yescharacteristic
407C4×C7#55911yescharacteristic
408Dic7—42no—
409Dic7—81no—
410Dic7—81no—
411C24—141no—
412C2×Q8—72no—
413C2×Q8—141no—
414C2×Q8—141no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×Q8—141no—
418C2×D4—141no—
419C2×D4—141no—
420C2×D4—141no—
421C2×D4—141no—
422C2×D4—141no—
423C2×D4—141no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C23×D711yescharacteristic
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—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—282no—
466C4⋊C4—561no—
467C4⋊C4#115712yes—
468C4⋊C4#115712yes—
469C4⋊C4#116112yes—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C22⋊C4—561no—
475C22⋊C4—561no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—81no—
484C42—81no—
485C42—282no—
486C2×C7#245811yescharacteristic
487C2×C7#311111yescharacteristic
488C2×C7#312411yescharacteristic
489C2×C7#330911yescharacteristic
490C2×C7#332711yescharacteristic
491C2×C7#337011yescharacteristic
492C2×C7#337711yescharacteristic
493D7—161no—
494C23—281no—
495C23—281no—
496C23—281no—
497C23—281no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23C22×C22⋊D711yescharacteristic
502Q8—142no—
503Q8—142no—
504Q8—281no—
505Q8—281no—
506D4—281no—
507D4—281no—
508D4—281no—
509D4—281no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
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—41no—
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—142no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4C2×#115711yescharacteristic
550C2×C4C2×#115711yescharacteristic
551C2×C4C2×#116111yescharacteristic
552C2×C4#655111yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C2×C4#655311yescharacteristic
555C8—281no—
556C8—281no—
557C8—281no—
558C8—281no—
559C7#2671911yescharacteristic, complemented, Sylow
560C22—561no—
561C22—561no—
562C22—561no—
563C22—561no—
564C22—561no—
565C22—561no—
566C22—561no—
567C22#9048211yescharacteristic
568C22#9050811yescharacteristic
569C22#9051311yescharacteristic
570C22#9051311yescharacteristic
571C22#9052011yescharacteristic
572C22#9052611yescharacteristic
573C22#9052911yescharacteristic
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—282no—
585C4—561no—
586C4—561no—
587C4#9027111yescharacteristic
588C4#9027211yescharacteristic
589C4#9027311yescharacteristic
590C4#9027611yescharacteristic
591C2—1121no—
592C2#1173994211yescharacteristic
593C2#1173997511yescharacteristic
594C2#1174001411yescharacteristic
595C2#1174105011yescharacteristic
596C2#1174106411yescharacteristic
597C2#1174108411yescharacteristic
598C2#1174114511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2370C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2370D7
hypocenterC7#26719
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C7×#2370C2semidirect
3. #11729498C2semidirect
5. #11727385C2semidirect
7. #11727376C2semidirect
10. #11727237C2semidirect
11. #11727229C2semidirect
12. #11727219C2semidirect
15. #11727215C2semidirect
122. #2370D7semidirect
559. C7#26719semidirect

Maximal subgroups

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

subgroupindex
C7×#23702
#117311182
#117294982
#117294972
#117273852
#117273822
#117273762
#117273712
#117272422
#117272372
#117272292
#117272192
#117272162
#117272152
#267197