Finite Group Databasebeta

← #13421808

Subgroups — #13421808

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×#2448C211yesmaximal, characteristic
2#11731104C211yesmaximal, characteristic, complemented
3#11729476C211yesmaximal, characteristic
4#11729475C211yesmaximal, characteristic
5#11729385C211yesmaximal, characteristic, complemented
6#11729382C211yesmaximal, characteristic, complemented
7#11727423C211yesmaximal, characteristic, complemented
8#11727421C211yesmaximal, characteristic, complemented
9#11727399C211yesmaximal, characteristic
10#11727393C211yesmaximal, characteristic
11#11727390C211yesmaximal, characteristic
12#11727390C211yesmaximal, characteristic
13#11727376C211yesmaximal, characteristic, complemented
14#11727357C211yesmaximal, characteristic, complemented
15#11727237C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×#6433C2211yescharacteristic
19C7×#399—21no—
20C7×#397—21no—
21C7×#391C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#388—21no—
26C7×#388—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89750—21no—
37#89744—21no—
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
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#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26719—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4C2311yescharacteristic
62C7×#124—22no—
63C7×#124—41no—
64C4×C7×Q8—22no—
65C4×C7×Q8—41no—
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×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97#6433—22no—
98#6433—22no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95D411yescharacteristic
103C7×#95D411yescharacteristic
104#6409—22no—
105#6409—22no—
106#6409—22no—
107#6409—22no—
108#6406—22no—
109#6406—22no—
110#6406—22no—
111#6406—41no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117#6381—22no—
118#6381—41no—
119C4⋊(C7⋊C8)—41no—
120#6377—41no—
121#2448D711yescharacteristic
122#2370—71no—
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—
136C24×C7—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8#4112yes—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—41no—
168C7×C4⋊C4—41no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4—81no—
171C7×C22⋊C4—42no—
172C7×C22⋊C4—42no—
173C7×C22⋊C4—42no—
174C7×C22⋊C4—42no—
175C7×C22⋊C4—81no—
176C7×C22⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—42no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7#4112yes—
190C4⋊Dic7#4112yes—
191C4⋊Dic7#4112yes—
192Dic7⋊C4—24no—
193Dic7⋊C4—24no—
194Dic7⋊C4—42no—
195Dic7⋊C4—81no—
196C4×Dic7—42no—
197C4×Dic7—81no—
198C4×Dic7—81no—
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×#125—71no—
212C2×#124C2×D711yescharacteristic
213C2×C4×Q8C2×D711yescharacteristic
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—21no—
227#397—21no—
228#391C2×D711yescharacteristic
229#390C2×D711yescharacteristic
230#390C2×D711yescharacteristic
231#389—71no—
232#389—71no—
233#389—141no—
234#389C2×D711yescharacteristic
235#388—21no—
236#388—21no—
237#387—141no—
238#387—141no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4—141no—
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×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7C22×D411yescharacteristic
262C7×Q8—22no—
263C7×Q8—22no—
264C7×Q8—41no—
265C7×Q8—41no—
266C7×D4—41no—
267C7×D4—41no—
268C7×D4—41no—
269C7×D4—41no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—22no—
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—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C7⋊C8—41no—
316C7⋊C8—41no—
317C7⋊C8—41no—
318C7⋊C8—41no—
319C22×Q8C22×D711yescharacteristic
320C22×D4C22×D711yescharacteristic
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—71no—
324C22×C8—71no—
325#125—142no—
326#125—142no—
327#124—22no—
328#124—41no—
329C4×Q8—22no—
330C4×Q8—41no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22⋊D711yescharacteristic
342C2×C4⋊C4C22⋊D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C42—21no—
352C2×C42—141no—
353C2×C42—141no—
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—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95C22⋊D711yescharacteristic
379#95C22⋊D711yescharacteristic
380C22×C7—81no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7#52511yescharacteristic
388C22×C7#54611yescharacteristic
389C22×C7#54811yescharacteristic
390C22×C7#57511yescharacteristic
391C22×C7#57511yescharacteristic
392C22×C7#57611yescharacteristic
393C22×C7#57811yescharacteristic
394C4×C7—41no—
395C4×C7—41no—
396C4×C7—42no—
397C4×C7—41no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—81no—
402C4×C7—81no—
403C4×C7#52911yescharacteristic
404C4×C7#54211yescharacteristic
405C4×C7#55211yescharacteristic
406C4×C7#55911yescharacteristic
407Dic7—42no—
408Dic7—42no—
409Dic7—42no—
410Dic7—81no—
411C24—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8#115912yes—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—141no—
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—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—72no—
458C4⋊C4—72no—
459C4⋊C4—72no—
460C4⋊C4—81no—
461C4⋊C4—144no—
462C4⋊C4—144no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—561no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—282no—
484C42—561no—
485C42—561no—
486C2×C7—161no—
487C2×C7#245811yescharacteristic
488C2×C7#311111yescharacteristic
489C2×C7#312411yescharacteristic
490C2×C7#330911yescharacteristic
491C2×C7#332711yescharacteristic
492C2×C7#337011yescharacteristic
493C2×C7#337711yescharacteristic
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×C22⋊D711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
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—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—281no—
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×#115911yescharacteristic
550C2×C4#654811yescharacteristic
551C2×C4D7×#4111yescharacteristic
552C2×C4D7×#4111yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C2×C4#655411yescharacteristic
555C8—281no—
556C8—281no—
557C8—281no—
558C8—281no—
559C7#2671911yescharacteristic, complemented, Sylow
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#9049711yescharacteristic
568C22#9051311yescharacteristic
569C22#9051311yescharacteristic
570C22#9051911yescharacteristic
571C22#9052211yescharacteristic
572C22#9052611yescharacteristic
573C22#9052911yescharacteristic
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—282no—
584C4—282no—
585C4—282no—
586C4—561no—
587C4#9034011yescharacteristic
588C4#9034111yescharacteristic
589C4#9035311yescharacteristic
590C4#9035511yescharacteristic
591C2—161no—
592C2#1174029611yescharacteristic
593C2#1174058711yescharacteristic
594C2#1174090611yescharacteristic
595C2#1174098611yescharacteristic
596C2#1174101711yescharacteristic
597C2#1174112311yescharacteristic
598C2#1174115911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2448C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2448D7
hypocenterC7#26719
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2448

Complements

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

NH ≅ G/Ntype
2. #11731104C2semidirect
5. #11729385C2semidirect
6. #11729382C2semidirect
7. #11727423C2semidirect
8. #11727421C2semidirect
13. #11727376C2semidirect
14. #11727357C2semidirect
15. #11727237C2semidirect
559. C7#26719semidirect

Maximal subgroups

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

subgroupindex
C7×#24482
#117311042
#117294762
#117294752
#117293852
#117293822
#117274232
#117274212
#117273992
#117273932
#117273902
#117273762
#117273572
#117272372
#267197