Finite Group Databasebeta

← #13411883

Subgroups — #13411883

Order 1792.

734 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
1#11740482C211yesmaximal, characteristic, complemented
2#11733747C211yesmaximal, characteristic
3C7×#2025C211yesmaximal, characteristic
4#11729492C211yesmaximal, characteristic, complemented
5#11727933C211yesmaximal, characteristic
6#11727918C211yesmaximal, characteristic
7#11727907C211yesmaximal, characteristic
8Dic7×#102C211yesmaximal, characteristic
9#11727406C211yesmaximal, characteristic
10#11726601C211yesmaximal, characteristic, complemented
11#11726556C211yesmaximal, characteristic, complemented
12#11726550C211yesmaximal, characteristic, complemented
13#11726542C211yesmaximal, characteristic, complemented
14#11726376C211yesmaximal, characteristic, complemented
15#11726371C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×#6453C2211yescharacteristic
19C2×#6440—21no—
20C2×C7×#102C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C7×#390C2211yescharacteristic
23#89993C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89755—21no—
31#89752—21no—
32#89748—21no—
33#89748C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745C2211yescharacteristic
38#89744—21no—
39C2×#6382C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42#89714—21no—
43C4×C4⋊Dic7C2211yescharacteristic
44C4×Dic7⋊C4C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#336C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26021—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×Dic7—21no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C7×#125—22no—
65C7×#125—41no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77D4×Dic7—22no—
78D4×Dic7Q814yes—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4D411yescharacteristic
92C2×Dic7⋊C4D411yescharacteristic
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7C2311yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100#6453—22no—
101#6453—41no—
102#6440—44no—
103C7×#102—22no—
104C7×#102—41no—
105#6406—22no—
106#6406—41no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95D411yescharacteristic
113C7⋊#95D411yescharacteristic
114#6391—22no—
115Dic7⋊C8—22no—
116Dic7⋊C8—41no—
117C8×Dic7—22no—
118#6382—22no—
119C4⋊(C7⋊C8)—41no—
120#2552—71no—
121#2354—71no—
122#2065—71no—
123#2065—71no—
124#2025—71no—
125#2025D711yescharacteristic
126#1989—71no—
127#1989—71no—
128#1948—71no—
129#1939—71no—
130#1919—71no—
131#1919—71no—
132#1834—71no—
133#1802—71no—
134C4×#102—71no—
135C24×C7—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C2×Q812yes—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—24no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—42no—
154C22×Dic7—42no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4#4112yes—
173C7×C22⋊C4—42no—
174C7×C22⋊C4—81no—
175C42×C7—21no—
176C42×C7—21no—
177C22⋊Dic7—24no—
178C22⋊Dic7—42no—
179C22⋊Dic7—81no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—42no—
187C4⋊Dic7C2×Q812yes—
188C4⋊Dic7#4112yes—
189Dic7⋊C4—22no—
190Dic7⋊C4—22no—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—81no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—42no—
201C4×Dic7—42no—
202C4×Dic7—42no—
203C4×Dic7C2×Q812yes—
204C4×Dic7#4112yes—
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×#125C2×D711yescharacteristic
213C2×C4×D4—71no—
214C2×#117—141no—
215C2×#106—71no—
216C2×#106—71no—
217C2×C4⋊C8—71no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8—71no—
220C2×#102—71no—
221C2×#102—71no—
222C2×#102C2×D711yescharacteristic
223C2×#102C2×D711yescharacteristic
224C2×C4×C8—71no—
225#398—141no—
226#398—141no—
227#390—71no—
228#390C2×D711yescharacteristic
229#389—71no—
230#389—71no—
231#389—141no—
232#388—141no—
233#388—141no—
234#384—71no—
235#384—71no—
236#384—141no—
237#380—141no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340C2×D711yescharacteristic
251#340C2×D711yescharacteristic
252#336—71no—
253#336C2×D711yescharacteristic
254C23×C7—22no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7C22×D411yescharacteristic
262C7×D4—22no—
263C7×D4—22no—
264C7×D4—41no—
265C7×D4—41no—
266C2×C4×C7—22no—
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—41no—
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×C7C22×Q811yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—22no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—22no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—48no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7C22⋊Q812yes—
309C2×Dic7C22⋊Q812yes—
310C2×Dic7#13512yes—
311C2×Dic7AGL(1,ℤ/8)12yes—
312C7×C8—41no—
313C7×C8—41no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C22×D4C22×D711yescharacteristic
317C23×C4—141no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#125—22no—
323#125—41no—
324#125—142no—
325#125—281no—
326C4×D4—74no—
327C4×D4—142no—
328#117—284no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
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⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42C22×D711yescharacteristic
359#106—142no—
360#106—142no—
361C4⋊C8—142no—
362C4⋊C8—281no—
363C4⋊C8—281no—
364#102—22no—
365#102—41no—
366#102—142no—
367#102—281no—
368C4×C8—142no—
369#95—71no—
370#95—71no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7C2×C22⋊Q811yescharacteristic
384C22×C7#54611yescharacteristic
385C22×C7#54811yescharacteristic
386C22×C7C2×#13511yescharacteristic
387C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C7#57711yescharacteristic
389C22×C7#57711yescharacteristic
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—42no—
393C4×C7—81no—
394C4×C7—81no—
395C4×C7#53311yescharacteristic
396C4×C7#54411yescharacteristic
397C4×C7#55511yescharacteristic
398C4×C7#55611yescharacteristic
399Dic7—22no—
400Dic7—42no—
401Dic7—41no—
402Dic7—42no—
403Dic7—41no—
404Dic7—81no—
405Dic7—81no—
406Dic7#47814yes—
407C24—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4Q8×D712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—144no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—282no—
427C22×C4—282no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—72no—
451C4⋊C4—72no—
452C4⋊C4—142no—
453C4⋊C4—142no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—282no—
463C4⋊C4—282no—
464C4⋊C4—561no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C4⋊C4—561no—
468C4⋊C4#115912yes—
469C22⋊C4—42no—
470C22⋊C4—81no—
471C22⋊C4—144no—
472C22⋊C4—282no—
473C22⋊C4—561no—
474C42—21no—
475C42—21no—
476C42—72no—
477C42—72no—
478C42—141no—
479C42—141no—
480C42—282no—
481C42—282no—
482C42—282no—
483C2×C7—82no—
484C2×C7#243911yescharacteristic
485C2×C7C2×#47811yescharacteristic
486C2×C7#308411yescharacteristic
487C2×C7#330311yescharacteristic
488C2×C7#333211yescharacteristic
489C2×C7#336011yescharacteristic
490C2×C7#336811yescharacteristic
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C2×D4×D711yescharacteristic
499D4—22no—
500D4—22no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—72no—
521C2×C4—72no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—142no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—142no—
532C2×C4—288no—
533C2×C4—281no—
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×C4C2×#115911yescharacteristic
544C2×C4#654811yescharacteristic
545C2×C4C2×Q8×D711yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C2×C4D7×#4111yescharacteristic
548C2×C4#655411yescharacteristic
549C8—41no—
550C8—41no—
551C8—281no—
552C8—281no—
553C7#2602111yescharacteristic, complemented, Sylow
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22#9028211yescharacteristic
562C22D7×C22⋊Q811yescharacteristic
563C22#9036811yescharacteristic
564C22D7×#13511yescharacteristic
565C22#9046111yescharacteristic
566C22D7×AGL(1,ℤ/8)11yescharacteristic
567C22#9046611yescharacteristic
568C4—41no—
569C4—42no—
570C4—41no—
571C4—74no—
572C4—81no—
573C4—81no—
574C4—142no—
575C4—282no—
576C4—281no—
577C4—281no—
578C4—282no—
579C4—561no—
580C4—561no—
581C4#9031311yescharacteristic
582C4#9031411yescharacteristic
583C4#9036111yescharacteristic
584C4#9036211yescharacteristic
585C2—82no—
586C2#1173422711yescharacteristic
587C2#1173754911yescharacteristic
588C2#1173756211yescharacteristic
589C2D7×#47811yescharacteristic
590C2#1173861611yescharacteristic
591C2#1173907711yescharacteristic
592C2#1173915411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2025C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2025D7
hypocenterC7#26021
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #11740482C2semidirect
4. #11729492C2semidirect
10. #11726601C2semidirect
11. #11726556C2semidirect
12. #11726550C2semidirect
13. #11726542C2semidirect
14. #11726376C2semidirect
15. #11726371C2semidirect
553. C7#26021semidirect

Maximal subgroups

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

subgroupindex
#117404822
#117337472
C7×#20252
#117294922
#117279332
#117279182
#117279072
Dic7×#1022
#117274062
#117266012
#117265562
#117265502
#117265422
#117263762
#117263712
#260217