Finite Group Databasebeta

← #13411968

Subgroups — #13411968

Order 1792.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740481C211yesmaximal, characteristic, complemented
2#11733748C211yesmaximal, characteristic
3C7×#2025C211yesmaximal, characteristic
4#11729492C211yesmaximal, characteristic, complemented
5#11727933C211yesmaximal, characteristic
6#11727917C211yesmaximal, characteristic
7#11727908C211yesmaximal, characteristic
8Dic7×#102C211yesmaximal, characteristic
9#11727407C211yesmaximal, characteristic
10#11726602C211yesmaximal, characteristic, complemented
11#11726554C211yesmaximal, characteristic, complemented
12#11726551C211yesmaximal, characteristic, complemented
13#11726544C211yesmaximal, characteristic, complemented
14#11726377C211yesmaximal, characteristic, complemented
15#11726371C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×#6451C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C7×#390C2211yescharacteristic
22#89993C2211yescharacteristic
23C2×#6406C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6391C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29#89752—21no—
30#89751—21no—
31#89749C2211yescharacteristic
32#89749C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35#89746—21no—
36#89745C2211yescharacteristic
37#89744—21no—
38C2×#6381C2211yescharacteristic
39#89717—21no—
40#89716—21no—
41#89715—21no—
42C22⋊C4×Dic7—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#26028—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×Dic7—21no—
62C22×Dic14—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C7×#125—22no—
66C7×#125—41no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7Q814yescomplemented
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⋊C4D411yescharacteristic
88C2×Dic7⋊C4D411yescharacteristic
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6451—22no—
97#6451—41no—
98C7×#102—22no—
99C7×#102—41no—
100#6406—22no—
101#6406—41no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113#6391—22no—
114Dic7⋊C8—22no—
115Dic7⋊C8—41no—
116C8×Dic7—22no—
117#6381—22no—
118C4⋊(C7⋊C8)—41no—
119#2550—71no—
120#2357—71no—
121#2067—71no—
122#2065—71no—
123#2025D711yescharacteristic
124#2023—71no—
125#1989—71no—
126#1988—71no—
127#1949—71no—
128#1939—71no—
129#1921—71no—
130#1920—71no—
131#1835—71no—
132#1804—71no—
133C4×#102—71no—
134C24×C7—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4C2×Q812yes—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—24no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—81no—
153C22×Dic7—81no—
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C2×Dic14—42no—
159C2×Dic14—42no—
160C2×Dic14—81no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4#4112yes—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C22⋊Dic7—24no—
180C22⋊Dic7—42no—
181C22⋊Dic7—81no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7C2×Q812yes—
189C4⋊Dic7#4112yes—
190Dic7⋊C4—22no—
191Dic7⋊C4—41no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—81no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—81no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C4×Dic7C2×Q812yes—
201C4×Dic7#4112yes—
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×C4⋊Q8—71no—
209C2×#125C2×D711yescharacteristic
210C2×C4×D4—71no—
211C2×#107—71no—
212C2×#106—71no—
213C2×C4⋊C8—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102C2×D711yescharacteristic
219C2×#102C2×D711yescharacteristic
220C2×C4×C8—71no—
221#398—141no—
222#395—141no—
223#391—141no—
224#390—71no—
225#390C2×D711yescharacteristic
226#389—71no—
227#389—71no—
228#389—141no—
229#388—141no—
230#388—141no—
231#382—71no—
232#382—71no—
233#381—141no—
234C4×C4⋊C4—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4—71no—
238C4×C22⋊C4—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#336—71no—
250#336C2×D711yescharacteristic
251C23×C7—22no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—22no—
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—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7C22×Q811yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—22no—
291C2×Dic7—21no—
292C2×Dic7—22no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—84no—
306C2×Dic7C22⋊Q812yes—
307C2×Dic7C22⋊Q812yes—
308C2×Dic7#13512yes—
309Dic14—84no—
310C7×C8—41no—
311C7×C8—41no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C22×Q8—141no—
315C22×D4C22×D711yescharacteristic
316C23×C4—141no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C4⋊Q8—142no—
322C4⋊Q8—281no—
323#125—22no—
324#125—41no—
325C4×D4—74no—
326C4×D4—142no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42—141no—
351C2×C42C22×D711yescharacteristic
352#107—142no—
353#106—142no—
354C4⋊C8—142no—
355C4⋊C8—281no—
356C4⋊C8—281no—
357#102—22no—
358#102—41no—
359#102—142no—
360#102—281no—
361C4×C8—142no—
362#95—71no—
363#95—71no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7C2×C22⋊Q811yescharacteristic
381C22×C7#54611yescharacteristic
382C22×C7#54811yescharacteristic
383C22×C7C2×#13511yescharacteristic
384C22×C7#57511yescharacteristic
385C22×C7#57611yescharacteristic
386C22×C7#57711yescharacteristic
387C4×C7—42no—
388C4×C7—41no—
389C4×C7—41no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7#53311yescharacteristic
393C4×C7#54411yescharacteristic
394C4×C7#55511yescharacteristic
395C4×C7#55611yescharacteristic
396Dic7—22no—
397Dic7—22no—
398Dic7—22no—
399Dic7—42no—
400Dic7—42no—
401Dic7—41no—
402Dic7—81no—
403Dic7—81no—
404C24—21no—
405C2×Q8—282no—
406C2×Q8—282no—
407C2×Q8—561no—
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—561no—
427C22×C4—561no—
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—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—281no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—561no—
463C4⋊C4#115912yes—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—144no—
467C22⋊C4—282no—
468C22⋊C4—561no—
469C42—21no—
470C42—21no—
471C42—72no—
472C42—72no—
473C42—141no—
474C42—141no—
475C42—561no—
476C42—561no—
477C42—561no—
478C2×C7—82no—
479C2×C7#243911yescharacteristic
480C2×C7#307611yescharacteristic
481C2×C7#308111yescharacteristic
482C2×C7#329711yescharacteristic
483C2×C7#332611yescharacteristic
484C2×C7#336511yescharacteristic
485C2×C7#337011yescharacteristic
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×D4×D711yescharacteristic
494Q8—564no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—72no—
515C2×C4—72no—
516C2×C4—72no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—142no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—142no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—564no—
539C2×C4C2×#115911yescharacteristic
540C2×C4#654811yescharacteristic
541C2×C4C2×Q8×D711yescharacteristic
542C2×C4D7×#4111yescharacteristic
543C2×C4D7×#4111yescharacteristic
544C2×C4#655411yescharacteristic
545C8—41no—
546C8—41no—
547C8—281no—
548C8—281no—
549C7#2602811yescharacteristic, complemented, Sylow
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#9028211yescharacteristic
558C22D7×C22⋊Q811yescharacteristic
559C22#9036811yescharacteristic
560C22D7×#13511yescharacteristic
561C22#9046111yescharacteristic
562C22#9046411yescharacteristic
563C22#9046511yescharacteristic
564C4—41no—
565C4—42no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—142no—
570C4—142no—
571C4—142no—
572C4—282no—
573C4—281no—
574C4—282no—
575C4—561no—
576C4—561no—
577C4#9031311yescharacteristic
578C4#9031411yescharacteristic
579C4#9036111yescharacteristic
580C4#9036211yescharacteristic
581C2—82no—
582C2#1173422711yescharacteristic
583C2#1173754711yescharacteristic
584C2#1173756011yescharacteristic
585C2#1173861011yescharacteristic
586C2#1173861311yescharacteristic
587C2#1173908011yescharacteristic
588C2#1173915511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2025C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2025D7
hypocenterC7#26028
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. #11740481C2semidirect
4. #11729492C2semidirect
10. #11726602C2semidirect
11. #11726554C2semidirect
12. #11726551C2semidirect
13. #11726544C2semidirect
14. #11726377C2semidirect
15. #11726371C2semidirect
79. D4×Dic7Q8semidirect
549. C7#26028semidirect

Maximal subgroups

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

subgroupindex
#117404812
#117337482
C7×#20252
#117294922
#117279332
#117279172
#117279082
Dic7×#1022
#117274072
#117266022
#117265542
#117265512
#117265442
#117263772
#117263712
#260287