Finite Group Databasebeta

← #13426628

Subgroups — #13426628

Order 1792.

678 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#11740193C211yesmaximal, characteristic
2#11731365C211yesmaximal, characteristic
3C7×#2035C211yesmaximal, characteristic
4#11729449C211yesmaximal, characteristic
5#11728318C211yesmaximal, characteristic, complemented
6#11728314C211yesmaximal, characteristic, complemented
7#11728001C211yesmaximal, characteristic, complemented
8#11727990C211yesmaximal, characteristic, complemented
9#11727988C211yesmaximal, characteristic, complemented
10Dic7×#103C211yesmaximal, characteristic, complemented
11#11727683C211yesmaximal, characteristic
12#11727667C211yesmaximal, characteristic
13#11727257C211yesmaximal, characteristic
14#11726574C211yesmaximal, characteristic, complemented
15#11726569C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C2×C7×#106C2211yescharacteristic
20C2×C7×#103C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C7×#380C2211yescharacteristic
24#90002C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27#89984C2211yescharacteristic
28C4×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89753—21no—
37#89749C2211yescharacteristic
38C4⋊C4×Dic7—21no—
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41#89745C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89725—21no—
44#89723—21no—
45#89723—21no—
46#89717—21no—
47#89432—21no—
48#89423—21no—
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#27042—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7D411yescharacteristic
75C2×C22⋊Dic7D411yescharacteristic
76Q8×Dic7—22no—
77Q8×Dic7—41no—
78#6489—22no—
79#6489—41no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C7×#106—22no—
97C7×#103—22no—
98C7×#103—22no—
99C7×C22⋊C8—22no—
100C7×#95—21no—
101#6409—22no—
102#6409—41no—
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⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115C22⋊(C7⋊C8)—41no—
116#6392—22no—
117#6391—41no—
118Dic7⋊C8—22no—
119C8×Dic7—41no—
120#6381—22no—
121#2495—71no—
122#2362—71no—
123#2074—71no—
124#2066—71no—
125#2036—71no—
126#2035D711yescharacteristic
127#1949—71no—
128#1941—71no—
129#1923—71no—
130#1921—71no—
131#1918—71no—
132#1863—71no—
133#1854—71no—
134#1825—71no—
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8#4112yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8#4112yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4#4112yes—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—81no—
178C22⋊Dic7—22no—
179C22⋊Dic7—41no—
180C22⋊Dic7—42no—
181C22⋊Dic7—42no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—42no—
190C4⋊Dic7—42no—
191C4⋊Dic7—81no—
192C4⋊Dic7#4112yes—
193Dic7⋊C4—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—42no—
199C4×Dic7—42no—
200C4×Dic7—42no—
201C4×Dic7—42no—
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×C22⋊Q8C2×D711yescharacteristic
209C2×C4×Q8—71no—
210C2×#117—71no—
211C2×#107—71no—
212C2×#107—71no—
213C2×#106—71no—
214C2×#106C2×D711yescharacteristic
215C2×C4⋊C8—71no—
216C2×#103—71no—
217C2×#103—71no—
218C2×#103C2×D711yescharacteristic
219C2×#103C2×D711yescharacteristic
220C2×C22⋊C8—71no—
221C2×C22⋊C8C2×D711yescharacteristic
222C2×C4×C8—71no—
223#400—141no—
224#397—141no—
225#391—71no—
226#389—71no—
227#389—71no—
228#389—71no—
229#388—141no—
230#388—141no—
231#388—141no—
232#387—141no—
233#383—141no—
234#382—71no—
235#380—71no—
236#380C2×D711yescharacteristic
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—141no—
240C4×C22⋊C4—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340C2×D711yescharacteristic
249#336—71no—
250#336—71no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C22×D411yescharacteristic
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—41no—
262C7×Q8—41no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
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×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7H(5,𝔽2)11yescharacteristic
287C2×C4×C72−1+411yescharacteristic
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—22no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7#13512yes—
312C7×C8—22no—
313C7×C8—41no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C22×Q8C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C4×Q8—142no—
325C4×Q8—281no—
326#117—142no—
327#117—281no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—71no—
343C2×C22⋊C4—71no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42—141no—
351C2×C42—141no—
352#107—142no—
353#107—142no—
354#106—22no—
355#106—281no—
356C4⋊C8—142no—
357#103—22no—
358#103—22no—
359#103—142no—
360#103—281no—
361C22⋊C8—22no—
362C22⋊C8—281no—
363C4×C8—281no—
364#95—21no—
365#95—71no—
366#95—71no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—41no—
383C22×C7—41no—
384C22×C7#52511yescharacteristic
385C22×C7#54611yescharacteristic
386C22×C7#54711yescharacteristic
387C22×C7C2×#13511yescharacteristic
388C22×C7#57511yescharacteristic
389C22×C7#57711yescharacteristic
390C22×C7#57811yescharacteristic
391C4×C7—42no—
392C4×C7—41no—
393C4×C7—42no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7#52911yescharacteristic
397C4×C7#53911yescharacteristic
398C4×C7#55311yescharacteristic
399C4×C7#55611yescharacteristic
400Dic7—22no—
401Dic7—41no—
402Dic7—41no—
403Dic7—41no—
404Dic7—42no—
405Dic7—81no—
406Dic7—81no—
407Dic7—81no—
408C24D4×D711yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8#115912yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4D4×D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8#115912yes—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—72no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—282no—
462C4⋊C4—282no—
463C4⋊C4—561no—
464C4⋊C4—561no—
465C4⋊C4#115912yes—
466C22⋊C4—24no—
467C22⋊C4—81no—
468C22⋊C4—142no—
469C22⋊C4—281no—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C42—141no—
473C42—141no—
474C42—141no—
475C42—141no—
476C42—282no—
477C42—282no—
478C42—282no—
479C42—282no—
480C2×C7—81no—
481C2×C7#246611yescharacteristic
482C2×C7#310211yescharacteristic
483C2×C7#312111yescharacteristic
484C2×C7#330011yescharacteristic
485C2×C7#330711yescharacteristic
486C2×C7#334011yescharacteristic
487C2×C7#335211yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×D4×D711yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—72no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—142no—
532C2×C4—281no—
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×C4C2×#115911yescharacteristic
545C2×C4C2×#115911yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C2×C4#655311yescharacteristic
548C2×C4#655411yescharacteristic
549C8—22no—
550C8—41no—
551C8—281no—
552C8—281no—
553C7#2704211yescharacteristic, complemented, Sylow
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#9027711yescharacteristic
562C22#9030211yescharacteristic
563C22#9034511yescharacteristic
564C22#9045311yescharacteristic
565C22D7×#13511yescharacteristic
566C22#9046811yescharacteristic
567C22#9047011yescharacteristic
568C4—42no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—142no—
574C4—281no—
575C4—281no—
576C4—281no—
577C4—282no—
578C4—561no—
579C4—561no—
580C4—561no—
581C4#9029111yescharacteristic
582C4#9030911yescharacteristic
583C4#9035611yescharacteristic
584C4#9036211yescharacteristic
585C2—81no—
586C2#1173436411yescharacteristic
587C2#1173747711yescharacteristic
588C2#1173765711yescharacteristic
589C2#1173809511yescharacteristic
590C2#1173834511yescharacteristic
591C2#1173887711yescharacteristic
592C2#1173896911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2035C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2035D7
hypocenterC7#27042
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
5. #11728318C2semidirect
6. #11728314C2semidirect
7. #11728001C2semidirect
8. #11727990C2semidirect
9. #11727988C2semidirect
10. Dic7×#103C2semidirect
14. #11726574C2semidirect
15. #11726569C2semidirect
553. C7#27042semidirect

Maximal subgroups

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

subgroupindex
#117401932
#117313652
C7×#20352
#117294492
#117283182
#117283142
#117280012
#117279902
#117279882
Dic7×#1032
#117276832
#117276672
#117272572
#117265742
#117265692
#270427