Finite Group Databasebeta

← #13426541

Subgroups — #13426541

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11730958C211yesmaximal, characteristic
2#11730910C211yesmaximal, characteristic
3C7×#1923C211yesmaximal, characteristic
4#11728243C211yesmaximal, characteristic, complemented
5#11728207C211yesmaximal, characteristic, complemented
6#11727691C211yesmaximal, characteristic
7#11727691C211yesmaximal, characteristic
8#11727681C211yesmaximal, characteristic
9#11727679C211yesmaximal, characteristic
10#11727524C211yesmaximal, characteristic, complemented
11C4×#6390C211yesmaximal, characteristic, complemented
12#11726437C211yesmaximal, characteristic, complemented
13#11726390C211yesmaximal, characteristic, complemented
14#11726378C211yesmaximal, characteristic, complemented
15#11726378C211yesmaximal, characteristic, complemented
16C22×C4⋊Dic7C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C4×C7×C22⋊C4C2211yescharacteristic
24#89984C2211yescharacteristic
25#89984C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34#89723—21no—
35#89720—21no—
36#89718—21no—
37C22⋊C4×Dic7—21no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89701C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89432—21no—
46#89427—21no—
47#89427—21no—
48#89419—21no—
49C7×#336C2211yescharacteristic
50C7×#336C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#27036—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C22⋊C4D411yescharacteristic
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—41no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86#6487—22no—
87#6487—22no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C4×Dic14—22no—
97C4×Dic14—41no—
98C7×C4⋊C8—22no—
99C7×C22⋊C8—22no—
100C7×C22⋊C8—41no—
101C4×C7×C8—41no—
102C7×#95D411yescharacteristic
103C7×#95D411yescharacteristic
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—
113#6392—22no—
114#6392—22no—
115#6392—22no—
116#6392—41no—
117#6390—22no—
118#6390—22no—
119#6390—22no—
120#6390—41no—
121#2497—71no—
122#2360—71no—
123#2072—71no—
124#2068—71no—
125#2036—71no—
126#2036—71no—
127#1947—71no—
128#1942—71no—
129#1923D711yescharacteristic
130#1921—71no—
131#1921—71no—
132#1863—71no—
133#1852—71no—
134#1824—71no—
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7—81no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14#4112yes—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8#4112yes—
178C7×C22⋊C4—41no—
179C7×C22⋊C4Q1614yes—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C22⋊Dic7—24no—
185C22⋊Dic7—81no—
186C22⋊Dic7—81no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—42no—
200C4⋊Dic7—81no—
201C4⋊Dic7—81no—
202C4⋊Dic7#4112yes—
203C4⋊Dic7#4112yes—
204Dic7⋊C4—24no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—81no—
208Dic7⋊C4—81no—
209C4×Dic7—42no—
210C4×Dic7—81no—
211C4×Dic7—81no—
212C2×C22⋊Q8—71no—
213C2×C4×Q8—71no—
214C22×C4⋊C4—71no—
215C2×#107—71no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107—71no—
219C2×C4⋊C8C2×D711yescharacteristic
220C2×#103—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×C22⋊C8C2×D711yescharacteristic
225C2×C22⋊C8C2×D711yescharacteristic
226C2×C4×C8C2×D711yescharacteristic
227#400—141no—
228#396—141no—
229#391—71no—
230#389—71no—
231#389—71no—
232#389—71no—
233#389—141no—
234#388—141no—
235#387—141no—
236#387—141no—
237#385—141no—
238#384—71no—
239#380—71no—
240#380—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4—71no—
243C4×C22⋊C4—141no—
244C4×C22⋊C4C2×D711yescharacteristic
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#336C2×D711yescharacteristic
254#336C2×D711yescharacteristic
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7C2×Q1612yes—
262C23×C7C22×D411yescharacteristic
263C2×C4×C7—21no—
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—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7C2×Q1612yes—
281C2×C4×C7C2×Q1612yes—
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—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
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—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—84no—
313Dic14—22no—
314Dic14—22no—
315Dic14—41no—
316Dic14—41no—
317C7×C8—22no—
318C7×C8—41no—
319C7×C8—41no—
320C7×C8—41no—
321C22×Q8—71no—
322C23×C4—141no—
323C23×C4C22×D711yescharacteristic
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C22×C8C22×D711yescharacteristic
327C22×C8C22×D711yescharacteristic
328C22⋊Q8—142no—
329C22⋊Q8—142no—
330C4×Q8—142no—
331C4×Q8—281no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—281no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4D2811yescharacteristic
353C2×C22⋊C4D2811yescharacteristic
354C2×C42—141no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42C22×D711yescharacteristic
358C2×C42C22×D711yescharacteristic
359#107—142no—
360#107—142no—
361#107—142no—
362#107—281no—
363C4⋊C8—22no—
364#103—142no—
365#103—142no—
366#103—142no—
367#103—281no—
368C22⋊C8—22no—
369C22⋊C8—41no—
370C4×C8—41no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95D2811yescharacteristic
381#95D2811yescharacteristic
382C22×C7—22no—
383C22×C7—22no—
384C22×C7—22no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—41no—
388C22×C7—41no—
389C22×C7#52511yescharacteristic
390C22×C7#54611yescharacteristic
391C22×C7#54711yescharacteristic
392C22×C7C22×Q1611yescharacteristic
393C22×C7#57511yescharacteristic
394C22×C7#57611yescharacteristic
395C22×C7#57711yescharacteristic
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7#52911yescharacteristic
402C4×C7#53911yescharacteristic
403C4×C7#55311yescharacteristic
404C4×C7#55611yescharacteristic
405Dic7—42no—
406Dic7—42no—
407Dic7—42no—
408Dic7—81no—
409Dic7—81no—
410Dic7—81no—
411Dic7—81no—
412Dic7—81no—
413C24C2×D2811yescharacteristic
414C2×Q8—72no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×Q8—141no—
418C2×Q8—141no—
419C2×Q8—141no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—282no—
435C22×C4—282no—
436C22×C4—561no—
437C22×C4C2×D2811yescharacteristic
438C22×C4C2×D2811yescharacteristic
439C22×C4C2×D2811yescharacteristic
440C22×C4C2×D2811yescharacteristic
441C22×C4C2×D2811yescharacteristic
442C22×C4C23×D711yescharacteristic
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8#115912yes—
455C4⋊C4—72no—
456C4⋊C4—72no—
457C4⋊C4—144no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—282no—
471C4⋊C4—282no—
472C4⋊C4—282no—
473C4⋊C4—561no—
474C4⋊C4—561no—
475C4⋊C4—561no—
476C4⋊C4—561no—
477C22⋊C4—41no—
478C22⋊C4—144no—
479C22⋊C4—561no—
480C22⋊C4—561no—
481C22⋊C4Dic2814yes—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—282no—
487C42—561no—
488C42—561no—
489C2×C7—42no—
490C2×C7#246611yescharacteristic
491C2×C7#310411yescharacteristic
492C2×C7#311911yescharacteristic
493C2×C7#329911yescharacteristic
494C2×C7#331911yescharacteristic
495C2×C7#333811yescharacteristic
496C2×C7#335111yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23C2×Dic2812yes—
504C23C22×D2811yescharacteristic
505Q8—142no—
506Q8—142no—
507Q8—281no—
508Q8—281no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—142no—
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—281no—
539C2×C4—281no—
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×C4—281no—
550C2×C4—564no—
551C2×C4C2×Dic2812yes—
552C2×C4C2×Dic2812yes—
553C2×C4C2×#115911yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4D7×#4111yescharacteristic
557C2×C4D7×#4111yescharacteristic
558C2×C4#655411yescharacteristic
559C8—22no—
560C8—41no—
561C8—41no—
562C8—41no—
563C7#2703611yescharacteristic, complemented, Sylow
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#9021511yescharacteristic
572C22#9024511yescharacteristic
573C22#9024611yescharacteristic
574C22C22×Dic2811yescharacteristic
575C22#9043911yescharacteristic
576C22#9044211yescharacteristic
577C22#9044311yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—282no—
584C4—282no—
585C4—282no—
586C4—561no—
587C4—561no—
588C4—561no—
589C4—561no—
590C4—561no—
591C4#9029211yescharacteristic
592C4#9035111yescharacteristic
593C4#9037111yescharacteristic
594C4#9039311yescharacteristic
595C2—42no—
596C2#1173296111yescharacteristic
597C2#1173675511yescharacteristic
598C2#1173682911yescharacteristic
599C2#1173686011yescharacteristic
600C2#1173689311yescharacteristic
601C2#1173695111yescharacteristic
602C2#1173697411yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1923C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1923D7
hypocenterC7#27036
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
4. #11728243C2semidirect
5. #11728207C2semidirect
10. #11727524C2semidirect
11. C4×#6390C2semidirect
12. #11726437C2semidirect
13. #11726390C2semidirect
14. #11726378C2semidirect
15. #11726378C2semidirect
563. C7#27036semidirect

Maximal subgroups

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

subgroupindex
#117309582
#117309102
C7×#19232
#117282432
#117282072
#117276912
#117276812
#117276792
#117275242
C4×#63902
#117264372
#117263902
#117263782
#270367