Finite Group Databasebeta

← #13358172

Subgroups — #13358172

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89482C211yesmaximal, characteristic
2#11730860C211yesmaximal, characteristic
3#11730831C211yesmaximal, characteristic, complemented
4C7×#1726C211yesmaximal, characteristic, complemented
5#11727580C211yesmaximal, characteristic
6#11727573C211yesmaximal, characteristic, complemented
7#11727572C211yesmaximal, characteristic
8#11727571C211yesmaximal, characteristic, complemented
9#11727568C211yesmaximal, characteristic, complemented
10#11727528C211yesmaximal, characteristic
11#11727524C211yesmaximal, characteristic, complemented
12#11727522C211yesmaximal, characteristic
13#11727521C211yesmaximal, characteristic, complemented
14C4×#6390C211yesmaximal, characteristic, complemented
15C4×#6395C211yesmaximal, characteristic
16C22×#1132C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C2×C7×#97C2211yescharacteristic
20C2×C7×#97C2211yescharacteristic
21C2×C7×#97C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C7×#376C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34C2×#6390C2211yescharacteristic
35#89711C2211yescharacteristic
36#89711C2211yescharacteristic
37C4×C4⋊Dic7C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39#89701C2211yescharacteristic
40#89701C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89482C418yescomplemented
46#89441—21no—
47#89441—21no—
48#89439—21no—
49#89436—21no—
50#89424—21no—
51#89424—21no—
52#89423—21no—
53#89422—21no—
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#21291—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×Dic14C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C42×C7C2311yescharacteristic
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×#1132—22no—
91C2×#1132—22no—
92C2×#1132C2×C414yes—
93C4×D28—22no—
94C4×D28C2×C414yes—
95C4×Dic14—22no—
96C4×Dic14C2×C414yes—
97C7×#97—22no—
98C7×#97—22no—
99C7×#97C2×C414yes—
100C4×C7×C8—22no—
101C4×C7×C8—22no—
102C4×C7×C8—22no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105C7×#95D411yescharacteristic
106C7×#95D411yescharacteristic
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117#6395—22no—
118#6395—22no—
119#6395C2×C414yes—
120#6392—22no—
121#6392C2×C414yes—
122#6391—22no—
123#6390—22no—
124#6390—22no—
125#6390C2×C414yes—
126C2×#440—71no—
127#2320—71no—
128#2319—71no—
129#1973—71no—
130#1968—71no—
131#1943—71no—
132#1942—71no—
133#1936—71no—
134#1935—71no—
135#1934—71no—
136#1933—71no—
137#1776—71no—
138C4×#103—71no—
139C4×#102—71no—
140#1726D711yescharacteristic, complemented
141C23×D7—21no—
142C22×C4×C7C2×D411yescharacteristic
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—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28C22×C412yes—
162C2×C4×D7—24no—
163C2×C4×D7—81no—
164C2×C4×D7—81no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14C22×C412yes—
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×C8C22×C412yes—
178C2×C7×C8C22×C412yes—
179C2×C7×C8#4112yes—
180C2×C7×C8#4112yes—
181C2×C7×C8#4112yes—
182C2×C7×C8#4112yes—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7C22×C412yes—
188C42×C7#4112yes—
189(C2×D7)⋊C4—24no—
190(C2×D7)⋊C4—81no—
191(C2×D7)⋊C4—81no—
192(C2×D7)⋊C4—81no—
193(C2×D7)⋊C4—81no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7C22×C412yes—
205C4⋊Dic7C22×C412yes—
206Dic7⋊C4—24no—
207Dic7⋊C4—81no—
208Dic7⋊C4—81no—
209Dic7⋊C4—81no—
210Dic7⋊C4—81no—
211C4×Dic7—24no—
212C4×Dic7—81no—
213C4×Dic7—81no—
214#1132—28no—
215C22×SD16—71no—
216C2×C4×Q8—71no—
217C2×C4×D4—71no—
218#440—78no—
219C2×#107—71no—
220C2×#107—71no—
221C2×#106—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#97C2×D711yescharacteristic
229C2×#97C2×D711yescharacteristic
230C2×#97C2×D711yescharacteristic
231C2×C4×C8C2×D711yescharacteristic
232C2×C4×C8C2×D711yescharacteristic
233C2×C4×C8C2×D711yescharacteristic
234#391—71no—
235#391—71no—
236#390—71no—
237#390—71no—
238#389—71no—
239#389—71no—
240#389—71no—
241#389—71no—
242#388—141no—
243#388—141no—
244#387—141no—
245#387—141no—
246#383—141no—
247#382—141no—
248#381—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252#376C2×D711yescharacteristic
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259C23×C7C22×D411yescharacteristic
260C22×D7—22no—
261C22×D7—41no—
262C22×D7—41no—
263C22×D7—41no—
264C22×D7—41no—
265C22×D7—41no—
266C22×D7—41no—
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—21no—
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×C7C4×D412yes—
284C2×C4×C7C4×D412yes—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—22no—
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—
312D28—22no—
313D28—22no—
314D28—41no—
315D28—41no—
316C4×D7—84no—
317Dic14—22no—
318Dic14—22no—
319Dic14—41no—
320Dic14—41no—
321C7×C8—22no—
322C7×C8—22no—
323C7×C8—22no—
324C7×C8#11714yes—
325C22×Q8—71no—
326C22×D4—71no—
327C23×C4—141no—
328C2×SD16—74no—
329C2×SD16—142no—
330C2×SD16—142no—
331C22×C8C22×D711yescharacteristic
332C22×C8C22×D711yescharacteristic
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335C4×Q8—74no—
336C4×Q8—142no—
337C4×D4—74no—
338C4×D4—142no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42C22×D711yescharacteristic
359C2×C42C22×D711yescharacteristic
360C2×C42C22×D711yescharacteristic
361#107—74no—
362#107—142no—
363#106—142no—
364#103—74no—
365#103—142no—
366#103—142no—
367#102—74no—
368#102—142no—
369#102—142no—
370#97—22no—
371#97—22no—
372#97C4×D714yes—
373C4×C8—22no—
374C4×C8—22no—
375C4×C8—22no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95D2811yescharacteristic
387#95D2811yescharacteristic
388#95D2811yescharacteristic
389#95D2811yescharacteristic
390C22×C7C2×C4×D411yescharacteristic
391C22×C7#53211yescharacteristic
392C22×C7#53211yescharacteristic
393C22×C7#57611yescharacteristic
394C22×C7#57711yescharacteristic
395C22×C7#57711yescharacteristic
396C22×C7#57811yescharacteristic
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—81no—
401C2×D7—81no—
402C2×D7—81no—
403C2×D7—81no—
404C4×C7—22no—
405C4×C7—22no—
406C4×C7—41no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7C2×#11711yescharacteristic
411C4×C7C4×#4111yescharacteristic
412C4×C7#52911yescharacteristic
413C4×C7#52911yescharacteristic
414Dic7—42no—
415Dic7—42no—
416Dic7—42no—
417Dic7—81no—
418Dic7—81no—
419Dic7—81no—
420Dic7—81no—
421C24—141no—
422C2×Q8—72no—
423C2×Q8—141no—
424C2×Q8—141no—
425C2×Q8—141no—
426C2×Q8—141no—
427C2×Q8—141no—
428C2×D4—72no—
429C2×D4—141no—
430C2×D4—141no—
431C2×D4—141no—
432C2×D4—141no—
433C2×D4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—144no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—561no—
443C22×C4—561no—
444C22×C4C2×D2811yescharacteristic
445C22×C4C2×D2811yescharacteristic
446C22×C4C2×D2811yescharacteristic
447C22×C4C2×D2811yescharacteristic
448C22×C4C2×D2811yescharacteristic
449C22×C4C2×D2811yescharacteristic
450C22×C4C23×D711yescharacteristic
451SD16—148no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8C2×C4×D712yes—
459C2×C8C2×C4×D712yes—
460C2×C8#115712yes—
461C2×C8#115712yes—
462C2×C8#115712yes—
463C2×C8#115712yes—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—144no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—561no—
478C4⋊C4—561no—
479C4⋊C4—561no—
480C4⋊C4—561no—
481C22⋊C4—144no—
482C22⋊C4—561no—
483C22⋊C4—561no—
484C22⋊C4—561no—
485C22⋊C4—561no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—144no—
491C42—561no—
492C42—561no—
493C42C2×C4×D712yes—
494C42#115712yes—
495C2×C7#229311yescharacteristic
496C2×C7#294611yescharacteristic
497C2×C7#294711yescharacteristic
498C2×C7#317411yescharacteristic
499C2×C7#317511yescharacteristic
500C2×C7#317811yescharacteristic
501C2×C7#317911yescharacteristic
502D7—82no—
503C23—142no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23—281no—
509C23—281no—
510C23C22×D2811yescharacteristic
511Q8—142no—
512Q8—142no—
513Q8—281no—
514Q8—281no—
515D4—142no—
516D4—142no—
517D4—281no—
518D4—281no—
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—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—142no—
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—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—564no—
557C2×C4C4×D2812yes—
558C2×C4C4×D2812yes—
559C2×C4C22×C4×D711yescharacteristic
560C2×C4C2×#115711yescharacteristic
561C2×C4C2×#115711yescharacteristic
562C2×C4C2×#115711yescharacteristic
563C2×C4C2×#115711yescharacteristic
564C2×C4C2×#115711yescharacteristic
565C8—22no—
566C8—22no—
567C8—22no—
568C8#643514yes—
569C7#2129111yescharacteristic, complemented, Sylow
570C22—282no—
571C22—282no—
572C22—282no—
573C22—561no—
574C22—561no—
575C22—561no—
576C22—561no—
577C22C2×C4×D2811yescharacteristic
578C22#9016811yescharacteristic
579C22#9016811yescharacteristic
580C22#9044211yescharacteristic
581C22#9044211yescharacteristic
582C22#9044311yescharacteristic
583C22#9044411yescharacteristic
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—282no—
591C4—282no—
592C4—282no—
593C4—561no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4C2×#643511yescharacteristic
598C4C4×#115711yescharacteristic
599C4#9017011yescharacteristic
600C4#9017011yescharacteristic
601C2—562no—
602C2#1173074411yescharacteristic
603C2#1173600511yescharacteristic
604C2#1173601211yescharacteristic
605C2#1173601311yescharacteristic
606C2#1173601511yescharacteristic
607C2#1173601611yescharacteristic
608C2#1173602311yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1726C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1726D7
hypocenterC7#21291
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
3. #11730831C2semidirect
4. C7×#1726C2semidirect
6. #11727573C2semidirect
8. #11727571C2semidirect
9. #11727568C2semidirect
11. #11727524C2semidirect
13. #11727521C2semidirect
14. C4×#6390C2semidirect
45. #89482C4semidirect
140. #1726D7semidirect
569. C7#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
#117308602
#117308312
C7×#17262
#117275802
#117275732
#117275722
#117275712
#117275682
#117275282
#117275242
#117275222
#117275212
C4×#63902
C4×#63952
#212917