Finite Group Databasebeta

← #13356874

Subgroups — #13356874

Order 1792.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89482C211yesmaximal, characteristic
2#11730857C211yesmaximal, characteristic
3#11730831C211yesmaximal, characteristic, complemented
4C7×#1746C211yesmaximal, characteristic, complemented
5#11727568C211yesmaximal, characteristic, complemented
6C4×#6390C211yesmaximal, characteristic, complemented
7C4×#6395C211yesmaximal, characteristic
8#11726426C211yesmaximal, characteristic
9#11726426C211yesmaximal, characteristic
10#11726425C211yesmaximal, characteristic
11#11726424C211yesmaximal, characteristic
12#11726382C211yesmaximal, characteristic, complemented
13#11726381C211yesmaximal, characteristic, complemented
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×#1132C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×#97C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C7×#376C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C4×C4⋊Dic7C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35C4×C4⋊Dic7C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89701C2211yescharacteristic
38#89701C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41#89482C418yescomplemented
42#89438—21no—
43#89438—21no—
44#89438—21no—
45D7×#95—21no—
46#89424—21no—
47#89424—21no—
48#89423—21no—
49#89422—21no—
50C7×#336C2211yescharacteristic
51C7×#336C2211yescharacteristic
52C7×#336C2211yescharacteristic
53C7×#336C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#21125—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
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×C4×Dic7—21no—
86C2×C4×Dic7—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×D28—22no—
95C4×D28—22no—
96C4×D28—22no—
97C4×D28—22no—
98C4×D28C2×C414yes—
99C4×Dic14—22no—
100C4×Dic14C2×C414yescomplemented
101C7×#97C2×C414yescomplemented
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×C414yescomplemented
122#6391—22no—
123#6390—22no—
124#6390—22no—
125#6390C2×C414yescomplemented
126C2×#440—71no—
127#2320—71no—
128#2275—71no—
129#1928—71no—
130#1927—71no—
131#1921—71no—
132#1918—71no—
133#1877—71no—
134#1877—71no—
135#1864—71no—
136#1862—71no—
137#1776—71no—
138C4×#103—71no—
139C4×#102—71no—
140#1746D711yescharacteristic, 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×D28C22×C412yes—
160C2×D28#4112yes—
161C2×D28#4112yes—
162C2×C4×D7—24no—
163C2×C4×D7—24no—
164C2×C4×D7—24no—
165C2×C4×D7—81no—
166C2×C4×D7—81no—
167C2×C4×D7—81no—
168C2×C4×D7—81no—
169C2×C4×D7—81no—
170C2×C4×D7—81no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14C22×C412yes—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8C22×C412yes—
188C2×C7×C8C22×C412yes—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7C22×C412yes—
194C42×C7#4112yes—
195(C2×D7)⋊C4—24no—
196(C2×D7)⋊C4—24no—
197(C2×D7)⋊C4—24no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7C22×C412yes—
207C4⋊Dic7C22×C412yes—
208C4⋊Dic7#4112yes—
209C4⋊Dic7#4112yes—
210Dic7⋊C4—24no—
211Dic7⋊C4—81no—
212Dic7⋊C4—81no—
213C4×Dic7—24no—
214C4×Dic7—81no—
215C4×Dic7—81no—
216C4×Dic7—81no—
217C4×Dic7—81no—
218#1132—28no—
219C22×SD16—71no—
220C2×C4×Q8—71no—
221C2×C4×D4—71no—
222C2×C4×D4—71no—
223C2×C4×D4—71no—
224#440—78no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#106—71no—
228C2×#103—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#102—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#97C2×D711yescharacteristic
235C2×C4×C8C2×D711yescharacteristic
236#391—71no—
237#391—71no—
238#389—71no—
239#389—71no—
240#387—141no—
241#387—141no—
242#383—141no—
243#382—141no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—71no—
248C4×C22⋊C4—141no—
249C4×C22⋊C4—141no—
250C4×C22⋊C4—141no—
251#376C2×D711yescharacteristic
252C2×#95—141no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#336C2×D711yescharacteristic
260#336C2×D711yescharacteristic
261#336C2×D711yescharacteristic
262#336C2×D711yescharacteristic
263C23×C7C22×D411yescharacteristic
264C22×D7—22no—
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
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×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7C4×D412yes—
288C2×C4×C7C4×D412yes—
289C2×C4×C7C23×C411yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
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—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316D28—22no—
317D28—22no—
318D28—22no—
319D28#11714yes—
320C4×D7—84no—
321C4×D7—84no—
322C4×D7—84no—
323Dic14—22no—
324Dic14—22no—
325Dic14—41no—
326Dic14—41no—
327C7×C8—22no—
328C7×C8—22no—
329C7×C8—41no—
330C7×C8—41no—
331C22×Q8—71no—
332C22×D4—71no—
333C23×C4—141no—
334C23×C4—141no—
335C23×C4—141no—
336C2×SD16—74no—
337C2×SD16—142no—
338C2×SD16—142no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C4×Q8—74no—
344C4×Q8—142no—
345C4×D4—74no—
346C4×D4—142no—
347C4×D4—142no—
348C4×D4—142no—
349C4×D4—142no—
350C4×D4—142no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C42—141no—
364C2×C42—141no—
365C2×C42—141no—
366C2×C42—141no—
367C2×C42—141no—
368C2×C42C22×D711yescharacteristic
369C2×C42C22×D711yescharacteristic
370C2×C42C22×D711yescharacteristic
371#107—74no—
372#107—142no—
373#106—142no—
374#103—74no—
375#103—142no—
376#103—142no—
377#102—74no—
378#102—142no—
379#102—142no—
380#97C4×D714yescomplemented
381C4×C8—22no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95D2811yescharacteristic
393#95D2811yescharacteristic
394#95D2811yescharacteristic
395#95D2811yescharacteristic
396C22×C7C2×C4×D411yescharacteristic
397C22×C7#52511yescharacteristic
398C22×C7#52511yescharacteristic
399C22×C7#57611yescharacteristic
400C22×C7#57711yescharacteristic
401C22×C7#57711yescharacteristic
402C22×C7#57811yescharacteristic
403C2×D7—24no—
404C2×D7—42no—
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C4×C7—22no—
411C4×C7—22no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7C2×#11711yescharacteristic
417C4×C7C4×#4111yescharacteristic
418C4×C7#52911yescharacteristic
419C4×C7#52911yescharacteristic
420Dic7—42no—
421Dic7—42no—
422Dic7—42no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427C24—141no—
428C2×Q8—72no—
429C2×Q8—141no—
430C2×Q8—141no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×D4—72no—
435C2×D4—72no—
436C2×D4—72no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C22×C4—144no—
441C22×C4—144no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—561no—
451C22×C4—561no—
452C22×C4—561no—
453C22×C4—561no—
454C22×C4—561no—
455C22×C4—561no—
456C22×C4C2×D2811yescharacteristic
457C22×C4C2×D2811yescharacteristic
458C22×C4C2×D2811yescharacteristic
459C22×C4C2×D2811yescharacteristic
460C22×C4C2×D2811yescharacteristic
461C22×C4C2×D2811yescharacteristic
462C22×C4C23×D711yescharacteristic
463SD16—148no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C4×D712yes—
475C2×C8C2×C4×D712yes—
476C4⋊C4—72no—
477C4⋊C4—72no—
478C4⋊C4—72no—
479C4⋊C4—72no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—561no—
490C4⋊C4—561no—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C22⋊C4—144no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—144no—
499C42—561no—
500C42—561no—
501C42—561no—
502C42—561no—
503C42C2×C4×D712yes—
504C42#115712yes—
505C2×C7#229011yescharacteristic
506C2×C7#294611yescharacteristic
507C2×C7#294711yescharacteristic
508C2×C7#311811yescharacteristic
509C2×C7#312011yescharacteristic
510C2×C7#312311yescharacteristic
511C2×C7#312411yescharacteristic
512D7—44no—
513C23—142no—
514C23—142no—
515C23—142no—
516C23—281no—
517C23—281no—
518C23—281no—
519C23—281no—
520C23C22×D2811yescharacteristic
521Q8—142no—
522Q8—142no—
523Q8—281no—
524Q8—281no—
525D4—74no—
526D4—142no—
527D4—142no—
528D4—142no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—564no—
567C2×C4—564no—
568C2×C4—564no—
569C2×C4C4×D2812yes—
570C2×C4C4×D2812yes—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×#115711yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C8—22no—
578C8—22no—
579C8—41no—
580C8—41no—
581C7#2112511yescharacteristic, complemented, Sylow
582C22—144no—
583C22—282no—
584C22—282no—
585C22—282no—
586C22—282no—
587C22—282no—
588C22—282no—
589C22C2×C4×D2811yescharacteristic
590C22#9021511yescharacteristic
591C22#9021511yescharacteristic
592C22#9044211yescharacteristic
593C22#9044211yescharacteristic
594C22#9044311yescharacteristic
595C22#9044411yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—282no—
603C4—282no—
604C4—282no—
605C4—282no—
606C4—282no—
607C4—561no—
608C4—561no—
609C4D7×#11711yescharacteristic
610C4#9021311yescharacteristic
611C4#9021911yescharacteristic
612C4#9021911yescharacteristic
613C2—284no—
614C2#1173085911yescharacteristic
615C2#1173600511yescharacteristic
616C2#1173601611yescharacteristic
617C2#1173682811yescharacteristic
618C2#1173683011yescharacteristic
619C2#1173683211yescharacteristic
620C2#1173683311yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1746C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1746D7
hypocenterC7#21125
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
3. #11730831C2semidirect
4. C7×#1746C2semidirect
5. #11727568C2semidirect
6. C4×#6390C2semidirect
12. #11726382C2semidirect
13. #11726381C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
41. #89482C4semidirect
100. C4×Dic14C2×C4semidirect
101. C7×#97C2×C4semidirect
121. #6392C2×C4semidirect
125. #6390C2×C4semidirect
140. #1746D7semidirect
380. #97C4×D7semidirect
581. C7#21125semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
#117308572
#117308312
C7×#17462
#117275682
C4×#63902
C4×#63952
#117264262
#117264252
#117264242
#117263822
#117263812
#117263782
#117263752
#211257