Finite Group Databasebeta

← #13356942

Subgroups — #13356942

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×#89560C211yesmaximal, characteristic
2#11733928C211yesmaximal, characteristic, complemented
3#11731196C211yesmaximal, characteristic
4C7×#1928C211yesmaximal, characteristic, complemented
5#11728392C211yesmaximal, characteristic
6Dic7×#107C211yesmaximal, characteristic, complemented
7#11728030C211yesmaximal, characteristic
8#11727978C211yesmaximal, characteristic, complemented
9#11727974C211yesmaximal, characteristic, complemented
10#11727441C211yesmaximal, characteristic
11#11727214C211yesmaximal, characteristic, complemented
12#11726543C211yesmaximal, characteristic, complemented
13#11726536C211yesmaximal, characteristic
14#11726424C211yesmaximal, characteristic
15#11726352C211yesmaximal, characteristic, complemented
16C2×Q8×Dic7C2211yescharacteristic
17C22×#1143C2211yescharacteristic
18C2×#6456C2211yescharacteristic
19C2×#6456C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26#90002C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89745C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C4×(C2×D7)⋊C4—21no—
40C4×(C2×D7)⋊C4—21no—
41#89704C2211yescharacteristic
42#89700C2211yescharacteristic
43#89560C418yescomplemented
44#89438—21no—
45D7×#95—21no—
46#89426—21no—
47#89425—21no—
48#89423—21no—
49#89422—21no—
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#21130—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7C2311yescharacteristic
73Q8×Dic7—22no—
74Q8×Dic7C2×C414yescomplemented
75C2×#1143—22no—
76C2×#1143—22no—
77C2×#1143C2×C414yes—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6456—22no—
96#6456—22no—
97#6456—22no—
98#6456C2×C414yes—
99C4×D28—22no—
100C4×D28—22no—
101C7×#107—22no—
102C7×#103—22no—
103C7×#103C2×C414yescomplemented
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107#6409—22no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95D411yescharacteristic
116C7⋊#95D411yescharacteristic
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6395—22no—
120#6395C2×C414yes—
121#6392—22no—
122#6389—22no—
123C8×Dic7C2×C414yescomplemented
124#6383—22no—
125#6382C2×C414yescomplemented
126C2×C4×SD16—71no—
127#2320—71no—
128#2275—71no—
129#1928—71no—
130#1928D711yescharacteristic, complemented
131#1920—71no—
132#1920—71no—
133#1877—71no—
134#1877—71no—
135#1862—71no—
136#1862—71no—
137C4×#107—71no—
138#1763—71no—
139#1762—71no—
140#1746—71no—
141C23×D7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C22×C412yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×D28—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28C22×C412yes—
166C2×D28#4112yes—
167C2×D28#4112yes—
168C2×C4×D7—24no—
169C2×C4×D7—24no—
170C2×C4×D7—24no—
171C2×C4×D7—81no—
172C2×C4×D7—81no—
173C2×C4×D7—81no—
174C2×C4×D7—81no—
175C2×C4×D7—81no—
176C2×C4×D7—81no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8C22×C412yes—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4C22×C412yes—
188C7×C4⋊C4#4112yes—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—81no—
192#1143—28no—
193(C2×D7)⋊C4—24no—
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—24no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—24no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C412yes—
202C4⋊Dic7#4112yes—
203Dic7⋊C4—81no—
204Dic7⋊C4—81no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—24no—
208C4×Dic7—81no—
209C4×Dic7—81no—
210C4×Dic7—81no—
211C4×Dic7C22×C412yes—
212C4×Dic7#4112yes—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8C22×C412yes—
219C22×SD16—71no—
220C2×C4×Q8—71no—
221C2×C4×D4—71no—
222C2×C4×D4—71no—
223C2×C4×D4—71no—
224C4×SD16—78no—
225C2×#107—71no—
226C2×#107C2×D711yescharacteristic
227C2×#106—71no—
228C2×#103—71no—
229C2×#103C2×D711yescharacteristic
230C2×#103C2×D711yescharacteristic
231C2×#102—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#97—71no—
235C2×C4×C8—71no—
236#391—71no—
237#391C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
248C4×C22⋊C4—141no—
249C4×C22⋊C4—141no—
250C4×C22⋊C4—141no—
251#376—71no—
252C2×#95—141no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340C2×D711yescharacteristic
259#336—71no—
260#336—71no—
261#336—71no—
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—
271C7×Q8—22no—
272C7×Q8—22no—
273C7×Q8—41no—
274C7×Q8—41no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—22no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—22no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7C4×D412yes—
319C2×Dic7C4×D412yes—
320D28—22no—
321D28—22no—
322D28—22no—
323D28#11714yes—
324C4×D7—84no—
325C4×D7—84no—
326C4×D7—84no—
327C7×C8—22no—
328C7×C8—41no—
329C7⋊C8—22no—
330C7⋊C8—41no—
331C22×Q8C22×D711yescharacteristic
332C22×D4—71no—
333C23×C4—141no—
334C23×C4—141no—
335C23×C4—141no—
336C2×SD16—74no—
337C2×SD16—142no—
338C2×SD16—142no—
339C22×C8—71no—
340C22×C8—71no—
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—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C42—21no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—141no—
367C2×C42—141no—
368C2×C42—141no—
369C2×C42—141no—
370C2×C42C22×D711yescharacteristic
371#107—22no—
372#107—142no—
373#106—74no—
374#103—22no—
375#103—142no—
376#103C4×D714yescomplemented
377#102—74no—
378#102—142no—
379#102—142no—
380#97—142no—
381C4×C8—74no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—71no—
386#95—71no—
387#95—71no—
388#95—71no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
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—41no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—81no—
416C4×C7C2×#11711yescharacteristic
417C4×C7C4×#4111yescharacteristic
418C4×C7#52911yescharacteristic
419C4×C7#52911yescharacteristic
420Dic7—22no—
421Dic7—22no—
422Dic7—41no—
423Dic7—42no—
424Dic7—41no—
425Dic7—42no—
426Dic7—81no—
427C24—141no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D712yes—
434C2×D4—72no—
435C2×D4—72no—
436C2×D4—72no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—144no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—561no—
455C22×C4—561no—
456C22×C4—561no—
457C22×C4—561no—
458C22×C4—561no—
459C22×C4—561no—
460C22×C4D4×D711yescharacteristic
461C22×C4D4×D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463SD16—148no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—72no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8C2×C4×D712yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—72no—
481C4⋊C4—72no—
482C4⋊C4—144no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—561no—
488C4⋊C4—561no—
489C4⋊C4C2×C4×D712yes—
490C4⋊C4#115912yes—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C22⋊C4—144no—
494C42—21no—
495C42—21no—
496C42—72no—
497C42—72no—
498C42—81no—
499C42—144no—
500C42—141no—
501C42—141no—
502C42—561no—
503C42—561no—
504C42—561no—
505C2×C7#229011yescharacteristic
506C2×C7#294911yescharacteristic
507C2×C7#295111yescharacteristic
508C2×C7#311811yescharacteristic
509C2×C7#311811yescharacteristic
510C2×C7#312011yescharacteristic
511C2×C7#312011yescharacteristic
512D7—44no—
513C23—142no—
514C23—142no—
515C23—142no—
516C23—281no—
517C23—281no—
518C23—281no—
519C23—281no—
520C23C2×D4×D711yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—74no—
526D4—142no—
527D4—142no—
528D4—142no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—72no—
548C2×C4—72no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—142no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—142no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—564no—
569C2×C4—564no—
570C2×C4—564no—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×#115911yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C8—22no—
578C8—41no—
579C8—142no—
580C8—281no—
581C7#2113011yescharacteristic, complemented, Sylow
582C22—144no—
583C22—282no—
584C22—282no—
585C22—282no—
586C22—282no—
587C22—282no—
588C22—282no—
589C22C4×D4×D711yescharacteristic
590C22#9029711yescharacteristic
591C22#9034711yescharacteristic
592C22#9045711yescharacteristic
593C22#9046011yescharacteristic
594C22#9046611yescharacteristic
595C22#9046911yescharacteristic
596C4—41no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—142no—
603C4—142no—
604C4—281no—
605C4—282no—
606C4—282no—
607C4—281no—
608C4—561no—
609C4D7×#11711yescharacteristic
610C4C4×#115911yescharacteristic
611C4#9037111yescharacteristic
612C4#9039711yescharacteristic
613C2—284no—
614C2#1173289411yescharacteristic
615C2#1173716611yescharacteristic
616C2#1173722211yescharacteristic
617C2#1173790511yescharacteristic
618C2#1173795311yescharacteristic
619C2#1173890111yescharacteristic
620C2#1173894111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1928C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1928D7
hypocenterC7#21130
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
2. #11733928C2semidirect
4. C7×#1928C2semidirect
6. Dic7×#107C2semidirect
8. #11727978C2semidirect
9. #11727974C2semidirect
11. #11727214C2semidirect
12. #11726543C2semidirect
15. #11726352C2semidirect
43. #89560C4semidirect
74. Q8×Dic7C2×C4semidirect
103. C7×#103C2×C4semidirect
123. C8×Dic7C2×C4semidirect
125. #6382C2×C4semidirect
130. #1928D7semidirect
376. #103C4×D7semidirect
581. C7#21130semidirect

Maximal subgroups

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

subgroupindex
C2×#895602
#117339282
#117311962
C7×#19282
#117283922
Dic7×#1072
#117280302
#117279782
#117279742
#117274412
#117272142
#117265432
#117265362
#117264242
#117263522
#211307