Finite Group Databasebeta

← #13388108

Subgroups — #13388108

Order 1792.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic, complemented
3C2×#89855C211yesmaximal, characteristic, complemented
4C4×#6493C211yesmaximal, characteristic
5#11729533C211yesmaximal, characteristic, complemented
6#11729532C211yesmaximal, characteristic, complemented
7#11729453C212yesmaximal
8#11729448C212yesmaximal, complemented
9#11729387C212yesmaximal, complemented
10#11729383C212yesmaximal, complemented
11#11729284C211yesmaximal, characteristic, complemented
12C7×#533—24no—
13C7×#533—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2212yes—
21C2×C7×#117—21no—
22C22×C42×C7C2211yescharacteristic
23C22×#1143C2212yes—
24C2×#6493C2211yescharacteristic
25C2×C4×C22⋊D7—21no—
26C2×C4×D28C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30#89985—21no—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2212yes—
33#89856—24no—
34#89855—24no—
35C2×#6383C2212yes—
36C2×#6382C2212yes—
37#89712—21no—
38#89711C2211yescharacteristic
39C4×(C2×D7)⋊C4—21no—
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24151—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8C2312yes—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C22×D28C2311yescharacteristic
55C22×C4×D7—21no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8D414yescomplemented
59C7×C4⋊Q8D414yescomplemented
60C7×#125—28no—
61C7×C22⋊Q8—28no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C4×C7×Q8—28no—
65C4×C7×Q8—24no—
66C4×C7×Q8—24no—
67C7×#117—28no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—22no—
75C2×C42×C7—22no—
76C2×C42×C7—22no—
77C2×C42×C7—22no—
78C2×C42×C7D411yescharacteristic
79C2×C42×C7D411yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C22⋊Dic7—21no—
83C2×#1143—24no—
84C2×#1143—24no—
85C2×#1143—42no—
86#6493—22no—
87#6493—22no—
88C4×C22⋊D7—44no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2312yes—
99C22×C7⋊C8C2312yes—
100C4×D28—22no—
101C4×D28—22no—
102#6409—24no—
103#6409—24no—
104#6409—42no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C22⋊(C7⋊C8)—42no—
109#6383—24no—
110#6382—24no—
111C4⋊(C7⋊C8)—22no—
112C4⋊(C7⋊C8)—22no—
113C2×#533D711yescharacteristic, complemented
114C2×#476—71no—
115C2×#461—71no—
116C4×#121—71no—
117#2035—72no—
118#2000—72no—
119#1987—71no—
120#1981—71no—
121#1880—72no—
122#1865—72no—
123#1797—71no—
124C24×C7C2×D411yescharacteristic
125C23×D7—21no—
126C2×C7×Q8—24no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—42no—
133C2×C7×Q8—42no—
134C2×C7×Q8C2×D414yes—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7SD1614yescomplemented
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C2×C22⋊D7—24no—
156C2×C22⋊D7—42no—
157C2×C22⋊D7—42no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C2×D28—21no—
162C2×D28—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28#4112yes—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—28no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4C2×D414yes—
177C7×C22⋊C4—28no—
178C42×C7—28no—
179C42×C7—22no—
180C42×C7—22no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7C2×D412yes—
184C42×C7C2×D412yes—
185C22⋊Dic7—42no—
186#1143—48no—
187(C2×D7)⋊C4—24no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—42no—
190(C2×D7)⋊C4—42no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7#4112yes—
197Dic7⋊C4—42no—
198C4×Dic7—42no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C22×SD16—72no—
206#533—24no—
207#533—24no—
208C2×C4⋊Q8C2×D711yescharacteristic
209C2×C4⋊Q8C2×D711yescharacteristic
210C2×#125—21no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8C2×D712yes—
213C2×#121—71no—
214C2×C4×Q8—21no—
215C2×C4×Q8C2×D712yes—
216C2×C4×D4—71no—
217C2×C4×D4—141no—
218C2×#117—21no—
219C22×C42C2×D711yescharacteristic
220#476—144no—
221#461—144no—
222C2×#106—72no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×#103—72no—
226C2×#103—72no—
227C2×#103—72no—
228C2×#102—72no—
229C2×C22⋊C8—72no—
230#390—71no—
231#389—71no—
232#388—141no—
233#386—141no—
234#380—71no—
235C4×C4⋊C4—71no—
236C4×C22⋊C4—141no—
237#340—72no—
238#340—72no—
239#336—72no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7C2×SD1612yes—
247C23×C7C22×D411yescharacteristic
248C22×D7—22no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7—41no—
253C22×D7—41no—
254C22×D7—41no—
255C7×Q8—24no—
256C7×Q8—24no—
257C7×Q8—48no—
258C7×Q8—42no—
259C7×Q8—42no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—28no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—44no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7#12012yes—
283C2×C4×C7#12012yes—
284C2×C4×C7C2×SD1612yes—
285C2×C4×C7C2×SD1612yes—
286C2×C4×C7#13712yes—
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C22⋊D7—48no—
294C2×Dic7—22no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303D28—22no—
304D28—22no—
305D28—41no—
306D28—41no—
307C4×D7—84no—
308C7⋊C8—42no—
309C7⋊C8—42no—
310C22×Q8—21no—
311C22×Q8C22×D712yes—
312C22×D4—71no—
313C22×D4—141no—
314C23×C4—141no—
315C23×C4C22⋊D711yescharacteristic
316C23×C4C22⋊D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C2×SD16—144no—
319C2×SD16—144no—
320C2×SD16—282no—
321C22×C8—72no—
322C22×C8—72no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8C22⋊D714yescomplemented
326C4⋊Q8C22⋊D714yescomplemented
327#125—28no—
328C22⋊Q8—28no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331#121—142no—
332#121—142no—
333C4×Q8—28no—
334C4×Q8—24no—
335C4×Q8—24no—
336C4×D4—142no—
337C4×D4—142no—
338C4×D4—284no—
339#117—28no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4C22×D712yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—141no—
360C2×C42C22⋊D711yescharacteristic
361C2×C42C22⋊D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364#106—144no—
365C4⋊C8—142no—
366C4⋊C8—142no—
367#103—144no—
368#103—144no—
369#103—282no—
370#102—144no—
371C22⋊C8—282no—
372#95—141no—
373#95—141no—
374#95—141no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7C2×#12011yescharacteristic
383C22×C7#52511yescharacteristic
384C22×C7#53711yescharacteristic
385C22×C7C22×SD1611yescharacteristic
386C22×C7C2×#13711yescharacteristic
387C22×C7#57511yescharacteristic
388C22×C7#57511yescharacteristic
389C2×D7—42no—
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—81no—
393C2×D7—81no—
394C2×D7—81no—
395C2×D7—81no—
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—41no—
399C4×C7—44no—
400C4×C7—41no—
401C4×C7—44no—
402C4×C7#44814yes—
403C4×C7#54611yescharacteristic
404C4×C7#54711yescharacteristic
405C4×C7#54811yescharacteristic
406C4×C7#55011yescharacteristic
407Dic7—42no—
408Dic7—81no—
409Dic7—81no—
410C24—141no—
411C24C2×C22⋊D711yescharacteristic
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×C22⋊D714yes—
421C2×D4—72no—
422C2×D4—144no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C2×D4—141no—
428C2×D4—282no—
429C2×D4—282no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4—282no—
449C22×C4#114314yescomplemented
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C23×D711yescharacteristic
456SD16—288no—
457C2×C8—142no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—142no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—72no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—282no—
476C4⋊C4C2×C22⋊D714yes—
477C22⋊C4—28no—
478C22⋊C4—144no—
479C22⋊C4—282no—
480C22⋊C4—282no—
481C22⋊C4—282no—
482C22⋊C4—282no—
483C42—28no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—21no—
488C42—282no—
489C42C2×C22⋊D712yes—
490C42C2×C22⋊D712yes—
491C2×C7—42no—
492C2×C7#240711yescharacteristic
493C2×C7C2×#44811yescharacteristic
494C2×C7#300511yescharacteristic
495C2×C7#310111yescharacteristic
496C2×C7#311611yescharacteristic
497C2×C7#324611yescharacteristic
498C2×C7#325211yescharacteristic
499D7—82no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—142no—
507C23—281no—
508C23—281no—
509C23—281no—
510C23—281no—
511C23—281no—
512C23—281no—
513C23C2×#114312yes—
514C23C22×C22⋊D711yescharacteristic
515Q8—24no—
516Q8—24no—
517Q8—48no—
518Q8—42no—
519Q8—42no—
520D4—142no—
521D4—142no—
522D4—288no—
523D4—281no—
524D4—281no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—142no—
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×C4C2×#114312yes—
558C2×C4C2×#114312yes—
559C2×C4#650512yes—
560C2×C4#651612yes—
561C2×C4#651612yes—
562C2×C4D7×#4111yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C2×C4#655311yescharacteristic
565C2×C4#655411yescharacteristic
566C2×C4C22×C22⋊D711yescharacteristic
567C2×C4C22×C22⋊D711yescharacteristic
568C8—282no—
569C8—282no—
570C7#2415111yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—282no—
579C22—282no—
580C22—282no—
581C22—561no—
582C22—561no—
583C22—561no—
584C22—561no—
585C22C22×#114311yescharacteristic
586C22C2×#650511yescharacteristic
587C22#9051311yescharacteristic
588C22#9051311yescharacteristic
589C22#9051911yescharacteristic
590C22#9052411yescharacteristic
591C22C2×#651611yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—44no—
596C4—41no—
597C4—44no—
598C4—282no—
599C4—561no—
600C4—561no—
601C4#8999914yes—
602C4#9051511yescharacteristic
603C4#9052011yescharacteristic
604C4#9052211yescharacteristic
605C4#9052511yescharacteristic
606C2—42no—
607C2—562no—
608C2#1174079611yescharacteristic
609C2#1174079911yescharacteristic
610C2#1174080811yescharacteristic
611C2#1174081011yescharacteristic
612C2C2×#8999911yescharacteristic
613C2#1174152611yescharacteristic
614C2#1174154211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#533C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC2×#533D7
hypocenterC7#24151
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
2. C2×#89856C2semidirect
3. C2×#89855C2semidirect
5. #11729533C2semidirect
6. #11729532C2semidirect
8. #11729448C2semidirect
9. #11729387C2semidirect
10. #11729383C2semidirect
11. #11729284C2semidirect
58. C7×C4⋊Q8D4semidirect
59. C7×C4⋊Q8D4semidirect
113. C2×#533D7semidirect
148. C22×C4×C7SD16semidirect
325. C4⋊Q8C22⋊D7semidirect
326. C4⋊Q8C22⋊D7semidirect
449. C22×C4#1143semidirect
570. C7#24151semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×#898562
C2×#898552
C4×#64932
#117295332
#117295322
#117294532
#117294482
#117293872
#117293832
#117292842
#241517