Finite Group Databasebeta

← #13386228

Subgroups — #13386228

Order 1792.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic
2C2×#89822C211yesmaximal, characteristic, complemented
3#11739794C211yesmaximal, characteristic
4C2×#89776C211yesmaximal, characteristic, complemented
5#11729498C212yesmaximal, complemented
6#11729454C212yesmaximal, complemented
7#11729451C211yesmaximal, characteristic, complemented
8#11729386C212yesmaximal, complemented
9#11729365C211yesmaximal, characteristic
10#11729329C211yesmaximal, characteristic, complemented
11#11729313C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C7×#552—24no—
14C7×#552—24no—
15C2×C7×#126—21no—
16C2×C7×#125—21no—
17C2×C7×#125C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117—21no—
24C2×C7×#117C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C22×#1144C2212yes—
27C2×#6487C2212yes—
28C22×#1136C2211yescharacteristic
29C2×#6409C2212yes—
30#89992—21no—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2212yes—
34#89822—24no—
35#89776—24no—
36#89751—21no—
37C2×#6384C2212yes—
38C2×#6384C2212yes—
39C2×#6384C2212yes—
40#89719—21no—
41#89718—21no—
42#89701C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#23983—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C23×Dic7—21no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14—22no—
57C22×Dic14—21no—
58C22×Dic14—42no—
59C22×Dic14C2312yes—
60C7×#126—28no—
61C7×#125—28no—
62C7×#125—22no—
63C7×#125—22no—
64C7×#123—28no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8D414yes—
69C4×C7×Q8—22no—
70C4×C7×Q8—22no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C7×#117—28no—
74C7×#117—22no—
75C7×#117D414yes—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4D411yescharacteristic
92C2×C7×C22⋊C4D411yescharacteristic
93C2×C42×C7—21no—
94C2×C42×C7—21no—
95C2×C42×C7C2311yescharacteristic
96C2×C22⋊Dic7—22no—
97C2×C22⋊Dic7—21no—
98C2×#1144—24no—
99C2×#1144—24no—
100C2×#1144—42no—
101C2×C4⋊Dic7C2312yes—
102#6487—24no—
103#6487—42no—
104C2×Dic7⋊C4—22no—
105C2×#1136—22no—
106C2×#1136—42no—
107C22×C7⋊C8C2312yes—
108C22×C7⋊C8C2312yes—
109#6409—42no—
110C7⋊#95—22no—
111C7⋊#95—22no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)—22no—
114C22⋊(C7⋊C8)—42no—
115#6384—24no—
116#6384—24no—
117#6384—24no—
118C2×#552D711yescharacteristic
119C2×#451—71no—
120C2×#419—71no—
121#2409—71no—
122#2026—72no—
123#2001—72no—
124#1894—71no—
125#1878—72no—
126#1872—72no—
127#1870—71no—
128#1825—71no—
129C24×C7C2×D411yescharacteristic
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—21no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8C2×D412yes—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—41no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—24no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—44no—
161C22×Dic7—42no—
162C2×Dic14—28no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—22no—
166C2×Dic14—22no—
167C2×Dic14—22no—
168C2×Dic14—48no—
169C2×Dic14—44no—
170C2×Dic14—44no—
171C2×Dic14—42no—
172C2×Dic14#4114yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4C2×D412yes—
184C7×C4⋊C4C2×D412yes—
185C7×C4⋊C4C2×D412yes—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C42×C7—24no—
192C42×C7—24no—
193C42×C7—21no—
194C42×C7—21no—
195C22⋊Dic7—44no—
196C22⋊Dic7—42no—
197#1144—48no—
198C4⋊Dic7—22no—
199C4⋊Dic7—22no—
200Dic7⋊C4—44no—
201#1136—44no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C23×Q8—71no—
209C22×Q16—72no—
210C22×M16—71no—
211#552—24no—
212#552—24no—
213C2×#126—21no—
214C2×#125—21no—
215C2×#125C2×D711yescharacteristic
216C2×#123—21no—
217C2×C22⋊Q8—72no—
218C2×C22⋊Q8C2×D711yescharacteristic
219C2×C22⋊Q8C2×D711yescharacteristic
220C2×C4×Q8C2×D711yescharacteristic
221C2×C4×D4C2×D711yescharacteristic
222C2×#117—21no—
223C2×#117C2×D711yescharacteristic
224C22×C4⋊C4C2×D711yescharacteristic
225#451—144no—
226#419—144no—
227C2×#103—72no—
228C2×#103—72no—
229C2×#103—72no—
230C2×#103—72no—
231C2×C22⋊C8—71no—
232C2×C22⋊C8—72no—
233#395—141no—
234#393—141no—
235#391—72no—
236#386—141no—
237#385—141no—
238#380—71no—
239#340—72no—
240#340—72no—
241#340—72no—
242#336—72no—
243C23×C7—21no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—21no—
249C23×C7—22no—
250C23×C7—22no—
251C23×C7#12012yes—
252C23×C7#13712yes—
253C23×C7C22×D411yescharacteristic
254C7×Q8—24no—
255C7×Q8—41no—
256C7×Q8—41no—
257C7×D4—44no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—22no—
273C2×C4×C7—48no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7#12012yes—
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4112yes—
287C2×C4×C7H(5,𝔽2)12yes—
288C2×Dic7—24no—
289C2×Dic7—48no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295Dic14—24no—
296Dic14—24no—
297Dic14—48no—
298Dic14—42no—
299Dic14—42no—
300C7⋊C8—42no—
301C7⋊C8—42no—
302C22×Q8—72no—
303C22×Q8—144no—
304C22×Q8—141no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—282no—
308C22×Q8C22×D711yescharacteristic
309C22×D4C22⋊D711yescharacteristic
310C23×C4—21no—
311C23×C4—141no—
312C23×C4C22×D711yescharacteristic
313C23×C4C22×D711yescharacteristic
314C2×Q16—144no—
315C2×Q16—144no—
316C2×Q16—282no—
317C2×M16—142no—
318C2×M16—282no—
319C22×C8—72no—
320C22×C8—72no—
321#126—28no—
322#125—28no—
323#125—22no—
324#125—22no—
325#123—28no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—144no—
330C22⋊Q8—282no—
331C22⋊Q8C22⋊D714yes—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×D4—22no—
335C4×D4—22no—
336#117—28no—
337#117—22no—
338#117C22⋊D714yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—72no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—142no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4C22⋊D711yescharacteristic
359C2×C22⋊C4C22⋊D711yescharacteristic
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C22×D711yescharacteristic
363#103—144no—
364#103—144no—
365#103—144no—
366#103—282no—
367C22⋊C8—142no—
368C22⋊C8—282no—
369#95—141no—
370#95—142no—
371#95—142no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—24no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7C2×#12011yescharacteristic
383C22×C7#54612yes—
384C22×C7C2×#13711yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57812yes—
387C4×C7—22no—
388C4×C7—41no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—41no—
394C4×C7—42no—
395C4×C7#52511yescharacteristic
396C4×C7#53511yescharacteristic
397C4×C7#54612yes—
398Dic7—44no—
399Dic7—82no—
400C24C2×C22⋊D711yescharacteristic
401C24C2×C22⋊D711yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—74no—
406C2×Q8—148no—
407C2×Q8—142no—
408C2×Q8—142no—
409C2×Q8—142no—
410C2×Q8—142no—
411C2×Q8—142no—
412C2×Q8—284no—
413C2×Q8—288no—
414C2×Q8—284no—
415C2×Q8—282no—
416C2×Q8C2×C22⋊D712yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—41no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—284no—
436C22×C4—282no—
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D712yes—
442C22×C4C2×C22⋊D712yes—
443C22×C4C23×D711yescharacteristic
444Q16—288no—
445M16—284no—
446C2×C8—142no—
447C2×C8—142no—
448C2×C8—142no—
449C2×C8—142no—
450C2×C8—142no—
451C2×C8—142no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—142no—
463C4⋊C4—142no—
464C4⋊C4—284no—
465C4⋊C4C2×C22⋊D712yes—
466C4⋊C4C2×C22⋊D712yes—
467C4⋊C4C2×C22⋊D712yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—284no—
474C22⋊C4—282no—
475C42—24no—
476C42—24no—
477C42—21no—
478C42—21no—
479C2×C7—42no—
480C2×C7—42no—
481C2×C7#240411yescharacteristic
482C2×C7#300711yescharacteristic
483C2×C7#301111yescharacteristic
484C2×C7#331212yes—
485C2×C7#332712yes—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23#651412yes—
495C23#651612yes—
496C23C22×C22⋊D711yescharacteristic
497Q8—24no—
498Q8—41no—
499Q8—41no—
500Q8—144no—
501Q8—144no—
502Q8—288no—
503Q8—282no—
504Q8—282no—
505D4—44no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—144no—
532C2×C4—288no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4#651612yes—
539C2×C4D7×#4112yes—
540C2×C4#655312yes—
541C2×C4C22×C22⋊D711yescharacteristic
542C2×C4C22×C22⋊D711yescharacteristic
543C8—282no—
544C8—282no—
545C7#2398311yescharacteristic, complemented, Sylow
546C22—22no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—22no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22#9051311yescharacteristic
557C22C2×#651411yescharacteristic
558C22#9052012yes—
559C22#9052912yes—
560C22C2×#651611yescharacteristic
561C4—22no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—284no—
570C4—562no—
571C4#9051911yescharacteristic
572C4#9052012yes—
573C4#9052111yescharacteristic
574C2—42no—
575C2—42no—
576C2#1174104812yes—
577C2#1174105012yes—
578C2#1174152911yescharacteristic
579C2#1174154111yescharacteristic
580C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#552C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×#552D7
hypocenterC7#23983
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#89822C2semidirect
4. C2×#89776C2semidirect
5. #11729498C2semidirect
6. #11729454C2semidirect
7. #11729451C2semidirect
8. #11729386C2semidirect
10. #11729329C2semidirect
11. #11729313C2semidirect
545. C7#23983semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×#898222
#117397942
C2×#897762
#117294982
#117294542
#117294512
#117293862
#117293652
#117293292
#117293132
#239837