Finite Group Databasebeta

← #13387640

Subgroups — #13387640

Order 1792.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic
2C2×#89815C211yesmaximal, characteristic, complemented
3#11739819C211yesmaximal, characteristic
4C2×#89767C211yesmaximal, characteristic, complemented
5#11729493C212yesmaximal, complemented
6#11729451C211yesmaximal, characteristic, complemented
7#11729449C212yesmaximal, complemented
8#11729383C212yesmaximal, complemented
9#11729365C211yesmaximal, characteristic
10#11729329C211yesmaximal, characteristic, complemented
11#11729310C212yesmaximal, complemented
12C7×#552—24no—
13C7×#552—24no—
14C2×C7×#126—21no—
15C2×C7×#125—21no—
16C2×C7×#125C2211yescharacteristic
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117—21no—
23C2×C7×#117C2211yescharacteristic
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×C4⋊Dic7C2211yescharacteristic
26C22×#1136C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29#89992—21no—
30#89984C2212yes—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2212yes—
34#89815—24no—
35#89767—24no—
36#89758—21no—
37C2×#6382C2212yes—
38C2×#6382C2212yes—
39C2×#6381C2212yes—
40#89721—21no—
41#89718—21no—
42#89700C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24111—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C23×Dic7—21no—
54C7×#126—28no—
55C7×#125—28no—
56C7×#125—22no—
57C7×#125—22no—
58C7×#123—28no—
59C7×C22⋊Q8—22no—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8D414yes—
63C4×C7×Q8—22no—
64C4×C7×Q8—22no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C7×#117—28no—
68C7×#117—22no—
69C7×#117D414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7—21no—
88C2×C42×C7—21no—
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—21no—
91C2×C22⋊Dic7—22no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2312yes—
96C2×C4⋊Dic7C2312yes—
97C2×Dic7⋊C4—22no—
98C2×#1136—22no—
99C2×#1136—24no—
100C22×C7⋊C8C2312yes—
101C22×C7⋊C8C2312yes—
102#6409—24no—
103#6409—24no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106C7⋊#95—21no—
107C22⋊(C7⋊C8)—22no—
108C22⋊(C7⋊C8)—24no—
109#6382—24no—
110#6382—24no—
111#6381—24no—
112C2×#552D711yescharacteristic
113C2×#483—71no—
114C2×#428—71no—
115#2487—71no—
116#2059—72no—
117#2036—72no—
118#1894—71no—
119#1870—71no—
120#1865—72no—
121#1853—72no—
122#1825—71no—
123C24×C7C2×D411yescharacteristic
124C24×C7C2×D411yescharacteristic
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8C2×D412yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—41no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—24no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—42no—
155C22×Dic7—82no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D412yes—
168C7×C4⋊C4C2×D412yes—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—24no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—22no—
174C42×C7—24no—
175C42×C7—24no—
176C42×C7—21no—
177C42×C7—21no—
178C22⋊Dic7—42no—
179C22⋊Dic7—82no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7—22no—
184C4⋊Dic7—42no—
185Dic7⋊C4—82no—
186#1136—28no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8#4114yes—
193C22×M16—71no—
194#552—24no—
195#552—24no—
196C2×#126—21no—
197C2×#125—21no—
198C2×#125C2×D711yescharacteristic
199C2×#123—21no—
200C2×C22⋊Q8C2×D711yescharacteristic
201C2×C22⋊Q8C2×D711yescharacteristic
202C2×C4×Q8C2×D711yescharacteristic
203C2×C4×D4C2×D711yescharacteristic
204C2×#117—21no—
205C2×#117C2×D711yescharacteristic
206C22×C4⋊C4—71no—
207C22×C4⋊C4C2×D711yescharacteristic
208#483—144no—
209#428—144no—
210C2×#107—72no—
211C2×#106—72no—
212C2×#106—72no—
213C2×#103—72no—
214C2×#103—72no—
215C2×C22⋊C8—71no—
216C2×C22⋊C8—72no—
217#400—141no—
218#399—141no—
219#389—72no—
220#386—141no—
221#385—141no—
222#380—71no—
223#380—72no—
224#340—72no—
225#340—72no—
226#340—72no—
227#336—72no—
228C23×C7—21no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7#12012yes—
237C23×C7#13712yes—
238C23×C7C22×D411yescharacteristic
239C7×Q8—24no—
240C7×Q8—41no—
241C7×Q8—41no—
242C7×D4—44no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—21no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—48no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7#12012yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C2×#4112yes—
272C2×C4×C72−1+412yes—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—84no—
280C7⋊C8—24no—
281C7⋊C8—42no—
282C22×Q8C22×D711yescharacteristic
283C22×D4C22⋊D711yescharacteristic
284C23×C4—21no—
285C23×C4—141no—
286C23×C4C22×D711yescharacteristic
287C23×C4C22×D711yescharacteristic
288C2×M16—144no—
289C2×M16—142no—
290C22×C8—72no—
291C22×C8—72no—
292#126—28no—
293#125—28no—
294#125—22no—
295#125—22no—
296#123—28no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8C22⋊D714yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—22no—
304C4×D4—22no—
305#117—28no—
306#117—22no—
307#117C22⋊D714yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—72no—
317C2×C4⋊C4—72no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4C22⋊D711yescharacteristic
323C2×C4⋊C4C22×D711yescharacteristic
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C4⋊C4C22×D711yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—141no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4C22⋊D711yescharacteristic
332C2×C22⋊C4C22⋊D711yescharacteristic
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D711yescharacteristic
336#107—144no—
337#106—144no—
338#106—144no—
339#103—144no—
340#103—144no—
341C22⋊C8—144no—
342C22⋊C8—142no—
343#95—142no—
344#95—141no—
345#95—142no—
346C22×C7—24no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—41no—
355C22×C7—41no—
356C22×C7C2×#12011yescharacteristic
357C22×C7#54712yes—
358C22×C7C2×#13711yescharacteristic
359C22×C7#57511yescharacteristic
360C22×C7#57712yes—
361C4×C7—22no—
362C4×C7—41no—
363C4×C7—42no—
364C4×C7—41no—
365C4×C7—42no—
366C4×C7—42no—
367C4×C7—41no—
368C4×C7—42no—
369C4×C7#52511yescharacteristic
370C4×C7#53611yescharacteristic
371C4×C7#54712yes—
372Dic7—82no—
373Dic7—82no—
374C24C2×C22⋊D711yescharacteristic
375C24C2×C22⋊D711yescharacteristic
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×C22⋊D712yes—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—41no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—282no—
399C22×C4—562no—
400C22×C4C2×C22⋊D711yescharacteristic
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C2×C22⋊D711yescharacteristic
403C22×C4C2×C22⋊D711yescharacteristic
404C22×C4C2×C22⋊D712yes—
405C22×C4C2×C22⋊D712yes—
406C22×C4C23×D711yescharacteristic
407M16—148no—
408C2×C8—74no—
409C2×C8—142no—
410C2×C8—142no—
411C2×C8—142no—
412C2×C8—142no—
413C2×C8—142no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—142no—
425C4⋊C4—142no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4—282no—
429C4⋊C4—562no—
430C4⋊C4C2×C22⋊D712yes—
431C4⋊C4C2×C22⋊D712yes—
432C4⋊C4C2×C22⋊D712yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—282no—
439C22⋊C4—562no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C2×C7—42no—
445C2×C7—42no—
446C2×C7#240611yescharacteristic
447C2×C7#300711yescharacteristic
448C2×C7#301011yescharacteristic
449C2×C7#334412yes—
450C2×C7#335112yes—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23#651412yes—
460C23#651612yes—
461C23C22×C22⋊D711yescharacteristic
462Q8—24no—
463Q8—41no—
464Q8—41no—
465D4—44no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—48no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—564no—
498C2×C4#651612yes—
499C2×C4D7×#4112yes—
500C2×C4#655412yes—
501C2×C4C22×C22⋊D711yescharacteristic
502C2×C4C22×C22⋊D711yescharacteristic
503C8—144no—
504C8—282no—
505C7#2411111yescharacteristic, complemented, Sylow
506C22—22no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—22no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22#9051311yescharacteristic
517C22C2×#651411yescharacteristic
518C22#9051512yes—
519C22#9052812yes—
520C22C2×#651611yescharacteristic
521C4—22no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—562no—
530C4—562no—
531C4#9051512yes—
532C4#9051611yescharacteristic
533C4#9051911yescharacteristic
534C2—42no—
535C2—42no—
536C2#1174103812yes—
537C2#1174104112yes—
538C2#1174152911yescharacteristic
539C2#1174153511yescharacteristic
540C2#1174155811yescharacteristic
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#24111
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×#89815C2semidirect
4. C2×#89767C2semidirect
5. #11729493C2semidirect
6. #11729451C2semidirect
7. #11729449C2semidirect
8. #11729383C2semidirect
10. #11729329C2semidirect
11. #11729310C2semidirect
505. C7#24111semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×#898152
#117398192
C2×#897672
#117294932
#117294512
#117294492
#117293832
#117293652
#117293292
#117293102
#241117