Finite Group Databasebeta

← #13410312

Subgroups — #13410312

Order 1792.

1344 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2C2×#89843C211yesmaximal, characteristic, complemented
3C2×#89794C211yesmaximal, characteristic, complemented
4#11739828C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729889C212yesmaximal, complemented
7#11729515C211yesmaximal, characteristic, complemented
8#11729434C212yesmaximal, complemented
9#11729433C212yesmaximal, complemented
10#11729385C211yesmaximal, characteristic, complemented
11#11729369C211yesmaximal, characteristic, complemented
12C22×C7×#41C2212yes—
13C7×#550—24no—
14C7×#550—24no—
15C2×C7×#127—21no—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—22no—
18C2×C7×#121C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6433C2211yescharacteristic
23C2×#6409C2211yescharacteristic
24C2×#6409C2211yescharacteristic
25C2×#6409C2212yes—
26#89994—21no—
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89984C2212yes—
32#89984C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89843—24no—
36#89794—24no—
37#89754—21no—
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89717—22no—
41C4×C4⋊Dic7C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#25893—71nomaximal, Sylow
46C2×C7×#41—24no—
47C2×C7×#41—24no—
48C2×C7×#41—24no—
49C2×C7×#41—24no—
50C2×C7×#41—42no—
51C2×C7×#41—42no—
52C22×C7×Q8C2311yescharacteristic
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2312yes—
59C23×C4×C7—22no—
60C23×C4×C7C2312yes—
61C7×#127—28no—
62C7×#127—22no—
63C7×#127D414yes—
64C7×#121—216no—
65C7×#121—24no—
66C7×#121—24no—
67C4×C7×Q8—22no—
68C4×C7×Q8D414yes—
69C4×C7×D4—28no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4D412yes—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7—22no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90#6433—22no—
91#6433—22no—
92#6409—22no—
93#6409—22no—
94#6409—24no—
95#6406—24no—
96#6406—24no—
97#6406—42no—
98#6406—42no—
99C7⋊#95—21no—
100C7⋊#95—22no—
101C7⋊#95—22no—
102C22⋊(C7⋊C8)—42no—
103C22⋊(C7⋊C8)—42no—
104#6382—22no—
105#6381—22no—
106C4⋊(C7⋊C8)—22no—
107#6377—22no—
108C2×#550D711yescharacteristic
109C2×#490—71no—
110C2×#479—71no—
111#2562—71no—
112#2034—72no—
113#2033—72no—
114#1982—71no—
115#1955—71no—
116#1788—72no—
117#1788—72no—
118#1763—71no—
119C24×C7—22no—
120C24×C7C2×D412yes—
121C7×#41—48no—
122C7×#41—48no—
123C7×#41—48no—
124C7×#41—48no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8C2×D412yes—
128C2×C7×Q8#4114yes—
129C2×C7×D4—28no—
130C2×C7×D4—28no—
131C2×C7×D4—22no—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—42no—
140C2×C7×D4—44no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—81no—
144C2×C7×D4—81no—
145C2×C7×D4—82no—
146C2×C7×D4—82no—
147C2×C7×D4C2×D414yes—
148C22×C4×C7—21no—
149C22×C4×C7—28no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—44no—
158C22×C4×C7—44no—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4C2×D412yes—
170C7×C22⋊C4—28no—
171C7×C22⋊C4—24no—
172C42×C7—24no—
173C42×C7C2×D412yes—
174C42×C7C2×D412yes—
175C22⋊Dic7—44no—
176C22⋊Dic7—82no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183Dic7⋊C4—42no—
184Dic7⋊C4—42no—
185C4×Dic7—81no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C22×#41C2×D712yes—
194#550—24no—
195#550—24no—
196C2×#127—21no—
197C2×#127C2×D711yescharacteristic
198C2×#125—71no—
199C2×#121—22no—
200C2×#121C2×D712yes—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—21no—
203C2×C4×D4C2×D711yescharacteristic
204#490—144no—
205#479—144no—
206C2×#107—71no—
207C2×#106—71no—
208C2×C4⋊C8—71no—
209C2×#103—71no—
210C2×#103—71no—
211C2×#103—72no—
212C2×#102—72no—
213C2×#102—72no—
214C2×#102—72no—
215C2×#102—72no—
216C2×C22⋊C8—72no—
217C2×C22⋊C8—72no—
218C2×#97—71no—
219#400—141no—
220#394—141no—
221#388—142no—
222#380—72no—
223#380—72no—
224C4×C4⋊C4—71no—
225#340—72no—
226C23×C7—22no—
227C23×C7—24no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7C22×D411yescharacteristic
241C7×Q8—22no—
242C7×Q8—24no—
243C7×Q8#12414yes—
244C7×D4—24no—
245C7×D4—24no—
246C7×D4—44no—
247C7×D4—48no—
248C7×D4—416no—
249C7×D4—42no—
250C7×D4—42no—
251C7×D4—84no—
252C7×D4—88no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—416no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12012yes—
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4112yes—
274C2×C4×C7H(5,𝔽2)12yes—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C7⋊C8—41no—
284C7⋊C8—41no—
285C7⋊C8—42no—
286C2×#41—24no—
287C2×#41—24no—
288C2×#41—24no—
289C2×#41—24no—
290C2×#41—42no—
291C2×#41—42no—
292C22×Q8C22×D711yescharacteristic
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4C22⋊D711yescharacteristic
298C22×D4C22×D712yes—
299C23×C4—22no—
300C23×C4C22×D712yes—
301C22×C8—72no—
302C22×C8—71no—
303C22×C8—71no—
304#127—28no—
305#127—22no—
306#127C22⋊D714yes—
307#125—142no—
308#125—142no—
309#121—216no—
310#121—24no—
311#121—24no—
312C4×Q8—22no—
313C4×Q8C22⋊D714yes—
314C4×D4—28no—
315C4×D4—22no—
316C4×D4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4C22⋊D711yescharacteristic
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C22⋊C4C22⋊D712yes—
329C2×C42—21no—
330C2×C42—141no—
331C2×C42C22×D711yescharacteristic
332#107—142no—
333#106—142no—
334C4⋊C8—142no—
335#103—144no—
336#103—142no—
337#103—142no—
338#102—144no—
339#102—144no—
340#102—282no—
341#102—282no—
342C22⋊C8—282no—
343C22⋊C8—282no—
344#97—142no—
345#95—141no—
346#95—142no—
347#95—142no—
348C22×C7—44no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—44no—
352C22×C7—42no—
353C22×C7—44no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—82no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7—82no—
362C22×C7C2×#12011yescharacteristic
363C22×C7#54612yes—
364C22×C7#57512yes—
365C22×C7#57611yescharacteristic
366C22×C7#57711yescharacteristic
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—22no—
370C4×C7—42no—
371C4×C7—42no—
372C4×C7C2×#12411yescharacteristic
373C4×C7#53511yescharacteristic
374C4×C7#54612yes—
375Dic7—81no—
376Dic7—81no—
377Dic7—82no—
378C24—22no—
379C24C2×C22⋊D712yes—
380#41—48no—
381#41—48no—
382#41—48no—
383#41—48no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8#115914yes—
387C2×Q8C2×C22⋊D712yes—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4C2×C22⋊D714yes—
407C22×C4—21no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—142no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4C2×C22⋊D712yes—
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C23×D711yescharacteristic
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8—141no—
431C2×C8—141no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—141no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—282no—
442C4⋊C4—282no—
443C4⋊C4C2×C22⋊D712yes—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—284no—
447C22⋊C4—562no—
448C42—24no—
449C42—561no—
450C42C2×C22⋊D712yes—
451C42C2×C22⋊D712yes—
452C2×C7—82no—
453C2×C7—84no—
454C2×C7#243511yescharacteristic
455C2×C7#300511yescharacteristic
456C2×C7#301211yescharacteristic
457C2×C7#330912yes—
458C2×C7#331712yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C22×C22⋊D711yescharacteristic
474Q8—22no—
475Q8—24no—
476Q8#649914yes—
477D4—24no—
478D4—24no—
479D4—44no—
480D4—48no—
481D4—416no—
482D4—42no—
483D4—42no—
484D4—84no—
485D4—88no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—416no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4#651612yes—
511C2×C4#651612yes—
512C2×C4C2×#115912yes—
513C2×C4#654812yes—
514C2×C4C22×C22⋊D711yescharacteristic
515C2×C4C22×C22⋊D711yescharacteristic
516C8—281no—
517C8—281no—
518C8—282no—
519C7#2589311yescharacteristic, complemented, Sylow
520C22—44no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22#9049712yes—
535C22#9051312yes—
536C22#9052611yescharacteristic
537C22#9052711yescharacteristic
538C22C2×#651611yescharacteristic
539C4—24no—
540C4—22no—
541C4—22no—
542C4—42no—
543C4—42no—
544C4—561no—
545C4—562no—
546C4—561no—
547C4C2×#649911yescharacteristic
548C4#9049611yescharacteristic
549C4#9049712yes—
550C2—82no—
551C2—84no—
552C2#1174101712yes—
553C2#1174101812yes—
554C2#1174149511yescharacteristic
555C2#1174152611yescharacteristic
556C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#550C2
socleC23×C7C22×D4
hypercenterC2×#550D7
hypocenterC7#25893
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#89843C2semidirect
3. C2×#89794C2semidirect
4. #11739828C2semidirect
5. #11729889C2semidirect
6. #11729889C2semidirect
7. #11729515C2semidirect
8. #11729434C2semidirect
9. #11729433C2semidirect
10. #11729385C2semidirect
11. #11729369C2semidirect
519. C7#25893semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×#898432
C2×#897942
#117398282
#117298892
#117295152
#117294342
#117294332
#117293852
#117293692
#258937