Finite Group Databasebeta

← #13410051

Subgroups — #13410051

Order 1792.

1238 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×#89853C211yesmaximal, characteristic, complemented
3C2×#89827C211yesmaximal, characteristic, complemented
4#11739709C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal
6#11729886C212yesmaximal, complemented
7#11729540C211yesmaximal, characteristic, complemented
8#11729535C212yesmaximal, complemented
9#11729479C211yesmaximal, characteristic, complemented
10#11729476C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#536—24no—
15C7×#536—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C7×#124C2211yescharacteristic
20C2×C7×#123—21no—
21C2×C7×#123—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8—21no—
24C2×C7×#117C2211yescharacteristic
25C2×#6432C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33C2×#6406C2212yes—
34#89984C2212yes—
35C2×C22⋊(C7⋊C8)C2212yes—
36#89853—24no—
37#89827—24no—
38#89758—21no—
39#89751—21no—
40#89721—21no—
41#89719—21no—
42#89708C2211yescharacteristic
43#89700C2212yes—
44C2×#6377C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46#89319C2212yes—
47#25866—71nomaximal, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—21no—
55C22×C7×Q8—24no—
56C22×C7×Q8—42no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7—21no—
64C23×C4×C7C2311yescharacteristic
65C22×Dic14—21no—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8D414yes—
70C7×#124—22no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#124D414yes—
74C7×#123—28no—
75C7×#123—28no—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—28no—
78C7×#117—22no—
79C7×#117—22no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4D411yescharacteristic
91C2×C7×C22⋊C4D411yescharacteristic
92C2×C42×C7C2311yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C2×C22⋊Dic7—22no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2312yes—
98C2×Dic7⋊C4—22no—
99C22×C7⋊C8C2312yes—
100C22×C7⋊C8C2312yes—
101#6432—22no—
102#6432—22no—
103#6409—24no—
104#6409—24no—
105#6409—24no—
106#6409—42no—
107#6409—42no—
108#6409—42no—
109#6406—24no—
110#6406—42no—
111C7⋊#95—22no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—22no—
115C22⋊(C7⋊C8)—24no—
116#6377—22no—
117C4×C7⋊C8—22no—
118C2×#536D711yescharacteristic
119C2×#489—71no—
120C2×#487—71no—
121#2628—71no—
122#2043—72no—
123#2042—72no—
124#1960—71no—
125#1958—71no—
126#1788—72no—
127#1787—72no—
128#1750—71no—
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C7×#41—48no—
132C7×#41—48no—
133C2×C7×Q8—24no—
134C2×C7×Q8—21no—
135C2×C7×Q8—21no—
136C2×C7×Q8—21no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—48no—
144C2×C7×Q8—48no—
145C2×C7×Q8—42no—
146C2×C7×Q8—42no—
147C2×C7×Q8C2×D412yes—
148C2×C7×Q8C2×D412yes—
149C2×C7×Q8C2×D412yes—
150C2×C7×D4—24no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—42no—
155C2×C7×D4—42no—
156C2×C7×D4C2×D412yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—42no—
166C22×C4×C7—42no—
167C22×C4×C7—42no—
168C22×C4×C7—42no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7#4114yes—
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—22no—
177C22×Dic7—22no—
178C2×Dic14—42no—
179C2×Dic14—82no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7C2×D412yes—
195C42×C7C2×D412yes—
196C22⋊Dic7—82no—
197C4⋊Dic7—22no—
198C4⋊Dic7—22no—
199C4⋊Dic7—22no—
200C4⋊Dic7—22no—
201C4⋊Dic7—42no—
202Dic7⋊C4—82no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C22×#41C2×D711yescharacteristic
210C23×Q8C2×D711yescharacteristic
211#536—24no—
212#536—24no—
213C2×C4⋊Q8—71no—
214C2×C4⋊Q8C2×D711yescharacteristic
215C2×C4⋊Q8C2×D711yescharacteristic
216C2×#124C2×D711yescharacteristic
217C2×#124C2×D711yescharacteristic
218C2×#123—21no—
219C2×#123—21no—
220C2×C22⋊Q8—21no—
221C2×C22⋊Q8—21no—
222C2×#117C2×D711yescharacteristic
223#489—144no—
224#487—144no—
225C2×#103—72no—
226C2×#103—72no—
227C2×#103—72no—
228C2×#103—72no—
229C2×#103—72no—
230C2×#103—72no—
231C2×#102—72no—
232C2×#102—72no—
233C2×C22⋊C8—72no—
234C2×#97—71no—
235C2×C4×C8—71no—
236#400—141no—
237#399—141no—
238#395—141no—
239#393—141no—
240#389—72no—
241#382—71no—
242#380—72no—
243#336—72no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—22no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7#12412yes—
254C23×C7C22×D411yescharacteristic
255C7×Q8—24no—
256C7×Q8—24no—
257C7×Q8—24no—
258C7×Q8—48no—
259C7×Q8—42no—
260C7×Q8—42no—
261C7×Q8—42no—
262C7×D4—24no—
263C7×D4—48no—
264C7×D4—42no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—48no—
279C2×C4×C7—48no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7#12012yes—
287C2×C4×C7#12012yes—
288C2×C4×C7#12412yes—
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C2×#4112yes—
292C2×C4×C7H(5,𝔽2)12yes—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299Dic14—84no—
300C7⋊C8—42no—
301C7⋊C8—42no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—24no—
305C2×#41—42no—
306C22×Q8—22no—
307C22×Q8—22no—
308C22×Q8—21no—
309C22×Q8—24no—
310C22×Q8—42no—
311C22×Q8—141no—
312C22×Q8C22×D711yescharacteristic
313C22×Q8C22×D711yescharacteristic
314C22×Q8C22×D711yescharacteristic
315C22×D4—21no—
316C22×D4C22×D711yescharacteristic
317C23×C4—21no—
318C23×C4—21no—
319C23×C4C22×D711yescharacteristic
320C22×C8—72no—
321C22×C8—72no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—142no—
326C4⋊Q8—142no—
327C4⋊Q8C22⋊D714yes—
328#124—22no—
329#124—22no—
330#124—22no—
331#124C22⋊D714yes—
332#123—28no—
333#123—28no—
334C22⋊Q8—28no—
335C22⋊Q8—28no—
336#117—22no—
337#117—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—72no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4C22⋊D711yescharacteristic
347C2×C4⋊C4C22⋊D711yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—142no—
353C2×C22⋊C4C22⋊D711yescharacteristic
354C2×C22⋊C4C22⋊D711yescharacteristic
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357#103—144no—
358#103—144no—
359#103—144no—
360#103—282no—
361#103—282no—
362#103—282no—
363#102—144no—
364#102—282no—
365C22⋊C8—144no—
366#97—142no—
367C4×C8—142no—
368#95—142no—
369#95—141no—
370#95—141no—
371#95—142no—
372C22×C7—24no—
373C22×C7—22no—
374C22×C7—42no—
375C22×C7—41no—
376C22×C7—41no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7C2×#12011yescharacteristic
383C22×C7C2×#12411yescharacteristic
384C22×C7#53511yescharacteristic
385C22×C7#57712yes—
386C22×C7#57812yes—
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—41no—
394C4×C7—41no—
395C4×C7#54612yes—
396C4×C7#54612yes—
397Dic7—82no—
398Dic7—82no—
399C24—21no—
400C24C2×C22⋊D711yescharacteristic
401#41—48no—
402#41—48no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—282no—
418C2×Q8—562no—
419C2×Q8C2×C22⋊D712yes—
420C2×Q8C2×C22⋊D712yes—
421C2×Q8C2×C22⋊D712yes—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×C22⋊D712yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4#115914yes—
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C23×D711yescharacteristic
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—142no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—142no—
463C4⋊C4—142no—
464C4⋊C4—142no—
465C4⋊C4—142no—
466C4⋊C4—282no—
467C4⋊C4—562no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—562no—
475C42—21no—
476C42—21no—
477C42C2×C22⋊D712yes—
478C42C2×C22⋊D712yes—
479C2×C7—42no—
480C2×C7—82no—
481C2×C7#243511yescharacteristic
482C2×C7#301011yescharacteristic
483C2×C7#301111yescharacteristic
484C2×C7#313712yes—
485C2×C7#322512yes—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23#649912yes—
496C23C22×C22⋊D711yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—24no—
500Q8—48no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—564no—
505D4—24no—
506D4—48no—
507D4—42no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4#649912yes—
536C2×C4#651612yes—
537C2×C4#651612yes—
538C2×C4C2×#115912yes—
539C2×C4#654812yes—
540C2×C4C22×C22⋊D711yescharacteristic
541C2×C4C22×C22⋊D711yescharacteristic
542C8—282no—
543C8—282no—
544C7#2586611yescharacteristic, complemented, Sylow
545C22—22no—
546C22—24no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22C2×#649911yescharacteristic
556C22#9049611yescharacteristic
557C22#9052812yes—
558C22#9052912yes—
559C22C2×#651611yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—562no—
569C4—562no—
570C4#9049712yes—
571C4#9049712yes—
572C2—42no—
573C2—82no—
574C2#1174084612yes—
575C2#1174085512yes—
576C2#1174149511yescharacteristic
577C2#1174155811yescharacteristic
578C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×#536D7
hypocenterC7#25866
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#89853C2semidirect
3. C2×#89827C2semidirect
4. #11739709C2semidirect
6. #11729886C2semidirect
7. #11729540C2semidirect
8. #11729535C2semidirect
9. #11729479C2semidirect
10. #11729476C2semidirect
11. #11729336C2semidirect
544. C7#25866semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×#898532
C2×#898272
#117397092
#117298892
#117298862
#117295402
#117295352
#117294792
#117294762
#117293362
#258667