Finite Group Databasebeta

← #13387738

Subgroups — #13387738

Order 1792.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic, complemented
3#11739821C211yesmaximal, characteristic, complemented
4C2×C4×#1142C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11729439C212yesmaximal, complemented
7#11729433C212yesmaximal, complemented
8#11729368C211yesmaximal, characteristic, complemented
9#11729358C211yesmaximal, characteristic, complemented
10#11729339C212yesmaximal, complemented
11#11729330C212yesmaximal, complemented
12C7×#555—24no—
13C7×#555—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#124—22no—
16C2×C7×#121—21no—
17C2×C7×#121C2212yes—
18C2×C4×C7×D4C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×C7×#117—21no—
21C2×C7×#117C2212yes—
22C22×#1142C2211yescharacteristic
23C2×#6487C2212yes—
24C2×C4×Dic14C2211yescharacteristic
25#89993—21no—
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89863—24no—
32C4×#1142—24no—
33#89745—21no—
34C2×#6384C2212yes—
35C2×#6384C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2212yes—
38#89720—22no—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24121—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×D4—22no—
48C22×C7×D4D411yescharacteristic
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7C2312yes—
52C22×Dic14C2311yescharacteristic
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yes—
55C7×#124—216no—
56C7×#121—28no—
57C7×#121—24no—
58C7×#121—24no—
59C4×C7×D4—22no—
60C4×C7×D4—22no—
61C4×C7×D4—22no—
62C4×C7×D4D414yes—
63C7×#117—28no—
64C7×#117—24no—
65C7×#117—24no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4D412yes—
74C2×C42×C7—22no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78C2×#1142—22no—
79C2×#1142—22no—
80C2×#1142—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2312yes—
85#6487—24no—
86#6487—42no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—22no—
89C2×C4×Dic7—21no—
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2312yes—
92C22×C7⋊C8C2311yescharacteristic
93C4×Dic14—22no—
94C4×Dic14—22no—
95#6406—22no—
96#6406—42no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6384—24no—
101#6384—22no—
102#6382—22no—
103#6381—24no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#555D711yescharacteristic
107C2×#477—71no—
108C2×C4×SD16—71no—
109#2538—71no—
110#2033—72no—
111#2000—72no—
112#1990—71no—
113#1954—71no—
114#1873—72no—
115#1854—72no—
116#1802—71no—
117C24×C7—21no—
118C24×C7C2×D412yes—
119C2×C7×Q8—24no—
120C2×C7×Q8—81no—
121C2×C7×Q8—81no—
122C2×C7×D4—28no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—82no—
132C2×C7×D4—82no—
133C2×C7×D4C2×D412yes—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14#4112yes—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4C2×D412yes—
163C7×C4⋊C4C2×D414yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—24no—
168C42×C7—28no—
169C42×C7C2×D412yes—
170C42×C7C2×D412yes—
171C22⋊Dic7—44no—
172C22⋊Dic7—82no—
173#1142—28no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—42no—
179C4⋊Dic7—81no—
180Dic7⋊C4—44no—
181Dic7⋊C4—42no—
182C4×Dic7—42no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8#4112yes—
192C22×SD16—71no—
193#555—24no—
194#555—24no—
195C2×C4⋊Q8C2×D711yescharacteristic
196C2×#124—22no—
197C2×C22⋊Q8—72no—
198C2×#121—21no—
199C2×#121C2×D712yes—
200C2×C4×Q8—71no—
201C2×C4×D4C2×D711yescharacteristic
202C2×C4×D4C2×D711yescharacteristic
203C2×#117—21no—
204C2×#117C2×D712yes—
205#477—144no—
206C4×SD16—144no—
207C2×#107—72no—
208C2×#106—71no—
209C2×C4⋊C8—71no—
210C2×#103—72no—
211C2×#103—71no—
212C2×#102—71no—
213C2×#102—72no—
214C2×C22⋊C8—72no—
215C2×C22⋊C8—72no—
216C2×C4×C8—71no—
217#396—142no—
218#390—141no—
219#389—141no—
220#384—71no—
221#380—72no—
222#340—72no—
223#340—72no—
224#340—72no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7C22×D411yescharacteristic
240C7×Q8—84no—
241C7×D4—22no—
242C7×D4—22no—
243C7×D4—41no—
244C7×D4—41no—
245C7×D4—44no—
246C7×D4—88no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—48no—
263C2×C4×C7#12012yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#13512yes—
266C2×C4×C7#13712yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7H(5,𝔽2)11yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C2×Dic7—22no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282Dic14—22no—
283Dic14—22no—
284Dic14—41no—
285Dic14—41no—
286C7⋊C8—22no—
287C7⋊C8—41no—
288C7⋊C8—42no—
289C22×Q8—21no—
290C22×Q8—71no—
291C22×D4—22no—
292C22×D4C22⋊D711yescharacteristic
293C22×D4C22×D711yescharacteristic
294C23×C4—21no—
295C23×C4C22×D712yes—
296C2×SD16—142no—
297C2×SD16—142no—
298C2×SD16—142no—
299C22×C8—71no—
300C22×C8—71no—
301C22×C8—72no—
302C4⋊Q8—22no—
303C4⋊Q8C22⋊D714yes—
304#124—216no—
305C22⋊Q8—144no—
306C22⋊Q8—282no—
307#121—28no—
308#121—24no—
309#121—24no—
310C4×Q8—142no—
311C4×Q8—142no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4C22⋊D714yes—
316#117—28no—
317#117—24no—
318#117—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—71no—
321C2×C4⋊C4—72no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4C22⋊D711yescharacteristic
327C2×C4⋊C4C22×D711yescharacteristic
328C2×C4⋊C4C22×D712yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—142no—
334C2×C22⋊C4C22⋊D712yes—
335C2×C42—22no—
336C2×C42—141no—
337C2×C42C22×D711yescharacteristic
338#107—144no—
339#106—142no—
340C4⋊C8—142no—
341#103—142no—
342#103—144no—
343#102—142no—
344#102—282no—
345C22⋊C8—144no—
346C22⋊C8—282no—
347C4×C8—142no—
348#95—142no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—81no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7C2×#12011yescharacteristic
364C22×C7#54611yescharacteristic
365C22×C7#54711yescharacteristic
366C22×C7C2×#13511yescharacteristic
367C22×C7C2×#13711yescharacteristic
368C22×C7#57611yescharacteristic
369C22×C7#57711yescharacteristic
370C4×C7—22no—
371C4×C7—42no—
372C4×C7—42no—
373C4×C7—44no—
374C4×C7#45214yes—
375C4×C7#52511yescharacteristic
376C4×C7#53711yescharacteristic
377C4×C7#54811yescharacteristic
378C4×C7#55011yescharacteristic
379Dic7—42no—
380Dic7—81no—
381Dic7—82no—
382C24—21no—
383C24C2×C22⋊D712yes—
384C2×Q8—24no—
385C2×Q8—72no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—141no—
389C2×Q8—141no—
390C2×Q8—141no—
391C2×Q8—141no—
392C2×Q8—141no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4C2×C22⋊D712yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—142no—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423SD16—148no—
424C2×C8—72no—
425C2×C8—141no—
426C2×C8—141no—
427C2×C8—142no—
428C2×C8—141no—
429C2×C8—142no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—142no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—284no—
442C4⋊C4—282no—
443C4⋊C4—282no—
444C4⋊C4—561no—
445C4⋊C4C2×C22⋊D712yes—
446C4⋊C4C2×C22⋊D714yes—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—284no—
452C22⋊C4—562no—
453C42—28no—
454C42—282no—
455C42C2×C22⋊D712yes—
456C42C2×C22⋊D712yes—
457C2×C7—82no—
458C2×C7—82no—
459C2×C7#240711yescharacteristic
460C2×C7C2×#45211yescharacteristic
461C2×C7#301211yescharacteristic
462C2×C7#330511yescharacteristic
463C2×C7#332911yescharacteristic
464C2×C7#334011yescharacteristic
465C2×C7#335711yescharacteristic
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C22×C22⋊D711yescharacteristic
481Q8—84no—
482Q8—142no—
483Q8—142no—
484Q8—281no—
485Q8—281no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—44no—
490D4—41no—
491D4—88no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—142no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—281no—
514C2×C4—282no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4#651312yes—
518C2×C4#651412yes—
519C2×C4#651612yes—
520C2×C4#651612yes—
521C2×C4D7×#4111yescharacteristic
522C2×C4D7×#4111yescharacteristic
523C2×C4#655311yescharacteristic
524C2×C4#655411yescharacteristic
525C2×C4C22×C22⋊D711yescharacteristic
526C2×C4C22×C22⋊D711yescharacteristic
527C8—142no—
528C8—281no—
529C8—282no—
530C7#2412111yescharacteristic, complemented, Sylow
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22C2×#651311yescharacteristic
546C22C2×#651411yescharacteristic
547C22#9051511yescharacteristic
548C22#9052011yescharacteristic
549C22#9052611yescharacteristic
550C22#9052711yescharacteristic
551C22C2×#651611yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—282no—
557C4—561no—
558C4—562no—
559C4#9001114yes—
560C4#9051911yescharacteristic
561C4#9052211yescharacteristic
562C4#9052411yescharacteristic
563C4#9052511yescharacteristic
564C2—82no—
565C2—82no—
566C2#1174109111yescharacteristic
567C2#1174109611yescharacteristic
568C2#1174110011yescharacteristic
569C2#1174110211yescharacteristic
570C2C2×#9001111yescharacteristic
571C2#1174154211yescharacteristic
572C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#555C2
socleC23×C7C22×D4
hypercenterC2×#555D7
hypocenterC7#24121
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. C2×#89863C2semidirect
3. #11739821C2semidirect
4. C2×C4×#1142C2semidirect
5. #11729541C2semidirect
6. #11729439C2semidirect
7. #11729433C2semidirect
8. #11729368C2semidirect
9. #11729358C2semidirect
10. #11729339C2semidirect
11. #11729330C2semidirect
530. C7#24121semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C2×#898632
#117398212
C2×C4×#11422
#117295412
#117294392
#117294332
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