Finite Group Databasebeta

← #13387705

Subgroups — #13387705

Order 1792.

1218 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×#554C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic, complemented
3C2×C4×#1143C211yesmaximal, characteristic, complemented
4#11739862C211yesmaximal, characteristic
5#11729541C211yesmaximal, characteristic, complemented
6#11729453C212yesmaximal, complemented
7#11729448C212yesmaximal, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11729368C211yesmaximal, characteristic, complemented
10#11729314C212yesmaximal, complemented
11#11729310C212yesmaximal, complemented
12C7×#554—24no—
13C7×#554—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—22no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C22×#1143C2211yescharacteristic
23C2×#6501—21no—
24C2×#6493C2212yes—
25C2×C4×D28C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89856—24no—
32C4×#1143—24no—
33#89759—21no—
34C2×#6383C2212yes—
35C2×#6383C2211yescharacteristic
36C2×#6382C2212yes—
37C2×#6382C2211yescharacteristic
38#89733—22no—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24118—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—22no—
50C23×C4×C7C2312yes—
51C22×C22⋊D7—22no—
52C22×D28C2311yescharacteristic
53C22×C4×D7—21no—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8D414yescomplemented
56C7×#125—28no—
57C7×C22⋊Q8—216no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8D414yescomplemented
62C4×C7×D4—28no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D412yes—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—22no—
79C2×#1143—22no—
80C2×#1143—22no—
81C2×#1143—22no—
82#6501—44no—
83#6493—24no—
84#6493—42no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2311yescharacteristic
94C4×D28—22no—
95C4×D28—22no—
96#6409—22no—
97#6409—42no—
98C7⋊#95—22no—
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)—42no—
101#6383—24no—
102#6383—22no—
103#6382—24no—
104#6382—22no—
105C4⋊(C7⋊C8)—22no—
106C4×C7⋊C8—22no—
107C2×#554D711yescharacteristic, complemented
108C2×#476—71no—
109C2×C4×SD16—71no—
110#2537—71no—
111#2035—72no—
112#2000—72no—
113#1990—71no—
114#1954—71no—
115#1874—72no—
116#1853—72no—
117#1803—71no—
118C24×C7C2×D412yes—
119C23×D7—21no—
120C2×C7×Q8—24no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—81no—
127C2×C7×Q8—81no—
128C2×C7×Q8C2×D412yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4SD1614yescomplemented
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—28no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—82no—
143C22×C4×C7—82no—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C2×C22⋊D7—44no—
149C2×C22⋊D7—44no—
150C2×C22⋊D7—82no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28#4112yes—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×C4×D7—42no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4C2×D412yes—
170C7×C4⋊C4C2×D414yes—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—24no—
173C42×C7—24no—
174C42×C7C2×D412yes—
175C42×C7C2×D412yes—
176C22⋊Dic7—44no—
177#1143—28no—
178(C2×D7)⋊C4—44no—
179(C2×D7)⋊C4—42no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—22no—
183C4⋊Dic7—22no—
184C4⋊Dic7—42no—
185C4⋊Dic7—81no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8#4112yes—
195C22×SD16—71no—
196#554—24no—
197#554—24no—
198C2×C4⋊Q8C2×D711yescharacteristic
199C2×#125—21no—
200C2×C22⋊Q8—22no—
201C2×C22⋊Q8C2×D712yes—
202C2×#121—72no—
203C2×#121—141no—
204C2×C4×Q8C2×D711yescharacteristic
205C2×C4×D4—21no—
206C2×C4×D4—71no—
207C2×C4×D4C2×D711yescharacteristic
208C22×C4⋊C4C2×D712yes—
209#476—144no—
210C4×SD16—144no—
211C2×#106—71no—
212C2×#106—72no—
213C2×C4⋊C8—71no—
214C2×#103—71no—
215C2×#103—72no—
216C2×#102—71no—
217C2×#102—72no—
218C2×C22⋊C8—72no—
219C2×C22⋊C8—72no—
220C2×C4×C8—71no—
221#394—142no—
222#384—71no—
223#380—72no—
224#380—141no—
225#340—72no—
226#340—72no—
227#340—72no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7C2×SD1614yes—
235C23×C7C22×D411yescharacteristic
236C22×D7—22no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7—41no—
243C7×Q8—22no—
244C7×Q8—22no—
245C7×Q8—41no—
246C7×Q8—41no—
247C7×Q8—84no—
248C7×D4—28no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—88no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7C2×SD1612yes—
271C2×C4×C7#13712yes—
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C72−1+411yescharacteristic
278C22⋊D7—88no—
279C2×Dic7—41no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—42no—
285D28—22no—
286D28—22no—
287D28—41no—
288D28—41no—
289C4×D7—84no—
290C7⋊C8—22no—
291C7⋊C8—41no—
292C7⋊C8—42no—
293C22×Q8—21no—
294C22×Q8C22×D711yescharacteristic
295C22×D4—71no—
296C22×D4—142no—
297C22×D4C22⋊D711yescharacteristic
298C23×C4—22no—
299C23×C4—141no—
300C23×C4C22×D712yes—
301C2×SD16—142no—
302C2×SD16—142no—
303C2×SD16—142no—
304C22×C8—71no—
305C22×C8—71no—
306C22×C8—72no—
307C4⋊Q8—22no—
308C4⋊Q8C22⋊D714yescomplemented
309#125—28no—
310C22⋊Q8—216no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313#121—144no—
314#121—282no—
315#121—284no—
316C4×Q8—22no—
317C4×Q8C22⋊D714yescomplemented
318C4×D4—28no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—142no—
322C4×D4—142no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—72no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4C22⋊D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D712yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—141no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4C22⋊D712yes—
341C2×C42—21no—
342C2×C42C22×D711yescharacteristic
343#106—142no—
344#106—144no—
345C4⋊C8—142no—
346#103—142no—
347#103—282no—
348#102—142no—
349#102—144no—
350C22⋊C8—144no—
351C22⋊C8—282no—
352C4×C8—142no—
353#95—142no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7C2×#12011yescharacteristic
362C22×C7#54611yescharacteristic
363C22×C7#54711yescharacteristic
364C22×C7C22×SD1611yescharacteristic
365C22×C7C2×#13711yescharacteristic
366C22×C7#57611yescharacteristic
367C22×C7#57711yescharacteristic
368C2×D7—42no—
369C2×D7—42no—
370C2×D7—42no—
371C2×D7—81no—
372C2×D7—81no—
373C2×D7—81no—
374C2×D7—81no—
375C4×C7—22no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—42no—
379C4×C7—44no—
380C4×C7#44814yes—
381C4×C7#52511yescharacteristic
382C4×C7#53711yescharacteristic
383C4×C7#54811yescharacteristic
384C4×C7#55011yescharacteristic
385Dic7—81no—
386Dic7—82no—
387C24—141no—
388C24C2×C22⋊D712yes—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8C2×C22⋊D712yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—72no—
401C2×D4—141no—
402C2×D4—141no—
403C2×D4—141no—
404C2×D4—141no—
405C2×D4—141no—
406C2×D4—284no—
407C2×D4—284no—
408C2×D4—562no—
409C2×D4#114314yescomplemented
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—141no—
423C22×C4—142no—
424C22×C4—282no—
425C22×C4—282no—
426C22×C4—282no—
427C22×C4C2×C22⋊D712yes—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431SD16—148no—
432C2×C8—72no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—142no—
436C2×C8—141no—
437C2×C8—142no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—142no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—142no—
449C4⋊C4—142no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—282no—
453C4⋊C4—561no—
454C4⋊C4C2×C22⋊D712yes—
455C4⋊C4C2×C22⋊D714yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—284no—
459C22⋊C4—284no—
460C22⋊C4—282no—
461C42—24no—
462C42C2×C22⋊D712yes—
463C42C2×C22⋊D712yes—
464C2×C7—44no—
465C2×C7#240711yescharacteristic
466C2×C7C2×#44811yescharacteristic
467C2×C7#301211yescharacteristic
468C2×C7#329511yescharacteristic
469C2×C7#333111yescharacteristic
470C2×C7#333511yescharacteristic
471C2×C7#335711yescharacteristic
472D7—82no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—142no—
480C23—281no—
481C23—281no—
482C23—281no—
483C23—281no—
484C23—281no—
485C23—281no—
486C23C2×#114314yes—
487C23C22×C22⋊D711yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492Q8—84no—
493D4—28no—
494D4—142no—
495D4—142no—
496D4—281no—
497D4—281no—
498D4—568no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—88no—
518C2×C4—281no—
519C2×C4—282no—
520C2×C4—281no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—281no—
524C2×C4—564no—
525C2×C4C2×#114312yes—
526C2×C4#650512yes—
527C2×C4#651612yes—
528C2×C4#651612yes—
529C2×C4D7×#4111yescharacteristic
530C2×C4D7×#4111yescharacteristic
531C2×C4#655311yescharacteristic
532C2×C4#655411yescharacteristic
533C2×C4C22×C22⋊D711yescharacteristic
534C2×C4C22×C22⋊D711yescharacteristic
535C8—142no—
536C8—281no—
537C8—282no—
538C7#2411811yescharacteristic, complemented, Sylow
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—282no—
547C22—282no—
548C22—282no—
549C22—561no—
550C22—561no—
551C22—561no—
552C22—561no—
553C22C22×#114311yescharacteristic
554C22C2×#650511yescharacteristic
555C22#9051511yescharacteristic
556C22#9052011yescharacteristic
557C22#9052611yescharacteristic
558C22#9052711yescharacteristic
559C22C2×#651611yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—561no—
566C4—562no—
567C4#8999914yes—
568C4#9051911yescharacteristic
569C4#9052211yescharacteristic
570C4#9052411yescharacteristic
571C4#9052511yescharacteristic
572C2—44no—
573C2—562no—
574C2#1174106711yescharacteristic
575C2#1174107211yescharacteristic
576C2#1174107911yescharacteristic
577C2#1174108111yescharacteristic
578C2C2×#8999911yescharacteristic
579C2#1174154211yescharacteristic
580C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radical#11739862C2
hypercenterC2×#554D7
hypocenterC7#24118
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. C2×#89856C2semidirect
3. C2×C4×#1143C2semidirect
5. #11729541C2semidirect
6. #11729453C2semidirect
7. #11729448C2semidirect
8. #11729380C2semidirect
9. #11729368C2semidirect
10. #11729314C2semidirect
11. #11729310C2semidirect
55. C7×C4⋊Q8D4semidirect
61. C4×C7×Q8D4semidirect
107. C2×#554D7semidirect
131. C2×C7×D4SD16semidirect
308. C4⋊Q8C22⋊D7semidirect
317. C4×Q8C22⋊D7semidirect
409. C2×D4#1143semidirect
538. C7#24118semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898562
C2×C4×#11432
#117398622
#117295412
#117294532
#117294482
#117293802
#117293682
#117293142
#117293102
#241187