Finite Group Databasebeta

← #13387707

Subgroups — #13387707

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×#90234C211yesmaximal, characteristic, complemented
2C7×#2537C211yesmaximal, characteristic
3C2×C4×#1142C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729439C212yesmaximal, complemented
6#11729431C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic, complemented
8#11727342C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727280C212yesmaximal, complemented
11#11727259C212yesmaximal, complemented
12C2×C7×#121—21no—
13C2×C7×#121C2212yes—
14C2×C4×C7×D4C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×D4×Dic7—21no—
17C22×#1142C2211yescharacteristic
18C22×C4⋊Dic7C2212yes—
19C2×#6487C2212yes—
20#90234—24no—
21#90234—24no—
22C2×#6453—21no—
23C2×#6441—22no—
24C2×#6432C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C7×#394—22no—
27C7×#384C2211yescharacteristic
28C7×#380—21no—
29C7×#380C2212yes—
30C2×#6406C2211yescharacteristic
31C2×#6406C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C4×#1142—24no—
35#89780—24no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2212yes—
38C2×#6382C2212yes—
39C2×#6382C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24118—71nomaximal, Sylow
46C22×C7×D4—22no—
47C22×C7×D4D411yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7C2312yes—
51C23×Dic7—22no—
52C22×Dic14—21no—
53C22×Dic14C2311yescharacteristic
54C7×#121—24no—
55C7×#121—42no—
56C7×#121—44no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C4×C7×D4—22no—
60C4×C7×D4—22no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2312yes—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4D412yes—
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—22no—
72D4×Dic7—28no—
73C2×#1142—22no—
74C2×#1142—22no—
75C2×#1142—22no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—24no—
79C2×C4⋊Dic7—24no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2312yes—
82#6487—24no—
83#6487—24no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—22no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2311yescharacteristic
90#6453—28no—
91#6441—216no—
92#6432—22no—
93#6432D414yescomplemented
94C4×Dic14—22no—
95C4×Dic14D414yescomplemented
96C7×#95—22no—
97#6406—24no—
98#6406—22no—
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)—42no—
101#6384—22no—
102#6384—42no—
103#6382—24no—
104#6382—22no—
105C4⋊(C7⋊C8)—22no—
106C4×C7⋊C8—22no—
107C2×#554—71no—
108C2×#476—71no—
109C2×C4×SD16—71no—
110#2537D711yescharacteristic
111#2035—72no—
112#2000—72no—
113#1990—71no—
114#1954—71no—
115#1874—72no—
116#1853—72no—
117#1803—71no—
118C24×C7—21no—
119C24×C7C2×D412yes—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—82no—
130C2×C7×D4SD1614yescomplemented
131C2×C7×D4#4112yes—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—28no—
151C22×Dic7—82no—
152C22×Dic7—82no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14—21no—
158C2×Dic14—24no—
159C2×Dic14—81no—
160C2×Dic14—81no—
161C2×Dic14C2×D412yes—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—42no—
168C7×C4⋊C4—81no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—44no—
171C7×C22⋊C4—44no—
172C7×C22⋊C4—42no—
173C42×C7C2×D412yes—
174C42×C7C2×D412yes—
175C22⋊Dic7—28no—
176#1142—28no—
177C4⋊Dic7—28no—
178C4⋊Dic7—24no—
179C4⋊Dic7—21no—
180C4⋊Dic7—22no—
181C4⋊Dic7C2×D412yes—
182C4⋊Dic7C2×D414yes—
183Dic7⋊C4—28no—
184Dic7⋊C4—24no—
185C4×Dic7—24no—
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—144no—
197#554—144no—
198C2×C4⋊Q8—71no—
199C2×#125—141no—
200C2×C22⋊Q8—72no—
201C2×C22⋊Q8—142no—
202C2×#121—21no—
203C2×#121C2×D712yes—
204C2×C4×Q8—71no—
205C2×C4×D4—141no—
206C2×C4×D4C2×D711yescharacteristic
207C2×C4×D4C2×D711yescharacteristic
208C22×C4⋊C4—72no—
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—22no—
222#384C2×D711yescharacteristic
223#380—21no—
224#380C2×D712yes—
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×C7—22no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7C2×SD1614yes—
242C23×C7C22×D411yescharacteristic
243C7×D4—22no—
244C7×D4—22no—
245C7×D4—28no—
246C7×D4—41no—
247C7×D4—41no—
248C7×D4—88no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—84no—
265C2×C4×C7#12012yes—
266C2×C4×C7#12012yes—
267C2×C4×C7C2×SD1612yes—
268C2×C4×C7#13712yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C72−1+411yescharacteristic
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—42no—
281C2×Dic7—41no—
282C2×Dic7—42no—
283C2×Dic7—41no—
284C2×Dic7—88no—
285Dic14—22no—
286Dic14—22no—
287Dic14—41no—
288Dic14—41no—
289Dic14—84no—
290C7⋊C8—22no—
291C7⋊C8—41no—
292C7⋊C8—42no—
293C22×Q8—71no—
294C22×Q8—141no—
295C22×D4—22no—
296C22×D4C22⋊D711yescharacteristic
297C22×D4C22×D711yescharacteristic
298C23×C4—21no—
299C23×C4—142no—
300C23×C4C22×D712yes—
301C2×SD16—142no—
302C2×SD16—142no—
303C2×SD16—142no—
304C22×C8—71no—
305C22×C8—71no—
306C22×C8—72no—
307C4⋊Q8—74no—
308C4⋊Q8—142no—
309#125—148no—
310C22⋊Q8—1416no—
311C22⋊Q8—144no—
312C22⋊Q8—144no—
313#121—24no—
314#121—42no—
315#121—44no—
316C4×Q8—74no—
317C4×Q8—142no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—148no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4—144no—
328C2×C4⋊C4—144no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4C22⋊D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D712yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4C22⋊D712yes—
341C2×C42—141no—
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—22no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—81no—
365C22×C7—81no—
366C22×C7—81no—
367C22×C7—81no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#54611yescharacteristic
370C22×C7#54711yescharacteristic
371C22×C7C22×SD1611yescharacteristic
372C22×C7C2×#13711yescharacteristic
373C22×C7#57611yescharacteristic
374C22×C7#57711yescharacteristic
375C4×C7—22no—
376C4×C7—42no—
377C4×C7—81no—
378C4×C7—82no—
379C4×C7#44814yes—
380C4×C7#52511yescharacteristic
381C4×C7#53711yescharacteristic
382C4×C7#54811yescharacteristic
383C4×C7#55011yescharacteristic
384Dic7—42no—
385Dic7—44no—
386Dic7—42no—
387C24—21no—
388C24C2×C22⋊D712yes—
389C2×Q8—72no—
390C2×Q8—144no—
391C2×Q8—141no—
392C2×Q8—141no—
393C2×Q8—141no—
394C2×Q8—141no—
395C2×Q8—141no—
396C2×Q8—561no—
397C2×Q8—561no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4#114214yescomplemented
409C2×D4#115912yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—142no—
424C22×C4—148no—
425C22×C4—562no—
426C22×C4—562no—
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—141no—
436C2×C8—142no—
437C2×C8—141no—
438C2×C8—142no—
439C2×C8—142no—
440C2×C8—141no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—72no—
448C4⋊C4—74no—
449C4⋊C4—81no—
450C4⋊C4—148no—
451C4⋊C4—148no—
452C4⋊C4—144no—
453C4⋊C4—144no—
454C4⋊C4—141no—
455C4⋊C4—142no—
456C22⋊C4—24no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—148no—
461C42—144no—
462C42D4×D712yes—
463C42D4×D712yes—
464C2×C7—44no—
465C2×C7—82no—
466C2×C7#240711yescharacteristic
467C2×C7C2×#44811yescharacteristic
468C2×C7#301211yescharacteristic
469C2×C7#329511yescharacteristic
470C2×C7#333111yescharacteristic
471C2×C7#333511yescharacteristic
472C2×C7#335711yescharacteristic
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C2×#114214yes—
487C23C22×C22⋊D711yescharacteristic
488Q8—142no—
489Q8—142no—
490Q8—281no—
491Q8—281no—
492Q8—564no—
493D4—22no—
494D4—22no—
495D4—28no—
496D4—41no—
497D4—41no—
498D4—88no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—144no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—282no—
522C2×C4—281no—
523C2×C4—282no—
524C2×C4—568no—
525C2×C4C2×#114212yes—
526C2×C4#650012yes—
527C2×C4#650012yes—
528C2×C4#651412yes—
529C2×C4C2×D4×D711yescharacteristic
530C2×C4C2×D4×D711yescharacteristic
531C2×C4C2×#115911yescharacteristic
532C2×C4#654811yescharacteristic
533C2×C4D7×#4111yescharacteristic
534C2×C4#655411yescharacteristic
535C8—142no—
536C8—282no—
537C8—281no—
538C7#2411811yescharacteristic, complemented, Sylow
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C22×#114211yescharacteristic
554C22C2×#650011yescharacteristic
555C22#9049711yescharacteristic
556C22C2×#651411yescharacteristic
557C22#9051511yescharacteristic
558C22#9052611yescharacteristic
559C22#9052811yescharacteristic
560C4—22no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—282no—
565C4—284no—
566C4—282no—
567C4#8981214yes—
568C4#9028911yescharacteristic
569C4#9029011yescharacteristic
570C4#9029711yescharacteristic
571C4#9029911yescharacteristic
572C2—44no—
573C2—82no—
574C2C2×#8981211yescharacteristic
575C2#1174014611yescharacteristic
576C2#1174086911yescharacteristic
577C2#1174092811yescharacteristic
578C2#1174096011yescharacteristic
579C2#1174096211yescharacteristic
580C2#1174109611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2537C2
socleC23×C7C22×D4
hypercenter#2537D7
hypocenterC7#24118
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#90234C2semidirect
3. C2×C4×#1142C2semidirect
4. C2×#89780C2semidirect
5. #11729439C2semidirect
6. #11729431C2semidirect
7. #11729368C2semidirect
8. #11727342C2semidirect
9. #11727336C2semidirect
10. #11727280C2semidirect
11. #11727259C2semidirect
93. #6432D4semidirect
95. C4×Dic14D4semidirect
130. C2×C7×D4SD16semidirect
408. C2×D4#1142semidirect
538. C7#24118semidirect

Maximal subgroups

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

subgroupindex
C2×#902342
C7×#25372
C2×C4×#11422
C2×#897802
#117294392
#117294312
#117293682
#117273422
#117273362
#117272802
#117272592
#241187