Finite Group Databasebeta

← #13380964

Subgroups — #13380964

Order 1792.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#137C211yesmaximal, characteristic
2C22×#6514C211yesmaximal, characteristic
3C22×#6507C211yesmaximal, characteristic, complemented
4#11741531C211yesmaximal, characteristic
5C2×#89997C211yesmaximal, characteristic, complemented
6C2×#89892C211yesmaximal, characteristic, complemented
7#11729827C214yesmaximal, complemented
8#11729811C214yesmaximal, complemented
9#11729733C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C23×Dic14C2211yescharacteristic
13C2×C7×#137—24no—
14C2×C7×#137—24no—
15C2×C7×#137—24no—
16C22×C7×Q16C2212yes—
17C22×C7×SD16C2212yes—
18C22×C7×M16C2211yescharacteristic
19C2×#6514—24no—
20C2×#6514—24no—
21C2×#6514—24no—
22C2×#6507—24no—
23C2×#6507—24no—
24C2×#6507—22no—
25C2×#6507C2214yes—
26C22×#1144C2212yes—
27C22×#1142C2212yes—
28C2×#6489C2211yescharacteristic
29C22×Dic7⋊C4—22no—
30C22×C4×Dic7C2211yescharacteristic
31C22×#1136C2211yescharacteristic
32#90002C2212yes—
33#89997—24no—
34C2×#6409C2214yes—
35#89993C2212yes—
36#89992—22no—
37C4×C22⋊Dic7—22no—
38#89892—24no—
39C2×#6390C2214yes—
40#89345C2214yes—
41#23432—71nomaximal, Sylow
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41D414yes—
48C22×C7×Q8—22no—
49C22×C7×Q8—21no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8D414yes—
55C22×C7×Q8C2312yes—
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4—21no—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C23×Dic7D412yes—
64C22×Dic14—22no—
65C22×Dic14—22no—
66C22×Dic14—22no—
67C22×Dic14—22no—
68C22×Dic14—21no—
69C22×Dic14—22no—
70C22×Dic14D414yes—
71C22×Dic14C2312yes—
72C7×#137—232no—
73C2×C7×Q16—24no—
74C2×C7×Q16—24no—
75C2×C7×Q16—24no—
76C2×C7×SD16—24no—
77C2×C7×SD16—24no—
78C2×C7×SD16—24no—
79C2×C7×M16—22no—
80C2×C7×M16—22no—
81C2×C7×M16D414yes—
82C22×C7×C8C2312yes—
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—24no—
85#6514—232no—
86#6507—216no—
87#6507—28no—
88#6507—28no—
89C2×#1144—24no—
90C2×#1144—24no—
91C2×#1144—24no—
92C2×#1142—24no—
93C2×#1142—24no—
94C2×#1142—24no—
95#6489—22no—
96#6489—22no—
97C2×C4⋊Dic7C2312yes—
98C2×Dic7⋊C4—216no—
99C2×Dic7⋊C4—28no—
100C2×C4×Dic7—22no—
101C2×C4×Dic7—22no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7D412yes—
104C2×C4×Dic7C2312yes—
105C2×#1136—22no—
106C2×#1136—22no—
107C2×#1136D414yes—
108C22×C7⋊C8C2312yes—
109#6409—28no—
110C7⋊#95—24no—
111#6390—28no—
112C22×#137—71no—
113C22×#137D711yescharacteristic
114C22×C4⋊Q8—71no—
115C2×#419—71no—
116C2×#419—71no—
117#2297—71no—
118#2016—74no—
119#2013—74no—
120#1728—71no—
121C24×C7—21no—
122C24×C7C2×D411yescharacteristic
123C7×#41—28no—
124C7×#41—28no—
125C7×#41—44no—
126C7×#41—44no—
127C2×C7×Q8—24no—
128C2×C7×Q8—22no—
129C2×C7×Q8—21no—
130C2×C7×Q8—28no—
131C2×C7×Q8—28no—
132C2×C7×Q8—22no—
133C2×C7×Q8—21no—
134C2×C7×Q8—21no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—44no—
141C2×C7×Q8—42no—
142C2×C7×Q8—44no—
143C2×C7×Q8—42no—
144C2×C7×Q8C2×D412yes—
145C2×C7×Q8C2×D414yes—
146C2×C7×D4—24no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—42no—
153C2×C7×D4—42no—
154C2×C7×D4C2×D412yes—
155C22×C4×C7—24no—
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—24no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7#4112yes—
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—24no—
172C22×Dic7—28no—
173C22×Dic7—24no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—42no—
177C22×Dic7—42no—
178C22×Dic7C2×D414yes—
179C2×Dic14—28no—
180C2×Dic14—28no—
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14—22no—
184C2×Dic14—22no—
185C2×Dic14—24no—
186C2×Dic14—22no—
187C2×Dic14—44no—
188C2×Dic14—44no—
189C2×Dic14—42no—
190C2×Dic14—42no—
191C2×Dic14C2×D414yes—
192C7×Q16—216no—
193C7×SD16—216no—
194C7×M16—28no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8C2×D414yes—
198C22⋊Dic7—42no—
199C22⋊Dic7—48no—
200#1144—216no—
201#1142—216no—
202C4⋊Dic7—22no—
203C4⋊Dic7—22no—
204Dic7⋊C4—232no—
205C4×Dic7—24no—
206C4×Dic7—22no—
207C4×Dic7—22no—
208C4×Dic7—42no—
209#1136—28no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8C2×D414yes—
213C22×#41C2×D711yescharacteristic
214C23×Q8—71no—
215C23×Q8C2×D711yescharacteristic
216C2×#137—24no—
217C2×#137—24no—
218C2×#137—24no—
219C2×#137—144no—
220C2×#137—144no—
221C2×#137—144no—
222C22×Q16—72no—
223C22×Q16C2×D712yes—
224C22×SD16—72no—
225C22×SD16C2×D712yes—
226C22×M16—71no—
227C22×M16C2×D711yescharacteristic
228C2×C4⋊Q8—74no—
229C2×C4⋊Q8—142no—
230C2×C4⋊Q8—144no—
231C2×C4⋊Q8—144no—
232C2×#117—71no—
233C22×C4⋊C4—142no—
234C22×C42—71no—
235#419—144no—
236#419—144no—
237C2×#103—74no—
238C2×#103—74no—
239#391—72no—
240#390—72no—
241#386—142no—
242C4×C22⋊C4—142no—
243#336—74no—
244C23×C7—24no—
245C23×C7—22no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—44no—
249C23×C7#12012yes—
250C23×C7#12112yes—
251C23×C7#12712yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—22no—
254C7×Q8—24no—
255C7×Q8—22no—
256C7×Q8—24no—
257C7×Q8—42no—
258C7×Q8—42no—
259C7×Q8—41no—
260C7×Q8—41no—
261C7×Q8—48no—
262C7×D4—22no—
263C7×D4—22no—
264C7×D4—48no—
265C7×D4—41no—
266C7×D4—41no—
267C2×C4×C7—24no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—48no—
274C2×C4×C7—48no—
275C2×C4×C7#12014yes—
276C2×C4×C7#12012yes—
277C2×C4×C7#12014yes—
278C2×C4×C7#12114yes—
279C2×C4×C7#12114yes—
280C2×C4×C7#12112yes—
281C2×C4×C7#12712yes—
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×Dic7—216no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—48no—
295C2×Dic7—42no—
296C2×Dic7—44no—
297C2×Dic7#13718yescomplemented
298Dic14—24no—
299Dic14—24no—
300Dic14—42no—
301Dic14—42no—
302Dic14—48no—
303C7×C8—24no—
304C7⋊C8—24no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41C22⋊D714yes—
311C22×Q8—22no—
312C22×Q8—21no—
313C22×Q8—22no—
314C22×Q8—22no—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8—74no—
318C22×Q8—72no—
319C22×Q8—141no—
320C22×Q8—142no—
321C22×Q8—142no—
322C22×Q8—142no—
323C22×Q8—142no—
324C22×Q8—142no—
325C22×Q8C22⋊D714yes—
326C22×Q8C22×D712yes—
327C22×Q8C22×D711yescharacteristic
328C22×D4—21no—
329C22×D4C22×D711yescharacteristic
330C23×C4—21no—
331C23×C4—21no—
332C23×C4—72no—
333C23×C4—141no—
334C23×C4C22×D711yescharacteristic
335#137—232no—
336#137—1432no—
337C2×Q16—24no—
338C2×Q16—24no—
339C2×Q16—24no—
340C2×Q16—144no—
341C2×Q16—144no—
342C2×Q16—144no—
343C2×SD16—24no—
344C2×SD16—24no—
345C2×SD16—24no—
346C2×SD16—144no—
347C2×SD16—144no—
348C2×SD16—144no—
349C2×M16—22no—
350C2×M16—22no—
351C2×M16—74no—
352C2×M16—142no—
353C2×M16—142no—
354C2×M16C22⋊D714yes—
355C22×C8—72no—
356C22×C8C22×D712yes—
357C4⋊Q8—148no—
358C4⋊Q8—1416no—
359C4⋊Q8—148no—
360#117—142no—
361#117—142no—
362C2×C4⋊C4—72no—
363C2×C4⋊C4—148no—
364C2×C4⋊C4—1416no—
365C2×C22⋊C4—144no—
366C2×C22⋊C4—141no—
367C2×C42—72no—
368C2×C42—72no—
369C2×C42—142no—
370C2×C42—142no—
371C2×C42—141no—
372#103—148no—
373#103—148no—
374#95—144no—
375C22×C7—24no—
376C22×C7—24no—
377C22×C7—24no—
378C22×C7—42no—
379C22×C7—44no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7#39214yes—
383C22×C7C2×#12011yescharacteristic
384C22×C7C2×#12111yescharacteristic
385C22×C7C2×#12711yescharacteristic
386C22×C7C2×#13714yes—
387C4×C7—42no—
388C4×C7—44no—
389C4×C7#39214yes—
390C4×C7C2×#12011yescharacteristic
391C4×C7C2×#12011yescharacteristic
392C4×C7C2×#12111yescharacteristic
393C4×C7C2×#12111yescharacteristic
394Dic7—28no—
395Dic7—44no—
396Dic7—82no—
397C24—21no—
398C24D4×D711yescharacteristic
399#41—28no—
400#41—28no—
401#41—44no—
402#41—44no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—74no—
421C2×Q8—144no—
422C2×Q8—142no—
423C2×Q8—148no—
424C2×Q8—148no—
425C2×Q8—142no—
426C2×Q8—142no—
427C2×Q8—142no—
428C2×Q8—142no—
429C2×Q8—282no—
430C2×Q8—284no—
431C2×Q8—282no—
432C2×Q8—284no—
433C2×Q8C2×C22⋊D714yes—
434C2×Q8C2×C22⋊D712yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C2×C22⋊D712yes—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—74no—
453C22×C4—148no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—282no—
459C22×C4—282no—
460C22×C4D4×D712yes—
461C22×C4D4×D711yescharacteristic
462C22×C4D4×D712yes—
463C22×C4#115912yes—
464C22×C4C2×C22⋊D712yes—
465C22×C4C2×C22⋊D712yes—
466C22×C4C2×C22⋊D712yes—
467C22×C4C23×D711yescharacteristic
468Q16—216no—
469Q16—1416no—
470SD16—216no—
471SD16—1416no—
472M16—28no—
473M16—148no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—74no—
477C2×C8—142no—
478C2×C8—142no—
479C2×C8C2×C22⋊D714yes—
480C4⋊C4—1432no—
481C4⋊C4—142no—
482C4⋊C4—142no—
483C22⋊C4—288no—
484C22⋊C4—282no—
485C42—144no—
486C42—142no—
487C42—142no—
488C42—282no—
489C2×C7—24no—
490C2×C7—82no—
491C2×C7C2×#39211yescharacteristic
492C2×C7#300412yes—
493C2×C7#303912yes—
494C2×C7#315212yes—
495C23—24no—
496C23—22no—
497C23—44no—
498C23—41no—
499C23—41no—
500C23#650012yes—
501C23#650212yes—
502C23#650312yes—
503C23C2×D4×D711yescharacteristic
504Q8—22no—
505Q8—24no—
506Q8—24no—
507Q8—22no—
508Q8—42no—
509Q8—42no—
510Q8—41no—
511Q8—41no—
512Q8—48no—
513Q8—144no—
514Q8—144no—
515Q8—282no—
516Q8—288no—
517Q8—282no—
518D4—22no—
519D4—22no—
520D4—48no—
521D4—41no—
522D4—41no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—78no—
532C2×C4—1416no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—288no—
539C2×C4—284no—
540C2×C4—282no—
541C2×C4#650014yes—
542C2×C4#650012yes—
543C2×C4#650114yes—
544C2×C4#650214yes—
545C2×C4#650212yes—
546C2×C4#650312yes—
547C2×C4#651614yes—
548C2×C4C2×D4×D711yescharacteristic
549C2×C4C2×D4×D711yescharacteristic
550C2×C4C2×#115911yescharacteristic
551C2×C4C22×C22⋊D711yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C2×C4C22×C22⋊D711yescharacteristic
554C8—24no—
555C8—144no—
556C7#2343211yescharacteristic, complemented, Sylow
557C22—24no—
558C22—24no—
559C22—24no—
560C22—42no—
561C22—44no—
562C22—82no—
563C22—82no—
564C22#8999514yes—
565C22D7×#13714yes—
566C22C2×#650011yescharacteristic
567C22C2×#650211yescharacteristic
568C22C2×#650311yescharacteristic
569C4—42no—
570C4—44no—
571C4—148no—
572C4—284no—
573C4—562no—
574C4#8999514yes—
575C4C2×#650011yescharacteristic
576C4C2×#650111yescharacteristic
577C4C2×#650211yescharacteristic
578C4C2×#651611yescharacteristic
579C2—24no—
580C2—82no—
581C2#1174141212yes—
582C2#1174141712yes—
583C2#1174142412yes—
584C2C2×#8999511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#137C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC22×#137D7
hypocenterC7#23432
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C22×#6507C2semidirect
5. C2×#89997C2semidirect
6. C2×#89892C2semidirect
7. #11729827C2semidirect
8. #11729811C2semidirect
9. #11729733C2semidirect
297. C2×Dic7#137semidirect
556. C7#23432semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C22×#65142
C22×#65072
#117415312
C2×#899972
C2×#898922
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