Finite Group Databasebeta

← #13387718

Subgroups — #13387718

Order 1792.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90233C211yesmaximal, characteristic, complemented
2C7×#2537C211yesmaximal, characteristic
3C2×C4×#1142C211yesmaximal, characteristic, complemented
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729439C212yesmaximal, complemented
6#11729432C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic, complemented
8#11727342C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727281C212yesmaximal, complemented
11#11727260C212yesmaximal, complemented
12C2×C7×#121—21no—
13C2×C7×#121C2212yes—
14C2×C4×C7×D4C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6499—21no—
17C22×#1142C2211yescharacteristic
18C2×#6489C2212yes—
19C2×#6487C2212yes—
20#90233—24no—
21#90233—24no—
22C2×#6450—21no—
23C2×#6442—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#89779—24no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2212yes—
38C2×#6382C2211yescharacteristic
39C2×#6381C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24119—71nomaximal, Sylow
46C22×C7×D4—22no—
47C22×C7×D4D411yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7C2312yes—
51C22×Dic14—21no—
52C22×Dic14C2311yescharacteristic
53C7×#121—24no—
54C7×#121—42no—
55C7×#121—44no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C4×C7×D4—22no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4D412yes—
69C2×C42×C7C2311yescharacteristic
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7—22no—
72#6499—28no—
73C2×#1142—22no—
74C2×#1142—22no—
75C2×#1142—22no—
76#6489—24no—
77#6489—24no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2312yes—
81#6487—24no—
82#6487—24no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—22no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—22no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2311yescharacteristic
90#6450—28no—
91#6442—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—22no—
104#6381—24no—
105C4⋊(C7⋊C8)—22no—
106C4×C7⋊C8—22no—
107C2×#556—71no—
108C2×#474—71no—
109C2×C4×SD16—71no—
110#2537D711yescharacteristic
111#2036—72no—
112#2000—72no—
113#1990—71no—
114#1954—71no—
115#1875—72no—
116#1854—72no—
117#1803—71no—
118C24×C7—21no—
119C24×C7C2×D412yes—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—82no—
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—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—24no—
155C2×Dic14—21no—
156C2×Dic14—42no—
157C2×Dic14—42no—
158C2×Dic14C2×D412yes—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—42no—
165C7×C4⋊C4—81no—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—44no—
168C7×C22⋊C4—44no—
169C7×C22⋊C4—42no—
170C42×C7C2×D412yes—
171C42×C7C2×D412yes—
172C22⋊Dic7—28no—
173C22⋊Dic7—28no—
174#1142—28no—
175C4⋊Dic7—24no—
176C4⋊Dic7—21no—
177C4⋊Dic7—22no—
178C4⋊Dic7C2×D412yes—
179C4⋊Dic7C2×D414yes—
180Dic7⋊C4—28no—
181Dic7⋊C4—24no—
182C4×Dic7—28no—
183C4×Dic7—24no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8#4112yes—
193C22×SD16—71no—
194#556—144no—
195#556—144no—
196C2×C4⋊Q8—71no—
197C2×#126—142no—
198C2×#124—141no—
199C2×C22⋊Q8—72no—
200C2×#121—21no—
201C2×#121C2×D712yes—
202C2×C4×Q8—71no—
203C2×C4×Q8—141no—
204C2×C4×D4C2×D711yescharacteristic
205C2×C4×D4C2×D711yescharacteristic
206C2×#117—72no—
207#474—144no—
208C4×SD16—144no—
209C2×#107—72no—
210C2×#106—71no—
211C2×C4⋊C8—71no—
212C2×#103—71no—
213C2×#103—72no—
214C2×#102—71no—
215C2×#102—72no—
216C2×C22⋊C8—72no—
217C2×C22⋊C8—72no—
218C2×C4×C8—71no—
219#394—22no—
220#384C2×D711yescharacteristic
221#380—21no—
222#380C2×D712yes—
223#340—72no—
224#340—72no—
225#340—72no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7C22×D411yescharacteristic
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—22no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—84no—
263C2×C4×C7#12012yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#13512yes—
266C2×C4×C7AGL(1,ℤ/8)12yes—
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—22no—
275C2×Dic7—24no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282Dic14—22no—
283Dic14—22no—
284Dic14—41no—
285Dic14—41no—
286Dic14—48no—
287C7⋊C8—22no—
288C7⋊C8—41no—
289C7⋊C8—42no—
290C22×Q8—71no—
291C22×Q8—141no—
292C22×D4—22no—
293C22×D4C22⋊D711yescharacteristic
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C23×C4C22×D712yes—
297C2×SD16—142no—
298C2×SD16—142no—
299C2×SD16—142no—
300C22×C8—71no—
301C22×C8—71no—
302C22×C8—72no—
303C4⋊Q8—74no—
304C4⋊Q8—142no—
305#126—1416no—
306#124—148no—
307C22⋊Q8—144no—
308C22⋊Q8—144no—
309#121—24no—
310#121—42no—
311#121—44no—
312C4×Q8—74no—
313C4×Q8—148no—
314C4×Q8—142no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319#117—144no—
320#117—144no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—72no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—142no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4C22⋊D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C4⋊C4C22×D712yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—142no—
336C2×C22⋊C4C22⋊D712yes—
337C2×C42—142no—
338C2×C42—141no—
339C2×C42C22×D711yescharacteristic
340#107—144no—
341#106—142no—
342C4⋊C8—142no—
343#103—142no—
344#103—282no—
345#102—142no—
346#102—144no—
347C22⋊C8—144no—
348C22⋊C8—282no—
349C4×C8—142no—
350#95—22no—
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—42no—
360C22×C7—42no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7—81no—
365C22×C7C2×#12011yescharacteristic
366C22×C7#54611yescharacteristic
367C22×C7#54711yescharacteristic
368C22×C7C2×#13511yescharacteristic
369C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C7#57711yescharacteristic
371C22×C7#57811yescharacteristic
372C4×C7—22no—
373C4×C7—42no—
374C4×C7—81no—
375C4×C7—82no—
376C4×C7#44914yes—
377C4×C7#52511yescharacteristic
378C4×C7#53711yescharacteristic
379C4×C7#54811yescharacteristic
380C4×C7#55011yescharacteristic
381Dic7—42no—
382Dic7—44no—
383Dic7—42no—
384C24—21no—
385C24C2×C22⋊D712yes—
386C2×Q8—72no—
387C2×Q8—144no—
388C2×Q8—141no—
389C2×Q8—141no—
390C2×Q8—141no—
391C2×Q8—141no—
392C2×Q8—141no—
393C2×Q8—282no—
394C2×Q8—282no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—82no—
406C2×D4#115912yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—141no—
419C22×C4—142no—
420C22×C4—141no—
421C22×C4C2×C22⋊D712yes—
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C23×D711yescharacteristic
425SD16—148no—
426C2×C8—72no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—142no—
431C2×C8—141no—
432C2×C8—142no—
433C2×C8—142no—
434C2×C8—141no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—72no—
442C4⋊C4—74no—
443C4⋊C4—81no—
444C4⋊C4—148no—
445C4⋊C4—144no—
446C4⋊C4—144no—
447C4⋊C4—141no—
448C4⋊C4—142no—
449C22⋊C4—24no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—148no—
454C22⋊C4—148no—
455C42—148no—
456C42—144no—
457C42D4×D712yes—
458C42D4×D712yes—
459C2×C7—82no—
460C2×C7—82no—
461C2×C7#240711yescharacteristic
462C2×C7C2×#44911yescharacteristic
463C2×C7#301311yescharacteristic
464C2×C7#330211yescharacteristic
465C2×C7#332411yescharacteristic
466C2×C7#334011yescharacteristic
467C2×C7#335311yescharacteristic
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C22×C22⋊D711yescharacteristic
483Q8—142no—
484Q8—142no—
485Q8—281no—
486Q8—281no—
487Q8—288no—
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
492D4—44no—
493D4—88no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—282no—
514C2×C4—281no—
515C2×C4—282no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4#649512yes—
520C2×C4#650012yes—
521C2×C4#650012yes—
522C2×C4#651312yes—
523C2×C4C2×D4×D711yescharacteristic
524C2×C4C2×D4×D711yescharacteristic
525C2×C4C2×#115911yescharacteristic
526C2×C4#654811yescharacteristic
527C2×C4D7×#4111yescharacteristic
528C2×C4#655411yescharacteristic
529C8—142no—
530C8—282no—
531C8—281no—
532C7#2411911yescharacteristic, complemented, Sylow
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#649511yescharacteristic
548C22C2×#650011yescharacteristic
549C22#9049711yescharacteristic
550C22C2×#651311yescharacteristic
551C22#9051511yescharacteristic
552C22#9052711yescharacteristic
553C22#9052911yescharacteristic
554C4—22no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—284no—
559C4—282no—
560C4—282no—
561C4#8980914yes—
562C4#9028911yescharacteristic
563C4#9029011yescharacteristic
564C4#9029711yescharacteristic
565C4#9029911yescharacteristic
566C2—82no—
567C2—82no—
568C2C2×#8980911yescharacteristic
569C2#1174014611yescharacteristic
570C2#1174087211yescharacteristic
571C2#1174093111yescharacteristic
572C2#1174095511yescharacteristic
573C2#1174096511yescharacteristic
574C2#1174109111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2537C2
socleC23×C7C22×D4
hypercenter#2537D7
hypocenterC7#24119
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×#90233C2semidirect
3. C2×C4×#1142C2semidirect
4. C2×#89779C2semidirect
5. #11729439C2semidirect
6. #11729432C2semidirect
7. #11729368C2semidirect
8. #11727342C2semidirect
9. #11727336C2semidirect
10. #11727281C2semidirect
11. #11727260C2semidirect
93. #6432D4semidirect
95. C4×Dic14D4semidirect
532. C7#24119semidirect

Maximal subgroups

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

subgroupindex
C2×#902332
C7×#25372
C2×C4×#11422
C2×#897792
#117294392
#117294322
#117293682
#117273422
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