Finite Group Databasebeta

← #13388158

Subgroups — #13388158

Order 1792.

1114 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×#90161C211yesmaximal, characteristic
2C4×C7×C22⋊Q8C211yesmaximal, characteristic
3C2×#89803C211yesmaximal, characteristic, complemented
4C2×#89793C211yesmaximal, characteristic, complemented
5#11729454C212yesmaximal
6#11729284C211yesmaximal, characteristic
7#11727444C212yesmaximal, complemented
8#11727385C212yesmaximal, complemented
9#11727281C212yesmaximal
10#11727243C211yesmaximal, characteristic, complemented
11#11727240C211yesmaximal, characteristic, complemented
12C2×C7×C22⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×#1144C2212yes—
16C2×#6489—21no—
17C2×#6487—21no—
18C2×#6487C2212yes—
19#90161—24no—
20#90161—24no—
21C2×#6433—21no—
22C2×#6432C2211yescharacteristic
23C2×#6432C2211yescharacteristic
24C2×C4×Dic14—21no—
25C2×C4×Dic14C2212yes—
26C7×#391C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#387—21no—
29C7×#385—21no—
30C7×#380C2211yescharacteristic
31C4×C7×C4⋊C4—21no—
32C4×C7×C4⋊C4C2211yescharacteristic
33C4×C7×C22⋊C4—21no—
34C2×#6409C2212yes—
35C2×C22⋊(C7⋊C8)C2212yes—
36#89803—24no—
37#89793—24no—
38C2×#6384C2212yes—
39C2×#6384C2212yes—
40C2×#6384C2212yes—
41C2×#6381C2212yes—
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24155—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C22×Dic14—21no—
53C22×Dic14C2312yes—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—22no—
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C42×C7—22no—
67C2×C42×C7—22no—
68C2×C42×C7—21no—
69C2×C42×C7—21no—
70C2×C42×C7D411yescharacteristic
71C2×C42×C7D411yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—22no—
75C2×#1144—24no—
76C2×#1144—24no—
77C2×#1144—42no—
78#6489—28no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2312yes—
83#6487—28no—
84#6487—24no—
85#6487—24no—
86C2×Dic7⋊C4—22no—
87C2×Dic7⋊C4—22no—
88C2×C4×Dic7—22no—
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2312yes—
91#6433—28no—
92#6432—22no—
93#6432—22no—
94#6432D414yes—
95#6432D414yes—
96C4×Dic14—28no—
97C4×Dic14—24no—
98C4×Dic14—24no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104#6409—24no—
105C22⋊(C7⋊C8)—42no—
106#6384—24no—
107#6384—24no—
108#6384—42no—
109#6381—24no—
110C4⋊(C7⋊C8)—22no—
111C4⋊(C7⋊C8)—22no—
112C2×#533—71no—
113C2×#477—71no—
114C2×#462—71no—
115C4×C22⋊Q8D711yescharacteristic
116#2036—72no—
117#2001—72no—
118#1987—71no—
119#1981—71no—
120#1878—72no—
121#1863—72no—
122#1797—71no—
123C24×C7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8#4112yes—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7Q1614yescomplemented
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—22no—
153C22×Dic7—22no—
154C2×Dic14—22no—
155C2×Dic14—22no—
156C2×Dic14—22no—
157C2×Dic14—22no—
158C2×Dic14—24no—
159C2×Dic14—22no—
160C2×Dic14—42no—
161C2×Dic14—42no—
162C2×Dic14C2×D414yes—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4#4112yes—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—42no—
174C42×C7—22no—
175C42×C7—22no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—42no—
179C42×C7—42no—
180C42×C7C2×D412yes—
181C42×C7C2×D412yes—
182C22⋊Dic7—28no—
183#1144—48no—
184C4⋊Dic7—24no—
185C4⋊Dic7—24no—
186C4⋊Dic7—24no—
187C4⋊Dic7—22no—
188C4⋊Dic7C2×D414yes—
189Dic7⋊C4—28no—
190Dic7⋊C4—28no—
191C4×Dic7—28no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C22×Q16—72no—
199#533—144no—
200#533—144no—
201C2×C4⋊Q8—71no—
202C2×C4⋊Q8—71no—
203C2×#125—141no—
204C2×C22⋊Q8—72no—
205C2×C22⋊Q8—141no—
206C2×C22⋊Q8C2×D711yescharacteristic
207C2×C4×Q8—72no—
208C2×C4×Q8—141no—
209C2×C4×Q8C2×D711yescharacteristic
210C2×#117—141no—
211C22×C42C2×D711yescharacteristic
212#477—144no—
213#462—144no—
214C2×#107—72no—
215C2×C4⋊C8—71no—
216C2×C4⋊C8—71no—
217C2×#103—72no—
218C2×#103—72no—
219C2×#103—72no—
220C2×#103—72no—
221C2×C22⋊C8—72no—
222#391C2×D711yescharacteristic
223#389C2×D711yescharacteristic
224#387—21no—
225#385—21no—
226#380C2×D711yescharacteristic
227C4×C4⋊C4—21no—
228C4×C4⋊C4C2×D711yescharacteristic
229C4×C22⋊C4—21no—
230#340—72no—
231#340—72no—
232#336—72no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7C2×Q1612yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—22no—
242C7×Q8—22no—
243C7×Q8—41no—
244C7×Q8—41no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—28no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—44no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12012yes—
275C2×C4×C7C2×Q1612yes—
276C2×C4×C7C2×Q1612yes—
277C2×C4×C7#13712yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290Dic14—24no—
291Dic14—24no—
292Dic14—48no—
293Dic14—42no—
294Dic14—42no—
295C7⋊C8—42no—
296C7⋊C8—42no—
297C22×Q8—72no—
298C22×Q8—141no—
299C22×Q8C22×D711yescharacteristic
300C23×C4C22⋊D711yescharacteristic
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22×D711yescharacteristic
303C2×Q16—144no—
304C2×Q16—144no—
305C2×Q16—282no—
306C22×C8—72no—
307C22×C8—72no—
308C4⋊Q8—74no—
309C4⋊Q8—74no—
310C4⋊Q8—142no—
311C4⋊Q8—142no—
312#125—148no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—144no—
316C22⋊Q8—144no—
317C22⋊Q8—148no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—144no—
321C4×Q8—144no—
322C4×Q8—148no—
323#117—148no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4C22×D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—142no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—142no—
344C2×C42C22⋊D711yescharacteristic
345C2×C42C22⋊D711yescharacteristic
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348#107—144no—
349C4⋊C8—142no—
350C4⋊C8—142no—
351#103—144no—
352#103—144no—
353#103—144no—
354#103—282no—
355C22⋊C8—282no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361C22×C7—22no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#52511yescharacteristic
370C22×C7#53711yescharacteristic
371C22×C7C22×Q1611yescharacteristic
372C22×C7C2×#13711yescharacteristic
373C22×C7#57511yescharacteristic
374C22×C7#57511yescharacteristic
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—42no—
378C4×C7—41no—
379C4×C7—42no—
380C4×C7—41no—
381C4×C7—81no—
382C4×C7—81no—
383C4×C7#45114yes—
384C4×C7#54611yescharacteristic
385C4×C7#54711yescharacteristic
386C4×C7#54811yescharacteristic
387C4×C7#55011yescharacteristic
388Dic7—44no—
389Dic7—44no—
390C24C2×C22⋊D711yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—74no—
397C2×Q8—144no—
398C2×Q8—142no—
399C2×Q8—142no—
400C2×Q8—142no—
401C2×Q8—142no—
402C2×Q8—142no—
403C2×Q8—282no—
404C2×Q8—282no—
405C2×Q8#115712yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4#114414yescomplemented
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C23×D711yescharacteristic
430Q16—288no—
431C2×C8—142no—
432C2×C8—142no—
433C2×C8—142no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—74no—
446C4⋊C4—148no—
447C4⋊C4—148no—
448C4⋊C4—144no—
449C4⋊C4—144no—
450C4⋊C4—144no—
451C4⋊C4—142no—
452C4⋊C4#115712yes—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—148no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—42no—
461C42—42no—
462C42—148no—
463C42D4×D712yes—
464C42D4×D712yes—
465C2×C7—42no—
466C2×C7#240711yescharacteristic
467C2×C7C2×#45111yescharacteristic
468C2×C7#300511yescharacteristic
469C2×C7#310411yescharacteristic
470C2×C7#311611yescharacteristic
471C2×C7#324911yescharacteristic
472C2×C7#325211yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C2×#114412yes—
480C23C22×C22⋊D711yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—144no—
486Q8—144no—
487Q8—288no—
488Q8—282no—
489Q8—282no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4#650012yes—
525C2×C4#650012yes—
526C2×C4C2×#114412yes—
527C2×C4C2×#114412yes—
528C2×C4#651412yes—
529C2×C4C2×#115711yescharacteristic
530C2×C4C2×#115711yescharacteristic
531C2×C4C2×D4×D711yescharacteristic
532C2×C4C2×D4×D711yescharacteristic
533C2×C4#655311yescharacteristic
534C2×C4#655411yescharacteristic
535C8—282no—
536C8—282no—
537C7#2415511yescharacteristic, complemented, Sylow
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#9048211yescharacteristic
546C22C2×#650011yescharacteristic
547C22C22×#114411yescharacteristic
548C22#9051311yescharacteristic
549C22#9051311yescharacteristic
550C22C2×#651411yescharacteristic
551C22#9052411yescharacteristic
552C4—22no—
553C4—22no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—284no—
561C4—284no—
562C4#8982214yes—
563C4#9024111yescharacteristic
564C4#9024211yescharacteristic
565C4#9024311yescharacteristic
566C4#9024411yescharacteristic
567C2—42no—
568C2#1173984711yescharacteristic
569C2#1173989211yescharacteristic
570C2#1173998011yescharacteristic
571C2C2×#8982211yescharacteristic
572C2#1174037311yescharacteristic
573C2#1174060111yescharacteristic
574C2#1174081811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×C22⋊Q8C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC4×C22⋊Q8D7
hypocenterC7#24155
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#89803C2semidirect
4. C2×#89793C2semidirect
7. #11727444C2semidirect
8. #11727385C2semidirect
10. #11727243C2semidirect
11. #11727240C2semidirect
145. C22×C4×C7Q16semidirect
423. C22×C4#1144semidirect
537. C7#24155semidirect

Maximal subgroups

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

subgroupindex
C2×#901612
C4×C7×C22⋊Q82
C2×#898032
C2×#897932
#117294542
#117292842
#117274442
#117273852
#117272812
#117272432
#117272402
#241557