Finite Group Databasebeta

← #13369437

Subgroups — #13369437

Order 1792.

1426 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2C2×#89997C212yesmaximal, complemented
3#11739782C211yesmaximal, characteristic
4C2×#89784C211yesmaximal, characteristic
5#11729886C212yesmaximal, complemented
6#11729539C211yesmaximal, characteristic, complemented
7#11729532C211yesmaximal, characteristic, complemented
8#11729451C212yesmaximal, complemented
9#11729447C212yesmaximal, complemented
10C4×#6409C211yesmaximal, characteristic, complemented
11#11729380C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2212yes—
13C7×D4×Q8—24no—
14C7×D4×Q8C418yescomplemented
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6489C2212yes—
23C22×#1136C2212yes—
24#89997—28no—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31C2×#6409C2211yescharacteristic
32#89993—21no—
33#89984C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89784—24no—
36C4⋊C4×Dic7—21no—
37#89723—22no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#89324C2211yescharacteristic
45#22430—71nomaximal, Sylow
46C22×C7×Q8—22no—
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—24no—
50C22×C7×Q8—42no—
51C22×C7×Q8—42no—
52C22×C7×Q8C2×C418yes—
53C22×C7×Q8C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7C2312yes—
58C7×C4⋊Q8—28no—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8C2×C414yes—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—216no—
63C7×C22⋊Q8C2×C418yes—
64C4×C7×Q8—22no—
65C4×C7×Q8C2×C414yes—
66C4×C7×D4—28no—
67C4×C7×D4C2×C414yes—
68C4×C7×D4D414yes—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4D412yes—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7—22no—
81#6489—24no—
82#6489—42no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2312yes—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—22no—
90C2×#1136—24no—
91C2×#1136—24no—
92C2×#1136—42no—
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2312yes—
95C22×C7⋊C8C2311yescharacteristic
96#6409—24no—
97#6409—22no—
98#6409—24no—
99#6409—24no—
100#6409—42no—
101#6409—42no—
102#6409D414yescomplemented
103C7⋊#95—21no—
104C7⋊#95—22no—
105C22⋊(C7⋊C8)—24no—
106C4⋊(C7⋊C8)—22no—
107C4×C7⋊C8—22no—
108C2×D4×Q8D711yescharacteristic
109C2×#436—71no—
110C2×#419—72no—
111#2354—71no—
112#1981—71no—
113#1956—71no—
114#1894—72no—
115#1873—72no—
116#1803—71no—
117#1787—72no—
118C4×#103—71no—
119C24×C7C2×D412yes—
120C2×C7×Q8—28no—
121C2×C7×Q8—21no—
122C2×C7×Q8—22no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—216no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—44no—
131C2×C7×Q8—44no—
132C2×C7×Q8—48no—
133C2×C7×Q8—48no—
134C2×C7×Q8—48no—
135C2×C7×Q8C22×C414yes—
136C2×C7×Q8C22×C412yes—
137C2×C7×Q8C2×D412yes—
138C2×C7×Q8#4112yes—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4C22⋊C414yes—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—28no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—44no—
151C22×C4×C7—44no—
152C22×C4×C7C22×C414yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—28no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4C22⋊C414yes—
167C7×C4⋊C4C22×C412yes—
168C7×C22⋊C4—28no—
169C7×C22⋊C4C22⋊C418yes—
170C42×C7—24no—
171C42×C7C22×C412yes—
172C42×C7C2×D412yes—
173C22⋊Dic7—44no—
174C22⋊Dic7—82no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7—81no—
182C4⋊Dic7C2×D412yes—
183C4×Dic7—44no—
184C4×Dic7—42no—
185#1136—48no—
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—21no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8C2×D412yes—
195C23×Q8C2×D712yes—
196C22×M16—72no—
197D4×Q8—24no—
198D4×Q8Dic718yescomplemented
199C2×C4⋊Q8—21no—
200C2×C4⋊Q8C2×D711yescharacteristic
201C2×C22⋊Q8—22no—
202C2×C22⋊Q8C2×D712yes—
203C2×C4×Q8C2×D711yescharacteristic
204C2×C4×D4—21no—
205C2×C4×D4C2×D711yescharacteristic
206C2×#117—72no—
207#436—144no—
208C2×C4⋊C8—71no—
209#419—148no—
210C2×#103—71no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103—72no—
214C2×#103—71no—
215C2×#103—72no—
216C2×#103—72no—
217C2×C22⋊C8—72no—
218C2×C4×C8—71no—
219#390—141no—
220#388—142no—
221#384—71no—
222#380—72no—
223C4×C4⋊C4—71no—
224C4×C4⋊C4—141no—
225#340—72no—
226#340—71no—
227#340—71no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7C2×C22⋊C414yes—
235C23×C7C22×D411yescharacteristic
236C7×Q8—22no—
237C7×Q8—22no—
238C7×Q8—22no—
239C7×Q8—24no—
240C7×Q8—24no—
241C7×Q8—416no—
242C7×Q8—42no—
243C7×Q8—42no—
244C7×Q8C4×D414yes—
245C7×D4—28no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—416no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7C2×C22⋊C412yes—
265C2×C4×C7C2×C22⋊C414yes—
266C2×C4×C7C2×C22⋊C412yes—
267C2×C4×C7#13712yes—
268C2×C4×C7#13712yes—
269C2×C4×C7C23×C411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C7H(5,𝔽2)11yescharacteristic
275C2×Dic7—22no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—41no—
284C7⋊C8—22no—
285C7⋊C8—42no—
286C7⋊C8—41no—
287C22×Q8—22no—
288C22×Q8—24no—
289C22×Q8—24no—
290C22×Q8—24no—
291C22×Q8—42no—
292C22×Q8—42no—
293C22×Q8C2×Dic718yes—
294C22×Q8C22×D712yes—
295C22×Q8C22×D711yescharacteristic
296C22×D4C22⋊D711yescharacteristic
297C23×C4—22no—
298C23×C4C22×D712yes—
299C2×M16—144no—
300C2×M16—144no—
301C2×M16—282no—
302C22×C8—71no—
303C22×C8—72no—
304C22×C8—71no—
305C4⋊Q8—28no—
306C4⋊Q8—22no—
307C4⋊Q8C2×Dic714yes—
308C22⋊Q8—24no—
309C22⋊Q8—216no—
310C22⋊Q8C2×Dic718yes—
311C4×Q8—22no—
312C4×Q8C2×Dic714yes—
313C4×D4—28no—
314C4×D4C2×Dic714yes—
315C4×D4C22⋊D714yes—
316#117—144no—
317#117—282no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—72no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4C22⋊D711yescharacteristic
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4—142no—
332C2×C22⋊C4C22⋊D712yes—
333C2×C42—21no—
334C2×C42—142no—
335C2×C42—141no—
336C2×C42C22×D711yescharacteristic
337C4⋊C8—142no—
338#103—74no—
339#103—144no—
340#103—144no—
341#103—144no—
342#103—142no—
343#103—282no—
344#103—282no—
345C22⋊C8—144no—
346C4×C8—142no—
347#95—141no—
348#95—142no—
349C22×C7—24no—
350C22×C7—24no—
351C22×C7—24no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7C22×C22⋊C411yescharacteristic
357C22×C7D4211yescharacteristic
358C22×C7#54611yescharacteristic
359C22×C7C2×#13711yescharacteristic
360C22×C7C2×#13711yescharacteristic
361C22×C7#57711yescharacteristic
362C22×C7#57811yescharacteristic
363C4×C7—24no—
364C4×C7—24no—
365C4×C7—22no—
366C4×C7—44no—
367C4×C7—42no—
368C4×C7#41914yescomplemented
369C4×C7C2×C4×D411yescharacteristic
370C4×C7#51811yescharacteristic
371C4×C7D4211yescharacteristic
372C4×C7#54611yescharacteristic
373Dic7—42no—
374Dic7—81no—
375Dic7—82no—
376C24C2×C22⋊D712yes—
377C2×Q8—28no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—216no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8D4×D712yes—
393C2×Q8#115912yes—
394C2×Q8C22×Dic714yes—
395C2×Q8C22×Dic712yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4C22⋊Dic714yes—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—142no—
412C22×C4C22×Dic714yes—
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D711yescharacteristic
415C22×C4C2×C22⋊D714yes—
416C22×C4C2×C22⋊D711yescharacteristic
417C22×C4C23×D711yescharacteristic
418M16—288no—
419C2×C8—72no—
420C2×C8—141no—
421C2×C8—141no—
422C2×C8—142no—
423C2×C8—141no—
424C2×C8—141no—
425C2×C8—142no—
426C2×C8—141no—
427C2×C8—142no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—21no—
433C4⋊C4—72no—
434C4⋊C4—141no—
435C4⋊C4—142no—
436C4⋊C4—141no—
437C4⋊C4—141no—
438C4⋊C4—142no—
439C4⋊C4—282no—
440C4⋊C4—561no—
441C4⋊C4C22⋊Dic714yes—
442C4⋊C4C22×Dic712yes—
443C22⋊C4—28no—
444C22⋊C4—284no—
445C22⋊C4—562no—
446C22⋊C4C22⋊Dic718yes—
447C42—24no—
448C42—284no—
449C42—282no—
450C42C22×Dic712yes—
451C42C2×C22⋊D712yes—
452C2×C7—44no—
453C2×C7D4×C22⋊C411yescharacteristic
454C2×C7C2×#41911yescharacteristic
455C2×C7#289811yescharacteristic
456C2×C7#327611yescharacteristic
457C2×C7#327811yescharacteristic
458C2×C7#332811yescharacteristic
459C2×C7#332911yescharacteristic
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23C2×C22⋊Dic714yes—
467C23C22×C22⋊D711yescharacteristic
468Q8—22no—
469Q8—22no—
470Q8—22no—
471Q8—24no—
472Q8—24no—
473Q8—416no—
474Q8—42no—
475Q8—42no—
476Q8D4×Dic714yes—
477D4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—416no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—142no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—282no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—282no—
503C2×C4—281no—
504C2×C4—282no—
505C2×C4#650512yes—
506C2×C4#650512yes—
507C2×C4C2×C22⋊Dic712yes—
508C2×C4C2×C22⋊Dic714yes—
509C2×C4C2×C22⋊Dic712yes—
510C2×C4C2×D4×D711yescharacteristic
511C2×C4C2×#115911yescharacteristic
512C2×C4#654811yescharacteristic
513C2×C4#654811yescharacteristic
514C2×C4C23×Dic711yescharacteristic
515C2×C4C22×C22⋊D711yescharacteristic
516C8—142no—
517C8—282no—
518C8—281no—
519C7#2243011yescharacteristic, complemented, Sylow
520C22—24no—
521C22—24no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22D4×C22⋊D711yescharacteristic
528C22#9049711yescharacteristic
529C22C2×#650511yescharacteristic
530C22C2×#650511yescharacteristic
531C22#9052811yescharacteristic
532C22#9052911yescharacteristic
533C22C22×C22⋊Dic711yescharacteristic
534C4—24no—
535C4—24no—
536C4—22no—
537C4—44no—
538C4—42no—
539C4—282no—
540C4—561no—
541C4—562no—
542C4#8999714yescomplemented
543C4C2×D4×Dic711yescharacteristic
544C4#9048911yescharacteristic
545C4D4×C22⋊D711yescharacteristic
546C4#9049711yescharacteristic
547C2—44no—
548C2#1174099911yescharacteristic
549C2#1174100011yescharacteristic
550C2#1174100511yescharacteristic
551C2#1174101111yescharacteristic
552C2D4×C22⋊Dic711yescharacteristic
553C2C2×#8999711yescharacteristic
554C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×D4×Q8C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
cosocleC22×C4×C7C24
hypercenterC2×D4×Q8D7
hypocenterC7#22430
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#89997C2semidirect
5. #11729886C2semidirect
6. #11729539C2semidirect
7. #11729532C2semidirect
8. #11729451C2semidirect
9. #11729447C2semidirect
10. C4×#6409C2semidirect
11. #11729380C2semidirect
14. C7×D4×Q8C4semidirect
102. #6409D4semidirect
198. D4×Q8Dic7semidirect
368. C4×C7#419semidirect
519. C7#22430semidirect
542. C4#89997semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
C2×#899972
#117397822
C2×#897842
#117298862
#117295392
#117295322
#117294512
#117294472
C4×#64092
#117293802
#224307