Finite Group Databasebeta

← #13359472

Subgroups — #13359472

Order 1792.

1436 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
1C22×C4×C7×Q8C211yesmaximal, characteristic
2C2×#90000C211yesmaximal, characteristic
3C2×#89802C212yesmaximal, complemented
4C4×#6487C211yesmaximal, characteristic
5#11729886C211yesmaximal, characteristic
6#11729386C212yesmaximal, complemented
7#11729385C212yesmaximal, complemented
8#11729383C212yesmaximal, complemented
9#11729313C211yesmaximal, characteristic
10#11729310C211yesmaximal, characteristic
11#11729283C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C2×C4×C7×Q8—28no—
14C2×C4×C7×Q8—21no—
15C2×C4×C7×Q8—21no—
16C2×C4×C7×Q8C2212yes—
17C2×C4×C7×Q8C2212yes—
18C22×C7×C4⋊C4—21no—
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C42×C7—21no—
21C22×C42×C7C2211yescharacteristic
22C22×#1144C2212yes—
23C2×#6487C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25#90000C418yescomplemented
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29#89984C2211yescharacteristic
30#89982—21no—
31C4×C22⋊Dic7—21no—
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34#89802—28no—
35C2×#6384C2212yes—
36C2×#6382C2212yes—
37C4×C4⋊Dic7C2211yescharacteristic
38#89705—21no—
39C4×Dic7⋊C4—21no—
40#89701C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×#6377C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89319C2212yes—
46#21455—71nomaximal, Sylow
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—21no—
50C22×C7×Q8—22no—
51C22×C7×Q8C2×C414yes—
52C22×C7×Q8C2312yes—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C22×Dic14C2311yescharacteristic
59C4×C7×Q8—28no—
60C4×C7×Q8—28no—
61C4×C7×Q8—24no—
62C4×C7×Q8—24no—
63C4×C7×Q8—24no—
64C4×C7×Q8D418yes—
65C2×C7×C4⋊C4—28no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4C2312yes—
71C2×C42×C7—28no—
72C2×C42×C7—22no—
73C2×C42×C7—22no—
74C2×C42×C7—24no—
75C2×C42×C7D411yescharacteristic
76C2×C42×C7D411yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×#1144—24no—
82C2×#1144—24no—
83C2×#1144C2×C418yescomplemented
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87#6487—22no—
88#6487C2×C414yes—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C22×C7⋊C8C2312yes—
95C22×C7⋊C8C2312yes—
96C4×Dic14—24no—
97#6409—24no—
98#6409—24no—
99#6409C2×C418yescomplemented
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C22⋊(C7⋊C8)—22no—
106C22⋊(C7⋊C8)C2×C414yes—
107#6384—24no—
108#6382—24no—
109#6377—24no—
110C22×C4×Q8D711yescharacteristic
111C2×#451—71no—
112C2×#441—72no—
113C4×C22⋊Q8—71no—
114#1878—72no—
115#1872—71no—
116#1865—72no—
117#1853—71no—
118#1787—71no—
119#1763—72no—
120#1749—71no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—24no—
123C2×C7×Q8—216no—
124C2×C7×Q8—24no—
125C2×C7×Q8—216no—
126C2×C7×Q8—24no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C22×C414yes—
131C2×C7×Q8C2×D414yes—
132C2×C7×Q8#4114yes—
133C22×C4×C7—24no—
134C22×C4×C7—24no—
135C22×C4×C7—24no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—41no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7#4112yes—
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—21no—
164C2×Dic14C22×C412yes—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4C2×D414yes—
170C42×C7—28no—
171C42×C7—28no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7—22no—
175C42×C7C2×D414yes—
176C22⋊Dic7—24no—
177C22⋊Dic7—42no—
178#1144—216no—
179C4⋊Dic7—21no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7C22×C412yes—
183Dic7⋊C4—24no—
184Dic7⋊C4—42no—
185Dic7⋊C4—42no—
186C4×Dic7—42no—
187C4×Dic7—42no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8C22×C414yes—
194C23×Q8C2×D711yescharacteristic
195C22×Q16—72no—
196C2×C22⋊Q8—71no—
197C2×C4×Q8—28no—
198C2×C4×Q8—21no—
199C2×C4×Q8—21no—
200C2×C4×Q8—71no—
201C2×C4×Q8C2×D712yes—
202C2×C4×Q8C2×D712yes—
203C22×C4⋊C4—21no—
204C22×C4⋊C4C2×D711yescharacteristic
205C22×C42—21no—
206C22×C42C2×D711yescharacteristic
207#451—78no—
208#441—148no—
209C2×#106—72no—
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103—72no—
214C2×C22⋊C8—71no—
215C2×C22⋊C8—71no—
216C2×#97—72no—
217#391—71no—
218#389—71no—
219#387—141no—
220#385—141no—
221#380—71no—
222C4×C4⋊C4—71no—
223C4×C4⋊C4—141no—
224C4×C22⋊C4—141no—
225#340—72no—
226#340—72no—
227#336—72no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7C4×D412yes—
233C23×C7#13712yes—
234C23×C7#13712yes—
235C23×C7C22×D411yescharacteristic
236C7×Q8—216no—
237C7×Q8—24no—
238C7×Q8—24no—
239C7×Q8—24no—
240C7×Q8C4×D418yes—
241C2×C4×C7—28no—
242C2×C4×C7—216no—
243C2×C4×C7—24no—
244C2×C4×C7—21no—
245C2×C4×C7—24no—
246C2×C4×C7—21no—
247C2×C4×C7—24no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—21no—
257C2×C4×C7—24no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—44no—
261C2×C4×C7—48no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7C4×D412yes—
265C2×C4×C7#12012yes—
266C2×C4×C7#12012yes—
267C2×C4×C7#13712yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C22×D412yes—
270C2×C4×C7C2×#4112yes—
271C2×C4×C7C2×#4111yescharacteristic
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284Dic14—24no—
285Dic14—42no—
286C7⋊C8—24no—
287C7⋊C8—42no—
288C22×Q8—24no—
289C22×Q8—24no—
290C22×Q8—22no—
291C22×Q8—21no—
292C22×Q8—71no—
293C22×Q8C4×D714yes—
294C22×Q8C22×D712yes—
295C23×C4—21no—
296C23×C4—21no—
297C23×C4C22⋊D711yescharacteristic
298C23×C4C22⋊D711yescharacteristic
299C23×C4C22×D711yescharacteristic
300C2×Q16—78no—
301C2×Q16—144no—
302C2×Q16—144no—
303C22×C8—72no—
304C22×C8—72no—
305C22⋊Q8—74no—
306C22⋊Q8—142no—
307C4×Q8—28no—
308C4×Q8—28no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—144no—
313C4×Q8C22⋊D718yes—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—71no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4C22×D712yes—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—141no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—141no—
333C2×C42—141no—
334C2×C42C22⋊D711yescharacteristic
335C2×C42C22⋊D711yescharacteristic
336C2×C42C22×D711yescharacteristic
337C2×C42C22×D711yescharacteristic
338#106—144no—
339#103—78no—
340#103—144no—
341#103—144no—
342#103—144no—
343C22⋊C8—74no—
344C22⋊C8—142no—
345#97—144no—
346#95—141no—
347#95—141no—
348#95—141no—
349#95—141no—
350#95—141no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7#44114yescomplemented
358C22×C7C2×C4×D411yescharacteristic
359C22×C7C2×#12011yescharacteristic
360C22×C7#52511yescharacteristic
361C22×C7C2×#13711yescharacteristic
362C22×C7C2×#13711yescharacteristic
363C22×C7C2×#13711yescharacteristic
364C22×C7#57511yescharacteristic
365C4×C7—22no—
366C4×C7—22no—
367C4×C7—22no—
368C4×C7—28no—
369C4×C7—22no—
370C4×C7—41no—
371C4×C7—44no—
372C4×C7C2×C4×D412yes—
373C4×C7#52512yes—
374Dic7—42no—
375Dic7—42no—
376Dic7—81no—
377Dic7—81no—
378C24C2×C22⋊D711yescharacteristic
379C2×Q8—24no—
380C2×Q8—216no—
381C2×Q8—24no—
382C2×Q8—216no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—72no—
388C2×Q8—141no—
389C2×Q8—142no—
390C2×Q8—142no—
391C2×Q8C2×C4×D714yes—
392C2×Q8#115714yes—
393C2×Q8C2×C22⋊D714yes—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4C2×C4×D712yes—
415C22×C4#115712yes—
416C22×C4C2×C22⋊D711yescharacteristic
417C22×C4C2×C22⋊D711yescharacteristic
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C23×D711yescharacteristic
422Q16—1416no—
423C2×C8—74no—
424C2×C8—142no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C2×C8—142no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—22no—
433C4⋊C4—72no—
434C4⋊C4—144no—
435C4⋊C4—141no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—282no—
439C4⋊C4—282no—
440C4⋊C4C2×C22⋊D714yes—
441C22⋊C4—144no—
442C22⋊C4—282no—
443C42—28no—
444C42—28no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—282no—
449C42—282no—
450C42C2×C22⋊D714yes—
451C2×C7—24no—
452C2×C7C4×#12011yescharacteristic
453C2×C7C2×#44111yescharacteristic
454C2×C7C4×#13711yescharacteristic
455C2×C7#300411yescharacteristic
456C2×C7#300711yescharacteristic
457C2×C7#311311yescharacteristic
458C2×C7#311711yescharacteristic
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23C4×C22⋊D712yes—
464C23#650512yes—
465C23#651412yes—
466C23C22×C22⋊D711yescharacteristic
467Q8—216no—
468Q8—24no—
469Q8—24no—
470Q8—24no—
471Q8—144no—
472Q8—282no—
473Q8C4×C22⋊D718yes—
474C2×C4—28no—
475C2×C4—216no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—142no—
498C2×C4—142no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4C4×C22⋊D712yes—
510C2×C4#650512yes—
511C2×C4#651612yes—
512C2×C4#651612yes—
513C2×C4C22×C4×D711yescharacteristic
514C2×C4C2×#115712yes—
515C2×C4C2×#115711yescharacteristic
516C2×C4C22×C22⋊D712yes—
517C8—144no—
518C8—282no—
519C7#2145511yescharacteristic, complemented, Sylow
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22#8980214yescomplemented
527C22C2×C4×C22⋊D711yescharacteristic
528C22#9048211yescharacteristic
529C22C2×#650511yescharacteristic
530C22C2×#650511yescharacteristic
531C22#9051311yescharacteristic
532C22C2×#651411yescharacteristic
533C22C2×#651611yescharacteristic
534C4—28no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—41no—
540C4—44no—
541C4—282no—
542C4—282no—
543C4—561no—
544C4—561no—
545C4C2×C4×C22⋊D712yes—
546C4#9048212yes—
547C2—24no—
548C2C2×#8980211yescharacteristic
549C2C4×#650511yescharacteristic
550C2#1173989011yescharacteristic
551C2#1173989311yescharacteristic
552C2C4×#651611yescharacteristic
553C2#1174150411yescharacteristic
554C2#1174152911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C4×C7×Q8C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC22×C4×Q8D7
hypocenterC7#21455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#89802C2semidirect
6. #11729386C2semidirect
7. #11729385C2semidirect
8. #11729383C2semidirect
25. #90000C4semidirect
83. C2×#1144C2×C4semidirect
99. #6409C2×C4semidirect
357. C22×C7#441semidirect
519. C7#21455semidirect
526. C22#89802semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×Q82
C2×#900002
C2×#898022
C4×#64872
#117298862
#117293862
#117293852
#117293832
#117293132
#117293102
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