Finite Group Databasebeta

← #13381293

Subgroups — #13381293

Order 1792.

1558 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×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×#89997C211yesmaximal, characteristic, complemented
3C2×#89997C211yesmaximal, characteristic, complemented
4#11741472C211yesmaximal, characteristic
5C2×#89979C211yesmaximal, characteristic
6C2×#89736C211yesmaximal, characteristic, complemented
7#11729449C214yesmaximal, complemented
8#11729448C214yesmaximal, complemented
9#11729312C211yesmaximal, characteristic, complemented
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C2×C7×C22⋊Q8—24no—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8—24no—
15C2×C7×C22⋊Q8C2214yes—
16C22×C7×C4⋊C4—21no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4—21no—
19C2×#6489C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23#90019—21no—
24#89997—24no—
25#89997—24no—
26C2×#6409C2212yescomplemented
27C2×#6409C2214yescomplemented
28C2×#6409C2212yescomplemented
29#89984C2212yes—
30#89984C2212yes—
31#89983—21no—
32C4×C22⋊Dic7—22no—
33#89979—24no—
34C2×C22⋊(C7⋊C8)C2214yes—
35#89736—24no—
36C2×#6382C2212yescomplemented
37C2×#6381C2212yescomplemented
38#89324C2214yescomplemented
39#23481—71nomaximal, Sylow
40C25×C7D411yescharacteristic
41C22×C7×Q8—22no—
42C22×C7×Q8—21no—
43C22×C7×Q8—22no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8D414yes—
48C22×C7×Q8C2312yes—
49C23×C4×C7—21no—
50C23×C4×C7—21no—
51C23×C4×C7—22no—
52C23×C4×C7—22no—
53C23×C4×C7—22no—
54C23×C4×C7—22no—
55C23×C4×C7—22no—
56C23×C4×C7—22no—
57C23×C4×C7D412yes—
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C7×C22⋊Q8—216no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—28no—
64C2×C7×C4⋊C4—28no—
65C2×C7×C4⋊C4—24no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4D414yes—
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—28no—
71C2×C7×C22⋊C4—24no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—22no—
76#6489—22no—
77#6489—22no—
78#6489—22no—
79#6489—22no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2312yes—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—21no—
85C2×#1136—22no—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136—22no—
90C2×#1136D414yes—
91C22×C7⋊C8C2312yes—
92C22×C7⋊C8C2312yes—
93#6409—24no—
94#6409—24no—
95#6409—28no—
96C7⋊#95—22no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—28no—
99#6382—24no—
100#6381—24no—
101C22×C22⋊Q8D711yescharacteristic
102C2×#428—71no—
103C2×#419—71no—
104C2×#419—71no—
105C2×#407—71no—
106#2280—71no—
107#2036—74no—
108#2035—74no—
109#1736—71no—
110C24×C7—22no—
111C24×C7—24no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—42no—
115C24×C7C2×D412yes—
116C24×C7C2×D411yescharacteristic
117C2×C7×Q8—24no—
118C2×C7×Q8—22no—
119C2×C7×Q8—28no—
120C2×C7×Q8—28no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—42no—
126C2×C7×Q8—44no—
127C2×C7×Q8—42no—
128C2×C7×Q8—44no—
129C2×C7×Q8C2×D414yes—
130C22×C4×C7—24no—
131C22×C4×C7—24no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—44no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—44no—
150C22×C4×C7—44no—
151C22×C4×C7—44no—
152C22×C4×C7—44no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7#4112yes—
159C22×C4×C7#4112yes—
160C22×C4×C7#4112yes—
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—22no—
165C7×C4⋊C4—216no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4C2×D414yes—
169C7×C22⋊C4—216no—
170C22⋊Dic7—44no—
171C22⋊Dic7—42no—
172C22⋊Dic7—42no—
173C22⋊Dic7—44no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—22no—
180C4×Dic7—42no—
181C4×Dic7—42no—
182#1136—28no—
183#1136—44no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8C2×D414yes—
190C23×Q8C2×D711yescharacteristic
191C24×C4C2×D711yescharacteristic
192C22×M16—71no—
193C22×M16—71no—
194C2×C22⋊Q8—24no—
195C2×C22⋊Q8—22no—
196C2×C22⋊Q8—24no—
197C2×C22⋊Q8C2×D714yes—
198C2×#117—71no—
199C2×#117—71no—
200C22×C4⋊C4—21no—
201C22×C4⋊C4C2×D711yescharacteristic
202C22×C22⋊C4—21no—
203#428—144no—
204C2×#107—72no—
205C2×#106—72no—
206#419—144no—
207#419—144no—
208C2×#103—72no—
209C2×#103—74no—
210C2×#103—72no—
211#407—144no—
212C2×C22⋊C8—74no—
213#381—141no—
214#380—72no—
215#380—72no—
216#379—141no—
217C4×C22⋊C4—142no—
218#340—74no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—28no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—48no—
229C23×C7—48no—
230C23×C7—44no—
231C23×C7—44no—
232C23×C7#12012yes—
233C23×C7#12112yes—
234C23×C7#12312yes—
235C23×C7#13714yes—
236C23×C7C22×D411yescharacteristic
237C7×Q8—24no—
238C7×Q8—24no—
239C7×Q8—42no—
240C7×Q8—42no—
241C7×Q8—48no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—48no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—44no—
251C2×C4×C7—44no—
252C2×C4×C7—48no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—48no—
258C2×C4×C7#12012yes—
259C2×C4×C7#12114yes—
260C2×C4×C7#12114yes—
261C2×C4×C7#12112yes—
262C2×C4×C7#12314yes—
263C2×C4×C7#12312yes—
264C2×C4×C7#12414yes—
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C7⋊C8—24no—
279C7⋊C8—42no—
280C25C22⋊D711yescharacteristic
281C22×Q8—22no—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—22no—
286C22×Q8—22no—
287C22×Q8C22⋊D714yes—
288C22×Q8C22×D712yes—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—22no—
292C23×C4—22no—
293C23×C4—22no—
294C23×C4—22no—
295C23×C4—22no—
296C23×C4—22no—
297C23×C4C22⋊D711yescharacteristic
298C23×C4C22⋊D712yes—
299C23×C4C22×D712yes—
300C23×C4C22×D711yescharacteristic
301C2×M16—74no—
302C2×M16—142no—
303C2×M16—142no—
304C2×M16—142no—
305C2×M16—142no—
306C2×M16—142no—
307C22×C8—72no—
308C22×C8—72no—
309C22⋊Q8—216no—
310C22⋊Q8—28no—
311C22⋊Q8—28no—
312#117—142no—
313#117—142no—
314#117—142no—
315#117—142no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4C22⋊D714yes—
324C2×C4⋊C4C22×D712yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—141no—
329C2×C22⋊C4—142no—
330C2×C22⋊C4—142no—
331C2×C42—141no—
332C2×C42—141no—
333#107—144no—
334#106—144no—
335#103—148no—
336#103—144no—
337#103—144no—
338C22⋊C8—148no—
339#95—142no—
340#95—142no—
341C22×C7—24no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—24no—
345C22×C7—24no—
346C22×C7—24no—
347C22×C7—24no—
348C22×C7—416no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7#39414yes—
354C22×C7C2×#12011yescharacteristic
355C22×C7C2×#12111yescharacteristic
356C22×C7C2×#12311yescharacteristic
357C22×C7C2×#13712yes—
358C22×C7#57512yes—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7—44no—
362C4×C7#39414yes—
363C4×C7C2×#12111yescharacteristic
364C4×C7C2×#12111yescharacteristic
365C4×C7C2×#12311yescharacteristic
366C4×C7C2×#12411yescharacteristic
367Dic7—81no—
368Dic7—81no—
369Dic7—82no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—42no—
375C24C2×C22⋊D712yes—
376C24C2×C22⋊D711yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8C2×C22⋊D714yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—142no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4D4×D712yes—
417C22×C4#115912yes—
418C22×C4#115912yes—
419C22×C4#115912yes—
420C22×C4C2×C22⋊D712yes—
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D712yes—
423C22×C4C2×C22⋊D712yes—
424C22×C4C23×D711yescharacteristic
425M16—148no—
426M16—284no—
427C2×C8—74no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8—142no—
431C2×C8—142no—
432C2×C8—142no—
433C4⋊C4—216no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—141no—
442C4⋊C4C2×C22⋊D714yes—
443C22⋊C4—216no—
444C22⋊C4—284no—
445C22⋊C4—282no—
446C22⋊C4—284no—
447C22⋊C4—282no—
448C42—282no—
449C42—282no—
450C2×C7—24no—
451C2×C7—44no—
452C2×C7C2×#39411yescharacteristic
453C2×C7#300411yescharacteristic
454C2×C7#300511yescharacteristic
455C2×C7#305612yes—
456C2×C7#309512yes—
457C23—22no—
458C23—22no—
459C23—28no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—48no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23#649812yes—
471C23#650212yes—
472C23#650514yes—
473C23#651612yes—
474C23C22×C22⋊D711yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—42no—
478Q8—42no—
479Q8—48no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4#649814yes—
504C2×C4#649812yes—
505C2×C4#649914yes—
506C2×C4#650114yes—
507C2×C4#650214yes—
508C2×C4#650212yes—
509C2×C4#651612yes—
510C2×C4C2×D4×D711yescharacteristic
511C2×C4C2×#115911yescharacteristic
512C2×C4C2×#115911yescharacteristic
513C2×C4C2×#115911yescharacteristic
514C2×C4C22×C22⋊D711yescharacteristic
515C2×C4C22×C22⋊D711yescharacteristic
516C8—144no—
517C8—282no—
518C7#2348111yescharacteristic, complemented, Sylow
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#8999414yes—
532C22C2×#649811yescharacteristic
533C22C2×#650211yescharacteristic
534C22C2×#650512yes—
535C22#9051312yes—
536C22C2×#651611yescharacteristic
537C4—42no—
538C4—44no—
539C4—44no—
540C4—562no—
541C4—561no—
542C4—561no—
543C4#8999414yes—
544C4C2×#649811yescharacteristic
545C4C2×#649911yescharacteristic
546C4C2×#650111yescharacteristic
547C4C2×#650211yescharacteristic
548C2—24no—
549C2—44no—
550C2#1174017212yes—
551C2#1174017712yes—
552C2C2×#8999411yescharacteristic
553C2#1174150411yescharacteristic
554C2#1174152611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×C22⋊Q8C2
socleC23×C7C22×D4
prime radicalC25×C7D4
hypercenterC22×C22⋊Q8D7
hypocenterC7#23481
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#89997C2semidirect
3. C2×#89997C2semidirect
6. C2×#89736C2semidirect
7. #11729449C2semidirect
8. #11729448C2semidirect
9. #11729312C2semidirect
26. C2×#6409C22semidirect
27. C2×#6409C22semidirect
28. C2×#6409C22semidirect
36. C2×#6382C22semidirect
37. C2×#6381C22semidirect
38. #89324C22semidirect
518. C7#23481semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C2×#899972
#117414722
C2×#899792
C2×#897362
#117294492
#117294482
#117293122
#234817