Finite Group Databasebeta

← #13410323

Subgroups — #13410323

Order 1792.

1260 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2C2×#89854C211yesmaximal, characteristic, complemented
3C2×#89794C211yesmaximal, characteristic, complemented
4#11739828C211yesmaximal, characteristic
5#11729886C212yesmaximal, complemented
6#11729886C212yesmaximal, complemented
7#11729532C211yesmaximal, characteristic, complemented
8#11729449C212yesmaximal, complemented
9#11729448C212yesmaximal, complemented
10#11729385C211yesmaximal, characteristic, complemented
11#11729369C211yesmaximal, characteristic
12C23×C7×Q8C2212yes—
13C7×D4×Q8—24no—
14C7×D4×Q8—24no—
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×#6433C2211yescharacteristic
23C2×#6409C2212yes—
24C2×#6409C2212yes—
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30#89994—21no—
31#89984C2212yes—
32#89984C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89854—24no—
36#89794—24no—
37#89754—21no—
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89717—22no—
41C4×C4⋊Dic7C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#25894—71nomaximal, Sylow
46C22×C7×Q8—24no—
47C22×C7×Q8—24no—
48C22×C7×Q8—22no—
49C22×C7×Q8—24no—
50C22×C7×Q8—24no—
51C22×C7×Q8—42no—
52C22×C7×Q8—42no—
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⋊Q8D414yes—
61C7×C22⋊Q8—216no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C4×C7×Q8—22no—
65C4×C7×Q8D414yes—
66C4×C7×D4—28no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
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—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90#6433—22no—
91#6433—22no—
92#6409—22no—
93#6409—22no—
94#6409—24no—
95#6409—24no—
96#6409—24no—
97#6409—42no—
98#6409—42no—
99C7⋊#95—21no—
100C7⋊#95—22no—
101C7⋊#95—22no—
102C22⋊(C7⋊C8)—42no—
103C22⋊(C7⋊C8)—42no—
104#6382—22no—
105#6381—22no—
106C4⋊(C7⋊C8)—22no—
107#6377—22no—
108C2×D4×Q8D711yescharacteristic
109C2×#491—71no—
110C2×#479—71no—
111#2562—71no—
112#2036—72no—
113#2035—72no—
114#1981—71no—
115#1955—71no—
116#1787—72no—
117#1787—72no—
118#1763—71no—
119C24×C7C2×D412yes—
120C2×C7×Q8—28no—
121C2×C7×Q8—21no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—44no—
129C2×C7×Q8—44no—
130C2×C7×Q8—48no—
131C2×C7×Q8—48no—
132C2×C7×Q8—48no—
133C2×C7×Q8—48no—
134C2×C7×Q8C2×D414yes—
135C2×C7×Q8C2×D412yes—
136C2×C7×Q8#4114yes—
137C2×C7×D4—24no—
138C2×C7×D4—22no—
139C22×C4×C7—21no—
140C22×C4×C7—28no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—44no—
150C22×C4×C7—44no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C7×C4⋊C4—28no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4C2×D412yes—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—24no—
167C42×C7—24no—
168C42×C7C2×D412yes—
169C42×C7C2×D412yes—
170C22⋊Dic7—44no—
171C22⋊Dic7—82no—
172C4⋊Dic7—21no—
173C4⋊Dic7—21no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178Dic7⋊C4—42no—
179Dic7⋊C4—42no—
180C4×Dic7—81no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C23×Q8C2×D712yes—
189D4×Q8—24no—
190D4×Q8—24no—
191C2×C4⋊Q8—21no—
192C2×C4⋊Q8C2×D711yescharacteristic
193C2×#125—71no—
194C2×C22⋊Q8—22no—
195C2×C22⋊Q8C2×D712yes—
196C2×C4×Q8C2×D711yescharacteristic
197C2×C4×D4—21no—
198C2×C4×D4C2×D711yescharacteristic
199#491—144no—
200#479—144no—
201C2×#107—71no—
202C2×#106—71no—
203C2×C4⋊C8—71no—
204C2×#103—71no—
205C2×#103—71no—
206C2×#103—72no—
207C2×#103—72no—
208C2×#103—72no—
209C2×#103—72no—
210C2×#103—72no—
211C2×C22⋊C8—72no—
212C2×C22⋊C8—72no—
213C2×#97—71no—
214#400—141no—
215#394—141no—
216#388—142no—
217#380—72no—
218#380—72no—
219C4×C4⋊C4—71no—
220#340—72no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7#13714yes—
228C23×C7C22×D411yescharacteristic
229C7×Q8—22no—
230C7×Q8—24no—
231C7×Q8—24no—
232C7×Q8—24no—
233C7×Q8—416no—
234C7×Q8—42no—
235C7×Q8—42no—
236C7×Q8#12414yes—
237C7×D4—28no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—22no—
244C2×C4×C7—24no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—416no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7#12012yes—
258C2×C4×C7#12012yes—
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C2×#4112yes—
262C2×C4×C7H(5,𝔽2)12yes—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C7⋊C8—41no—
272C7⋊C8—41no—
273C7⋊C8—42no—
274C22×Q8—24no—
275C22×Q8—24no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—42no—
280C22×Q8—42no—
281C22×Q8C22×D712yes—
282C22×Q8C22×D711yescharacteristic
283C22×D4C22⋊D711yescharacteristic
284C23×C4—22no—
285C23×C4C22×D712yes—
286C22×C8—72no—
287C22×C8—71no—
288C22×C8—71no—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8C22⋊D714yes—
292#125—142no—
293#125—142no—
294C22⋊Q8—216no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C4×Q8—22no—
298C4×Q8C22⋊D714yes—
299C4×D4—28no—
300C4×D4—22no—
301C4×D4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—72no—
309C2×C4⋊C4—141no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4C22⋊D711yescharacteristic
312C2×C4⋊C4C22×D711yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—142no—
315C2×C22⋊C4—142no—
316C2×C22⋊C4C22⋊D712yes—
317C2×C42—21no—
318C2×C42—141no—
319C2×C42C22×D711yescharacteristic
320#107—142no—
321#106—142no—
322C4⋊C8—142no—
323#103—144no—
324#103—142no—
325#103—142no—
326#103—144no—
327#103—144no—
328#103—282no—
329#103—282no—
330C22⋊C8—282no—
331C22⋊C8—282no—
332#97—142no—
333#95—141no—
334#95—142no—
335#95—142no—
336C22×C7—24no—
337C22×C7—24no—
338C22×C7—24no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7C2×#12011yescharacteristic
344C22×C7#54612yes—
345C22×C7C2×#13712yes—
346C22×C7#57711yescharacteristic
347C22×C7#57811yescharacteristic
348C4×C7—24no—
349C4×C7—22no—
350C4×C7—22no—
351C4×C7—42no—
352C4×C7—44no—
353C4×C7—42no—
354C4×C7C2×#12411yescharacteristic
355C4×C7#53511yescharacteristic
356C4×C7#54612yes—
357Dic7—81no—
358Dic7—81no—
359Dic7—82no—
360C24C2×C22⋊D712yes—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8#115914yes—
376C2×Q8C2×C22⋊D714yes—
377C2×Q8C2×C22⋊D712yes—
378C2×D4—24no—
379C2×D4—22no—
380C22×C4—21no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—142no—
393C22×C4—141no—
394C22×C4—141no—
395C22×C4C2×C22⋊D712yes—
396C22×C4C2×C22⋊D711yescharacteristic
397C22×C4C2×C22⋊D711yescharacteristic
398C22×C4C23×D711yescharacteristic
399C2×C8—142no—
400C2×C8—142no—
401C2×C8—142no—
402C2×C8—142no—
403C2×C8—142no—
404C2×C8—141no—
405C2×C8—141no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—142no—
413C4⋊C4—142no—
414C4⋊C4—141no—
415C4⋊C4—141no—
416C4⋊C4—141no—
417C4⋊C4—141no—
418C4⋊C4—282no—
419C4⋊C4—282no—
420C4⋊C4C2×C22⋊D712yes—
421C22⋊C4—28no—
422C22⋊C4—24no—
423C22⋊C4—284no—
424C22⋊C4—562no—
425C42—24no—
426C42—561no—
427C42C2×C22⋊D712yes—
428C42C2×C22⋊D712yes—
429C2×C7—44no—
430C2×C7#243511yescharacteristic
431C2×C7#300411yescharacteristic
432C2×C7#301311yescharacteristic
433C2×C7#331212yes—
434C2×C7#332012yes—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23#650514yes—
442C23C22×C22⋊D711yescharacteristic
443Q8—22no—
444Q8—24no—
445Q8—24no—
446Q8—24no—
447Q8—416no—
448Q8—42no—
449Q8—42no—
450Q8#649914yes—
451D4—28no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—416no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—281no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4#651612yes—
480C2×C4#651612yes—
481C2×C4C2×#115912yes—
482C2×C4#654812yes—
483C2×C4C22×C22⋊D711yescharacteristic
484C2×C4C22×C22⋊D711yescharacteristic
485C8—281no—
486C8—281no—
487C8—282no—
488C7#2589411yescharacteristic, complemented, Sylow
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22#9049712yes—
497C22C2×#650512yes—
498C22#9052811yescharacteristic
499C22#9052911yescharacteristic
500C22C2×#651611yescharacteristic
501C4—24no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—561no—
508C4—562no—
509C4—561no—
510C4C2×#649911yescharacteristic
511C4#9049611yescharacteristic
512C4#9049712yes—
513C2—44no—
514C2#1174099712yes—
515C2#1174099812yes—
516C2#1174149511yescharacteristic
517C2#1174150411yescharacteristic
518C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×D4×Q8C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×D4×Q8D7
hypocenterC7#25894
LCSG ▷ C22×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×#89854C2semidirect
3. C2×#89794C2semidirect
5. #11729886C2semidirect
6. #11729886C2semidirect
7. #11729532C2semidirect
8. #11729449C2semidirect
9. #11729448C2semidirect
10. #11729385C2semidirect
488. C7#25894semidirect

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×#898542
C2×#897942
#117398282
#117298862
#117295322
#117294492
#117294482
#117293852
#117293692
#258947