Finite Group Databasebeta

← #13395124

Subgroups — #13395124

Order 1792.

828 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6509C211yesmaximal, characteristic
2C2×#89793C211yesmaximal, characteristic, complemented
3C2×#89607C211yesmaximal, characteristic
4#11731821C211yesmaximal, characteristic
5C7×#1803C211yesmaximal, characteristic, complemented
6#11728213C211yesmaximal, characteristic, complemented
7#11728028C212yesmaximal
8#11728009C212yesmaximal, complemented
9#11727322C211yesmaximal, characteristic, complemented
10#11726643C212yesmaximal
11#11726615C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×#6460—21no—
15C2×C42×D7C2211yescharacteristic
16C2×#6432C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#103C2212yes—
19C7×#384C2211yescharacteristic
20#90002C2211yescharacteristic
21C2×#6409C2212yes—
22C2×(C2×D7)⋊C8C2212yes—
23C2×#6389C2212yes—
24#89793—24no—
25#89761—21no—
26#89759—21no—
27#89759C2211yescharacteristic
28#89759C2212yes—
29#89757—21no—
30#89757C2212yes—
31#89750—21no—
32#89746—21no—
33#89745C2211yescharacteristic
34C2×#6382C2212yes—
35C2×#6381C2212yes—
36C4×(C2×D7)⋊C4—21no—
37C4×C4⋊Dic7C2211yescharacteristic
38C4×Dic7⋊C4—21no—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40#89607—24no—
41C7×#340C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#24711—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C4×D7D411yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7C2311yescharacteristic
50C22×Dic14—21no—
51C22×C7×C8C2312yes—
52C4×C7×Q8—22no—
53C4×C7×Q8—22no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C7×C4⋊C4C2312yes—
59C2×C42×C7—21no—
60C2×C42×C7C2311yescharacteristic
61#6509—22no—
62#6509—22no—
63C2×(C2×D7)⋊C4—21no—
64C2×(C2×D7)⋊C4—22no—
65C2×(C2×D7)⋊C4—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7C2312yes—
69C2×C4⋊Dic7C2311yescharacteristic
70C2×C4⋊Dic7C2311yescharacteristic
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×C4×Dic7—21no—
74C2×C4×Dic7D411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7C2311yescharacteristic
77C22×C7⋊C8C2312yes—
78#6460—44no—
79C42×D7—22no—
80C42×D7—22no—
81#6432—22no—
82#6432—22no—
83C7×C4⋊C8—22no—
84C7×#103—24no—
85#6409—24no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93(C2×D7)⋊C8—24no—
94#6389—24no—
95#6382—24no—
96#6381—24no—
97C4⋊(C7⋊C8)D414yes—
98C2×#477—71no—
99C2×#432—71no—
100#2857—71no—
101C4×C22⋊Q8—71no—
102#2075—72no—
103#2035—72no—
104#1987—71no—
105#1948—71no—
106#1944—72no—
107#1935—72no—
108#1803D711yescharacteristic, complemented
109C23×D7C2×D411yescharacteristic
110C2×C7×Q8—21no—
111C2×C7×Q8—21no—
112C2×C7×Q8—21no—
113C2×C7×Q8—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8#4112yes—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2411yescharacteristic
122C22×Dic7—21no—
123C22×Dic7—21no—
124C22×Dic7—22no—
125C22×Dic7C2×D411yescharacteristic
126C22×Dic7C2×D411yescharacteristic
127C22×Dic7C2×D411yescharacteristic
128C2×C4×D7—22no—
129C2×C4×D7—22no—
130C2×C4×D7—22no—
131C2×C4×D7—22no—
132C2×C4×D7—22no—
133C2×C4×D7—21no—
134C2×C4×D7—21no—
135C2×C4×D7—21no—
136C2×C4×D7—21no—
137C2×C4×D7—21no—
138C2×C4×D7—21no—
139C2×C4×D7—41no—
140C2×Dic14—42no—
141C2×Dic14—42no—
142C2×Dic14—42no—
143C2×C7×C8—22no—
144C2×C7×C8—22no—
145C2×C7×C8—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—42no—
151C7×C4⋊C4—42no—
152C7×C4⋊C4—42no—
153C42×C7—42no—
154C42×C7C2×D412yes—
155C42×C7#4112yes—
156(C2×D7)⋊C4—24no—
157(C2×D7)⋊C4—44no—
158(C2×D7)⋊C4—42no—
159C4⋊Dic7—21no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7—21no—
163C4⋊Dic7—21no—
164C4⋊Dic7—44no—
165C4⋊Dic7—42no—
166C4⋊Dic7#4112yes—
167Dic7⋊C4—24no—
168Dic7⋊C4—42no—
169C4×Dic7—22no—
170C4×Dic7—22no—
171C4×Dic7—21no—
172C4×Dic7—21no—
173C4×Dic7—42no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8C2×D414yes—
177C2×C4⋊Q8—71no—
178C2×C22⋊Q8—71no—
179C2×C22⋊Q8—141no—
180C2×C4×Q8C2×D711yescharacteristic
181C22×C42—71no—
182#477—144no—
183#432—144no—
184C2×#107—72no—
185C2×#106—72no—
186C2×C4⋊C8—71no—
187C2×C4⋊C8C2×D711yescharacteristic
188C2×#103—72no—
189C2×#103C2×D712yes—
190C2×C22⋊C8—72no—
191C2×#97—72no—
192#391—71no—
193#391—141no—
194#389—72no—
195#389—71no—
196#389—141no—
197#387—141no—
198#385—141no—
199#384C2×D711yescharacteristic
200#380—71no—
201#380—72no—
202#380—141no—
203C4×C4⋊C4—71no—
204C4×C4⋊C4—141no—
205C4×C22⋊C4—141no—
206#340—72no—
207#340—72no—
208#340—72no—
209#340C2×D712yes—
210C23×C7C22×D411yescharacteristic
211C22×D7—21no—
212C22×D7—21no—
213C22×D7—21no—
214C22×D7—21no—
215C22×D7—21no—
216C22×D7—21no—
217C22×D7AGL(1,ℤ/8)12yes—
218C7×Q8—22no—
219C7×Q8—22no—
220C7×Q8—41no—
221C7×Q8—41no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—22no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—41no—
233C2×C4×C7#12112yes—
234C2×C4×C7#12112yes—
235C2×C4×C7#13712yes—
236C2×C4×C7#13712yes—
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C2×C4×C7C2×#4111yescharacteristic
241C2×C4×C72−1+411yescharacteristic
242C2×C4×C72−1+411yescharacteristic
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—22no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—21no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—42no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—42no—
261C4×D7—44no—
262C4×D7—41no—
263C4×D7—41no—
264C4×D7—41no—
265C4×D7—41no—
266C4×D7—44no—
267Dic14—84no—
268C7×C8—42no—
269C7⋊C8—24no—
270C22×Q8—141no—
271C22×Q8C22×D711yescharacteristic
272C23×C4—71no—
273C23×C4—71no—
274C23×C4—71no—
275C22×C8—72no—
276C22×C8C22×D712yes—
277C4⋊Q8—142no—
278C4⋊Q8—142no—
279C22⋊Q8—142no—
280C22⋊Q8—142no—
281C22⋊Q8—284no—
282C4×Q8—22no—
283C4×Q8—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—72no—
289C2×C4⋊C4—71no—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—142no—
292C2×C4⋊C4—141no—
293C2×C4⋊C4—141no—
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C4⋊C4C22×D712yes—
296C2×C22⋊C4—141no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—141no—
299C2×C42—21no—
300C2×C42—71no—
301C2×C42—71no—
302C2×C42—71no—
303C2×C42—142no—
304C2×C42—142no—
305C2×C42—141no—
306C2×C42C22×D711yescharacteristic
307#107—144no—
308#106—144no—
309C4⋊C8—22no—
310C4⋊C8—74no—
311#103—24no—
312#103—144no—
313C22⋊C8—144no—
314#97—144no—
315#95—141no—
316#95—141no—
317#95—141no—
318#95—141no—
319#95—142no—
320#95—142no—
321#95—141no—
322C22×C7C2×#12111yescharacteristic
323C22×C7#52511yescharacteristic
324C22×C7#53611yescharacteristic
325C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C7C2×#13711yescharacteristic
327C22×C7C2×#13711yescharacteristic
328C22×C7#57511yescharacteristic
329C2×D7—22no—
330C2×D7—22no—
331C2×D7—22no—
332C2×D7—41no—
333C2×D7—41no—
334C2×D7—41no—
335C2×D7—41no—
336C4×C7—22no—
337C4×C7—42no—
338C4×C7—82no—
339C4×C7—81no—
340C4×C7#47014yes—
341C4×C7#54711yescharacteristic
342C4×C7#54811yescharacteristic
343C4×C7#55211yescharacteristic
344C4×C7#55611yescharacteristic
345Dic7—42no—
346Dic7—41no—
347Dic7—41no—
348Dic7—41no—
349Dic7—81no—
350Dic7—82no—
351C24—71no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—282no—
358C2×Q8—282no—
359C2×Q8—282no—
360C2×Q8#115912yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—71no—
365C22×C4—71no—
366C22×C4—71no—
367C22×C4—142no—
368C22×C4—141no—
369C22×C4—142no—
370C22×C4—142no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—141no—
374C22×C4—141no—
375C22×C4—141no—
376C22×C4—141no—
377C22×C4—141no—
378C22×C4—141no—
379C22×C4—141no—
380C22×C4—142no—
381C22×C4—281no—
382C22×C4D4×D711yescharacteristic
383C22×C4D4×D711yescharacteristic
384C22×C4C23×D711yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—74no—
389C2×C8—142no—
390C2×C8—142no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—72no—
399C4⋊C4—144no—
400C4⋊C4—141no—
401C4⋊C4—142no—
402C4⋊C4—142no—
403C4⋊C4—141no—
404C4⋊C4—141no—
405C4⋊C4—282no—
406C4⋊C4—284no—
407C4⋊C4—282no—
408C22⋊C4—144no—
409C22⋊C4—282no—
410C22⋊C4—284no—
411C42—42no—
412C42—142no—
413C42—142no—
414C42—141no—
415C42—141no—
416C42—282no—
417C42#115712yes—
418C42D4×D712yes—
419C2×C7#241511yescharacteristic
420C2×C7C2×#47011yescharacteristic
421C2×C7#305511yescharacteristic
422C2×C7#311211yescharacteristic
423C2×C7#311611yescharacteristic
424C2×C7#323711yescharacteristic
425C2×C7#323811yescharacteristic
426D7—42no—
427C23—72no—
428C23—141no—
429C23—141no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23C2×D4×D711yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—564no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—141no—
452C2×C4—141no—
453C2×C4—141no—
454C2×C4—141no—
455C2×C4—141no—
456C2×C4—141no—
457C2×C4—141no—
458C2×C4—142no—
459C2×C4—141no—
460C2×C4—141no—
461C2×C4—284no—
462C2×C4—281no—
463C2×C4—281no—
464C2×C4—281no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—281no—
468C2×C4—284no—
469C2×C4—282no—
470C2×C4—281no—
471C2×C4—282no—
472C2×C4—281no—
473C2×C4—282no—
474C2×C4—281no—
475C2×C4#644712yes—
476C2×C4#644712yes—
477C2×C4#647812yes—
478C2×C4#648112yes—
479C2×C4C2×#115711yescharacteristic
480C2×C4C2×D4×D711yescharacteristic
481C2×C4C2×#115911yescharacteristic
482C2×C4#655111yescharacteristic
483C2×C4D7×#4111yescharacteristic
484C2×C4#655411yescharacteristic
485C8—42no—
486C8—144no—
487C7#2471111yescharacteristic, complemented, Sylow
488C22—142no—
489C22—142no—
490C22—142no—
491C22—281no—
492C22—281no—
493C22—281no—
494C22—281no—
495C22C2×#644711yescharacteristic
496C22#9029711yescharacteristic
497C22#9031911yescharacteristic
498C22C2×#647811yescharacteristic
499C22C2×#648111yescharacteristic
500C22D7×AGL(1,ℤ/8)11yescharacteristic
501C22#9046811yescharacteristic
502C4—22no—
503C4—42no—
504C4—81no—
505C4—82no—
506C4—281no—
507C4—281no—
508C4—282no—
509C4—281no—
510C4—562no—
511C4—561no—
512C4#8958814yes—
513C4#9020411yescharacteristic
514C4#9022411yescharacteristic
515C4#9028911yescharacteristic
516C4#9032211yescharacteristic
517C2—282no—
518C2#1173146511yescharacteristic
519C2C2×#8958811yescharacteristic
520C2#1173633311yescharacteristic
521C2#1173780711yescharacteristic
522C2#1173790411yescharacteristic
523C2#1173846211yescharacteristic
524C2#1173856011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1803C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#1803D7
hypocenterC7#24711
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#89793C2semidirect
5. C7×#1803C2semidirect
6. #11728213C2semidirect
8. #11728009C2semidirect
9. #11727322C2semidirect
11. #11726615C2semidirect
108. #1803D7semidirect
487. C7#24711semidirect

Maximal subgroups

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

subgroupindex
C4×#65092
C2×#897932
C2×#896072
#117318212
C7×#18032
#117282132
#117280282
#117280092
#117273222
#117266432
#117266152
#247117