Finite Group Databasebeta

← #13395118

Subgroups — #13395118

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
1C2×C7×#477C211yesmaximal, characteristic, complemented
2#11740714C211yesmaximal, characteristic
3C4×#6509C211yesmaximal, characteristic
4C2×#89607C211yesmaximal, characteristic
5#11729533C211yesmaximal, characteristic, complemented
6#11729380C211yesmaximal, characteristic, complemented
7#11728390C212yesmaximal
8#11728368C212yesmaximal, complemented
9#11728208C211yesmaximal, characteristic, complemented
10#11728010C212yesmaximal, complemented
11#11727997C212yesmaximal
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×#6509—21no—
15C2×#6509C2211yescharacteristic
16C2×C42×D7C2211yescharacteristic
17C7×#477—24no—
18C2×C7×#107C2212yes—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103C2212yes—
21#90002—21no—
22#90002C2211yescharacteristic
23C2×#6409C2212yes—
24C2×(C2×D7)⋊C8C2212yes—
25C2×#6391C2212yes—
26C2×#6389C2212yes—
27#89761—21no—
28#89759—21no—
29#89759C2211yescharacteristic
30#89759C2212yes—
31#89750—21no—
32#89745—21no—
33#89745C2211yescharacteristic
34#89745C2212yes—
35C4×(C2×D7)⋊C4—21no—
36#89707C2211yescharacteristic
37C4×C4⋊Dic7C2211yescharacteristic
38C4×Dic7⋊C4—21no—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40#89607—24no—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24711—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7D411yescharacteristic
50C22×C4×D7C2311yescharacteristic
51C22×C7×C8C2312yes—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8—22no—
54C4×C7×Q8—22no—
55C4×C7×Q8—22no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2312yes—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62#6509—22no—
63#6509—22no—
64#6509—44no—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—22no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2312yes—
72C2×C4⋊Dic7C2311yescharacteristic
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C22×C7⋊C8C2312yes—
81C42×D7—22no—
82C42×D7—22no—
83C7×#107—24no—
84C7×C4⋊C8D414yes—
85C7×#103—24no—
86#6409—24no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94(C2×D7)⋊C8—24no—
95#6391—24no—
96#6389—24no—
97C4⋊(C7⋊C8)—22no—
98C2×#477D711yescharacteristic, complemented
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#1803—71no—
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—42no—
116C2×C7×Q8—42no—
117C2×C7×Q8—42no—
118C2×C7×Q8#4112yes—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2411yescharacteristic
125C22×Dic7—21no—
126C22×Dic7—21no—
127C22×Dic7—22no—
128C22×Dic7C2×D411yescharacteristic
129C22×Dic7C2×D411yescharacteristic
130C22×Dic7C2×D411yescharacteristic
131C2×C4×D7—22no—
132C2×C4×D7—22no—
133C2×C4×D7—22no—
134C2×C4×D7—22no—
135C2×C4×D7—22no—
136C2×C4×D7—21no—
137C2×C4×D7—21no—
138C2×C4×D7—21no—
139C2×C4×D7—21no—
140C2×C4×D7—21no—
141C2×C4×D7—21no—
142C2×C4×D7—41no—
143C2×C7×C8—22no—
144C2×C7×C8—22no—
145C2×C7×C8C2×D414yes—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—42no—
151C7×C4⋊C4—42no—
152C42×C7—42no—
153C42×C7C2×D412yes—
154C42×C7#4112yes—
155(C2×D7)⋊C4—24no—
156(C2×D7)⋊C4—42no—
157(C2×D7)⋊C4—44no—
158C4⋊Dic7—21no—
159C4⋊Dic7—22no—
160C4⋊Dic7—22no—
161C4⋊Dic7—21no—
162C4⋊Dic7—21no—
163C4⋊Dic7—42no—
164C4⋊Dic7—42no—
165C4⋊Dic7#4112yes—
166Dic7⋊C4—24no—
167Dic7⋊C4—42no—
168Dic7⋊C4—44no—
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⋊C8—22no—
177C2×C4⋊Q8C2×D711yescharacteristic
178C2×C22⋊Q8—71no—
179C2×C22⋊Q8—141no—
180C2×C4×Q8C2×D711yescharacteristic
181C22×C42—71no—
182#477—24no—
183#432—144no—
184C2×#107C2×D712yes—
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#384—71no—
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#340—72no—
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—
222C7×Q8—84no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—22no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7#12112yes—
235C2×C4×C7#12112yes—
236C2×C4×C7#13712yes—
237C2×C4×C7#13712yes—
238C2×C4×C7C22×D411yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C2×C4×C7C2×#4111yescharacteristic
241C2×C4×C7C2×#4111yescharacteristic
242C2×C4×C72−1+411yescharacteristic
243C2×C4×C72−1+411yescharacteristic
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—22no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—21no—
253C2×Dic7—21no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—42no—
257C2×Dic7—41no—
258C2×Dic7—42no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—42no—
262C4×D7—44no—
263C4×D7—41no—
264C4×D7—41no—
265C4×D7—41no—
266C4×D7—41no—
267C4×D7—44no—
268C7×C8—24no—
269C7⋊C8—42no—
270C22×Q8—21no—
271C22×Q8C22×D711yescharacteristic
272C23×C4—71no—
273C23×C4—71no—
274C23×C4—71no—
275C22×C8—72no—
276C22×C8C22×D712yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
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—71no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—72no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4—142no—
291C2×C4⋊C4—141no—
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—24no—
308#106—144no—
309C4⋊C8—142no—
310C4⋊C8C22⋊D714yes—
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—81no—
339C4×C7—82no—
340C4×C7#47014yes—
341C4×C7#54711yescharacteristic
342C4×C7#54811yescharacteristic
343C4×C7#55211yescharacteristic
344C4×C7#55611yescharacteristic
345Dic7—42no—
346Dic7—41no—
347Dic7—41no—
348Dic7—41no—
349Dic7—82no—
350Dic7—81no—
351C24—71no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8#116112yes—
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—142no—
388C2×C8—142no—
389C2×C8—142no—
390C2×C8C2×C22⋊D714yes—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—72no—
398C4⋊C4—144no—
399C4⋊C4—141no—
400C4⋊C4—142no—
401C4⋊C4—141no—
402C4⋊C4—142no—
403C4⋊C4—141no—
404C4⋊C4—282no—
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#115912yes—
418C42C2×C22⋊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—84no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
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—284no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—281no—
468C2×C4—281no—
469C2×C4—281no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4—281no—
473C2×C4—282no—
474C2×C4—281no—
475C2×C4#648112yes—
476C2×C4#648112yes—
477C2×C4#650112yes—
478C2×C4#650112yes—
479C2×C4C2×#115911yescharacteristic
480C2×C4C2×#116111yescharacteristic
481C2×C4#655111yescharacteristic
482C2×C4D7×#4111yescharacteristic
483C2×C4#655411yescharacteristic
484C2×C4C22×C22⋊D711yescharacteristic
485C8—24no—
486C8—282no—
487C7#2471111yescharacteristic, complemented, Sylow
488C22—142no—
489C22—142no—
490C22—142no—
491C22—281no—
492C22—281no—
493C22—281no—
494C22—281no—
495C22#9031811yescharacteristic
496C22#9031911yescharacteristic
497C22C2×#648111yescharacteristic
498C22C2×#648111yescharacteristic
499C22D7×AGL(1,ℤ/8)11yescharacteristic
500C22#9046411yescharacteristic
501C22C2×#650111yescharacteristic
502C4—22no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—281no—
507C4—281no—
508C4—282no—
509C4—281no—
510C4—562no—
511C4—561no—
512C4#8996114yes—
513C4#9033211yescharacteristic
514C4#9033311yescharacteristic
515C4#9050811yescharacteristic
516C4#9051511yescharacteristic
517C2—282no—
518C2#1173853911yescharacteristic
519C2#1173854011yescharacteristic
520C2#1173855911yescharacteristic
521C2#1173856011yescharacteristic
522C2#1174025811yescharacteristic
523C2C2×#8996111yescharacteristic
524C2#1174139511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#477C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#477D7
hypocenterC7#24711
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×C7×#477C2semidirect
5. #11729533C2semidirect
6. #11729380C2semidirect
8. #11728368C2semidirect
9. #11728208C2semidirect
10. #11728010C2semidirect
98. C2×#477D7semidirect
487. C7#24711semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4772
#117407142
C4×#65092
C2×#896072
#117295332
#117293802
#117283902
#117283682
#117282082
#117280102
#117279972
#247117