Finite Group Databasebeta

← #13383786

Subgroups — #13383786

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#121C211yesmaximal, characteristic
2#11739507C211yesmaximal, characteristic, complemented
3#11729433C212yesmaximal
4C4×#6406C212yesmaximal, complemented
5C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11727409C211yesmaximal, characteristic, complemented
7#11727394C211yesmaximal, characteristic, complemented
8#11727365C211yesmaximal, characteristic, complemented
9#11727361C211yesmaximal, characteristic, complemented
10#11727272C212yesmaximal, complemented
11C4×#6381C212yesmaximal, complemented
12C2×C7×#121C2211yescharacteristic
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C2×#6489—21no—
17C7×#390C2211yescharacteristic
18C7×#389C2211yescharacteristic
19C7×#388—21no—
20C7×#386—21no—
21C7×#380C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23C4×C7×C22⋊C4—21no—
24C2×#6406C2212yes—
25C2×#6406C2212yes—
26#89984—21no—
27#89984C2212yes—
28C2×C22⋊(C7⋊C8)C2212yes—
29C2×#6381C2212yes—
30C2×#6381C2212yes—
31#89708—21no—
32#89707C2211yescharacteristic
33#89707C2211yescharacteristic
34C4×C4⋊Dic7—21no—
35C4×C4⋊Dic7C2212yes—
36C2×C4×C7⋊C8C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89319C2211yescharacteristic
42#89319C2211yescharacteristic
43#23754—71nomaximal, Sylow
44C22×C7×D4—21no—
45C22×C7×D4C2311yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C7×#121—22no—
51C7×#121—22no—
52C4×C7×D4—24no—
53C4×C7×D4—44no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C42×C7—24no—
64C2×C42×C7—21no—
65C2×C42×C7D411yescharacteristic
66C2×C42×C7D411yescharacteristic
67C2×C42×C7C2311yescharacteristic
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70#6489—44no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2312yes—
76C2×C4⋊Dic7C2312yes—
77C2×C4×Dic7—22no—
78C22×C7⋊C8C2312yes—
79C22×C7⋊C8C2312yes—
80C7×#95—21no—
81C7×#95—21no—
82C7×#95—21no—
83#6406—24no—
84#6406—24no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C22⋊(C7⋊C8)—24no—
88#6381—24no—
89#6381—24no—
90C4×C7⋊C8—24no—
91C4×#121D711yescharacteristic
92#2285—71no—
93#2099—71no—
94#2093—71no—
95#2083—71no—
96#2075—72no—
97#2071—71no—
98#2033—72no—
99C4×#107—72no—
100C4×#102—72no—
101C4×C22⋊C8—71no—
102C24×C7—21no—
103C24×C7C2×D411yescharacteristic
104C2×C7×D4—24no—
105C2×C7×D4—22no—
106C2×C7×D4—22no—
107C2×C7×D4—21no—
108C2×C7×D4—42no—
109C2×C7×D4—42no—
110C2×C7×D4#4112yes—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—41no—
123C22×C4×C7—42no—
124C22×C4×C7—42no—
125C22×C4×C7—42no—
126C22×C4×C7D814yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2411yescharacteristic
133C22×Dic7—22no—
134C22×Dic7—22no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4#4112yes—
140C7×C22⋊C4—24no—
141C7×C22⋊C4—42no—
142C7×C22⋊C4—42no—
143C7×C22⋊C4—42no—
144C7×C22⋊C4—42no—
145C42×C7—22no—
146C42×C7—22no—
147C42×C7—22no—
148C42×C7—42no—
149C42×C7#4114yes—
150C22⋊Dic7—44no—
151C4⋊Dic7—22no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—42no—
156C4⋊Dic7—42no—
157C4⋊Dic7—42no—
158C4⋊Dic7—42no—
159C4×Dic7—44no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—22no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—22no—
164C2×C7⋊C8—22no—
165C2×C7⋊C8#4114yes—
166C2×#121C2×D711yescharacteristic
167C2×C4×D4—21no—
168C2×C4×D4C2×D711yescharacteristic
169C2×#117—141no—
170C22×C42C2×D711yescharacteristic
171C2×#107—72no—
172C2×#107—72no—
173C2×#102—72no—
174C2×#102—72no—
175C2×C22⋊C8—72no—
176C2×C4×C8—72no—
177#390C2×D711yescharacteristic
178#389C2×D711yescharacteristic
179#388—21no—
180#386—21no—
181#384—71no—
182#384—71no—
183#382—141no—
184#380—72no—
185#380—141no—
186#380C2×D711yescharacteristic
187C4×C4⋊C4—72no—
188C4×C4⋊C4—141no—
189C4×C4⋊C4C2×D711yescharacteristic
190C4×C22⋊C4—21no—
191#340—72no—
192#340—72no—
193#340—72no—
194#340—72no—
195#336—71no—
196#336—71no—
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—22no—
204C23×C7—41no—
205C23×C7—41no—
206C23×C7—41no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7C2×D812yes—
211C23×C7C22×D411yescharacteristic
212C7×D4—24no—
213C7×D4—48no—
214C7×D4—42no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—28no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—22no—
225C2×C4×C7—21no—
226C2×C4×C7—44no—
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—41no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—84no—
238C2×C4×C7#12312yes—
239C2×C4×C7#12312yes—
240C2×C4×C7C2×D812yes—
241C2×C4×C7C2×D812yes—
242C2×C4×C7#13512yes—
243C2×C4×C7#13712yes—
244C2×C4×C7C2×#4111yescharacteristic
245C2×C4×C7C2×#4111yescharacteristic
246C2×C4×C7C2×#4112yes—
247C2×C4×C7C2×#4112yes—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C7⋊C8—24no—
255C7⋊C8—42no—
256C22×D4—21no—
257C22×D4C22×D711yescharacteristic
258C23×C4—21no—
259C23×C4C22⋊D711yescharacteristic
260C23×C4C22⋊D711yescharacteristic
261C23×C4C22×D711yescharacteristic
262C22×C8—72no—
263C22×C8—72no—
264#121—22no—
265#121—22no—
266C4×D4—24no—
267C4×D4—44no—
268#117—284no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—72no—
271C2×C4⋊C4—72no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4C22×D711yescharacteristic
277C2×C4⋊C4C22×D711yescharacteristic
278C2×C4⋊C4C22×D711yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—142no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—142no—
288C2×C42C22⋊D711yescharacteristic
289C2×C42C22⋊D711yescharacteristic
290C2×C42C22×D711yescharacteristic
291C2×C42C22×D711yescharacteristic
292#107—144no—
293#107—144no—
294#102—144no—
295#102—144no—
296C22⋊C8—144no—
297C4×C8—144no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—142no—
302#95—142no—
303C22×C7—22no—
304C22×C7—22no—
305C22×C7—22no—
306C22×C7—41no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—42no—
311C22×C7—42no—
312C22×C7—42no—
313C22×C7—81no—
314C22×C7—81no—
315C22×C7—81no—
316C22×C7—81no—
317C22×C7C2×#12311yescharacteristic
318C22×C7#52511yescharacteristic
319C22×C7#53211yescharacteristic
320C22×C7C22×D811yescharacteristic
321C22×C7C2×#13511yescharacteristic
322C22×C7C2×#13711yescharacteristic
323C22×C7#57511yescharacteristic
324C4×C7—22no—
325C4×C7—22no—
326C4×C7—22no—
327C4×C7—42no—
328C4×C7—41no—
329C4×C7—81no—
330C4×C7—81no—
331C4×C7#48014yes—
332C4×C7#52912yes—
333C4×C7#52912yes—
334Dic7—82no—
335Dic7—82no—
336C24—21no—
337C24C2×C22⋊D711yescharacteristic
338C2×D4—24no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4#115712yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—142no—
361C22×C4—142no—
362C22×C4#114114yes—
363C22×C4C2×C22⋊D711yescharacteristic
364C22×C4C2×C22⋊D711yescharacteristic
365C22×C4C2×C22⋊D711yescharacteristic
366C22×C4C2×C22⋊D711yescharacteristic
367C22×C4C2×C22⋊D711yescharacteristic
368C22×C4C23×D711yescharacteristic
369C2×C8—74no—
370C2×C8—142no—
371C2×C8—142no—
372C2×C8—142no—
373C2×C8—142no—
374C2×C8—142no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—142no—
380C4⋊C4—142no—
381C4⋊C4—142no—
382C4⋊C4—142no—
383C4⋊C4—282no—
384C4⋊C4—282no—
385C4⋊C4—282no—
386C4⋊C4—282no—
387C4⋊C4#115712yes—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—284no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—42no—
398C42—284no—
399C42#115914yes—
400C2×C7—42no—
401C2×C7—82no—
402C2×C7#239511yescharacteristic
403C2×C7C2×#48011yescharacteristic
404C2×C7#309111yescharacteristic
405C2×C7#310011yescharacteristic
406C2×C7#310711yescharacteristic
407C2×C7#316911yescharacteristic
408C2×C7#317211yescharacteristic
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×#114112yes—
423C23C22×C22⋊D711yescharacteristic
424D4—24no—
425D4—48no—
426D4—42no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—28no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4—282no—
453C2×C4—282no—
454C2×C4—282no—
455C2×C4—282no—
456C2×C4C2×#114112yes—
457C2×C4C2×#114112yes—
458C2×C4#649812yes—
459C2×C4#649812yes—
460C2×C4#651312yes—
461C2×C4#651412yes—
462C2×C4C2×#115711yescharacteristic
463C2×C4C2×#115711yescharacteristic
464C2×C4C2×#115912yes—
465C2×C4D7×#4112yes—
466C8—144no—
467C8—282no—
468C7#2375411yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22#9048211yescharacteristic
484C22C22×#114111yescharacteristic
485C22C2×#649811yescharacteristic
486C22C2×#651311yescharacteristic
487C22#9051311yescharacteristic
488C22C2×#651411yescharacteristic
489C22#9052311yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—41no—
495C4—81no—
496C4—81no—
497C4—562no—
498C4—562no—
499C4#8980514yes—
500C4#9022912yes—
501C4#9024812yes—
502C2—42no—
503C2—82no—
504C2#1173975311yescharacteristic
505C2#1173979111yescharacteristic
506C2#1173997911yescharacteristic
507C2C2×#8980511yescharacteristic
508C2#1174035811yescharacteristic
509C2#1174058111yescharacteristic
510C2#1174073611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#121C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC4×#121D7
hypocenterC7#23754
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. #11739507C2semidirect
4. C4×#6406C2semidirect
5. C4×C22⋊(C7⋊C8)C2semidirect
6. #11727409C2semidirect
7. #11727394C2semidirect
8. #11727365C2semidirect
9. #11727361C2semidirect
10. #11727272C2semidirect
11. C4×#6381C2semidirect
468. C7#23754semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1212
#117395072
#117294332
C4×#64062
C4×C22⋊(C7⋊C8)2
#117274092
#117273942
#117273652
#117273612
#117272722
C4×#63812
#237547