Finite Group Databasebeta

← #13406009

Subgroups — #13406009

Order 1792.

1132 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×#533C211yesmaximal, characteristic
2C2×#89823C212yesmaximal
3#11739520C211yesmaximal, characteristic
4#11729548C211yesmaximal, characteristic, complemented
5#11729547C211yesmaximal, characteristic, complemented
6#11729536C211yesmaximal, characteristic, complemented
7#11729535C211yesmaximal, characteristic, complemented
8#11729450C212yesmaximal
9#11729386C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11C2×#89319C211yesmaximal, characteristic
12C7×#533—28no—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C7×#117—21no—
21C22×C42×C7C2211yescharacteristic
22C22×#1144C2212yes—
23C2×#6487C2211yescharacteristic
24C23×C7⋊C8C2212yes—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28#89984C2211yescharacteristic
29#89983—21no—
30#89982—21no—
31#89823—28no—
32C2×#6384C2212yes—
33C2×#6381C2212yes—
34#89705—21no—
35#89705—21no—
36#89701C2211yescharacteristic
37#89701C2211yescharacteristic
38#89700C2211yescharacteristic
39#89700C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2211yescharacteristic
43#89319C2211yescharacteristic
44#89319C2212yes—
45#25498—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2312yes—
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C22×Dic14C2311yescharacteristic
52C7×C4⋊Q8—24no—
53C7×C4⋊Q8—24no—
54C7×#125—28no—
55C7×C22⋊Q8—28no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8D418yes—
58C4×C7×Q8—28no—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C7×#117—28no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4C2312yes—
68C2×C7×C22⋊C4—22no—
69C2×C42×C7—24no—
70C2×C42×C7—22no—
71C2×C42×C7D411yescharacteristic
72C2×C42×C7D411yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×#1144—24no—
78C2×#1144—24no—
79C2×#1144—42no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83#6487—24no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C22×C7⋊C8—24no—
88C22×C7⋊C8—24no—
89C22×C7⋊C8—42no—
90C22×C7⋊C8C2312yes—
91C22×C7⋊C8C2312yes—
92#6409—24no—
93#6409—24no—
94#6409—42no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102#6384—24no—
103#6381—24no—
104C2×#533D711yescharacteristic
105C2×#467—72no—
106#2855—71no—
107#2057—71no—
108#2043—71no—
109#2024—71no—
110#2016—71no—
111#1892—72no—
112#1878—72no—
113#1863—72no—
114C2×#336—71no—
115C24×C7C2×D411yescharacteristic
116C2×C7×Q8—24no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—42no—
123C2×C7×Q8—42no—
124C2×C7×Q8C2×D414yes—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—41no—
135C22×C4×C7Q1614yescomplemented
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D414yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—28no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4C2×D414yes—
158C7×C22⋊C4—28no—
159C42×C7—28no—
160C42×C7—22no—
161C42×C7—22no—
162C42×C7—22no—
163C42×C7—22no—
164C22⋊Dic7—42no—
165C22⋊Dic7—42no—
166#1144—48no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171Dic7⋊C4—42no—
172Dic7⋊C4—42no—
173Dic7⋊C4—42no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—48no—
180C2×C7⋊C8#4114yes—
181C22×Q16—72no—
182C23×C8—72no—
183#533—28no—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×C4⋊Q8C2×D711yescharacteristic
186C2×#125—21no—
187C2×C22⋊Q8—21no—
188C2×C22⋊Q8—71no—
189C2×C22⋊Q8C2×D712yes—
190C2×C4×Q8—21no—
191C2×C4×Q8C2×D712yes—
192C2×#117—21no—
193C22×C42C2×D711yescharacteristic
194#467—148no—
195C2×#107—72no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—72no—
199C2×#103—72no—
200#391—71no—
201#391—71no—
202#389—71no—
203#389—71no—
204#387—141no—
205#387—141no—
206#385—141no—
207#381—141no—
208#380—71no—
209#340—72no—
210#340—72no—
211#336—71no—
212#336—72no—
213#336—71no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—21no—
218C23×C7#12012yes—
219C23×C7C2×Q1612yes—
220C23×C7#13712yes—
221C23×C7C22×D411yescharacteristic
222C7×Q8—24no—
223C7×Q8—24no—
224C7×Q8—48no—
225C7×Q8—42no—
226C7×Q8—42no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—28no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—21no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—44no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7#12012yes—
249C2×C4×C7C2×Q1612yes—
250C2×C4×C7C2×Q1612yes—
251C2×C4×C7#13512yes—
252C2×C4×C7C22×D412yes—
253C2×C4×C7C2×#4112yes—
254C2×C4×C7H(5,𝔽2)11yescharacteristic
255C2×C4×C72−1+411yescharacteristic
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268Dic14—42no—
269Dic14—42no—
270C7⋊C8—24no—
271C7⋊C8—42no—
272C22×Q8—21no—
273C22×Q8—71no—
274C22×Q8C22×D712yes—
275C23×C4C22⋊D711yescharacteristic
276C23×C4C22⋊D711yescharacteristic
277C23×C4C22×D711yescharacteristic
278C2×Q16—144no—
279C2×Q16—144no—
280C2×Q16—282no—
281C22×C8—72no—
282C22×C8—72no—
283C22×C8—144no—
284C22×C8—144no—
285C22×C8—282no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288#125—28no—
289C22⋊Q8—28no—
290C22⋊Q8—24no—
291C22⋊Q8—144no—
292C22⋊Q8C22⋊D718yes—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296#117—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—71no—
303C2×C4⋊C4—71no—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4C22×D712yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—141no—
311C2×C22⋊C4—141no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42C22⋊D711yescharacteristic
315C2×C42C22⋊D711yescharacteristic
316C2×C42C22×D711yescharacteristic
317C2×C42C22×D711yescharacteristic
318#107—144no—
319#103—144no—
320#103—144no—
321#103—144no—
322#103—282no—
323#95—141no—
324#95—141no—
325#95—141no—
326#95—141no—
327#95—141no—
328#95—141no—
329#95—141no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7—22no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7#45114yes—
337C22×C7C2×#12011yescharacteristic
338C22×C7#53211yescharacteristic
339C22×C7#53711yescharacteristic
340C22×C7C22×Q1611yescharacteristic
341C22×C7C2×#13511yescharacteristic
342C22×C7C2×#13711yescharacteristic
343C22×C7#57511yescharacteristic
344C4×C7—22no—
345C4×C7—22no—
346C4×C7—22no—
347C4×C7—22no—
348C4×C7—44no—
349C4×C7—41no—
350C4×C7—44no—
351C4×C7#54612yes—
352C4×C7#54712yes—
353Dic7—81no—
354Dic7—81no—
355Dic7—81no—
356Dic7—81no—
357C24C2×C22⋊D711yescharacteristic
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—142no—
367C2×Q8—141no—
368C2×Q8—141no—
369C2×Q8—142no—
370C2×Q8C2×C22⋊D714yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—141no—
382C22×C4—141no—
383C22×C4—141no—
384C22×C4—141no—
385C22×C4#114414yescomplemented
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388C22×C4C2×C22⋊D711yescharacteristic
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C2×C22⋊D711yescharacteristic
391C22×C4C2×C22⋊D714yes—
392C22×C4C23×D711yescharacteristic
393Q16—288no—
394C2×C8—74no—
395C2×C8—142no—
396C2×C8—142no—
397C2×C8—142no—
398C2×C8—142no—
399C2×C8—142no—
400C2×C8—288no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—22no—
407C4⋊C4—142no—
408C4⋊C4—141no—
409C4⋊C4—141no—
410C4⋊C4—142no—
411C4⋊C4—282no—
412C4⋊C4—282no—
413C4⋊C4—282no—
414C4⋊C4C2×C22⋊D714yes—
415C22⋊C4—28no—
416C22⋊C4—282no—
417C22⋊C4—282no—
418C42—28no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C2×C7—24no—
424C2×C7#243011yescharacteristic
425C2×C7C2×#45111yescharacteristic
426C2×C7#300211yescharacteristic
427C2×C7#317011yescharacteristic
428C2×C7#318111yescharacteristic
429C2×C7#324811yescharacteristic
430C2×C7#324911yescharacteristic
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23#650512yes—
436C23C2×#114412yes—
437C23#651612yes—
438C23C22×C22⋊D711yescharacteristic
439Q8—24no—
440Q8—24no—
441Q8—48no—
442Q8—42no—
443Q8—42no—
444Q8—282no—
445Q8—282no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—281no—
468C2×C4—281no—
469C2×C4—281no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4C2×#114412yes—
480C2×C4C2×#114412yes—
481C2×C4#651312yes—
482C2×C4#651612yes—
483C2×C4D7×#4112yes—
484C2×C4#655311yescharacteristic
485C2×C4#655411yescharacteristic
486C2×C4C22×C22⋊D712yes—
487C8—144no—
488C8—282no—
489C7#2549811yescharacteristic, complemented, Sylow
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22#9000014yes—
497C22C2×#650511yescharacteristic
498C22C22×#114411yescharacteristic
499C22C2×#651311yescharacteristic
500C22#9051311yescharacteristic
501C22#9052311yescharacteristic
502C22#9052411yescharacteristic
503C22C2×#651611yescharacteristic
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—44no—
510C4—44no—
511C4—561no—
512C4—561no—
513C4—561no—
514C4—561no—
515C4#9051512yes—
516C4#9052012yes—
517C2—24no—
518C2#1174078911yescharacteristic
519C2#1174079111yescharacteristic
520C2#1174080911yescharacteristic
521C2#1174081811yescharacteristic
522C2C2×#9000011yescharacteristic
523C2#1174152511yescharacteristic
524C2#1174153611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#533C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#533D7
hypocenterC7#25498
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #11729548C2semidirect
5. #11729547C2semidirect
6. #11729536C2semidirect
7. #11729535C2semidirect
9. #11729386C2semidirect
10. #11729382C2semidirect
135. C22×C4×C7Q16semidirect
385. C22×C4#1144semidirect
489. C7#25498semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×#898232
#117395202
#117295482
#117295472
#117295362
#117295352
#117294502
#117293862
#117293822
C2×#893192
#254987