Finite Group Databasebeta

← #13517103

Subgroups — #13517103

Order 1792.

1898 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C7×#2331C211yesmaximal, characteristic, complemented
3#11740673C211yesmaximal, characteristic, complemented
4#11733932C214yesmaximal
5#11733928C214yesmaximal, complemented
6#11733434C212yesmaximal, complemented
7#11733425C212yesmaximal
8#11732759C214yesmaximal
9#11732409C214yesmaximal, complemented
10#11731599C211yesmaximal, characteristic, complemented
11#11731592C211yesmaximal, characteristic
12#11731105C212yesmaximal, complemented
13#11731078C212yesmaximal
14#11730858C211yesmaximal, characteristic
15#11730836C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×#6509C2212yes—
18C2×Q8×Dic7C2212yes—
19#90419C4116yescomplemented
20C2×#6460C2212yes—
21C2×#6455C2212yes—
22C2×#6453C2212yes—
23C2×#6450C2212yes—
24C2×C42×D7C2211yescharacteristic
25C2×#6432C2211yescharacteristic
26C7×#391C2214yes—
27C7×#389C2212yes—
28C7×#387C2214yes—
29C7×#382C2211yescharacteristic
30C4×C7×C4⋊C4C2212yes—
31C7×#376C2211yescharacteristic
32#90002C2214yes—
33#89762C2214yes—
34#89759C2212yes—
35#89757C2212yes—
36#89753C2214yes—
37#89750C2214yes—
38C4⋊C4×Dic7C2212yes—
39C4⋊C4×Dic7C2212yes—
40C4⋊C4×Dic7C2212yes—
41C4⋊C4×Dic7C2214yes—
42#89745C2214yes—
43#89745C2212yes—
44#89712C2214yes—
45C4×(C2×D7)⋊C4C2211yescharacteristic
46C4×(C2×D7)⋊C4C2211yescharacteristic
47#89708C2211yescharacteristic
48C4×C4⋊Dic7C2212yes—
49#89704C2211yescharacteristic
50#89704C2214yes—
51C4×Dic7⋊C4C2211yescharacteristic
52C4×Dic7⋊C4C2214yes—
53C4×Dic7⋊C4C2211yescharacteristic
54#89700C2214yes—
55#89700C2212yes—
56#89442C2214yes—
57#89441C2214yes—
58#89439C2212yes—
59#89438C2212yes—
60#89436C2212yes—
61#89436C2212yes—
62#89427C2214yes—
63#89426C2214yes—
64#89426C2214yes—
65#89425C2214yes—
66#89425C2214yes—
67#89423C2212yes—
68#89423C2214yes—
69#89423C2212yes—
70#89422C2214yes—
71#89422C2212yes—
72#89421C2214yes—
73#89420C2212yes—
74#89418C2214yes—
75#89418C2214yes—
76#34403—71nomaximal, Sylow
77C22×C7×Q8C2312yes—
78C22×C4×D7C2311yescharacteristic
79C22×C4×D7C2311yescharacteristic
80C22×C4×D7C2311yescharacteristic
81C22×Dic14C2312yes—
82C7×C4⋊Q8C2×C418yes—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2314yes—
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C4⋊C4C2312yes—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2314yes—
90C2×C42×C7C2311yescharacteristic
91#6509C2×C4116yes—
92Q8×Dic7C2×C4116yes—
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2312yes—
95C2×(C2×D7)⋊C4C2312yes—
96C2×(C2×D7)⋊C4C2314yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2314yes—
99C2×C4⋊Dic7C2312yes—
100C2×C4⋊Dic7C2312yes—
101C2×C4⋊Dic7C2314yes—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2312yes—
105C2×Dic7⋊C4C2312yes—
106C2×Dic7⋊C4C2314yes—
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2314yes—
110C2×C4×Dic7C2314yes—
111C2×C4×Dic7C2314yes—
112C2×C4×Dic7C2314yes—
113C2×C4×Dic7C2311yescharacteristic
114C2×C4×Dic7C2311yescharacteristic
115C2×C4×Dic7C2311yescharacteristic
116C2×C4×Dic7C2314yes—
117#6460C2×C4116yes—
118#6455C2×C4116yes—
119#6453C2×C4116yes—
120#6450C2×C4116yes—
121C42×D7C2×C418yes—
122#6432C2×C418yes—
123C7×#95C2312yes—
124C7×#95C2312yes—
125C7×#95C2314yes—
126C7×#95C2312yes—
127C7×#95C2314yes—
128C7×#95C2312yes—
129C7⋊#95C2312yes—
130C7⋊#95C2312yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2312yes—
137C7⋊#95C2312yes—
138C7⋊#95C2312yes—
139C7⋊#95C2312yes—
140C7⋊#95C2314yes—
141C7⋊#95C2314yes—
142C7⋊#95C2312yes—
143C7⋊#95C2314yes—
144C7⋊#95C2312yes—
145C2×#533—71no—
146#2378—74no—
147#2370—72no—
148#2370—72no—
149#2356—74no—
150#2347—72no—
151#2346—72no—
152#2332—74no—
153#2331—71no—
154#2331—71no—
155#2331—71no—
156#2331D711yescharacteristic, complemented
157#2320—74no—
158#2283—71no—
159#2283—71no—
160C23×D7C2411yescharacteristic
161C2×C7×Q8—44no—
162C2×C7×Q8C22×C418yes—
163C22×C4×C7C2412yes—
164C22×C4×C7C2414yes—
165C22×C4×C7C2412yes—
166C22×C4×C7C2411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×C4×C7C2414yes—
169C22×C4×C7C2411yescharacteristic
170C22×Dic7C2412yes—
171C22×Dic7C2412yes—
172C22×Dic7C2414yes—
173C22×Dic7C2411yescharacteristic
174C22×Dic7C2411yescharacteristic
175C22×Dic7C2414yes—
176C22×Dic7C2411yescharacteristic
177C2×C4×D7—22no—
178C2×C4×D7—22no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7C22×C414yes—
182C2×C4×D7C22×C414yes—
183C2×C4×D7C22×C414yes—
184C2×Dic14—44no—
185C2×Dic14C22×C418yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—44no—
188C7×C4⋊C4C22×C418yes—
189C7×C4⋊C4C22×C418yes—
190C42×C7—22no—
191C42×C7—22no—
192C42×C7—44no—
193C42×C7C22×C414yes—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—44no—
196(C2×D7)⋊C4—42no—
197(C2×D7)⋊C4C22×C4116yes—
198C4⋊Dic7—24no—
199C4⋊Dic7—44no—
200C4⋊Dic7C22×C418yes—
201C4⋊Dic7C22×C418yes—
202C4⋊Dic7C22×C4116yes—
203Dic7⋊C4—44no—
204Dic7⋊C4—44no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4C22×C4116yes—
208C4×Dic7—22no—
209C4×Dic7—22no—
210C4×Dic7—44no—
211C4×Dic7—44no—
212C4×Dic7—44no—
213C4×Dic7—44no—
214C4×Dic7C22×C414yes—
215C4×Dic7C22×C414yes—
216C4×Dic7C22×C414yes—
217C4×Dic7C22×C4116yes—
218#533—716no—
219C2×C4⋊Q8—71no—
220C2×C4⋊Q8C2×D711yescharacteristic
221C2×#125—72no—
222C2×C22⋊Q8—72no—
223C2×C22⋊Q8—72no—
224C2×C4×Q8—72no—
225C2×C4×Q8—72no—
226C2×#117—72no—
227C22×C42—71no—
228#391—74no—
229#391—74no—
230#391—74no—
231#391C2×D714yes—
232#389—74no—
233#389—72no—
234#389—72no—
235#389—74no—
236#389—72no—
237#389C2×D712yes—
238#388—74no—
239#388—74no—
240#388—74no—
241#388—74no—
242#387—74no—
243#387—74no—
244#387—74no—
245#387—74no—
246#387—74no—
247#387—74no—
248#387—74no—
249#387C2×D714yes—
250#383—72no—
251#383—74no—
252#383—72no—
253#382—71no—
254#382—72no—
255#382—74no—
256#382C2×D711yescharacteristic
257#381—72no—
258#381—72no—
259#380—72no—
260#380—72no—
261C4×C4⋊C4—74no—
262C4×C4⋊C4—72no—
263C4×C4⋊C4—72no—
264C4×C4⋊C4—72no—
265C4×C4⋊C4—72no—
266C4×C4⋊C4—74no—
267C4×C4⋊C4—71no—
268C4×C4⋊C4—71no—
269C4×C4⋊C4—74no—
270C4×C4⋊C4C2×D712yes—
271C4×C22⋊C4—71no—
272C4×C22⋊C4—71no—
273C4×C22⋊C4—72no—
274#376—74no—
275#376—71no—
276#376—72no—
277#376C2×D711yescharacteristic
278C23×C7C2511yescharacteristic
279C22×D7—21no—
280C22×D7—21no—
281C22×D7—22no—
282C22×D7—22no—
283C22×D7C23×C412yes—
284C7×Q8—48no—
285C2×C4×C7—24no—
286C2×C4×C7—24no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—24no—
292C2×C4×C7—22no—
293C2×C4×C7—24no—
294C2×C4×C7—24no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7C23×C418yes—
299C2×C4×C7C23×C412yes—
300C2×C4×C7C23×C412yes—
301C2×C4×C7C23×C412yes—
302C2×C4×C72−1+412yes—
303C2×C4×C72−1+412yes—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—24no—
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—22no—
314C2×Dic7—24no—
315C2×Dic7—22no—
316C2×Dic7—24no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7C23×C412yes—
320C2×Dic7C23×C412yes—
321C2×Dic7C23×C412yes—
322C2×Dic7C23×C418yes—
323C4×D7—28no—
324C4×D7—44no—
325C4×D7—44no—
326Dic14—48no—
327C22×Q8—72no—
328C22×Q8C22×D712yes—
329C23×C4—71no—
330C23×C4—71no—
331C23×C4—71no—
332C4⋊Q8—78no—
333C4⋊Q8C4×D718yes—
334#125—716no—
335C22⋊Q8—716no—
336C22⋊Q8—716no—
337C4×Q8—716no—
338C4×Q8—716no—
339#117—716no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—72no—
345C2×C4⋊C4—72no—
346C2×C4⋊C4—72no—
347C2×C4⋊C4—74no—
348C2×C4⋊C4—74no—
349C2×C4⋊C4—74no—
350C2×C4⋊C4C22×D712yes—
351C2×C4⋊C4C22×D714yes—
352C2×C4⋊C4C22×D712yes—
353C2×C4⋊C4C22×D712yes—
354C2×C22⋊C4—74no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—72no—
357C2×C22⋊C4—72no—
358C2×C42—74no—
359C2×C42—71no—
360C2×C42—71no—
361C2×C42—71no—
362C2×C42—78no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—74no—
366C2×C42—74no—
367C2×C42—74no—
368C2×C42—74no—
369C2×C42C22×D711yescharacteristic
370C2×C42C22×D711yescharacteristic
371C2×C42C22×D714yes—
372C2×C42C22×D711yescharacteristic
373#95—72no—
374#95—74no—
375#95—72no—
376#95—72no—
377#95—72no—
378#95—74no—
379#95—74no—
380#95—72no—
381#95—72no—
382#95—72no—
383#95—72no—
384#95—74no—
385#95—74no—
386#95—74no—
387#95—74no—
388#95—74no—
389#95C22×D712yes—
390#95C22×D712yes—
391#95C22×D712yes—
392#95C22×D712yes—
393#95C22×D714yes—
394#95C22×D714yes—
395C22×C7C24×C411yescharacteristic
396C22×C7C2×2−1+411yescharacteristic
397C22×C7C2×2−1+411yescharacteristic
398C22×C7#58512yes—
399C22×C7#58512yes—
400C2×D7—22no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7—42no—
404C2×D7—42no—
405C4×C7—22no—
406C4×C7—22no—
407C4×C7—28no—
408C4×C7—42no—
409C4×C7—44no—
410C4×C7—42no—
411C4×C7#51914yes—
412Dic7—22no—
413Dic7—28no—
414Dic7—22no—
415Dic7—22no—
416Dic7—42no—
417Dic7—42no—
418Dic7—44no—
419C24—71no—
420C2×Q8—44no—
421C2×Q8—78no—
422C2×Q8—284no—
423C2×Q8C2×C4×D718yes—
424C22×C4—71no—
425C22×C4—71no—
426C22×C4—74no—
427C22×C4—71no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—74no—
431C22×C4—72no—
432C22×C4—74no—
433C22×C4—72no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4C23×D712yes—
439C22×C4C23×D714yes—
440C22×C4C23×D712yes—
441C22×C4C23×D711yescharacteristic
442C22×C4C23×D711yescharacteristic
443C22×C4C23×D714yes—
444C22×C4C23×D711yescharacteristic
445C4⋊C4—24no—
446C4⋊C4—44no—
447C4⋊C4—716no—
448C4⋊C4—78no—
449C4⋊C4—716no—
450C4⋊C4—78no—
451C4⋊C4—144no—
452C4⋊C4—282no—
453C4⋊C4—284no—
454C4⋊C4—282no—
455C4⋊C4—284no—
456C4⋊C4—284no—
457C4⋊C4C2×C4×D718yes—
458C4⋊C4C2×C4×D718yes—
459C22⋊C4—716no—
460C22⋊C4—282no—
461C22⋊C4—282no—
462C22⋊C4—284no—
463C42—22no—
464C42—22no—
465C42—44no—
466C42—74no—
467C42—74no—
468C42—74no—
469C42—716no—
470C42—142no—
471C42—142no—
472C42—284no—
473C42—284no—
474C42—284no—
475C42—284no—
476C42C2×C4×D714yes—
477C2×C7C2×#51911yescharacteristic
478C2×C7#342811yescharacteristic
479C2×C7#342811yescharacteristic
480C2×C7#356512yes—
481C2×C7#356512yes—
482D7—42no—
483C23—72no—
484C23—141no—
485C23—141no—
486C23—142no—
487C23—142no—
488C23C24×D711yescharacteristic
489Q8—48no—
490Q8—288no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—72no—
505C2×C4—72no—
506C2×C4—72no—
507C2×C4—78no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—144no—
514C2×C4—148no—
515C2×C4—142no—
516C2×C4—144no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—144no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—144no—
523C2×C4—142no—
524C2×C4—284no—
525C2×C4—284no—
526C2×C4C22×C4×D712yes—
527C2×C4C22×C4×D712yes—
528C2×C4C22×C4×D718yes—
529C2×C4C22×C4×D712yes—
530C2×C4#655112yes—
531C2×C4#655412yes—
532C7#3440311yescharacteristic, complemented, Sylow
533C22—142no—
534C22—142no—
535C22—142no—
536C22—282no—
537C22—282no—
538C22C23×C4×D711yescharacteristic
539C22C2×#655111yescharacteristic
540C22C2×#655411yescharacteristic
541C22#9061812yes—
542C22#9062012yes—
543C4—22no—
544C4—28no—
545C4—22no—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—142no—
550C4—142no—
551C4—142no—
552C4—148no—
553C4—282no—
554C4—282no—
555C4—284no—
556C4#9026314yes—
557C2—282no—
558C2C2×#9026311yescharacteristic
559C2#1174292411yescharacteristic
560C2#1174294011yescharacteristic
561C2#1174360912yes—
562C2#1174366212yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2331C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2331D7
hypocenterC7#34403
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
2. C7×#2331C2semidirect
3. #11740673C2semidirect
5. #11733928C2semidirect
6. #11733434C2semidirect
9. #11732409C2semidirect
10. #11731599C2semidirect
12. #11731105C2semidirect
15. #11730836C2semidirect
19. #90419C4semidirect
156. #2331D7semidirect
532. C7#34403semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
C7×#23312
#117406732
#117339322
#117339282
#117334342
#117334252
#117327592
#117324092
#117315992
#117315922
#117311052
#117310782
#117308582
#117308362
#344037