Finite Group Databasebeta

← #13514898

Subgroups — #13514898

Order 1792.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90263C211yesmaximal, characteristic
2C7×#2351C211yesmaximal, characteristic, complemented
3Q8×(C2×D7)⋊C4C211yesmaximal, characteristic
4Q8×(C2×D7)⋊C4C211yesmaximal, characteristic
5Q8×C4⋊Dic7C211yesmaximal, characteristic, complemented
6#11739895C211yesmaximal, characteristic, complemented
7C4×Q8×Dic7C211yesmaximal, characteristic, complemented
8#11733425C213yesmaximal
9#11733424C213yesmaximal
10#11733415C213yesmaximal, complemented
11#11733407C213yesmaximal, complemented
12#11731669C213yesmaximal
13#11731604C213yesmaximal, complemented
14#11731085C213yesmaximal, complemented
15#11731065C213yesmaximal
16C22×Q8×D7C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23#90263C4116yescomplemented
24C2×#6455C2213yes—
25C2×#6450C2213yes—
26C2×#6435C2213yes—
27C2×C4×Dic14C2213yes—
28C7×#389C2213yes—
29C7×#389C2213yes—
30C7×#382C2213yes—
31C4×C7×C4⋊C4C2213yes—
32#89762C2213yes—
33#89762C2213yes—
34#89759C2213yes—
35#89759C2213yes—
36#89757C2213yes—
37#89757C2213yes—
38#89750C2213yes—
39#89750C2213yes—
40#89749C2216yes—
41C4⋊C4×Dic7C2216yes—
42C4⋊C4×Dic7C2213yes—
43C4⋊C4×Dic7C2213yes—
44C4⋊C4×Dic7C2213yes—
45C4⋊C4×Dic7C2213yes—
46#89746C2213yes—
47#89746C2213yes—
48C4×(C2×D7)⋊C4C2213yes—
49C4×(C2×D7)⋊C4C2213yes—
50#89707C2213yes—
51C4×C4⋊Dic7C2213yes—
52C4×C4⋊Dic7C2213yes—
53C4×C4⋊Dic7C2213yes—
54#89704C2213yes—
55C42×Dic7C2213yes—
56#89701C2213yes—
57#89701C2213yes—
58#89442C2211yescharacteristic
59#89442C2213yes—
60#89442C2216yes—
61#89427C2213yes—
62#89427C2213yes—
63#89427C2216yes—
64#89427C2216yes—
65#89423C2216yes—
66#89420C2216yes—
67#34262—71nomaximal, Sylow
68C22×C7×Q8C2311yescharacteristic
69C2×Q8×D7—24no—
70C2×Q8×D7—24no—
71C2×Q8×D7C2×C418yes—
72C22×C4×D7C2313yes—
73C22×Dic14C2313yes—
74C4×C7×Q8—24no—
75C4×C7×Q8—24no—
76C4×C7×Q8C2×C418yes—
77C2×C7×C4⋊C4C2313yes—
78C2×C7×C4⋊C4C2313yes—
79C2×C7×C4⋊C4C2313yes—
80C2×C7×C4⋊C4C2313yes—
81C2×C7×C4⋊C4C2313yes—
82C2×C42×C7C2313yes—
83C2×C42×C7C2313yes—
84C2×C42×C7C2313yes—
85Q8×Dic7—24no—
86Q8×Dic7—24no—
87Q8×Dic7C2×C418yes—
88C2×(C2×D7)⋊C4C2311yescharacteristic
89C2×(C2×D7)⋊C4C2311yescharacteristic
90C2×(C2×D7)⋊C4C2313yes—
91C2×(C2×D7)⋊C4C2313yes—
92C2×(C2×D7)⋊C4C2311yescharacteristic
93C2×(C2×D7)⋊C4C2313yes—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2313yes—
96C2×C4⋊Dic7C2313yes—
97C2×C4⋊Dic7C2313yes—
98C2×C4⋊Dic7C2316yes—
99C2×C4⋊Dic7C2313yes—
100C2×Dic7⋊C4C2313yes—
101C2×Dic7⋊C4C2313yes—
102C2×C4×Dic7C2313yes—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2313yes—
105C2×C4×Dic7C2313yes—
106C2×C4×Dic7C2316yes—
107C2×C4×Dic7C2313yes—
108C2×C4×Dic7C2313yes—
109C2×C4×Dic7C2316yes—
110C2×C4×Dic7C2313yes—
111#6455C2×C4124yes—
112#6450C2×C4124yes—
113#6435C2×C4124yes—
114C4×Dic14C2×C4124yes—
115C7×#95C2316yes—
116C7×#95C2311yescharacteristic
117C7×#95C2313yes—
118C7⋊#95C2313yes—
119C7⋊#95C2313yes—
120C7⋊#95C2316yes—
121C7⋊#95C2313yes—
122C7⋊#95C2313yes—
123C7⋊#95C2313yes—
124C7⋊#95C2316yes—
125C7⋊#95C2316yes—
126C7⋊#95C2313yes—
127C7⋊#95C2313yes—
128C7⋊#95C2313yes—
129C7⋊#95C2313yes—
130C7⋊#95C2311yescharacteristic
131C2×#519—71no—
132#2356—73no—
133#2355—73no—
134#2351D711yescharacteristic, complemented
135Q8×C4⋊C4—71no—
136#2347—73no—
137#2346—73no—
138Q8×C22⋊C4—71no—
139Q8×C22⋊C4—71no—
140#2320—73no—
141#2314—73no—
142#2298—73no—
143#2295—73no—
144#2276—71no—
145C42×Q8—71no—
146C23×D7C2411yescharacteristic
147C2×C7×Q8C22×C414yes—
148C2×C7×Q8#4114yes—
149C2×C7×Q8#4114yes—
150C22×C4×C7C2411yescharacteristic
151C22×C4×C7C2413yes—
152C22×C4×C7C2413yes—
153C22×C4×C7C2413yes—
154C22×C4×C7C2411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×C4×C7C2413yes—
157C22×Dic7C2411yescharacteristic
158C22×Dic7C2413yes—
159C22×Dic7C2413yes—
160C22×Dic7C2413yes—
161C22×Dic7C2411yescharacteristic
162C22×Dic7C2413yes—
163C22×Dic7C2411yescharacteristic
164Q8×D7—232no—
165C2×C4×D7—26no—
166C2×C4×D7—26no—
167C2×C4×D7C22×C4112yes—
168C2×Dic14—26no—
169C2×Dic14—26no—
170C2×Dic14C22×C4112yes—
171C7×C4⋊C4—26no—
172C7×C4⋊C4—26no—
173C7×C4⋊C4—26no—
174C7×C4⋊C4—43no—
175C7×C4⋊C4C22×C4112yes—
176C42×C7—26no—
177C42×C7—26no—
178C42×C7C22×C4112yes—
179(C2×D7)⋊C4—22no—
180(C2×D7)⋊C4—22no—
181(C2×D7)⋊C4—43no—
182(C2×D7)⋊C4—43no—
183(C2×D7)⋊C4C22×C414yes—
184(C2×D7)⋊C4C22×C4112yes—
185C4⋊Dic7—26no—
186C4⋊Dic7—26no—
187C4⋊Dic7—22no—
188C4⋊Dic7—43no—
189C4⋊Dic7—46no—
190C4⋊Dic7C22×C4112yes—
191Dic7⋊C4C22×C4112yes—
192Dic7⋊C4C22×C4112yes—
193C4×Dic7—26no—
194C4×Dic7—26no—
195C4×Dic7—22no—
196C4×Dic7—26no—
197C4×Dic7—46no—
198C4×Dic7—43no—
199C4×Dic7—43no—
200C4×Dic7C22×C4124yes—
201C4×Dic7C22×C4112yes—
202C23×Q8—71no—
203#519—716no—
204C2×C4×Q8—71no—
205C2×C4×Q8—71no—
206C2×C4×Q8—71no—
207C2×C4×Q8—73no—
208C2×C4×Q8—73no—
209C2×C4×Q8C2×D711yescharacteristic
210C2×C4×Q8C2×D711yescharacteristic
211C2×C4×Q8C2×D711yescharacteristic
212C2×#117—73no—
213C2×#117—73no—
214#391—73no—
215#391—73no—
216#391—73no—
217#391—73no—
218#389—73no—
219#389—73no—
220#389C2×D713yes—
221#389C2×D713yes—
222#388—73no—
223#388—73no—
224#388—73no—
225#388—76no—
226#388—71no—
227#387—73no—
228#387—73no—
229#387—76no—
230#387—76no—
231#387—73no—
232#387—73no—
233#384—73no—
234#383—76no—
235#382—76no—
236#382C2×D713yes—
237#380—73no—
238#380—73no—
239C4×C4⋊C4—73no—
240C4×C4⋊C4—73no—
241C4×C4⋊C4—73no—
242C4×C4⋊C4—76no—
243C4×C4⋊C4—73no—
244C4×C4⋊C4—73no—
245C4×C4⋊C4—73no—
246C4×C4⋊C4—73no—
247C4×C4⋊C4C2×D713yes—
248C4×C22⋊C4—73no—
249C4×C22⋊C4—73no—
250#376—73no—
251#376—76no—
252C43—73no—
253C23×C7C2511yescharacteristic
254C22×D7—21no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7—21no—
259C22×D7—21no—
260C22×D7C23×C412yes—
261C7×Q8#117116yes—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—23no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—23no—
268C2×C4×C7—21no—
269C2×C4×C7—23no—
270C2×C4×C7—23no—
271C2×C4×C7—23no—
272C2×C4×C7—21no—
273C2×C4×C7—23no—
274C2×C4×C7—23no—
275C2×C4×C7—23no—
276C2×C4×C7—21no—
277C2×C4×C7C23×C416yes—
278C2×C4×C7C23×C412yes—
279C2×C4×C7C23×C416yes—
280C2×C4×C7C2×#4116yes—
281C2×C4×C7C2×#4116yes—
282C2×C4×C72−1+412yes—
283C2×Dic7—21no—
284C2×Dic7—23no—
285C2×Dic7—23no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—23no—
289C2×Dic7—21no—
290C2×Dic7—23no—
291C2×Dic7—21no—
292C2×Dic7—23no—
293C2×Dic7—23no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—23no—
297C2×Dic7—23no—
298C2×Dic7—23no—
299C2×Dic7—23no—
300C2×Dic7C23×C412yes—
301C2×Dic7C23×C416yes—
302C2×Dic7C23×C416yes—
303C2×Dic72−1+412yes—
304C4×D7—224no—
305Dic14—224no—
306C22×Q8—78no—
307C22×Q8—73no—
308C22×Q8—144no—
309C22×Q8—144no—
310C22×Q8C22×D711yescharacteristic
311C23×C4—73no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—78no—
315C4×Q8—724no—
316C4×Q8—724no—
317C4×Q8—144no—
318C4×Q8—144no—
319C4×Q8C4×D718yes—
320#117—724no—
321#117—724no—
322C2×C4⋊C4—73no—
323C2×C4⋊C4—73no—
324C2×C4⋊C4—73no—
325C2×C4⋊C4—73no—
326C2×C4⋊C4—73no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—76no—
329C2×C4⋊C4—73no—
330C2×C4⋊C4C22×D713yes—
331C2×C4⋊C4C22×D713yes—
332C2×C4⋊C4C22×D713yes—
333C2×C4⋊C4C22×D713yes—
334C2×C4⋊C4C22×D713yes—
335C2×C22⋊C4—73no—
336C2×C22⋊C4—71no—
337C2×C22⋊C4—71no—
338C2×C22⋊C4—71no—
339C2×C22⋊C4—73no—
340C2×C22⋊C4—73no—
341C2×C42—73no—
342C2×C42—76no—
343C2×C42—73no—
344C2×C42—73no—
345C2×C42—71no—
346C2×C42—73no—
347C2×C42—73no—
348C2×C42—73no—
349C2×C42—76no—
350C2×C42C22×D713yes—
351C2×C42C22×D713yes—
352C2×C42C22×D713yes—
353#95—71no—
354#95—73no—
355#95—73no—
356#95—73no—
357#95—73no—
358#95—73no—
359#95—76no—
360#95—73no—
361#95—73no—
362#95—73no—
363#95—76no—
364#95—76no—
365#95—73no—
366#95C22×D716yes—
367#95C22×D711yescharacteristic
368#95C22×D713yes—
369C22×C7C24×C411yescharacteristic
370C22×C7C22×#4111yescharacteristic
371C22×C7C22×#4111yescharacteristic
372C22×C7C2×2−1+411yescharacteristic
373C22×C7C2×2−1+411yescharacteristic
374C22×C7#58511yescharacteristic
375C22×C7#58511yescharacteristic
376C2×D7—22no—
377C2×D7—22no—
378C2×D7—22no—
379C2×D7—41no—
380C2×D7—41no—
381C2×D7—41no—
382C2×D7—41no—
383C4×C7—26no—
384C4×C7—22no—
385C4×C7—22no—
386C4×C7—43no—
387C4×C7—41no—
388C4×C7—43no—
389C4×C7C2×#117112yes—
390Dic7—26no—
391Dic7—22no—
392Dic7—26no—
393Dic7—22no—
394Dic7—41no—
395Dic7—43no—
396Dic7—43no—
397C24—71no—
398C2×Q8—712no—
399C2×Q8—1432no—
400C2×Q8—146no—
401C2×Q8—146no—
402C2×Q8C2×C4×D714yes—
403C2×Q8#115914yes—
404C2×Q8#116114yes—
405C22×C4—73no—
406C22×C4—71no—
407C22×C4—71no—
408C22×C4—73no—
409C22×C4—712no—
410C22×C4—71no—
411C22×C4—73no—
412C22×C4—73no—
413C22×C4—146no—
414C22×C4—146no—
415C22×C4C23×D711yescharacteristic
416C22×C4C23×D713yes—
417C22×C4C23×D713yes—
418C22×C4C23×D713yes—
419C22×C4C23×D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421C22×C4C23×D713yes—
422C4⋊C4—26no—
423C4⋊C4—26no—
424C4⋊C4—26no—
425C4⋊C4—43no—
426C4⋊C4—712no—
427C4⋊C4—712no—
428C4⋊C4—712no—
429C4⋊C4—142no—
430C4⋊C4—146no—
431C4⋊C4—146no—
432C4⋊C4—286no—
433C4⋊C4—283no—
434C4⋊C4C2×C4×D7112yes—
435C22⋊C4—712no—
436C22⋊C4—74no—
437C22⋊C4—142no—
438C22⋊C4—142no—
439C22⋊C4—283no—
440C22⋊C4—283no—
441C42—26no—
442C42—26no—
443C42—712no—
444C42—724no—
445C42—146no—
446C42—146no—
447C42—146no—
448C42—142no—
449C42—283no—
450C42—283no—
451C42—286no—
452C42C2×C4×D7112yes—
453C2×C7C22×#11711yescharacteristic
454C2×C7C4×2−1+411yescharacteristic
455C2×C7C4×2−1+411yescharacteristic
456C2×C7#351611yescharacteristic
457C2×C7#351811yescharacteristic
458C2×C7#352311yescharacteristic
459C2×C7#352411yescharacteristic
460D7—42no—
461C23—72no—
462C23—141no—
463C23—141no—
464C23—141no—
465C23—141no—
466C23—141no—
467C23—141no—
468C23C24×D711yescharacteristic
469Q8—1424no—
470Q8#6455116yes—
471C2×C4—21no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—21no—
483C2×C4—23no—
484C2×C4—21no—
485C2×C4—23no—
486C2×C4—72no—
487C2×C4—76no—
488C2×C4—76no—
489C2×C4—72no—
490C2×C4—143no—
491C2×C4—141no—
492C2×C4—143no—
493C2×C4—141no—
494C2×C4—141no—
495C2×C4—141no—
496C2×C4—143no—
497C2×C4—143no—
498C2×C4—1424no—
499C2×C4—141no—
500C2×C4—143no—
501C2×C4—143no—
502C2×C4—141no—
503C2×C4—143no—
504C2×C4—143no—
505C2×C4—141no—
506C2×C4—143no—
507C2×C4—143no—
508C2×C4C22×C4×D716yes—
509C2×C4C22×C4×D712yes—
510C2×C4C22×C4×D716yes—
511C2×C4C2×#115916yes—
512C2×C4C2×#116116yes—
513C2×C4#655412yes—
514C7#3426211yescharacteristic, complemented, Sylow
515C22—142no—
516C22—142no—
517C22—142no—
518C22—281no—
519C22—281no—
520C22—281no—
521C22—281no—
522C22C23×C4×D711yescharacteristic
523C22C22×#115911yescharacteristic
524C22C22×#116111yescharacteristic
525C22C2×#655411yescharacteristic
526C22#9061611yescharacteristic
527C22D7×2−1+411yescharacteristic
528C22#9062011yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—43no—
533C4—41no—
534C4—43no—
535C4—146no—
536C4—142no—
537C4—142no—
538C4—146no—
539C4—281no—
540C4—283no—
541C4—283no—
542C4C2×#6455112yes—
543C2—282no—
544C2C22×#645511yescharacteristic
545C2C4×#655411yescharacteristic
546C2#1174294111yescharacteristic
547C2#1174378611yescharacteristic
548C2#1174378811yescharacteristic
549C2#1174409211yescharacteristic
550C2#1174409411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2351C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2351D7
hypocenterC7#34262
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C7×#2351C2semidirect
5. Q8×C4⋊Dic7C2semidirect
6. #11739895C2semidirect
7. C4×Q8×Dic7C2semidirect
10. #11733415C2semidirect
11. #11733407C2semidirect
13. #11731604C2semidirect
14. #11731085C2semidirect
23. #90263C4semidirect
134. #2351D7semidirect
514. C7#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#902632
C7×#23512
Q8×(C2×D7)⋊C42
Q8×C4⋊Dic72
#117398952
C4×Q8×Dic72
#117334252
#117334242
#117334152
#117334072
#117316692
#117316042
#117310852
#117310652
#342627