Finite Group Databasebeta

← #13387749

Subgroups — #13387749

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic, complemented
3C2×C4×#1144C211yesmaximal, characteristic, complemented
4#11739821C211yesmaximal, characteristic
5#11729541C211yesmaximal, characteristic, complemented
6#11729454C212yesmaximal, complemented
7#11729449C212yesmaximal, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11729368C211yesmaximal, characteristic
10#11729313C212yesmaximal, complemented
11#11729309C212yesmaximal, complemented
12C7×#554—24no—
13C7×#554—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—22no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C22×#1144C2211yescharacteristic
23C2×#6487C2212yes—
24C2×C4×Dic14C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2212yes—
27#89993—21no—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89863—24no—
32C4×#1144—24no—
33#89745—21no—
34C2×#6384C2212yes—
35C2×#6384C2211yescharacteristic
36C2×#6381C2212yes—
37C2×#6381C2211yescharacteristic
38#89720—22no—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24122—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—22no—
50C23×C4×C7C2312yes—
51C22×Dic14C2311yescharacteristic
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8D414yes—
54C7×#125—28no—
55C7×C22⋊Q8—216no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—24no—
58C4×C7×Q8—22no—
59C4×C7×Q8D414yes—
60C4×C7×D4—28no—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—24no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4D412yes—
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78C2×#1144—22no—
79C2×#1144—22no—
80C2×#1144—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2312yes—
85#6487—24no—
86#6487—42no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—22no—
89C2×C4×Dic7—21no—
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2312yes—
92C22×C7⋊C8C2311yescharacteristic
93C4×Dic14—22no—
94C4×Dic14—22no—
95#6409—22no—
96#6409—42no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6384—24no—
101#6384—22no—
102#6381—24no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#554D711yescharacteristic
107C2×#477—71no—
108C2×C4×Q16—71no—
109#2538—71no—
110#2036—72no—
111#2001—72no—
112#1990—71no—
113#1954—71no—
114#1872—72no—
115#1852—72no—
116#1803—71no—
117C24×C7C2×D412yes—
118C2×C7×Q8—24no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—81no—
125C2×C7×Q8—81no—
126C2×C7×Q8C2×D412yes—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4Q1614yescomplemented
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—28no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—82no—
141C22×C4×C7—82no—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2411yescharacteristic
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—22no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14#4112yes—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4C2×D412yes—
163C7×C4⋊C4C2×D414yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—24no—
166C42×C7—24no—
167C42×C7C2×D412yes—
168C42×C7C2×D412yes—
169C22⋊Dic7—44no—
170C22⋊Dic7—82no—
171#1144—28no—
172C4⋊Dic7—21no—
173C4⋊Dic7—21no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—42no—
177C4⋊Dic7—81no—
178Dic7⋊C4—44no—
179Dic7⋊C4—42no—
180C4×Dic7—42no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8#4112yes—
190C22×Q16—71no—
191#554—24no—
192#554—24no—
193C2×C4⋊Q8C2×D711yescharacteristic
194C2×#125—21no—
195C2×C22⋊Q8—22no—
196C2×C22⋊Q8—72no—
197C2×C22⋊Q8C2×D712yes—
198C2×C4×Q8—71no—
199C2×C4×Q8C2×D711yescharacteristic
200C2×C4×D4—21no—
201C2×C4×D4C2×D711yescharacteristic
202C22×C4⋊C4C2×D712yes—
203#477—144no—
204C4×Q16—144no—
205C2×#107—72no—
206C2×#107—71no—
207C2×C4⋊C8—71no—
208C2×#103—71no—
209C2×#103—72no—
210C2×#103—71no—
211C2×#103—72no—
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×C4×C8—71no—
215#396—142no—
216#390—141no—
217#389—141no—
218#384—71no—
219#380—72no—
220#340—72no—
221#340—72no—
222#340—72no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7C2×Q1614yes—
230C23×C7C22×D411yescharacteristic
231C7×Q8—22no—
232C7×Q8—22no—
233C7×Q8—41no—
234C7×Q8—41no—
235C7×Q8—84no—
236C7×D4—28no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—42no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—88no—
256C2×C4×C7#12012yes—
257C2×C4×C7#12012yes—
258C2×C4×C7C2×Q1612yes—
259C2×C4×C7#13712yes—
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C2×#4111yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7H(5,𝔽2)11yescharacteristic
265C2×C4×C72−1+411yescharacteristic
266C2×Dic7—22no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275Dic14—22no—
276Dic14—22no—
277Dic14—41no—
278Dic14—41no—
279C7⋊C8—22no—
280C7⋊C8—41no—
281C7⋊C8—42no—
282C22×Q8—21no—
283C22×Q8—71no—
284C22×Q8C22×D711yescharacteristic
285C22×D4C22⋊D711yescharacteristic
286C23×C4—22no—
287C23×C4C22×D712yes—
288C2×Q16—142no—
289C2×Q16—142no—
290C2×Q16—142no—
291C22×C8—71no—
292C22×C8—71no—
293C22×C8—72no—
294C4⋊Q8—22no—
295C4⋊Q8C22⋊D714yes—
296#125—28no—
297C22⋊Q8—216no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—144no—
301C22⋊Q8—282no—
302C4×Q8—22no—
303C4×Q8—142no—
304C4×Q8—142no—
305C4×Q8C22⋊D714yes—
306C4×D4—28no—
307C4×D4—22no—
308C4×D4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—72no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4C22⋊D711yescharacteristic
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C4⋊C4C22×D712yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4C22⋊D712yes—
328C2×C42—21no—
329C2×C42—141no—
330C2×C42C22×D711yescharacteristic
331#107—142no—
332#107—144no—
333C4⋊C8—142no—
334#103—142no—
335#103—142no—
336#103—144no—
337#103—282no—
338C22⋊C8—144no—
339C22⋊C8—282no—
340C4×C8—142no—
341#95—142no—
342C22×C7—24no—
343C22×C7—24no—
344C22×C7—24no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7C2×#12011yescharacteristic
350C22×C7#54611yescharacteristic
351C22×C7#54711yescharacteristic
352C22×C7C22×Q1611yescharacteristic
353C22×C7C2×#13711yescharacteristic
354C22×C7#57611yescharacteristic
355C22×C7#57711yescharacteristic
356C4×C7—22no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7#45114yes—
362C4×C7#52511yescharacteristic
363C4×C7#53711yescharacteristic
364C4×C7#54811yescharacteristic
365C4×C7#55011yescharacteristic
366Dic7—42no—
367Dic7—82no—
368Dic7—81no—
369C24C2×C22⋊D712yes—
370C2×Q8—24no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—72no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8—141no—
384C2×Q8C2×C22⋊D712yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4#114414yescomplemented
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—142no—
403C22×C4C2×C22⋊D712yes—
404C22×C4C2×C22⋊D711yescharacteristic
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407Q16—148no—
408C2×C8—72no—
409C2×C8—141no—
410C2×C8—141no—
411C2×C8—142no—
412C2×C8—141no—
413C2×C8—142no—
414C2×C8—141no—
415C2×C8—141no—
416C2×C8—142no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—142no—
425C4⋊C4—142no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4—284no—
429C4⋊C4—282no—
430C4⋊C4—282no—
431C4⋊C4—561no—
432C4⋊C4C2×C22⋊D712yes—
433C4⋊C4C2×C22⋊D714yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—284no—
437C22⋊C4—562no—
438C42—24no—
439C42—282no—
440C42C2×C22⋊D712yes—
441C42C2×C22⋊D712yes—
442C2×C7—44no—
443C2×C7#240711yescharacteristic
444C2×C7C2×#45111yescharacteristic
445C2×C7#301211yescharacteristic
446C2×C7#329911yescharacteristic
447C2×C7#333311yescharacteristic
448C2×C7#333811yescharacteristic
449C2×C7#335611yescharacteristic
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23C2×#114414yes—
457C23C22×C22⋊D711yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—41no—
461Q8—41no—
462Q8—84no—
463Q8—142no—
464Q8—142no—
465Q8—281no—
466Q8—281no—
467D4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4—142no—
488C2×C4—281no—
489C2×C4—281no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—281no—
495C2×C4—281no—
496C2×C4#650512yes—
497C2×C4C2×#114412yes—
498C2×C4#651612yes—
499C2×C4#651612yes—
500C2×C4D7×#4111yescharacteristic
501C2×C4D7×#4111yescharacteristic
502C2×C4#655311yescharacteristic
503C2×C4#655411yescharacteristic
504C2×C4C22×C22⋊D711yescharacteristic
505C2×C4C22×C22⋊D711yescharacteristic
506C8—142no—
507C8—281no—
508C8—282no—
509C7#2412211yescharacteristic, complemented, Sylow
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#650511yescharacteristic
518C22C22×#114411yescharacteristic
519C22#9051511yescharacteristic
520C22#9052011yescharacteristic
521C22#9052611yescharacteristic
522C22#9052711yescharacteristic
523C22C2×#651611yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—44no—
529C4—282no—
530C4—561no—
531C4—562no—
532C4#9000014yes—
533C4#9051911yescharacteristic
534C4#9052211yescharacteristic
535C4#9052411yescharacteristic
536C4#9052511yescharacteristic
537C2—44no—
538C2#1174106811yescharacteristic
539C2#1174107111yescharacteristic
540C2#1174108511yescharacteristic
541C2#1174108711yescharacteristic
542C2C2×#9000011yescharacteristic
543C2#1174154211yescharacteristic
544C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7#24122
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#89863C2semidirect
3. C2×C4×#1144C2semidirect
5. #11729541C2semidirect
6. #11729454C2semidirect
7. #11729449C2semidirect
8. #11729380C2semidirect
10. #11729313C2semidirect
11. #11729309C2semidirect
129. C2×C7×D4Q16semidirect
387. C2×D4#1144semidirect
509. C7#24122semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898632
C2×C4×#11442
#117398212
#117295412
#117294542
#117294492
#117293802
#117293682
#117293132
#117293092
#241227