Finite Group Databasebeta

← #13387751

Subgroups — #13387751

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×#90234C211yesmaximal, characteristic
2C7×#2538C211yesmaximal, characteristic
3C2×#89793C211yesmaximal, characteristic, complemented
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5#11729454C212yesmaximal, complemented
6#11729446C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic
8#11727342C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727281C212yesmaximal, complemented
11#11727258C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2212yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C22×#1144C2211yescharacteristic
16C2×D4×Dic7—21no—
17C22×C4⋊Dic7C2212yes—
18C2×#6487C2212yes—
19#90234—24no—
20#90234—24no—
21C2×#6453—21no—
22C2×#6441—22no—
23C2×#6432C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#396—22no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#384C2211yescharacteristic
29C7×#380C2212yes—
30C2×#6409C2211yescharacteristic
31C2×#6409C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C4×#1144—24no—
35#89793—24no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2212yes—
38C2×#6381C2211yescharacteristic
39C2×#6381C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24122—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7C2312yes—
49C23×Dic7—22no—
50C22×Dic14—21no—
51C22×Dic14C2311yescharacteristic
52C7×C22⋊Q8—24no—
53C7×C22⋊Q8—42no—
54C4×C7×Q8—22no—
55C4×C7×Q8—22no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4D412yes—
68C2×C42×C7—21no—
69C2×C42×C7C2311yescharacteristic
70C2×C22⋊Dic7—22no—
71C2×#1144—22no—
72C2×#1144—22no—
73C2×#1144—22no—
74D4×Dic7—28no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—24no—
78C2×C4⋊Dic7—24no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2312yes—
81#6487—24no—
82#6487—24no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—22no—
85C2×C4×Dic7—21no—
86C22×C7⋊C8C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89#6453—28no—
90#6441—216no—
91#6432—22no—
92#6432D414yes—
93C4×Dic14—22no—
94C4×Dic14D414yes—
95C7×#95—22no—
96#6409—24no—
97#6409—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6384—22no—
101#6384—42no—
102#6381—24no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105C4×C7⋊C8—22no—
106C2×#554—71no—
107C2×#477—71no—
108C2×C4×Q16—71no—
109#2538D711yescharacteristic
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—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8#4112yes—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4Q1614yescomplemented
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—28no—
144C22×Dic7—82no—
145C22×Dic7—82no—
146C2×Dic14—21no—
147C2×Dic14—21no—
148C2×Dic14—21no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14—24no—
152C2×Dic14—81no—
153C2×Dic14—81no—
154C2×Dic14C2×D412yes—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—44no—
161C7×C4⋊C4—42no—
162C7×C4⋊C4—42no—
163C7×C4⋊C4—81no—
164C7×C22⋊C4—24no—
165C7×C22⋊C4—44no—
166C7×C22⋊C4—82no—
167C42×C7—42no—
168C42×C7C2×D412yes—
169C42×C7C2×D412yes—
170C22⋊Dic7—28no—
171#1144—28no—
172C4⋊Dic7—28no—
173C4⋊Dic7—24no—
174C4⋊Dic7—21no—
175C4⋊Dic7—22no—
176C4⋊Dic7C2×D412yes—
177C4⋊Dic7C2×D414yes—
178Dic7⋊C4—28no—
179Dic7⋊C4—24no—
180C4×Dic7—24no—
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—144no—
192#554—144no—
193C2×C4⋊Q8—71no—
194C2×#125—141no—
195C2×C22⋊Q8—72no—
196C2×C22⋊Q8—142no—
197C2×C22⋊Q8C2×D712yes—
198C2×C4×Q8—71no—
199C2×C4×Q8C2×D711yescharacteristic
200C2×C4×D4—141no—
201C2×C4×D4C2×D711yescharacteristic
202C22×C4⋊C4—72no—
203#477—144no—
204C4×Q16—144no—
205C2×#107—72no—
206C2×#107—71no—
207C2×C4⋊C8—71no—
208C2×#103—72no—
209C2×#103—71no—
210C2×#103—71no—
211C2×#103—72no—
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×C4×C8—71no—
215#396—22no—
216#390—21no—
217#389—21no—
218#384C2×D711yescharacteristic
219#380C2×D712yes—
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×D4—28no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7#12012yes—
255C2×C4×C7#12012yes—
256C2×C4×C7C2×Q1612yes—
257C2×C4×C7#13712yes—
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C2×#4111yescharacteristic
261C2×C4×C7C2×#4111yescharacteristic
262C2×C4×C7H(5,𝔽2)11yescharacteristic
263C2×C4×C72−1+411yescharacteristic
264C2×Dic7—22no—
265C2×Dic7—22no—
266C2×Dic7—24no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—41no—
271C2×Dic7—42no—
272C2×Dic7—41no—
273C2×Dic7—88no—
274Dic14—22no—
275Dic14—22no—
276Dic14—41no—
277Dic14—41no—
278Dic14—84no—
279C7⋊C8—22no—
280C7⋊C8—41no—
281C7⋊C8—42no—
282C22×Q8—71no—
283C22×Q8—141no—
284C22×Q8C22×D711yescharacteristic
285C22×D4C22⋊D711yescharacteristic
286C23×C4—142no—
287C23×C4C22×D712yes—
288C2×Q16—142no—
289C2×Q16—142no—
290C2×Q16—142no—
291C22×C8—71no—
292C22×C8—71no—
293C22×C8—72no—
294C4⋊Q8—74no—
295C4⋊Q8—142no—
296#125—148no—
297C22⋊Q8—24no—
298C22⋊Q8—42no—
299C22⋊Q8—1416no—
300C22⋊Q8—144no—
301C22⋊Q8—144no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—74no—
305C4×Q8—142no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—148no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—71no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—144no—
316C2×C4⋊C4—144no—
317C2×C4⋊C4—142no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4C22⋊D711yescharacteristic
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C4⋊C4C22×D712yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
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—22no—
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—82no—
360C4×C7—81no—
361C4×C7#45114yes—
362C4×C7#52511yescharacteristic
363C4×C7#53711yescharacteristic
364C4×C7#54811yescharacteristic
365C4×C7#55011yescharacteristic
366Dic7—42no—
367Dic7—44no—
368Dic7—42no—
369C24C2×C22⋊D712yes—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—72no—
376C2×Q8—144no—
377C2×Q8—141no—
378C2×Q8—141no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—561no—
383C2×Q8—561no—
384C2×Q8#115912yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4#114414yescomplemented
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—141no—
398C22×C4—141no—
399C22×C4—142no—
400C22×C4—148no—
401C22×C4—562no—
402C22×C4—562no—
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—141no—
412C2×C8—142no—
413C2×C8—141no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—141no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—72no—
426C4⋊C4—74no—
427C4⋊C4—81no—
428C4⋊C4—148no—
429C4⋊C4—148no—
430C4⋊C4—144no—
431C4⋊C4—144no—
432C4⋊C4—141no—
433C4⋊C4—142no—
434C22⋊C4—24no—
435C22⋊C4—44no—
436C22⋊C4—82no—
437C22⋊C4—148no—
438C42—42no—
439C42—144no—
440C42D4×D712yes—
441C42D4×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—142no—
463Q8—142no—
464Q8—281no—
465Q8—281no—
466Q8—564no—
467D4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—144no—
489C2×C4—281no—
490C2×C4—281no—
491C2×C4—281no—
492C2×C4—282no—
493C2×C4—281no—
494C2×C4—282no—
495C2×C4—568no—
496C2×C4#650012yes—
497C2×C4#650012yes—
498C2×C4C2×#114412yes—
499C2×C4#651412yes—
500C2×C4C2×D4×D711yescharacteristic
501C2×C4C2×D4×D711yescharacteristic
502C2×C4C2×#115911yescharacteristic
503C2×C4#654811yescharacteristic
504C2×C4D7×#4111yescharacteristic
505C2×C4#655411yescharacteristic
506C8—142no—
507C8—282no—
508C8—281no—
509C7#2412211yescharacteristic, complemented, Sylow
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#650011yescharacteristic
518C22#9049711yescharacteristic
519C22C22×#114411yescharacteristic
520C22C2×#651411yescharacteristic
521C22#9051511yescharacteristic
522C22#9052611yescharacteristic
523C22#9052811yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—81no—
529C4—282no—
530C4—284no—
531C4—282no—
532C4#8982214yes—
533C4#9028911yescharacteristic
534C4#9029011yescharacteristic
535C4#9029711yescharacteristic
536C4#9029911yescharacteristic
537C2—44no—
538C2#1174014611yescharacteristic
539C2C2×#8982211yescharacteristic
540C2#1174086911yescharacteristic
541C2#1174099611yescharacteristic
542C2#1174102011yescharacteristic
543C2#1174106811yescharacteristic
544C2#1174110811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenter#2538D7
hypocenterC7#24122
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
3. C2×#89793C2semidirect
4. C2×C4×#1144C2semidirect
5. #11729454C2semidirect
6. #11729446C2semidirect
8. #11727342C2semidirect
9. #11727336C2semidirect
10. #11727281C2semidirect
11. #11727258C2semidirect
126. 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×#902342
C7×#25382
C2×#897932
C2×C4×#11442
#117294542
#117294462
#117293682
#117273422
#117273362
#117272812
#117272582
#241227