Finite Group Databasebeta

← #13416759

Subgroups — #13416759

Order 1792.

788 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2451C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic
3C2×C4×#1144C211yesmaximal, characteristic
4#11731082C211yesmaximal, characteristic
5#11729536C211yesmaximal, characteristic
6#11729383C211yesmaximal, characteristic
7#11727444C211yesmaximal, characteristic
8#11727444C211yesmaximal, characteristic
9#11727440C211yesmaximal, characteristic
10#11727436C211yesmaximal, characteristic
11#11727386C211yesmaximal, characteristic
12#11727385C211yesmaximal, characteristic
13#11727372C211yesmaximal, characteristic
14#11727363C211yesmaximal, characteristic
15#11727236C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×#1144C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#398—21no—
23C7×#395—21no—
24C7×#391—21no—
25C7×#391C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#387—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89866—24no—
35C4×#1144—24no—
36#89744—21no—
37#89744—21no—
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89708C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89422—21no—
52#89421—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26369—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×Q8C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—41no—
67C4×C7×Q8—22no—
68C4×C7×Q8—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×#1144—22no—
83C2×#1144—22no—
84C2×#1144Q814yescomplemented
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C4×Dic14—22no—
101C4×Dic14—22no—
102C4×Dic14—22no—
103C4×Dic14—22no—
104C4×Dic14—41no—
105C4×Dic14—41no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6409—22no—
115#6409—22no—
116#6409—41no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6384—22no—
122#6384—22no—
123#6384D414yes—
124#6382—22no—
125#6382—22no—
126#6382—22no—
127#6381—22no—
128#6381—41no—
129C4⋊(C7⋊C8)—22no—
130C4×C7⋊C8—22no—
131C2×#475—71no—
132C2×C4×Q16—71no—
133#2451D711yescharacteristic
134#2351—71no—
135#2097—71no—
136#2085—71no—
137#2066—71no—
138#2057—71no—
139#1941—71no—
140#1916—71no—
141#1878—71no—
142#1878—71no—
143#1865—71no—
144#1865—71no—
145#1863—71no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—42no—
152C2×C7×Q8—42no—
153C2×C7×Q8—81no—
154C2×C7×Q8C2×Q812yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2411yescharacteristic
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14C2×D412yes—
174C2×Dic14C2×Q812yes—
175C2×Dic14#4112yes—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—81no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4C2×D412yes—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—42no—
195#1144—28no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7#4112yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—24no—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C2×D412yes—
221C2×C7⋊C8C2×Q812yes—
222C22×Q16—71no—
223C2×C4⋊Q8C2×D711yescharacteristic
224C2×C4×Q8—71no—
225C2×C4×Q8—71no—
226C2×C4×Q8—71no—
227C2×C4×Q8C2×D711yescharacteristic
228#475—144no—
229C4×Q16—144no—
230C2×#107—71no—
231C2×#107—71no—
232C2×#106—71no—
233C2×#106—71no—
234C2×#106—71no—
235C2×C4⋊C8—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×C4×C8—71no—
243#398—21no—
244#395—21no—
245#391—21no—
246#391C2×D711yescharacteristic
247#391C2×D711yescharacteristic
248#389—71no—
249#389—71no—
250#389—141no—
251#389—141no—
252#389C2×D711yescharacteristic
253#389C2×D711yescharacteristic
254#387—21no—
255#382—71no—
256#382—141no—
257C4×C4⋊C4—141no—
258C4×C4⋊C4C2×D711yescharacteristic
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265#336—71no—
266#336—71no—
267C23×C7C22×D411yescharacteristic
268C7×Q8—22no—
269C7×Q8—22no—
270C7×Q8—41no—
271C7×Q8—41no—
272C7×Q8—84no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7#13512yes—
302C2×C4×C7#13712yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×Q811yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7H(5,𝔽2)11yescharacteristic
308C2×C4×C72−1+411yescharacteristic
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321Dic14—22no—
322Dic14—22no—
323Dic14—22no—
324Dic14C22⋊Q814yes—
325C7⋊C8—22no—
326C7⋊C8—22no—
327C7⋊C8—41no—
328C7⋊C8—41no—
329C22×Q8—21no—
330C22×Q8—71no—
331C22×Q8C22×D711yescharacteristic
332C2×Q16—74no—
333C2×Q16—142no—
334C2×Q16—142no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8—71no—
339C4⋊Q8—22no—
340C4⋊Q8—41no—
341C4×Q8—22no—
342C4×Q8—41no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×Q8—142no—
346C4×Q8—142no—
347C4×Q8—281no—
348C4×Q8—281no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22⋊D711yescharacteristic
362C2×C4⋊C4C22⋊D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C42—21no—
367C2×C42—141no—
368C2×C42—141no—
369C2×C42—141no—
370C2×C42C22×D711yescharacteristic
371C2×C42C22×D711yescharacteristic
372C2×C42C22×D711yescharacteristic
373#107—142no—
374#107—281no—
375#106—142no—
376#106—142no—
377#106—142no—
378C4⋊C8—142no—
379#103—74no—
380#103—142no—
381#103—142no—
382#103—142no—
383#103—142no—
384#103—281no—
385C4×C8—142no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7#52511yescharacteristic
399C22×C7D4211yescharacteristic
400C22×C7D4×Q811yescharacteristic
401C22×C7C2×#13511yescharacteristic
402C22×C7C2×#13711yescharacteristic
403C22×C7#57711yescharacteristic
404C22×C7#57811yescharacteristic
405C4×C7—22no—
406C4×C7—22no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—42no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7C2×C22⋊Q811yescharacteristic
416C4×C7#53911yescharacteristic
417C4×C7#54711yescharacteristic
418C4×C7#55411yescharacteristic
419Dic7—24no—
420Dic7—42no—
421Dic7—81no—
422Dic7—81no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—72no—
431C2×Q8—72no—
432C2×Q8—72no—
433C2×Q8—81no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×Q8—141no—
437C2×Q8Q8×D712yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C23×D711yescharacteristic
453Q16—148no—
454C2×C8—72no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—72no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—144no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4D4×D712yes—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—144no—
497C42—282no—
498C42—282no—
499C2×C7#244911yescharacteristic
500C2×C7#310711yescharacteristic
501C2×C7#312111yescharacteristic
502C2×C7#327011yescharacteristic
503C2×C7#328211yescharacteristic
504C2×C7#338311yescharacteristic
505C2×C7#338711yescharacteristic
506C23C22×C22⋊D711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511Q8—74no—
512Q8—84no—
513Q8—142no—
514Q8—142no—
515Q8—142no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4#650512yes—
557C2×C4#651312yes—
558C2×C4C2×D4×D711yescharacteristic
559C2×C4#654811yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4#655111yescharacteristic
562C2×C4D7×#4111yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C8—142no—
565C8—142no—
566C8—281no—
567C8—281no—
568C7#2636911yescharacteristic, complemented, Sylow
569C22D4×C22⋊D711yescharacteristic
570C22C2×#650511yescharacteristic
571C22Q8×C22⋊D711yescharacteristic
572C22C2×#651311yescharacteristic
573C22#9051911yescharacteristic
574C22#9052711yescharacteristic
575C22#9052911yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—144no—
587C4—282no—
588C4—561no—
589C4—561no—
590C4D7×C22⋊Q811yescharacteristic
591C4#9032211yescharacteristic
592C4#9032711yescharacteristic
593C4#9032911yescharacteristic
594C2#1174026211yescharacteristic
595C2#1174079711yescharacteristic
596C2#1174091211yescharacteristic
597C2#1174095411yescharacteristic
598C2#1174101011yescharacteristic
599C2#1174112711yescharacteristic
600C2#1174115411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2451C2
socle, prime radicalC23×C7C22×D4
hypercenter#2451D7
hypocenterC7#26369
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2451

Complements

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

NH ≅ G/Ntype
84. C2×#1144Q8semidirect
568. C7#26369semidirect

Maximal subgroups

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

subgroupindex
C7×#24512
C2×#898662
C2×C4×#11442
#117310822
#117295362
#117293832
#117274442
#117274402
#117274362
#117273862
#117273852
#117273722
#117273632
#117272362
#263697