Finite Group Databasebeta

← #13416939

Subgroups — #13416939

Order 1792.

796 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2451C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic
3C2×C4×#1144C211yesmaximal, characteristic
4C4⋊C4×Dic14C211yesmaximal, characteristic
5#11729536C211yesmaximal, characteristic
6#11729382C211yesmaximal, characteristic
7#11727444C211yesmaximal, characteristic
8#11727444C211yesmaximal, characteristic
9#11727438C211yesmaximal, characteristic
10#11727437C211yesmaximal, characteristic
11#11727385C211yesmaximal, characteristic
12#11727385C211yesmaximal, characteristic
13#11727373C211yesmaximal, characteristic
14#11727361C211yesmaximal, 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#89863—24no—
35C4×#1144—24no—
36#89748—21no—
37C4⋊C4×Dic7—21no—
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89707C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89419—21no—
52#89419—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26381—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⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C4×Dic14—22no—
103C4×Dic14—22no—
104C4×Dic14—22no—
105C4×Dic14—22no—
106C4×Dic14—41no—
107C4×Dic14—41no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95D411yescharacteristic
115C7×#95D411yescharacteristic
116#6409—22no—
117#6409—22no—
118#6409—41no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6384—22no—
122#6384—22no—
123#6384D414yes—
124#6381—22no—
125#6381—22no—
126#6381—22no—
127#6381—22no—
128#6381—41no—
129C4⋊(C7⋊C8)—22no—
130C4×C7⋊C8—22no—
131C2×#477—71no—
132C2×C4×Q16—71no—
133#2451D711yescharacteristic
134Q8×C4⋊C4—71no—
135#2095—71no—
136#2083—71no—
137#2068—71no—
138#2057—71no—
139#1942—71no—
140#1916—71no—
141#1878—71no—
142#1878—71no—
143#1863—71no—
144#1863—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—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—41no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—81no—
185C7×C4⋊C4—81no—
186C7×C4⋊C4Q1614yescomplemented
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—42no—
202C4⋊Dic7—81no—
203C4⋊Dic7#4112yes—
204Dic7⋊C4—24no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—81no—
208Dic7⋊C4—81no—
209C4×Dic7—24no—
210C4×Dic7—42no—
211C4×Dic7—42no—
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⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8C2×D412yes—
223C2×C7⋊C8C2×Q812yes—
224C22×Q16—71no—
225C2×C4⋊Q8C2×D711yescharacteristic
226C2×C4×Q8—71no—
227C2×C4×Q8—71no—
228C2×C4×Q8—71no—
229C2×C4×Q8C2×D711yescharacteristic
230#477—144no—
231C4×Q16—144no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#107—71no—
237C2×C4⋊C8—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×C4×C8—71no—
245#398—21no—
246#395—21no—
247#391—21no—
248#391C2×D711yescharacteristic
249#391C2×D711yescharacteristic
250#389—71no—
251#389—71no—
252#389—141no—
253#389—141no—
254#389C2×D711yescharacteristic
255#389C2×D711yescharacteristic
256#387—21no—
257#384—71no—
258#384—141no—
259C4×C4⋊C4—141no—
260C4×C4⋊C4C2×D711yescharacteristic
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#336—71no—
266#336—71no—
267#336—71no—
268#336—71no—
269C23×C7C22×D411yescharacteristic
270C7×Q8—22no—
271C7×Q8—22no—
272C7×Q8—41no—
273C7×Q8—41no—
274C7×Q8—84no—
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—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
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—41no—
302C2×C4×C7C2×Q1612yes—
303C2×C4×C7C2×Q1612yes—
304C2×C4×C7C2×Q1612yes—
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×Q811yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7H(5,𝔽2)11yescharacteristic
310C2×C4×C72−1+411yescharacteristic
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323Dic14—22no—
324Dic14—22no—
325Dic14—22no—
326Dic14C22⋊Q814yes—
327C7⋊C8—22no—
328C7⋊C8—22no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×Q8—21no—
332C22×Q8—71no—
333C22×Q8C22×D711yescharacteristic
334C2×Q16—74no—
335C2×Q16—142no—
336C2×Q16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C4⋊Q8—22no—
342C4⋊Q8—41no—
343C4×Q8—22no—
344C4×Q8—41no—
345C4×Q8—142no—
346C4×Q8—142no—
347C4×Q8—142no—
348C4×Q8—142no—
349C4×Q8—281no—
350C4×Q8—281no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4C22⋊D711yescharacteristic
366C2×C4⋊C4C22⋊D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C42—21no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377#107—142no—
378#107—142no—
379#107—142no—
380#107—142no—
381#107—281no—
382C4⋊C8—142no—
383#103—74no—
384#103—142no—
385#103—142no—
386#103—142no—
387#103—142no—
388#103—281no—
389C4×C8—142no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—141no—
397#95—141no—
398#95C22⋊D711yescharacteristic
399#95C22⋊D711yescharacteristic
400C22×C7#52511yescharacteristic
401C22×C7D4211yescharacteristic
402C22×C7D4×Q811yescharacteristic
403C22×C7C22×Q1611yescharacteristic
404C22×C7#57511yescharacteristic
405C22×C7#57611yescharacteristic
406C22×C7#57811yescharacteristic
407C4×C7—22no—
408C4×C7—22no—
409C4×C7—22no—
410C4×C7—41no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—42no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7C2×C22⋊Q811yescharacteristic
418C4×C7#53911yescharacteristic
419C4×C7#54711yescharacteristic
420C4×C7#55411yescharacteristic
421Dic7—24no—
422Dic7—42no—
423Dic7—81no—
424Dic7—81no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—72no—
433C2×Q8—72no—
434C2×Q8—72no—
435C2×Q8—81no—
436C2×Q8—141no—
437C2×Q8—141no—
438C2×Q8—141no—
439C2×Q8Q8×D712yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C23×D711yescharacteristic
455Q16—148no—
456C2×C8—72no—
457C2×C8—72no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—72no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—144no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—561no—
489C4⋊C4—561no—
490C4⋊C4—561no—
491C4⋊C4#114414yescomplemented
492C4⋊C4D4×D712yes—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—144no—
501C42—282no—
502C42—282no—
503C2×C7#244911yescharacteristic
504C2×C7#310411yescharacteristic
505C2×C7#312411yescharacteristic
506C2×C7#326811yescharacteristic
507C2×C7D4×Q1611yescharacteristic
508C2×C7Q8×Q1611yescharacteristic
509C2×C7#339011yescharacteristic
510C23C22×C22⋊D711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515Q8—74no—
516Q8—84no—
517Q8—142no—
518Q8—142no—
519Q8—142no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4C2×#114412yes—
560C2×C4C2×#114412yes—
561C2×C4C2×#114412yes—
562C2×C4C2×D4×D711yescharacteristic
563C2×C4#654811yescharacteristic
564C2×C4C2×Q8×D711yescharacteristic
565C2×C4#655111yescharacteristic
566C2×C4D7×#4111yescharacteristic
567C2×C4D7×#4111yescharacteristic
568C8—142no—
569C8—142no—
570C8—281no—
571C8—281no—
572C7#2638111yescharacteristic, complemented, Sylow
573C22D4×C22⋊D711yescharacteristic
574C22C22×#114411yescharacteristic
575C22Q8×C22⋊D711yescharacteristic
576C22#9051311yescharacteristic
577C22#9051911yescharacteristic
578C22#9052611yescharacteristic
579C22#9052911yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—144no—
591C4—282no—
592C4—561no—
593C4—561no—
594C4D7×C22⋊Q811yescharacteristic
595C4#9032211yescharacteristic
596C4#9032711yescharacteristic
597C4#9032911yescharacteristic
598C2#1174026211yescharacteristic
599C2#1174081311yescharacteristic
600C2#1174090611yescharacteristic
601C2#1174095211yescharacteristic
602C2D4×#114411yescharacteristic
603C2#1174112911yescharacteristic
604C2Q8×#114411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2451C2
socle, prime radicalC23×C7C22×D4
hypercenter#2451D7
hypocenterC7#26381
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
186. C7×C4⋊C4Q16semidirect
491. C4⋊C4#1144semidirect
572. C7#26381semidirect

Maximal subgroups

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

subgroupindex
C7×#24512
C2×#898632
C2×C4×#11442
C4⋊C4×Dic142
#117295362
#117293822
#117274442
#117274382
#117274372
#117273852
#117273732
#117273612
#117272362
#263817