Finite Group Databasebeta

← #13372173

Subgroups — #13372173

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89645C211yesmaximal, characteristic
2C2×#89625C211yesmaximal, characteristic
3C2×#89488C211yesmaximal, characteristic
4#11731768C211yesmaximal, characteristic
5#11731602C211yesmaximal, characteristic
6C7×#1847C211yesmaximal, characteristic
7#11728396C211yesmaximal, characteristic
8#11728396C211yesmaximal, characteristic
9#11728368C211yesmaximal, characteristic
10#11728342C211yesmaximal, characteristic
11#11728342C211yesmaximal, characteristic
12#11728313C211yesmaximal, characteristic
13#11727564C211yesmaximal, characteristic
14#11727342C211yesmaximal, characteristic
15#11727313C211yesmaximal, characteristic
16C22×Dic28C2211yescharacteristic
17C2×#6450C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×#6450C2211yescharacteristic
20C2×#6450C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#107C2211yescharacteristic
25C2×C7×#106C2211yescharacteristic
26C2×C7×#106C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C7×#384C2211yescharacteristic
29C7×#382C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35#89748—21no—
36C4⋊C4×Dic7—21no—
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89745C2211yescharacteristic
40#89745C2211yescharacteristic
41#89744—21no—
42#89744—21no—
43C2×#6384C2211yescharacteristic
44C2×#6384C2211yescharacteristic
45C2×#6384C2211yescharacteristic
46C2×#6384C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49#89645—24no—
50#89625—24no—
51#89488C418yescomplemented
52#89422—21no—
53#89421—21no—
54#89419—21no—
55#89419—21no—
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22679—71nomaximal, Sylow
63C22×Dic14C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C4⋊Dic7D411yescharacteristic
75C2×C4⋊Dic7D411yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C2×Dic28C2×C414yes—
99C2×Dic28D414yes—
100C2×Dic28Q814yes—
101#6450—22no—
102#6450—22no—
103#6450—22no—
104#6450—22no—
105#6450—41no—
106#6450—41no—
107#6450—41no—
108#6450—41no—
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14C2×C414yes—
112C4×Dic14C2×C414yes—
113C7×#107—22no—
114C7×#107—41no—
115C7×#106—22no—
116C7×#106—41no—
117C7×#97C2×C414yes—
118C7×#95—21no—
119C7×#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95D411yescharacteristic
125C7⋊#95D411yescharacteristic
126#6391C2×C414yes—
127#6390C2×C414yes—
128#6390C2×C414yes—
129#6389—22no—
130C8×Dic7—22no—
131#6384—22no—
132#6384—22no—
133#6384—22no—
134#6384—22no—
135C4⋊(C7⋊C8)—22no—
136C2×#441—71no—
137C2×#441—71no—
138C2×C4×Q16—71no—
139#2351—71no—
140Q8×C4⋊C4—71no—
141#1944—71no—
142#1941—71no—
143#1865—71no—
144#1865—71no—
145#1863—71no—
146#1863—71no—
147#1847D711yescharacteristic
148#1805—71no—
149#1803—71no—
150#1771—71no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14C22×C412yes—
173C2×Dic14C22×C412yes—
174C2×Dic14C2×D412yes—
175C2×Dic14C2×D412yes—
176C2×Dic14C2×Q812yes—
177C2×Dic14C2×Q812yes—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8C22×C412yes—
181C2×C7×C8C22×C412yes—
182C2×C7×C8C2×D412yes—
183C2×C7×C8C2×Q812yes—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4—81no—
194C42×C7C22×C412yes—
195C42×C7#4112yes—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—22no—
199C4⋊Dic7—41no—
200C4⋊Dic7C22×C412yes—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—24no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—81no—
211Dic7⋊C4—81no—
212C4×Dic7—24no—
213C4×Dic7—24no—
214C4×Dic7—21no—
215C4×Dic7—21no—
216C4×Dic7—21no—
217C4×Dic7—21no—
218C4×Dic7—42no—
219C4×Dic7—42no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228Dic28C4⋊C4116yescomplemented
229C22×Q16—71no—
230C2×C4×Q8—71no—
231C2×C4×Q8—71no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×Q8—71no—
235C2×C4×Q8—71no—
236#441—78no—
237#441—144no—
238C4×Q16—144no—
239C2×#107C2×D711yescharacteristic
240C2×#107C2×D711yescharacteristic
241C2×#106—71no—
242C2×#106C2×D711yescharacteristic
243C2×#106C2×D711yescharacteristic
244C2×C4⋊C8—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#97—71no—
252C2×#97C2×D711yescharacteristic
253C2×C4×C8—71no—
254#389—71no—
255#389—71no—
256#389—71no—
257#389—71no—
258#389—141no—
259#389—141no—
260#389—141no—
261#389—141no—
262#384—141no—
263#384C2×D711yescharacteristic
264#382—141no—
265#382C2×D711yescharacteristic
266C4×C4⋊C4—71no—
267C4×C4⋊C4—141no—
268C4×C4⋊C4—141no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#336—71no—
274#336—71no—
275C23×C7C22×D411yescharacteristic
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7C4×D412yes—
293C2×C4×C7C4×D412yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—21no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—21no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7#13712yes—
324Dic14—22no—
325Dic14—22no—
326Dic14—22no—
327Dic14—22no—
328Dic14—22no—
329Dic14—22no—
330Dic14C2×C4⋊C414yes—
331Dic14C2×C4⋊C414yes—
332C7×C8—22no—
333C7×C8C2×C4⋊C414yes—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8—71no—
337C22×Q8—71no—
338C2×Q16—74no—
339C2×Q16—74no—
340C2×Q16—74no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—74no—
346C4×Q8—74no—
347C4×Q8—142no—
348C4×Q8—142no—
349C4×Q8—142no—
350C4×Q8—142no—
351C4×Q8—142no—
352C4×Q8—142no—
353C4×Q8—281no—
354C4×Q8—281no—
355C4×Q8—281no—
356C4×Q8—281no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C4⋊C4C22×D711yescharacteristic
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42C22×D711yescharacteristic
386#107—22no—
387#107—41no—
388#106—22no—
389#106—41no—
390#106—74no—
391C4⋊C8—142no—
392#103—74no—
393#103—74no—
394#103—142no—
395#103—142no—
396#103—142no—
397#103—142no—
398#97—142no—
399#97C4×D714yes—
400C4×C8—142no—
401#95—21no—
402#95—21no—
403#95—71no—
404#95—71no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409C22×C7C2×C4×D411yescharacteristic
410C22×C7D4211yescharacteristic
411C22×C7D4×Q811yescharacteristic
412C22×C7C2×#13711yescharacteristic
413C22×C7#57511yescharacteristic
414C22×C7#57711yescharacteristic
415C22×C7#57811yescharacteristic
416C4×C7—22no—
417C4×C7—22no—
418C4×C7—81no—
419C4×C7—81no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7C22×C4⋊C411yescharacteristic
423C4×C7#52011yescharacteristic
424C4×C7#54711yescharacteristic
425C4×C7#55411yescharacteristic
426Dic7—24no—
427Dic7—24no—
428Dic7—22no—
429Dic7—42no—
430Dic7—42no—
431Dic7—41no—
432Dic7—41no—
433Dic7—41no—
434C2×Q8—72no—
435C2×Q8—72no—
436C2×Q8—72no—
437C2×Q8—72no—
438C2×Q8—72no—
439C2×Q8—72no—
440C2×Q8—141no—
441C2×Q8—141no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8—141no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4C23×D711yescharacteristic
461Q16—716no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8C2×C4×D712yes—
471C2×C8C2×C4×D712yes—
472C2×C8D4×D712yes—
473C2×C8Q8×D712yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—72no—
484C4⋊C4—72no—
485C4⋊C4—81no—
486C4⋊C4—142no—
487C4⋊C4—144no—
488C4⋊C4—144no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—281no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4—561no—
499C4⋊C4—561no—
500C42—144no—
501C42—144no—
502C42—141no—
503C42—141no—
504C42—141no—
505C42—141no—
506C42—282no—
507C42—282no—
508C42—282no—
509C42—282no—
510C42C2×C4×D712yes—
511C42#116112yes—
512C2×C7D4×C4⋊C411yescharacteristic
513C2×C7C4×#13711yescharacteristic
514C2×C7#294711yescharacteristic
515C2×C7#327011yescharacteristic
516C2×C7#327111yescharacteristic
517C2×C7#338711yescharacteristic
518C2×C7#339011yescharacteristic
519C23C2×D4×D711yescharacteristic
520Q8—74no—
521Q8—74no—
522Q8—142no—
523Q8—142no—
524Q8—142no—
525Q8—142no—
526Q8—142no—
527Q8—142no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
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—72no—
545C2×C4—141no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—141no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4#645612yes—
569C2×C4#645612yes—
570C2×C4C22×C4×D711yescharacteristic
571C2×C4C2×D4×D711yescharacteristic
572C2×C4#654811yescharacteristic
573C2×C4C2×Q8×D711yescharacteristic
574C2×C4C2×#116111yescharacteristic
575C2×C4#655111yescharacteristic
576C8—22no—
577C8—281no—
578C8—281no—
579C8D7×C4⋊C414yes—
580C7#2267911yescharacteristic, complemented, Sylow
581C22C2×#645611yescharacteristic
582C22#9040811yescharacteristic
583C22#9041811yescharacteristic
584C22#9046411yescharacteristic
585C22#9046611yescharacteristic
586C22D7×#13711yescharacteristic
587C22#9047011yescharacteristic
588C4—22no—
589C4—22no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—144no—
595C4—144no—
596C4—142no—
597C4—282no—
598C4—282no—
599C4—281no—
600C4—281no—
601C4—281no—
602C4C2×D7×C4⋊C411yescharacteristic
603C4#9019511yescharacteristic
604C4#9041711yescharacteristic
605C4#9042111yescharacteristic
606C2#1173180511yescharacteristic
607C2#1173664511yescharacteristic
608C2#1173664711yescharacteristic
609C2#1173930411yescharacteristic
610C2#1173932911yescharacteristic
611C2#1173941211yescharacteristic
612C2#1173941611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1847C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1847D7
hypocenterC7#22679
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
51. #89488C4semidirect
228. Dic28C4⋊C4semidirect
580. C7#22679semidirect

Maximal subgroups

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

subgroupindex
C2×#896452
C2×#896252
C2×#894882
#117317682
#117316022
C7×#18472
#117283962
#117283682
#117283422
#117283132
#117275642
#117273422
#117273132
#226797