Finite Group Databasebeta

← #33103566

Subgroups — #33103566

Order 18688.

780 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988648C211yesmaximal, characteristic
2C2×#17988644C211yesmaximal, characteristic
3#22610655C211yesmaximal, characteristic, complemented
4#22610650C211yesmaximal, characteristic, complemented
5#22610497C211yesmaximal, characteristic, complemented
6#22610497C211yesmaximal, characteristic, complemented
7#22610497C211yesmaximal, characteristic, complemented
8#22610490C211yesmaximal, characteristic, complemented
9#22610002C211yesmaximal, characteristic
10#22609999C211yesmaximal, characteristic
11#22609915C211yesmaximal, characteristic
12#22609806C211yesmaximal, characteristic, complemented
13#22609800C211yesmaximal, characteristic, complemented
14#22609697C211yesmaximal, characteristic
15C73×#2296C211yesmaximal, characteristic
16#17988648—24no—
17#17988644—24no—
18C2×C4⋊#10596831C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×#14556502C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988342C2211yescharacteristic
35#17988342C2211yescharacteristic
36#17988342C2211yescharacteristic
37#17988342C2211yescharacteristic
38#17988342C2211yescharacteristic
39#17988342C2211yescharacteristic
40#17988342C2211yescharacteristic
41#17988342C2211yescharacteristic
42#17988342C2211yescharacteristic
43#17988342C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46C2×#117×C73C2211yescharacteristic
47C2×#117×C73C2211yescharacteristic
48C22×C42×C73C2211yescharacteristic
49#389×C73C411yescharacteristic
50#389×C73C411yescharacteristic
51#389×C73C411yescharacteristic
52#389×C73C411yescharacteristic
53#385×C73C411yescharacteristic
54#385×C73C411yescharacteristic
55#385×C73C411yescharacteristic
56#385×C73C411yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C4⋊#10596831—22no—
70C4⋊#10596831—22no—
71C4⋊#10596831—22no—
72C4⋊#10596831—22no—
73C4⋊#10596831—22no—
74C4⋊#10596831—22no—
75C22⋊#10596831—22no—
76C22⋊#10596831—22no—
77C22⋊#10596831—22no—
78C22⋊#10596831D414yes—
79#14556502—22no—
80#14556502—22no—
81#14556502—22no—
82#14556502—22no—
83C4×#10596831—22no—
84C4×#10596831—22no—
85C23×C4×C73C2×C411yescharacteristic
86C23×C4×C73C2×C411yescharacteristic
87C23×C4×C73C2311yescharacteristic
88#117×C73—22no—
89#117×C73—22no—
90#117×C73—22no—
91#117×C73—22no—
92C2×C4⋊C4×C73C2×C411yescharacteristic
93C2×C4⋊C4×C73C2×C411yescharacteristic
94C2×C4⋊C4×C73C2×C411yescharacteristic
95C2×C4⋊C4×C73C2×C411yescharacteristic
96C2×C4⋊C4×C73C2×C411yescharacteristic
97C2×C4⋊C4×C73C2×C411yescharacteristic
98C2×C4⋊C4×C73C2×C411yescharacteristic
99C2×C4⋊C4×C73C2×C411yescharacteristic
100C2×C4⋊C4×C73C2×C411yescharacteristic
101C2×C4⋊C4×C73C2×C411yescharacteristic
102C2×C4⋊C4×C73C2×C411yescharacteristic
103C2×C4⋊C4×C73C2×C411yescharacteristic
104C2×C22⋊C4×C73C2×C411yescharacteristic
105C2×C22⋊C4×C73C2×C411yescharacteristic
106C2×C22⋊C4×C73C2×C411yescharacteristic
107C2×C22⋊C4×C73C2×C411yescharacteristic
108C2×C42×C73—22no—
109C2×C42×C73—22no—
110C2×C42×C73C2×C411yescharacteristic
111C2×C42×C73C2×C411yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73C2×C411yescharacteristic
119#95×C73C2×C411yescharacteristic
120#95×C73C2×C411yescharacteristic
121#95×C73C2×C411yescharacteristic
122#95×C73C2×C411yescharacteristic
123#95×C73C2×C411yescharacteristic
124#95×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831C2×D412yes—
149C2×#10596831C2×D412yes—
150C24×C73C22×C411yescharacteristic
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—41no—
159C22×C4×C73—41no—
160C22×C4×C73—41no—
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C22×C411yescharacteristic
163C22×C4×C73C22×C411yescharacteristic
164C22×C4×C73C22×C411yescharacteristic
165C22×C4×C73C22×C411yescharacteristic
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C22×C411yescharacteristic
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C22×C411yescharacteristic
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C2×D412yes—
175C22×C4×C73#4112yes—
176C22×C4×C73C2411yescharacteristic
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—41no—
184C4⋊C4×C73—41no—
185C4⋊C4×C73—41no—
186C4⋊C4×C73—41no—
187C4⋊C4×C73—41no—
188C4⋊C4×C73—41no—
189C22⋊C4×C73—22no—
190C22⋊C4×C73—22no—
191C22⋊C4×C73—22no—
192C22⋊C4×C73—22no—
193C42×C73—22no—
194C42×C73—22no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73#4112yes—
206C42×C73#4112yes—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C4×D412yes—
214C23×C73C23×C411yescharacteristic
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—44no—
255C2×C4×C73—44no—
256C2×C4×C73#11712yes—
257C2×C4×C73#11712yes—
258C2×C4×C73C4×D412yes—
259C2×C4×C73#13112yes—
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73C2×#4111yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C732−1+411yescharacteristic
265C2×C4×C732−1+411yescharacteristic
266#10596831—22no—
267#10596831—22no—
268#10596831—41no—
269#10596831—41no—
270#10596831—41no—
271#10596831—41no—
272#10596831—41no—
273#10596831—41no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73C2×#11711yescharacteristic
282C22×C73C2×C4×D411yescharacteristic
283C22×C73#51911yescharacteristic
284C22×C73C2×#13111yescharacteristic
285C22×C73#56811yescharacteristic
286C22×C73#56811yescharacteristic
287C22×C73#56811yescharacteristic
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73#54711yescharacteristic
302C4×C73#54711yescharacteristic
303C4×C73#55211yescharacteristic
304C4×C73#55211yescharacteristic
305#22605—731nomaximal, Sylow
306C2×C73—42no—
307C2×C73#234411yescharacteristic
308C2×C73#292311yescharacteristic
309C2×C73#292511yescharacteristic
310C2×C73#293011yescharacteristic
311C2×C73#293311yescharacteristic
312C2×C73#298511yescharacteristic
313C2×C73#298711yescharacteristic
314C2×#433—731no—
315C2×#432—731no—
316#2296D7311yescharacteristic
317#1940—731no—
318#1939—731no—
319#1917—731no—
320#1917—731no—
321#1917—731no—
322#1916—731no—
323#1818—731no—
324#1817—731no—
325#1798—731no—
326#1771—731no—
327#1770—731no—
328#1750—731no—
329C73#2260511yescharacteristic, complemented, Sylow
330C2×#117C2×D7311yescharacteristic
331C2×#117C2×D7311yescharacteristic
332C22×C42C2×D7311yescharacteristic
333#433—1464no—
334#432—1464no—
335C2×C4⋊C8—731no—
336C2×C4⋊C8—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×C22⋊C8—731no—
342C2×C22⋊C8—731no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×#97—731no—
346C2×#97—731no—
347C2×#97—731no—
348C2×#97—731no—
349C2×C4×C8—731no—
350C2×C4×C8—731no—
351#389Dic7311yescharacteristic
352#389Dic7311yescharacteristic
353#389Dic7311yescharacteristic
354#389Dic7311yescharacteristic
355#385Dic7311yescharacteristic
356#385Dic7311yescharacteristic
357#385Dic7311yescharacteristic
358#385Dic7311yescharacteristic
359C4×C4⋊C4C2×D7311yescharacteristic
360C4×C4⋊C4C2×D7311yescharacteristic
361C4×C4⋊C4C2×D7311yescharacteristic
362C4×C4⋊C4C2×D7311yescharacteristic
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369#336—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374#336—731no—
375C23×C4C2×Dic7311yescharacteristic
376C23×C4C2×Dic7311yescharacteristic
377C23×C4C22×D7311yescharacteristic
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386#117—22no—
387#117—22no—
388#117—22no—
389#117—22no—
390C2×C4⋊C4C2×Dic7311yescharacteristic
391C2×C4⋊C4C2×Dic7311yescharacteristic
392C2×C4⋊C4C2×Dic7311yescharacteristic
393C2×C4⋊C4C2×Dic7311yescharacteristic
394C2×C4⋊C4C2×Dic7311yescharacteristic
395C2×C4⋊C4C2×Dic7311yescharacteristic
396C2×C4⋊C4C2×Dic7311yescharacteristic
397C2×C4⋊C4C2×Dic7311yescharacteristic
398C2×C4⋊C4C2×Dic7311yescharacteristic
399C2×C4⋊C4C2×Dic7311yescharacteristic
400C2×C4⋊C4C2×Dic7311yescharacteristic
401C2×C4⋊C4C2×Dic7311yescharacteristic
402C2×C22⋊C4C2×Dic7311yescharacteristic
403C2×C22⋊C4C2×Dic7311yescharacteristic
404C2×C22⋊C4C2×Dic7311yescharacteristic
405C2×C22⋊C4C2×Dic7311yescharacteristic
406C2×C42—22no—
407C2×C42—22no—
408C2×C42C2×Dic7311yescharacteristic
409C2×C42C2×Dic7311yescharacteristic
410C2×C42C22×D7311yescharacteristic
411C2×C42C22×D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413C2×C42C22×D7311yescharacteristic
414C2×C42C22×D7311yescharacteristic
415C2×C42C22×D7311yescharacteristic
416C4⋊C8—1462no—
417C4⋊C8—1462no—
418C4⋊C8—1462no—
419C4⋊C8—1462no—
420C4⋊C8—1462no—
421C4⋊C8—1462no—
422C22⋊C8—734no—
423C22⋊C8—1462no—
424C22⋊C8—1462no—
425C22⋊C8—1462no—
426#97—1462no—
427#97—1462no—
428#97—1462no—
429#97—1462no—
430C4×C8—1462no—
431C4×C8—1462no—
432#95C2×Dic7311yescharacteristic
433#95C2×Dic7311yescharacteristic
434#95C2×Dic7311yescharacteristic
435#95C2×Dic7311yescharacteristic
436#95C2×Dic7311yescharacteristic
437#95C2×Dic7311yescharacteristic
438#95C2×Dic7311yescharacteristic
439#95C2×Dic7311yescharacteristic
440C24C22×Dic7311yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4C22×Dic7311yescharacteristic
452C22×C4C22×Dic7311yescharacteristic
453C22×C4C22×Dic7311yescharacteristic
454C22×C4C22×Dic7311yescharacteristic
455C22×C4C22×Dic7311yescharacteristic
456C22×C4C22×Dic7311yescharacteristic
457C22×C4C22×Dic7311yescharacteristic
458C22×C4C22×Dic7311yescharacteristic
459C22×C4C22×Dic7311yescharacteristic
460C22×C4C22×Dic7311yescharacteristic
461C22×C4C22×Dic7311yescharacteristic
462C22×C4C22×Dic7311yescharacteristic
463C22×C4C22×Dic7311yescharacteristic
464C22×C4D4×D7312yes—
465C22×C4#1192402712yes—
466C22×C4C23×D7311yescharacteristic
467C2×C8—732no—
468C2×C8—732no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42#1192402812yes—
520C42#1192402812yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23D4×Dic7312yes—
528C23C23×Dic7311yescharacteristic
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4#1455655012yes—
571C2×C4#1455655012yes—
572C2×C4D4×Dic7312yes—
573C2×C4#1455659812yes—
574C2×C4C2×D4×D7311yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C2×C4C2×#1192402811yescharacteristic
577C2×C4C2×#1192402811yescharacteristic
578C2×C4#1455663911yescharacteristic
579C2×C4#1455663911yescharacteristic
580C8—2921no—
581C8—2921no—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C8—1462no—
587C8—1462no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#1455655011yescharacteristic
596C22C2×D4×Dic7311yescharacteristic
597C22#1798906511yescharacteristic
598C22C2×#1455659811yescharacteristic
599C22#1798935411yescharacteristic
600C22#1798935411yescharacteristic
601C22#1798935411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4#1798923811yescharacteristic
616C4#1798923811yescharacteristic
617C4#1798927211yescharacteristic
618C4#1798927211yescharacteristic
619C2—42no—
620C2#2261280011yescharacteristic
621C2#2261858211yescharacteristic
622C2#2261859711yescharacteristic
623C2#2261863511yescharacteristic
624C2#2261865811yescharacteristic
625C2#2261902611yescharacteristic
626C2#2261903711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#2296C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#2296D73
hypocenterC73#22605
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
3. #22610655C2semidirect
4. #22610650C2semidirect
5. #22610497C2semidirect
6. #22610497C2semidirect
7. #22610497C2semidirect
8. #22610490C2semidirect
12. #22609806C2semidirect
13. #22609800C2semidirect
329. C73#22605semidirect

Maximal subgroups

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

subgroupindex
C2×#179886482
C2×#179886442
#226106552
#226106502
#226104972
#226104902
#226100022
#226099992
#226099152
#226098062
#226098002
#226096972
C73×#22962
#2260573