Finite Group Databasebeta

← #33098009

Subgroups — #33098009

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988714C211yesmaximal, characteristic, complemented
2C2×#17988645C211yesmaximal, characteristic
3#22613205C211yesmaximal, characteristic
4#22610865C211yesmaximal, characteristic, complemented
5#22610651C211yesmaximal, characteristic, complemented
6#22610649C211yesmaximal, characteristic, complemented
7#22610648C211yesmaximal, characteristic, complemented
8#22610547C211yesmaximal, characteristic
9#22610547C211yesmaximal, characteristic
10#22610539C211yesmaximal, characteristic
11#22610494C211yesmaximal, characteristic, complemented
12#22610486C211yesmaximal, characteristic, complemented
13#22609913C211yesmaximal, characteristic
14#22609805C211yesmaximal, characteristic, complemented
15C73×#1923C211yesmaximal, characteristic
16C2×C4×Dic146C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18#17988714—24no—
19#17988645—24no—
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556520C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34#17988529C411yescharacteristic
35#17988523C411yescharacteristic
36#17988521C411yescharacteristic
37#17988519C411yescharacteristic
38#17988510C411yescharacteristic
39#17988510C411yescharacteristic
40#17988508C411yescharacteristic
41#17988499C411yescharacteristic
42C4×#11924007C2211yescharacteristic
43C4×#11924007C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45C4×C22⋊Dic73C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146C2×C411yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C4×Dic146—22no—
67C4×Dic146—22no—
68#14556550—22no—
69#14556550—22no—
70C2×C4⋊Dic73C2×C411yescharacteristic
71C2×C4⋊Dic73C2×C411yescharacteristic
72C2×C4⋊Dic73C2×C411yescharacteristic
73C2×C4⋊Dic73C2×C411yescharacteristic
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×C22⋊Dic73C2×C411yescharacteristic
82C2×C22⋊Dic73C2×C411yescharacteristic
83C2×C22⋊Dic73C2×C411yescharacteristic
84C2×C22⋊Dic73C2×C411yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89C4⋊#10596831—41no—
90#14556520—22no—
91#14556520—22no—
92#14556520—22no—
93#14556520—22no—
94#14556520—22no—
95#14556520—41no—
96C22⋊#10596831—22no—
97C22⋊#10596831—41no—
98#14556501—22no—
99#14556501—41no—
100#14556501D414yes—
101C4×#10596831—22no—
102C8×Dic73—22no—
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111C23×C4×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C2×C22⋊C4×C73C2×C411yescharacteristic
117C2×C22⋊C4×C73C2×C411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C4⋊C8×C73—22no—
121C22⋊C8×C73—22no—
122C22⋊C8×C73—41no—
123C4×C8×C73—41no—
124#95×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
126C2×Dic146—22no—
127C2×Dic146—22no—
128C2×Dic146—41no—
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
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—
149C4⋊Dic73—22no—
150C4⋊Dic73—22no—
151C4⋊Dic73—22no—
152C4⋊Dic73—22no—
153C4⋊Dic73—41no—
154#11924007—22no—
155#11924007—22no—
156#11924007—22no—
157#11924007—22no—
158#11924007—41no—
159#11924007—41no—
160C22⋊Dic73—22no—
161C22⋊Dic73—22no—
162C22⋊Dic73—41no—
163C22⋊Dic73—41no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73C2×D412yes—
171C4×Dic73#4112yes—
172C24×C73C22×C411yescharacteristic
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73C2×D412yes—
197C22⋊C4×C73—22no—
198C22⋊C4×C73—22no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73C23×C411yescharacteristic
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
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—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73#13112yes—
229C2×C4×C73#13112yes—
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×C4×C732−1+411yescharacteristic
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73C4×D412yes—
259C2×Dic73C4×D412yes—
260Dic146—44no—
261C8×C73—22no—
262C8×C73—41no—
263C8×C73—41no—
264C8×C73—41no—
265#10596831—22no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73#51911yescharacteristic
278C22×C73#52011yescharacteristic
279C22×C73C2×#13111yescharacteristic
280C22×C73C2×#13111yescharacteristic
281C22×C73#56811yescharacteristic
282C22×C73#56811yescharacteristic
283C4×C73—22no—
284C4×C73—22no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73#53611yescharacteristic
289C4×C73#53711yescharacteristic
290C4×C73#54111yescharacteristic
291C4×C73#54211yescharacteristic
292Dic73—22no—
293Dic73—22no—
294Dic73—41no—
295Dic73—41no—
296Dic73—41no—
297Dic73—41no—
298Dic73—41no—
299Dic73—41no—
300#22109—731nomaximal, Sylow
301C2×C73—81no—
302C2×C73#232311yescharacteristic
303C2×C73#293311yescharacteristic
304C2×C73#293511yescharacteristic
305C2×C73#298611yescharacteristic
306C2×C73#298711yescharacteristic
307C2×C73#299211yescharacteristic
308C2×C73#299311yescharacteristic
309C2×#446—731no—
310C2×#433—731no—
311#2396—731no—
312#1984—731no—
313#1940—731no—
314#1939—731no—
315#1939—731no—
316#1924—731no—
317#1924—731no—
318#1923—731no—
319#1923D7311yescharacteristic
320#1917—731no—
321#1916—731no—
322#1798—731no—
323#1771—731no—
324C73#2210911yescharacteristic, complemented, Sylow
325C2×C4×Q8—731no—
326C2×#117—731no—
327#446—1464no—
328#433—1464no—
329C2×C4⋊C8—731no—
330C2×C4⋊C8—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×C4⋊C8—731no—
336C2×C4⋊C8C2×D7311yescharacteristic
337C2×C22⋊C8—731no—
338C2×C22⋊C8—731no—
339C2×C22⋊C8C2×D7311yescharacteristic
340C2×C22⋊C8C2×D7311yescharacteristic
341C2×#97—731no—
342C2×#97—731no—
343C2×#97—731no—
344C2×C4×C8—731no—
345C2×C4×C8—731no—
346C2×C4×C8C2×D7311yescharacteristic
347#400—731no—
348#399—731no—
349#398—731no—
350#398—731no—
351#396—731no—
352#396—731no—
353#395—731no—
354#393—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C22⋊C4—731no—
359C4×C22⋊C4C2×D7311yescharacteristic
360#336—731no—
361#336—731no—
362#336—731no—
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336C2×D7311yescharacteristic
369#336C2×D7311yescharacteristic
370C22×Q8—731no—
371C23×C4C22×D7311yescharacteristic
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8C22×D7311yescharacteristic
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C4×Q8—1462no—
381C4×Q8—1462no—
382#117—1462no—
383#117—1462no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C22⋊C4—731no—
396C2×C22⋊C4—731no—
397C2×C22⋊C4—731no—
398C2×C22⋊C4—731no—
399C2×C22⋊C4C4×D7311yescharacteristic
400C2×C22⋊C4C4×D7311yescharacteristic
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42C22×D7311yescharacteristic
406C2×C42C22×D7311yescharacteristic
407C4⋊C8—22no—
408C4⋊C8—2921no—
409C4⋊C8—2921no—
410C4⋊C8—1462no—
411C4⋊C8—1462no—
412C4⋊C8—1462no—
413C4⋊C8—1462no—
414C4⋊C8—1462no—
415C22⋊C8—22no—
416C22⋊C8—41no—
417C22⋊C8—2921no—
418C22⋊C8—1462no—
419#97—2921no—
420#97—734no—
421#97—1462no—
422C4×C8—41no—
423C4×C8—1462no—
424C4×C8—1462no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—731no—
429#95—731no—
430#95—731no—
431#95—731no—
432#95—731no—
433#95C4×D7311yescharacteristic
434#95C4×D7311yescharacteristic
435C24C2×C4×D7311yescharacteristic
436C2×Q8—2921no—
437C2×Q8—1462no—
438C2×Q8—1462no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4C2×C4×D7311yescharacteristic
454C22×C4C2×C4×D7311yescharacteristic
455C22×C4C2×C4×D7311yescharacteristic
456C22×C4C2×C4×D7311yescharacteristic
457C22×C4C2×C4×D7311yescharacteristic
458C22×C4C23×D7311yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—732no—
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×C8D4×D7312yes—
483C4⋊C4—2921no—
484C4⋊C4—2921no—
485C4⋊C4—2921no—
486C4⋊C4—1462no—
487C4⋊C4—1462no—
488C4⋊C4—1462no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C4⋊C4—1462no—
492C4⋊C4—1462no—
493C4⋊C4—1462no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—2921no—
497C22⋊C4—2921no—
498C22⋊C4—1462no—
499C22⋊C4—1462no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—732no—
505C42—732no—
506C42—1461no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C22×C4×D7311yescharacteristic
520Q8—2924no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—732no—
539C2×C4—732no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4#1455659712yes—
563C2×C4#1455659912yes—
564C2×C4C2×D4×D7311yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4#1455663611yescharacteristic
567C2×C4#1455663811yescharacteristic
568C2×C4#1455663811yescharacteristic
569C2×C4#1455663811yescharacteristic
570C8—22no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—2921no—
575C8—2921no—
576C8—2921no—
577C8—1462no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C4×D4×D7311yescharacteristic
586C22#1798906411yescharacteristic
587C22#1798906611yescharacteristic
588C22C2×#1455659711yescharacteristic
589C22C2×#1455659911yescharacteristic
590C22#1798935211yescharacteristic
591C22#1798935211yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—2921no—
598C4—2921no—
599C4—2921no—
600C4—2921no—
601C4—2921no—
602C4—2921no—
603C4—1462no—
604C4—1462no—
605C4#1798915711yescharacteristic
606C4#1798916411yescharacteristic
607C4#1798919411yescharacteristic
608C4#1798920511yescharacteristic
609C2—81no—
610C2#2261260611yescharacteristic
611C2#2261865111yescharacteristic
612C2#2261867611yescharacteristic
613C2#2261903211yescharacteristic
614C2#2261903511yescharacteristic
615C2#2261907111yescharacteristic
616C2#2261908111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1923C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#1923D73
hypocenterC73#22109
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
1. C2×#17988714C2semidirect
4. #22610865C2semidirect
5. #22610651C2semidirect
6. #22610649C2semidirect
7. #22610648C2semidirect
11. #22610494C2semidirect
12. #22610486C2semidirect
14. #22609805C2semidirect
324. C73#22109semidirect

Maximal subgroups

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

subgroupindex
C2×#179887142
C2×#179886452
#226132052
#226108652
#226106512
#226106492
#226106482
#226105472
#226105392
#226104942
#226104862
#226099132
#226098052
C73×#19232
#2210973