Finite Group Databasebeta

← #33130453

Subgroups — #33130453

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614228C211yesmaximal, characteristic
2#22611488C211yesmaximal, characteristic, complemented
3#22611403C211yesmaximal, characteristic, complemented
4#22611362C211yesmaximal, characteristic, complemented
5#22611357C211yesmaximal, characteristic, complemented
6#22610523C211yesmaximal, characteristic, complemented
7#22610509C211yesmaximal, characteristic, complemented
8#22610224C211yesmaximal, characteristic
9#22610130C211yesmaximal, characteristic, complemented
10#22610130C211yesmaximal, characteristic, complemented
11#22610026C211yesmaximal, characteristic, complemented
12#22610023C211yesmaximal, characteristic, complemented
13#22609958C211yesmaximal, characteristic, complemented
14#22609647C211yesmaximal, characteristic, complemented
15C73×#2354C211yesmaximal, characteristic
16C23×#10596831C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25#17988515—21no—
26#17988501—21no—
27#17988485C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988475—21no—
30#17988468—21no—
31#17988447C2211yescharacteristic
32#17988447C2211yescharacteristic
33#17988447C2211yescharacteristic
34#17988349C2211yescharacteristic
35#17988349C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73C2211yescharacteristic
49C2×#117×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51#390×C73—21no—
52#388×C73—21no—
53#388×C73—21no—
54#384×C73C2211yescharacteristic
55#380×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73—21no—
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×#10596831—22no—
60C22×#10596831—22no—
61C22×#10596831—41no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556550—22no—
67#14556550—22no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79#14556524—22no—
80#14556524—41no—
81#14556524—41no—
82#14556522—22no—
83C22⋊#10596831—22no—
84C22⋊#10596831—22no—
85C22⋊#10596831—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73D411yescharacteristic
92C23×C4×C73C2311yescharacteristic
93C23×C4×C73C2311yescharacteristic
94C4×D4×C73—22no—
95C4×D4×C73—22no—
96#117×C73—22no—
97#117×C73—22no—
98#117×C73—41no—
99#117×C73—41no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73—21no—
114C2×C42×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C2×#10596831—21no—
125C2×#10596831—21no—
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—44no—
136C2×#10596831#4112yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145#11924007—42no—
146#11924007—81no—
147C22⋊Dic73—42no—
148C22⋊Dic73—42no—
149C22⋊Dic73—81no—
150C22⋊Dic73—81no—
151C4×Dic73—42no—
152C24×C73C2×D411yescharacteristic
153C24×C73C2×D411yescharacteristic
154C2×D4×C73—22no—
155C2×D4×C73—22no—
156C2×D4×C73—41no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73#4112yes—
176C22×C4×C73#4112yes—
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—22no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73—81no—
188C4⋊C4×C73#4112yes—
189C22⋊C4×C73—22no—
190C22⋊C4×C73—22no—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—81no—
194C22⋊C4×C73—81no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—42no—
198C42×C73—42no—
199C42×C73—42no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73#12312yes—
214C23×C73C22×D411yescharacteristic
215D4×C73—44no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73#12312yes—
247C2×C4×C73#13512yes—
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×C4×C732−1+411yescharacteristic
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266#10596831—22no—
267#10596831—41no—
268#10596831—41no—
269#10596831—41no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73C2×#12311yescharacteristic
285C22×C73#54611yescharacteristic
286C22×C73#54711yescharacteristic
287C22×C73C2×#13511yescharacteristic
288C22×C73#57511yescharacteristic
289C22×C73#57611yescharacteristic
290C22×C73#57811yescharacteristic
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73—81no—
299C4×C73#52911yescharacteristic
300C4×C73#53911yescharacteristic
301C4×C73#55111yescharacteristic
302C4×C73#55211yescharacteristic
303Dic73—81no—
304Dic73—81no—
305Dic73—81no—
306Dic73—81no—
307#24942—731nomaximal, Sylow
308C2×C73—42no—
309C2×C73—81no—
310C2×C73#241911yescharacteristic
311C2×C73#309111yescharacteristic
312C2×C73#309811yescharacteristic
313C2×C73#330511yescharacteristic
314C2×C73#331511yescharacteristic
315C2×C73#334211yescharacteristic
316C2×C73#334711yescharacteristic
317#2485—731no—
318#2354D7311yescharacteristic
319#2099—731no—
320#2083—731no—
321#2076—731no—
322#2075—731no—
323#1921—731no—
324#1919—731no—
325#1869—731no—
326#1854—731no—
327#1854—731no—
328#1825—731no—
329#1825—731no—
330#1808—731no—
331#1735—731no—
332C73#2494211yescharacteristic, complemented, Sylow
333C23×C8—731no—
334C2×C4×D4C2×D7311yescharacteristic
335C2×#117—731no—
336C2×#117C2×D7311yescharacteristic
337C2×#117C2×D7311yescharacteristic
338C2×#107—731no—
339C2×#107—731no—
340C2×#107—731no—
341C2×#106—731no—
342C2×C22⋊C8—731no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345#397—1461no—
346#394—1461no—
347#390—21no—
348#389—731no—
349#389—731no—
350#388—21no—
351#388—21no—
352#388—1461no—
353#387—1461no—
354#384C2×D7311yescharacteristic
355#380—731no—
356#380—731no—
357#380—731no—
358#380C2×D7311yescharacteristic
359#380C2×D7311yescharacteristic
360C4×C4⋊C4—21no—
361C4×C4⋊C4C2×D7311yescharacteristic
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#336—731no—
375#336—731no—
376C22×D4C22⋊D7311yescharacteristic
377C23×C4C22×D7311yescharacteristic
378C23×C4C22×D7311yescharacteristic
379C22×C8—2921no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—1462no—
385C22×C8—1462no—
386C4×D4—22no—
387C4×D4—22no—
388#117—22no—
389#117—22no—
390#117—41no—
391#117—41no—
392#117—1462no—
393#117—1462no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4C22⋊D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4C22⋊D7311yescharacteristic
416C2×C22⋊C4C22⋊D7311yescharacteristic
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—1461no—
421C2×C42C22×D7311yescharacteristic
422#107—2921no—
423#107—2921no—
424#107—1462no—
425#106—1462no—
426C22⋊C8—2921no—
427C22⋊C8—1462no—
428C22⋊C8—1462no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437C24C2×C22⋊D7311yescharacteristic
438C24C2×C22⋊D7311yescharacteristic
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—41no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4#1192402812yes—
465C22×C4#1192402812yes—
466C22×C4C23×D7311yescharacteristic
467C2×C8—2924no—
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—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—2922no—
491C4⋊C4—5841no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4#1192402712yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—2922no—
508C22⋊C4—2922no—
509C22⋊C4—5841no—
510C22⋊C4—5841no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—2922no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1455657412yes—
531C23C22×C22⋊D7311yescharacteristic
532D4—44no—
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—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4#1455657412yes—
576C2×C4#1455661112yes—
577C2×C4#41×D7311yescharacteristic
578C2×C4C2×#1192402711yescharacteristic
579C2×C4C2×#1192402811yescharacteristic
580C2×C4C2×#1192402811yescharacteristic
581C2×C4#1455663611yescharacteristic
582C2×C4#1455663811yescharacteristic
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C8—1462no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22C2×#1455657411yescharacteristic
602C22#1798923111yescharacteristic
603C22#1798923611yescharacteristic
604C22C2×#1455661111yescharacteristic
605C22#1798938511yescharacteristic
606C22#1798939011yescharacteristic
607C22#1798940211yescharacteristic
608C4—22no—
609C4—22no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—5841no—
617C4—5841no—
618C4—5841no—
619C4—5841no—
620C4#1798911711yescharacteristic
621C4#1798917911yescharacteristic
622C4#1798926611yescharacteristic
623C4#1798927711yescharacteristic
624C2—42no—
625C2—81no—
626C2#2261340211yescharacteristic
627C2#2261984411yescharacteristic
628C2#2261990911yescharacteristic
629C2#2262188111yescharacteristic
630C2#2262202711yescharacteristic
631C2#2262235711yescharacteristic
632C2#2262243111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2354C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#2354D73
hypocenterC73#24942
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #22611488C2semidirect
3. #22611403C2semidirect
4. #22611362C2semidirect
5. #22611357C2semidirect
6. #22610523C2semidirect
7. #22610509C2semidirect
9. #22610130C2semidirect
10. #22610130C2semidirect
11. #22610026C2semidirect
12. #22610023C2semidirect
13. #22609958C2semidirect
14. #22609647C2semidirect
332. C73#24942semidirect

Maximal subgroups

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

subgroupindex
#226142282
#226114882
#226114032
#226113622
#226113572
#226105232
#226105092
#226102242
#226101302
#226100262
#226100232
#226099582
#226096472
C73×#23542
#2494273