Finite Group Databasebeta

← #33097295

Subgroups — #33097295

Order 18688.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988743C211yesmaximal, characteristic
2#22612613C211yesmaximal, characteristic
3#22610801C211yesmaximal, characteristic, complemented
4#22610791C211yesmaximal, characteristic, complemented
5#22610624C211yesmaximal, characteristic, complemented
6#22610610C211yesmaximal, characteristic, complemented
7#22610569C211yesmaximal, characteristic, complemented
8#22610562C211yesmaximal, characteristic, complemented
9#22610275C211yesmaximal, characteristic, complemented
10#22610275C211yesmaximal, characteristic, complemented
11#22610243C211yesmaximal, characteristic
12#22610130C211yesmaximal, characteristic
13#22609866C211yesmaximal, characteristic
14#22609697C211yesmaximal, characteristic
15C73×#2310C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924018C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988743C418yescomplemented
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×C22⋊#10596831C2211yescharacteristic
33C2×#14556502C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35#17988493C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988482—21no—
38#17988481—21no—
39#17988475—21no—
40#17988468—21no—
41#17988447C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C23×Q8×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51C2×#117×C73C2211yescharacteristic
52#391×C73—21no—
53#391×C73—21no—
54#391×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#391×C73C2211yescharacteristic
57#391×C73C2211yescharacteristic
58#381×C73—21no—
59#381×C73—21no—
60C22×Dic146C2311yescharacteristic
61C2×#11924018—41no—
62C2×#11924018—41no—
63C2×#11924018—41no—
64C2×#11924018—41no—
65C2×#11924018C2×C414yes—
66C2×#11924018C2×C414yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556570—22no—
72#14556570C2×C414yes—
73C4×Dic146—41no—
74C4×Dic146—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85#14556524—41no—
86#14556522—41no—
87#14556515—41no—
88#14556515—41no—
89#14556515—41no—
90#14556515—41no—
91#14556515C2×C414yes—
92#14556515C2×C414yes—
93#14556514—41no—
94#14556514—41no—
95C22⋊#10596831—22no—
96C22⋊#10596831C2×C414yes—
97#14556502—41no—
98C4×#10596831—41no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104C22×Q8×C73—21no—
105C22×Q8×C73—22no—
106C22×Q8×C73—41no—
107C22×Q8×C73—41no—
108C22×Q8×C73—41no—
109C22×Q8×C73—41no—
110C22×Q8×C73C2×C414yes—
111C22×Q8×C73C2311yescharacteristic
112C22×Q8×C73C2311yescharacteristic
113C23×C4×C73—21no—
114C23×C4×C73C2311yescharacteristic
115#117×C73—41no—
116#117×C73—41no—
117#117×C73—41no—
118#117×C73—41no—
119C2×C4⋊C4×C73D411yescharacteristic
120C2×C4⋊C4×C73D411yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C22⋊C4×C73—21no—
124C2×C22⋊C4×C73D411yescharacteristic
125C2×C22⋊C4×C73D411yescharacteristic
126C2×C42×C73—21no—
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2311yescharacteristic
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135#95×C73—21no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147#11924018—44no—
148#11924018—44no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831C22×C412yes—
160C2×#10596831C22×C412yes—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73C22×C412yes—
167#11924007—24no—
168#11924007—42no—
169#11924007—81no—
170C22⋊Dic73—24no—
171C22⋊Dic73—81no—
172C4×Dic73—42no—
173C4×Dic73—81no—
174C24×C73C2×D411yescharacteristic
175C2×Q8×C73—24no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—44no—
187C2×Q8×C73—44no—
188C2×Q8×C73—42no—
189C2×Q8×C73—42no—
190C2×Q8×C73—44no—
191C2×Q8×C73—44no—
192C2×Q8×C73C22×C412yes—
193C2×Q8×C73C22×C412yes—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—24no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—42no—
204C22×C4×C73—42no—
205C22×C4×C73C22×C412yes—
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73C2×D411yescharacteristic
208C22×C4×C73C2×D411yescharacteristic
209C22×C4×C73C2×D411yescharacteristic
210C22×C4×C73C2×D411yescharacteristic
211C22×C4×C73#4112yes—
212C22×C4×C73C2411yescharacteristic
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—41no—
218C4⋊C4×C73—41no—
219C22⋊C4×C73—41no—
220C22⋊C4×C73—41no—
221C22⋊C4×C73—42no—
222C42×C73—21no—
223C42×C73—21no—
224C42×C73—21no—
225C42×C73—21no—
226C42×C73—42no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73C4×D412yes—
234C23×C73C22×D411yescharacteristic
235Q8×C73—22no—
236Q8×C73—22no—
237Q8×C73—22no—
238Q8×C73—22no—
239Q8×C73—48no—
240Q8×C73—41no—
241Q8×C73—41no—
242Q8×C73—41no—
243Q8×C73—41no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—21no—
263C2×C4×C73—21no—
264C2×C4×C73—48no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—41no—
270C2×C4×C73—41no—
271C2×C4×C73—41no—
272C2×C4×C73—41no—
273C2×C4×C73—41no—
274C2×C4×C73—41no—
275C2×C4×C73C4×D412yes—
276C2×C4×C73C23×C411yescharacteristic
277C2×C4×C73C2×#4111yescharacteristic
278C2×C4×C73H(5,𝔽2)11yescharacteristic
279C2×C4×C73H(5,𝔽2)11yescharacteristic
280C2×C4×C73H(5,𝔽2)11yescharacteristic
281C2×C4×C73H(5,𝔽2)11yescharacteristic
282C2×Dic73—22no—
283C2×Dic73—22no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292C2×Dic73—41no—
293C2×Dic73—41no—
294Dic146—22no—
295Dic146—22no—
296Dic146—41no—
297Dic146—41no—
298#10596831—22no—
299#10596831—22no—
300#10596831—41no—
301#10596831—41no—
302C22×C73—22no—
303C22×C73—22no—
304C22×C73—22no—
305C22×C73—41no—
306C22×C73—41no—
307C22×C73—41no—
308C22×C73—41no—
309C22×C73C2×C4×D411yescharacteristic
310C22×C73#53411yescharacteristic
311C22×C73#53511yescharacteristic
312C22×C73#57711yescharacteristic
313C22×C73#57811yescharacteristic
314C22×C73#57811yescharacteristic
315C22×C73#57811yescharacteristic
316C4×C73—22no—
317C4×C73—42no—
318C4×C73—42no—
319C4×C73—41no—
320C4×C73—41no—
321C4×C73—41no—
322C4×C73—41no—
323C4×C73—81no—
324C4×C73—81no—
325C4×C73#51811yescharacteristic
326C4×C73#51811yescharacteristic
327C4×C73#53811yescharacteristic
328C4×C73#53811yescharacteristic
329Dic73—42no—
330Dic73—42no—
331Dic73—81no—
332Dic73—81no—
333#22054—731nomaximal, Sylow
334C2×C73—42no—
335C2×C73#232111yescharacteristic
336C2×C73#294711yescharacteristic
337C2×C73#295011yescharacteristic
338C2×C73#320111yescharacteristic
339C2×C73#320211yescharacteristic
340C2×C73#322311yescharacteristic
341C2×C73#322511yescharacteristic
342C2×#451—731no—
343#2323—731no—
344#2310D7311yescharacteristic
345#1971—731no—
346#1969—731no—
347#1935—731no—
348#1933—731no—
349#1928—731no—
350#1927—731no—
351#1878—731no—
352#1878—731no—
353#1873—731no—
354#1854—731no—
355#1787—731no—
356#1750—731no—
357C73#2205411yescharacteristic, complemented, Sylow
358C23×Q8C2×D7311yescharacteristic
359C22×Q16—731no—
360C22×Q16—731no—
361C2×C22⋊Q8—731no—
362C2×C4×Q8—731no—
363C2×#117C2×D7311yescharacteristic
364C2×#117C2×D7311yescharacteristic
365#451—738no—
366C2×#107—731no—
367C2×#106—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×C22⋊C8—731no—
377C2×C22⋊C8—731no—
378C2×#97—731no—
379C2×C4×C8—731no—
380#391—21no—
381#391—21no—
382#391—731no—
383#391C2×D7311yescharacteristic
384#391C2×D7311yescharacteristic
385#391C2×D7311yescharacteristic
386#391C2×D7311yescharacteristic
387#389—731no—
388#389—1461no—
389#388—1461no—
390#388—1461no—
391#387—1461no—
392#381—21no—
393#381—21no—
394#380—731no—
395C4×C4⋊C4—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400#336—731no—
401#336—731no—
402C22×Q8—21no—
403C22×Q8—22no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8—41no—
408C22×Q8—731no—
409C22×Q8C4×D7314yes—
410C22×Q8C22×D7311yescharacteristic
411C22×Q8C22×D7311yescharacteristic
412C23×C4—21no—
413C23×C4C22×D7311yescharacteristic
414C2×Q16—2921no—
415C2×Q16—2921no—
416C2×Q16—2921no—
417C2×Q16—2921no—
418C2×Q16—734no—
419C2×Q16—734no—
420C22×C8—731no—
421C22×C8—731no—
422C22×C8—731no—
423C22×C8—731no—
424C22⋊Q8—734no—
425C22⋊Q8—1462no—
426C4×Q8—2921no—
427C4×Q8—2921no—
428#117—41no—
429#117—41no—
430#117—41no—
431#117—41no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4C22⋊D7311yescharacteristic
439C2×C4⋊C4C22⋊D7311yescharacteristic
440C2×C4⋊C4C22×D7311yescharacteristic
441C2×C4⋊C4C22×D7311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1461no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4C22⋊D7311yescharacteristic
446C2×C22⋊C4C22⋊D7311yescharacteristic
447C2×C42—21no—
448C2×C42—1461no—
449C2×C42—1461no—
450C2×C42C22×D7311yescharacteristic
451C2×C42C22×D7311yescharacteristic
452#107—2921no—
453#106—2921no—
454#103—2921no—
455#103—2921no—
456#103—2921no—
457#103—2921no—
458#103—2921no—
459#103—2921no—
460#103—734no—
461#103—734no—
462C22⋊C8—734no—
463C22⋊C8—1462no—
464#97—2921no—
465C4×C8—2921no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479C24C2×C22⋊D7311yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—732no—
498C2×Q8—1461no—
499C2×Q8—1461no—
500C2×Q8—1461no—
501C2×Q8—1461no—
502C2×Q8—1461no—
503C2×Q8C2×C4×D7312yes—
504C2×Q8C2×C4×D7312yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4—1461no—
520C22×C4C2×C4×D7312yes—
521C22×C4C2×C22⋊D7311yescharacteristic
522C22×C4C2×C22⋊D7311yescharacteristic
523C22×C4C2×C22⋊D7311yescharacteristic
524C22×C4C2×C22⋊D7311yescharacteristic
525C22×C4C2×C22⋊D7311yescharacteristic
526C22×C4#1192402812yes—
527C22×C4C23×D7311yescharacteristic
528Q16—2924no—
529Q16—2924no—
530C2×C8—732no—
531C2×C8—732no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C2×C8—1461no—
539C2×C8—1461no—
540C2×C8—1461no—
541C2×C8—1461no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—2922no—
549C4⋊C4—5841no—
550C4⋊C4—732no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1464no—
556C4⋊C4—1461no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—42no—
560C22⋊C4—5841no—
561C22⋊C4—1464no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—2922no—
568C42—5841no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D7312yes—
576C23C22×C22⋊D7311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—48no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—2921no—
587Q8—2921no—
588Q8—1462no—
589Q8—1462no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—2921no—
622C2×C4—2921no—
623C2×C4—2921no—
624C2×C4—2921no—
625C2×C4—2921no—
626C2×C4—2921no—
627C2×C4—2921no—
628C2×C4—2921no—
629C2×C4—2921no—
630C2×C4—2921no—
631C2×C4—1462no—
632C2×C4—1462no—
633C2×C4C4×C22⋊D7312yes—
634C2×C4C22×C4×D7311yescharacteristic
635C2×C4C2×#1192402811yescharacteristic
636C2×C4#1455663611yescharacteristic
637C2×C4#1455663611yescharacteristic
638C2×C4#1455663611yescharacteristic
639C2×C4#1455663611yescharacteristic
640C8—2921no—
641C8—2921no—
642C8—1462no—
643C8—1462no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D7311yescharacteristic
652C22#1798914611yescharacteristic
653C22#1798915411yescharacteristic
654C22#1798939511yescharacteristic
655C22#1798940211yescharacteristic
656C22#1798940211yescharacteristic
657C22#1798940211yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—2922no—
668C4—2922no—
669C4—5841no—
670C4—5841no—
671C4#1798906011yescharacteristic
672C4#1798906011yescharacteristic
673C4#1798917511yescharacteristic
674C4#1798917511yescharacteristic
675C2—42no—
676C2#2261258911yescharacteristic
677C2#2261877411yescharacteristic
678C2#2261879711yescharacteristic
679C2#2262081311yescharacteristic
680C2#2262082211yescharacteristic
681C2#2262101411yescharacteristic
682C2#2262104011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2310C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2310D73
hypocenterC73#22054
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #22610801C2semidirect
4. #22610791C2semidirect
5. #22610624C2semidirect
6. #22610610C2semidirect
7. #22610569C2semidirect
8. #22610562C2semidirect
9. #22610275C2semidirect
10. #22610275C2semidirect
20. #17988743C4semidirect
357. C73#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#179887432
#226126132
#226108012
#226107912
#226106242
#226106102
#226105692
#226105622
#226102752
#226102432
#226101302
#226098662
#226096972
C73×#23102
#2205473