Finite Group Databasebeta

← #33103229

Subgroups — #33103229

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988586C211yesmaximal, characteristic
2#22612771C211yesmaximal, characteristic
3#22610627C211yesmaximal, characteristic, complemented
4#22610613C211yesmaximal, characteristic, complemented
5#22610558C211yesmaximal, characteristic, complemented
6#22610551C211yesmaximal, characteristic, complemented
7#22610198C211yesmaximal, characteristic, complemented
8#22610184C211yesmaximal, characteristic, complemented
9#22610017C211yesmaximal, characteristic, complemented
10#22609962C211yesmaximal, characteristic, complemented
11#22609869C211yesmaximal, characteristic
12#22609764C211yesmaximal, characteristic, complemented
13C4×#14556514C211yesmaximal, characteristic, complemented
14#22609715C211yesmaximal, characteristic, complemented
15C73×#2345C211yesmaximal, characteristic, complemented
16C22×#11924028C2211yescharacteristic
17C22×#11924012C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22#17988586C418yescomplemented
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556511C2211yescharacteristic
30C2×#14556511C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988489C2211yescharacteristic
35#17988489C2211yescharacteristic
36#17988487—21no—
37#17988487—21no—
38#17988447C2211yescharacteristic
39#17988446—21no—
40#17988434—21no—
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C22×C4⋊C4×C73C2211yescharacteristic
50#389×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#387×C73—21no—
53#387×C73—21no—
54#380×C73—21no—
55#380×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C4×C22⋊C4×C73C2211yescharacteristic
60C2×#11924028—22no—
61C2×#11924028—22no—
62C2×#11924028—22no—
63C2×#11924028—41no—
64C2×#11924028—41no—
65C2×#11924028C2×C414yes—
66C22×Dic146C2311yescharacteristic
67C22×D292C2311yescharacteristic
68C22×C22⋊D73—21no—
69C22×C4×D73—21no—
70C2×#11924012—41no—
71C2×#11924012—41no—
72C2×#11924012C2×C414yes—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76C22×#10596831C2311yescharacteristic
77C4×Dic146—22no—
78C4×Dic146—22no—
79C4×Dic146—41no—
80C4×Dic146—41no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×C22⋊Dic73—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92#14556524—41no—
93#14556522—41no—
94#14556514—22no—
95#14556514—22no—
96#14556514C2×C414yes—
97#14556514C2×C414yes—
98#14556511—41no—
99#14556511—41no—
100#14556511C2×C414yes—
101#14556511C2×C414yes—
102C22⋊#10596831—22no—
103#14556502—41no—
104C4×#10596831—41no—
105#14556498—21no—
106C23×C4×C73—21no—
107C23×C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—22no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73C2×C414yes—
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C23×D73—21no—
132#11924028—28no—
133#11924028—44no—
134#11924028—44no—
135#11924028—44no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146C22×C412yes—
140C2×Dic146#4112yes—
141C2×Dic146#4112yes—
142C2×D292—21no—
143C2×D292—21no—
144C2×D292—21no—
145C2×D292—21no—
146C2×D292—21no—
147C2×D292C22×C412yes—
148C2×C22⋊D73—24no—
149C2×C22⋊D73—42no—
150C2×C22⋊D73—42no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154C2×C4×D73—24no—
155C2×C4×D73—42no—
156C2×C4×D73—42no—
157#11924012—44no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831C22×C412yes—
169C2×#10596831C22×C412yes—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174#11924007—42no—
175#11924007—42no—
176C22⋊Dic73—81no—
177#11924005—42no—
178#11924005—42no—
179#11924005—81no—
180#11924005—81no—
181C4×Dic73—42no—
182C4×Dic73—42no—
183C24×C73C2×D411yescharacteristic
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—24no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—81no—
194C22×C4×C73—81no—
195C22×C4×C73C22×C412yes—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73#4112yes—
202C22×C4×C73C2411yescharacteristic
203C4⋊C4×C73—24no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—81no—
211C4⋊C4×C73—81no—
212C4⋊C4×C73C22×C412yes—
213C4⋊C4×C73C22×C412yes—
214C22⋊C4×C73—22no—
215C22⋊C4×C73—41no—
216C22⋊C4×C73—42no—
217C42×C73—21no—
218C42×C73—21no—
219C42×C73—21no—
220C42×C73—21no—
221C42×C73—81no—
222C42×C73—81no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73C4×D412yes—
230C23×C73C22×D411yescharacteristic
231C22×D73—22no—
232C22×D73—41no—
233C22×D73—41no—
234C22×D73—41no—
235C22×D73—41no—
236C22×D73—41no—
237C22×D73—41no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
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—21no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—84no—
268C2×C4×C73C4×D412yes—
269C2×C4×C73#13712yes—
270C2×C4×C73C23×C411yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C73C2×#4111yescharacteristic
274C2×C4×C73H(5,𝔽2)11yescharacteristic
275C2×C4×C73H(5,𝔽2)11yescharacteristic
276C22⋊D73—48no—
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286D292—22no—
287D292—22no—
288D292—41no—
289D292—41no—
290C4×D73—48no—
291Dic146—22no—
292Dic146—22no—
293Dic146—22no—
294Dic146#11714yes—
295#10596831—22no—
296#10596831—22no—
297#10596831—41no—
298#10596831—41no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73C2×C4×D411yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73#54611yescharacteristic
309C22×C73C2×#13711yescharacteristic
310C22×C73#57511yescharacteristic
311C22×C73#57611yescharacteristic
312C22×C73#57711yescharacteristic
313C2×D73—42no—
314C2×D73—42no—
315C2×D73—42no—
316C2×D73—81no—
317C2×D73—81no—
318C2×D73—81no—
319C2×D73—81no—
320C4×C73—22no—
321C4×C73—22no—
322C4×C73—41no—
323C4×C73—42no—
324C4×C73—42no—
325C4×C73—41no—
326C4×C73—41no—
327C4×C73—81no—
328C4×C73—81no—
329C4×C73C2×#11711yescharacteristic
330C4×C73#51811yescharacteristic
331C4×C73#55111yescharacteristic
332C4×C73#55111yescharacteristic
333Dic73—24no—
334Dic73—81no—
335Dic73—81no—
336#22579—731nomaximal, Sylow
337C2×C73—42no—
338C2×C73#234211yescharacteristic
339C2×C73C4×#13711yescharacteristic
340C2×C73#294611yescharacteristic
341C2×C73#330911yescharacteristic
342C2×C73#331611yescharacteristic
343C2×C73#332611yescharacteristic
344C2×C73#333311yescharacteristic
345D73—82no—
346C2×#417—731no—
347#2345D7311yescharacteristic, complemented
348#2341—731no—
349#1936—731no—
350#1934—731no—
351#1926—731no—
352#1925—731no—
353#1865—731no—
354#1863—731no—
355#1823—731no—
356#1809—731no—
357#1788—731no—
358#1763—731no—
359C4×#103—731no—
360#1753—731no—
361C73#2257911yescharacteristic, complemented, Sylow
362C22×#41—731no—
363C22×M16—731no—
364C2×C4×Q8—731no—
365C2×C4×Q8—731no—
366C22×C4⋊C4C2×D7311yescharacteristic
367C2×#107—731no—
368C2×#106—731no—
369#417—738no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#102—731no—
375C2×#102—731no—
376C2×#102—731no—
377C2×#102—731no—
378C2×C22⋊C8—731no—
379C2×#97—731no—
380C2×C4×C8—731no—
381#390—731no—
382#390—731no—
383#390—1461no—
384#390—1461no—
385#389C2×D7311yescharacteristic
386#389C2×D7311yescharacteristic
387#387—21no—
388#387—21no—
389#380—21no—
390#380—731no—
391#380—1461no—
392#380C2×D7311yescharacteristic
393C4×C4⋊C4C2×D7311yescharacteristic
394C4×C4⋊C4C2×D7311yescharacteristic
395C4×C22⋊C4—21no—
396C4×C22⋊C4—1461no—
397C4×C22⋊C4C2×D7311yescharacteristic
398#340—731no—
399#340—731no—
400#340—731no—
401#340—731no—
402#340—731no—
403#340—731no—
404#336—731no—
405#336—731no—
406C2×#41—2921no—
407C2×#41—2921no—
408C2×#41—734no—
409C2×#41—1462no—
410C2×#41—1462no—
411C2×#41—1462no—
412C22×Q8—731no—
413C22×D4—731no—
414C22×D4—1461no—
415C23×C4—21no—
416C23×C4—1461no—
417C23×C4C22×D7311yescharacteristic
418C2×M16—2921no—
419C2×M16—2921no—
420C2×M16—734no—
421C22×C8—731no—
422C22×C8—731no—
423C22×C8—731no—
424C22×C8—731no—
425C4×Q8—2921no—
426C4×Q8—2921no—
427C4×Q8—1462no—
428C4×Q8—1462no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4C4×D7314yes—
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⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4—1461no—
448C2×C22⋊C4C22⋊D7311yescharacteristic
449C2×C22⋊C4C22⋊D7311yescharacteristic
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—1461no—
453C2×C42—1461no—
454C2×C42C22×D7311yescharacteristic
455C2×C42C22×D7311yescharacteristic
456#107—2921no—
457#106—2921no—
458#103—734no—
459#103—734no—
460#103—1462no—
461#103—1462no—
462#102—2921no—
463#102—2921no—
464#102—734no—
465#102—734no—
466C22⋊C8—1462no—
467#97—2921no—
468C4×C8—2921no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1461no—
475#95C22⋊D7311yescharacteristic
476#95C22⋊D7311yescharacteristic
477C24—1461no—
478C24C2×C22⋊D7311yescharacteristic
479#41—2924no—
480#41—2924no—
481#41—2924no—
482#41—1468no—
483C2×Q8—732no—
484C2×Q8—732no—
485C2×Q8—732no—
486C2×Q8—1461no—
487C2×Q8—1461no—
488C2×Q8—1461no—
489C2×D4—2922no—
490C2×D4—2922no—
491C2×D4—732no—
492C2×D4—1464no—
493C2×D4—1461no—
494C2×D4—1461no—
495C2×D4—1461no—
496C2×D4—1461no—
497C2×D4—1461no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—24no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—2922no—
510C22×C4—2922no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1464no—
514C22×C4—1461no—
515C22×C4C2×C4×D7312yes—
516C22×C4C2×C22⋊D7311yescharacteristic
517C22×C4C2×C22⋊D7311yescharacteristic
518C22×C4C2×C22⋊D7311yescharacteristic
519C22×C4C2×C22⋊D7311yescharacteristic
520C22×C4C2×C22⋊D7311yescharacteristic
521C22×C4#1192402812yes—
522C22×C4C23×D7311yescharacteristic
523M16—2924no—
524C2×C8—732no—
525C2×C8—732no—
526C2×C8—1461no—
527C2×C8—1461no—
528C2×C8—1461no—
529C2×C8—1461no—
530C2×C8—1461no—
531C2×C8—1461no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8—1461no—
536C4⋊C4—24no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4—2922no—
546C4⋊C4—2922no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4C2×C4×D7312yes—
552C4⋊C4C2×C4×D7312yes—
553C22⋊C4—22no—
554C22⋊C4—41no—
555C22⋊C4—42no—
556C22⋊C4—2922no—
557C22⋊C4—2922no—
558C22⋊C4—5841no—
559C22⋊C4—5841no—
560C22⋊C4—5841no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—81no—
566C42—81no—
567C42—2922no—
568C42—2922no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—2921no—
576C23—2921no—
577C23—2921no—
578C23—2921no—
579C23—2921no—
580C23—2921no—
581C23—1462no—
582C23C4×C22⋊D7312yes—
583C23C22×C22⋊D7311yescharacteristic
584Q8—734no—
585Q8—1462no—
586Q8—1462no—
587Q8—1462no—
588D4—2928no—
589D4—2921no—
590D4—2921no—
591D4—1462no—
592D4—1462no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
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—41no—
622C2×C4—84no—
623C2×C4—2928no—
624C2×C4—2921no—
625C2×C4—2921no—
626C2×C4—2921no—
627C2×C4—2921no—
628C2×C4—2921no—
629C2×C4—2921no—
630C2×C4—1462no—
631C2×C4—1462no—
632C2×C4—1462no—
633C2×C4C4×C22⋊D7312yes—
634C2×C4#1455662212yes—
635C2×C4C22×C4×D7311yescharacteristic
636C2×C4#41×D7311yescharacteristic
637C2×C4#41×D7311yescharacteristic
638C2×C4C2×#1192402811yescharacteristic
639C2×C4#1455663711yescharacteristic
640C2×C4#1455663711yescharacteristic
641C8—2921no—
642C8—2921no—
643C8—1462no—
644C8—1462no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—2922no—
653C22—2922no—
654C22—2922no—
655C22—5841no—
656C22—5841no—
657C22—5841no—
658C22—5841no—
659C22C2×C4×C22⋊D7311yescharacteristic
660C22#1798922511yescharacteristic
661C22#1798922511yescharacteristic
662C22C2×#1455662211yescharacteristic
663C22#1798938511yescharacteristic
664C22#1798939011yescharacteristic
665C22#1798939811yescharacteristic
666C4—22no—
667C4—22no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—5841no—
676C4—5841no—
677C4—1464no—
678C4#117×D7311yescharacteristic
679C4#1798906111yescharacteristic
680C4#1798926111yescharacteristic
681C4#1798926111yescharacteristic
682C2—42no—
683C2—5842no—
684C2#2261278111yescharacteristic
685C2C4×#1455662211yescharacteristic
686C2#2261876311yescharacteristic
687C2#2262193111yescharacteristic
688C2#2262203611yescharacteristic
689C2#2262217811yescharacteristic
690C2#2262226311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2345C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenter#2345D73
hypocenterC73#22579
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #22610627C2semidirect
4. #22610613C2semidirect
5. #22610558C2semidirect
6. #22610551C2semidirect
7. #22610198C2semidirect
8. #22610184C2semidirect
9. #22610017C2semidirect
10. #22609962C2semidirect
12. #22609764C2semidirect
13. C4×#14556514C2semidirect
14. #22609715C2semidirect
15. C73×#2345C2semidirect
22. #17988586C4semidirect
347. #2345D73semidirect
361. C73#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#179885862
#226127712
#226106272
#226106132
#226105582
#226105512
#226101982
#226101842
#226100172
#226099622
#226098692
#226097642
C4×#145565142
#226097152
C73×#23452
#2257973