Finite Group Databasebeta

← #33136352

Subgroups — #33136352

Order 18688.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988894C211yesmaximal, characteristic, complemented
2C2×#17988659C211yesmaximal, characteristic
3#22615097C211yesmaximal, characteristic
4#22611367C211yesmaximal, characteristic, complemented
5#22611362C211yesmaximal, characteristic, complemented
6#22611124C211yesmaximal, characteristic, complemented
7#22611124C211yesmaximal, characteristic, complemented
8#22610899C211yesmaximal, characteristic, complemented
9#22610692C211yesmaximal, characteristic, complemented
10#22610365C211yesmaximal, characteristic, complemented
11#22610355C211yesmaximal, characteristic, complemented
12#22609962C211yesmaximal, characteristic, complemented
13#22609954C211yesmaximal, characteristic, complemented
14#22609761C211yesmaximal, characteristic, complemented
15C73×#2538C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556583C2211yescharacteristic
19#17988894—24no—
20#17988659—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33#17988525—21no—
34#17988501—21no—
35#17988481—21no—
36#17988481—21no—
37#17988447C2211yescharacteristic
38#17988447C2211yescharacteristic
39#17988447C2211yescharacteristic
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×C22⋊Q8×C73C2211yescharacteristic
51C2×C22⋊Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54#396×C73—21no—
55#396×C73—21no—
56#390×C73—21no—
57#389×C73—21no—
58#384×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#380×C73C2211yescharacteristic
61C2×#11924012—22no—
62C2×#11924012—22no—
63C2×#11924012—41no—
64C22×#10596831—22no—
65C22×#10596831—22no—
66C22×#10596831—41no—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556583—22no—
72#14556583—22no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84#14556524—22no—
85#14556522—22no—
86#14556522—41no—
87#14556522—41no—
88C4⋊#10596831D414yes—
89#14556515—22no—
90#14556515—22no—
91#14556515—22no—
92#14556515—22no—
93C22⋊#10596831—22no—
94C22⋊#10596831—22no—
95#14556502—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×Q8×C73C2311yescharacteristic
102C22×D4×C73D411yescharacteristic
103C23×C4×C73C2311yescharacteristic
104C23×C4×C73C2311yescharacteristic
105C22⋊Q8×C73—22no—
106C22⋊Q8×C73—22no—
107C22⋊Q8×C73—41no—
108C22⋊Q8×C73—41no—
109C4×Q8×C73—22no—
110C4×Q8×C73—22no—
111C4×D4×C73—22no—
112C4×D4×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73D411yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C22⋊C4×C73—21no—
125C2×C22⋊C4×C73—21no—
126C2×C22⋊C4×C73D411yescharacteristic
127C2×C22⋊C4×C73D411yescharacteristic
128C2×C42×C73—21no—
129C2×C42×C73C2311yescharacteristic
130#95×C73—21no—
131#95×C73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136#11924012—44no—
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—44no—
147C2×#10596831C2×D412yes—
148C2×#10596831C2×D412yes—
149C2×#10596831#4112yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158#11924007—42no—
159#11924007—42no—
160C22⋊Dic73—42no—
161C22⋊Dic73—42no—
162C22⋊Dic73—81no—
163C22⋊Dic73—81no—
164C4×Dic73—81no—
165C24×C73C2×D411yescharacteristic
166C24×C73C2×D411yescharacteristic
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73#4112yes—
173C2×D4×C73—22no—
174C2×D4×C73—22no—
175C2×D4×C73—41no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
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—21no—
191C22×C4×C73—21no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2411yescharacteristic
197C4⋊C4×C73—22no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73—42no—
206C4⋊C4×C73—42no—
207C4⋊C4×C73—42no—
208C4⋊C4×C73—81no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—42no—
213C22⋊C4×C73—81no—
214C22⋊C4×C73—81no—
215C42×C73—42no—
216C42×C73C2×D412yes—
217C42×C73#4112yes—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73#13712yes—
232C23×C73C22×D411yescharacteristic
233Q8×C73—22no—
234Q8×C73—22no—
235Q8×C73—41no—
236Q8×C73—41no—
237D4×C73—44no—
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—22no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
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—41no—
268C2×C4×C73#12112yes—
269C2×C4×C73#12112yes—
270C2×C4×C73C22×D411yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C73C2×#4111yescharacteristic
274C2×C4×C73H(5,𝔽2)11yescharacteristic
275C2×C4×C732−1+411yescharacteristic
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288#10596831—22no—
289#10596831—22no—
290#10596831—22no—
291#10596831—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73C2×#12111yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73#54711yescharacteristic
309C22×C73C2×#13711yescharacteristic
310C22×C73#57511yescharacteristic
311C22×C73#57711yescharacteristic
312C22×C73#57711yescharacteristic
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—41no—
316C4×C73—41no—
317C4×C73—42no—
318C4×C73—81no—
319C4×C73—81no—
320C4×C73—81no—
321C4×C73#52911yescharacteristic
322C4×C73#54211yescharacteristic
323C4×C73#54611yescharacteristic
324C4×C73#54811yescharacteristic
325Dic73—81no—
326Dic73—81no—
327Dic73—81no—
328Dic73—81no—
329#25357—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73—81no—
332C2×C73#242711yescharacteristic
333C2×C73#305611yescharacteristic
334C2×C73#306011yescharacteristic
335C2×C73#331711yescharacteristic
336C2×C73#332011yescharacteristic
337C2×C73#334411yescharacteristic
338C2×C73#335011yescharacteristic
339C2×#479—731no—
340C2×#435—731no—
341#2562—731no—
342#2538D7311yescharacteristic
343#2077—731no—
344#2076—731no—
345#2035—731no—
346#2035—731no—
347#1992—731no—
348#1948—731no—
349#1894—731no—
350#1892—731no—
351#1809—731no—
352#1807—731no—
353#1763—731no—
354C73#2535711yescharacteristic, complemented, Sylow
355C22×M16—731no—
356C23×C8—731no—
357C2×#125—731no—
358C2×C22⋊Q8C2×D7311yescharacteristic
359C2×C22⋊Q8C2×D7311yescharacteristic
360C2×C4×Q8C2×D7311yescharacteristic
361C2×C4×D4C2×D7311yescharacteristic
362#479—1464no—
363#435—1464no—
364C2×#107—731no—
365C2×#106—731no—
366C2×#106—731no—
367C2×#106—731no—
368C2×C4⋊C8—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×C22⋊C8—731no—
374C2×C22⋊C8—731no—
375C2×#97—731no—
376#400—1461no—
377#396—21no—
378#396—21no—
379#394—1461no—
380#390—21no—
381#389—21no—
382#388—1461no—
383#388—1461no—
384#384C2×D7311yescharacteristic
385#380—731no—
386#380—731no—
387#380—731no—
388#380—731no—
389#380C2×D7311yescharacteristic
390#380C2×D7311yescharacteristic
391C4×C4⋊C4—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400C22×Q8C22×D7311yescharacteristic
401C22×D4C22⋊D7311yescharacteristic
402C23×C4C22×D7311yescharacteristic
403C23×C4C22×D7311yescharacteristic
404C2×M16—2921no—
405C2×M16—1462no—
406C2×M16—1462no—
407C22×C8—2921no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—1462no—
413C22×C8—1462no—
414#125—1462no—
415#125—1462no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—41no—
419C22⋊Q8—41no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4C22⋊D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C4⋊C4C22×D7311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1461no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4C22⋊D7311yescharacteristic
448C2×C22⋊C4C22⋊D7311yescharacteristic
449C2×C42—21no—
450C2×C42—1461no—
451C2×C42C22×D7311yescharacteristic
452#107—1462no—
453#106—2921no—
454#106—2921no—
455#106—1462no—
456C4⋊C8—734no—
457#103—1462no—
458#103—1462no—
459#103—1462no—
460#103—1462no—
461C22⋊C8—1462no—
462C22⋊C8—1462no—
463#97—1462no—
464#95—21no—
465#95—21no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24C2×C22⋊D7311yescharacteristic
472C24C2×C22⋊D7311yescharacteristic
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8#1192402712yes—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C23×D7311yescharacteristic
507M16—2924no—
508C2×C8—2924no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—732no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—2922no—
534C4⋊C4—2922no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—2922no—
550C22⋊C4—2922no—
551C22⋊C4—5841no—
552C22⋊C4—5841no—
553C42—42no—
554C42—5841no—
555C42D4×D7312yes—
556C42#1192402712yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1455662212yes—
571C23C22×C22⋊D7311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4#1455656312yes—
620C2×C4#1455656312yes—
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4C2×#1192402711yescharacteristic
624C2×C4C2×#1192402711yescharacteristic
625C2×C4#1455663611yescharacteristic
626C2×C4#1455663811yescharacteristic
627C8—2921no—
628C8—1462no—
629C8—1462no—
630C8—1462no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1455656311yescharacteristic
646C22#1798923111yescharacteristic
647C22#1798923611yescharacteristic
648C22C2×#1455662211yescharacteristic
649C22#1798938511yescharacteristic
650C22#1798939511yescharacteristic
651C22#1798939811yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—5841no—
661C4—5841no—
662C4—5841no—
663C4—5841no—
664C4#1798911811yescharacteristic
665C4#1798920411yescharacteristic
666C4#1798923411yescharacteristic
667C4#1798924311yescharacteristic
668C2—42no—
669C2—81no—
670C2#2261351011yescharacteristic
671C2#2261955711yescharacteristic
672C2#2261959011yescharacteristic
673C2#2262205711yescharacteristic
674C2#2262210211yescharacteristic
675C2#2262238011yescharacteristic
676C2#2262247611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2538C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#2538D73
hypocenterC73#25357
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#17988894C2semidirect
4. #22611367C2semidirect
5. #22611362C2semidirect
6. #22611124C2semidirect
7. #22611124C2semidirect
8. #22610899C2semidirect
9. #22610692C2semidirect
10. #22610365C2semidirect
11. #22610355C2semidirect
12. #22609962C2semidirect
13. #22609954C2semidirect
14. #22609761C2semidirect
354. C73#25357semidirect

Maximal subgroups

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

subgroupindex
C2×#179888942
C2×#179886592
#226150972
#226113672
#226113622
#226111242
#226108992
#226106922
#226103652
#226103552
#226099622
#226099542
#226097612
C73×#25382
#2535773