Finite Group Databasebeta

← #33149533

Subgroups — #33149533

Order 18688.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613633C211yesmaximal, characteristic, complemented
2#22612945C211yesmaximal, characteristic
3#22611490C211yesmaximal, characteristic, complemented
4#22611419C211yesmaximal, characteristic, complemented
5#22611318C211yesmaximal, characteristic, complemented
6#22611068C211yesmaximal, characteristic, complemented
7#22610882C211yesmaximal, characteristic, complemented
8#22610503C211yesmaximal, characteristic, complemented
9#22610359C211yesmaximal, characteristic
10#22610246C211yesmaximal, characteristic
11#22610181C211yesmaximal, characteristic, complemented
12#22610121C211yesmaximal, characteristic, complemented
13#22609932C211yesmaximal, characteristic, complemented
14C2×#17988350C211yesmaximal, characteristic, complemented
15C73×#1850C211yesmaximal, characteristic, complemented
16C2×#14556591C2211yescharacteristic
17C2×#14556568—21no—
18C2×#14556565C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×#14556551—21no—
21C22×C4⋊Dic73C2211yescharacteristic
22C22×C4⋊Dic73C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29#17988519—21no—
30#17988508—21no—
31#17988489C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988479—21no—
36#17988464—21no—
37#17988462—21no—
38#17988461—21no—
39#17988457C2211yescharacteristic
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×D292C2311yescharacteristic
61C22×C22⋊D73—21no—
62C23×Dic73—21no—
63C23×Dic73—21no—
64C22×C4×D73—21no—
65#14556591—22no—
66#14556591—41no—
67#14556568—44no—
68#14556565—22no—
69#14556565Q814yescomplemented
70C4×D292—22no—
71C4×D292—41no—
72#14556551—44no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73—21no—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73—41no—
81C2×C4⋊Dic73—41no—
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—21no—
92C2×C22⋊Dic73—21no—
93C2×#11924005—21no—
94C2×#11924005—21no—
95C2×#11924005—21no—
96C2×#11924005—21no—
97C2×C4×Dic73—21no—
98#14556525—22no—
99#14556525—41no—
100#14556513—22no—
101#14556513—41no—
102#14556513—41no—
103#14556513—41no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109C23×C4×C73C2311yescharacteristic
110C22×C8×C73—22no—
111C22×C8×C73—41no—
112C22×C8×C73—41no—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C4⋊C8×C73—41no—
122C4⋊C8×C73—41no—
123C22⋊C8×C73—22no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C23×D73—21no—
127C2×Dic146—42no—
128C2×Dic146—42no—
129C2×Dic146—81no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292—21no—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292C2×Q812yes—
136C2×C22⋊D73—24no—
137C2×C22⋊D73—42no—
138C2×C22⋊D73—42no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—42no—
147C22×Dic73—42no—
148C22×Dic73—42no—
149C22×Dic73—42no—
150C22×Dic73—81no—
151C22×Dic73—81no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73—81no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—42no—
167C4⋊Dic73—42no—
168C4⋊Dic73—42no—
169C4⋊Dic73—42no—
170C4⋊Dic73—81no—
171C4⋊Dic73C2×Q812yes—
172C4⋊Dic73#4112yes—
173C4⋊Dic73#4112yes—
174#11924007—42no—
175#11924007—42no—
176#11924007—42no—
177C22⋊Dic73—42no—
178#11924005—24no—
179#11924005—42no—
180#11924005—42no—
181#11924005—81no—
182C4×Dic73—42no—
183C24×C73C2×D411yescharacteristic
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×Q812yes—
194C22×C4×C73#4112yes—
195C22×C4×C73C2411yescharacteristic
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—44no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73D814yes—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73C22⋊Q812yes—
220C23×C73C2×D812yes—
221C23×C73#13712yes—
222C23×C73C22×D411yescharacteristic
223C22×D73—22no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73C22⋊Q812yes—
247C2×C4×C73C2×D812yes—
248C2×C4×C73C2×D812yes—
249C2×C4×C73C22×Q811yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×C4×C73H(5,𝔽2)11yescharacteristic
255C22⋊D73—48no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—84no—
278C2×Dic73—84no—
279D292—22no—
280D292—22no—
281D292—41no—
282D292—41no—
283C4×D73—84no—
284Dic146—84no—
285C8×C73—41no—
286C8×C73—41no—
287C8×C73—41no—
288C8×C73—41no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73#47414yes—
296C22×C73C2×C22⋊Q811yescharacteristic
297C22×C73#54611yescharacteristic
298C22×C73#54611yescharacteristic
299C22×C73C22×D811yescharacteristic
300C22×C73C2×#13711yescharacteristic
301C22×C73#57711yescharacteristic
302C22×C73#57711yescharacteristic
303C2×D73—42no—
304C2×D73—42no—
305C2×D73—42no—
306C2×D73—81no—
307C2×D73—81no—
308C2×D73—81no—
309C2×D73—81no—
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—22no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73#52911yescharacteristic
316C4×C73#53811yescharacteristic
317C4×C73#55411yescharacteristic
318C4×C73#55411yescharacteristic
319Dic73—42no—
320Dic73—42no—
321Dic73—42no—
322Dic73—81no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326#26408—731nomaximal, Sylow
327C2×C73—24no—
328C2×C73#245011yescharacteristic
329C2×C73C2×#47411yescharacteristic
330C2×C73#308411yescharacteristic
331C2×C73#329411yescharacteristic
332C2×C73#329411yescharacteristic
333C2×C73#331611yescharacteristic
334C2×C73#332011yescharacteristic
335D73—82no—
336#2440—731no—
337#2361—731no—
338#2099—731no—
339#2086—731no—
340#2067—731no—
341#2023—731no—
342#1988—731no—
343#1918—731no—
344#1893—731no—
345#1874—731no—
346#1862—731no—
347#1852—731no—
348#1850D7311yescharacteristic, complemented
349#1802—731no—
350C2×#340—731no—
351C73#2640811yescharacteristic, complemented, Sylow
352C23×C8C2×D7311yescharacteristic
353C2×C4⋊Q8—731no—
354C2×C22⋊Q8—1461no—
355C2×#121—731no—
356C2×C4×D4—731no—
357C2×C4×D4—1461no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C2×#107—731no—
361C2×#107—731no—
362C2×C4⋊C8C2×D7311yescharacteristic
363C2×C4⋊C8C2×D7311yescharacteristic
364C2×#102—731no—
365C2×#102—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×C22⋊C8C2×D7311yescharacteristic
369#398—1461no—
370#395—1461no—
371#390—731no—
372#389—731no—
373#389—731no—
374#389—731no—
375#388—1461no—
376#386—1461no—
377#385—1461no—
378#385—1461no—
379#384—731no—
380#380—731no—
381C4×C4⋊C4—731no—
382C4×C22⋊C4C2×D7311yescharacteristic
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#336C2×D7311yescharacteristic
394#336C2×D7311yescharacteristic
395C22×Q8—1461no—
396C22×D4—731no—
397C22×D4—1461no—
398C23×C4—1461no—
399C23×C4—1461no—
400C23×C4—1461no—
401C23×C4C22×D7311yescharacteristic
402C22×C8—22no—
403C22×C8—41no—
404C22×C8—41no—
405C22×C8C22×D7311yescharacteristic
406C22×C8C22×D7311yescharacteristic
407C22×C8C22×D7311yescharacteristic
408C22×C8C22×D7311yescharacteristic
409C4⋊Q8—2921no—
410C4⋊Q8—1462no—
411C22⋊Q8—2924no—
412#121—734no—
413#121—1462no—
414C4×D4—2921no—
415C4×D4—2924no—
416C4×D4—1462no—
417C2×C4⋊C4—2921no—
418C2×C4⋊C4—2921no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4—1462no—
435C2×C4⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4D29211yescharacteristic
442C2×C22⋊C4D29211yescharacteristic
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446#107—2921no—
447#107—1462no—
448C4⋊C8—41no—
449C4⋊C8—41no—
450#102—2921no—
451#102—2921no—
452#102—2921no—
453#102—1462no—
454C22⋊C8—22no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95D29211yescharacteristic
461#95D29211yescharacteristic
462C24—1461no—
463C24C2×D29211yescharacteristic
464C2×Q8—2922no—
465C2×Q8—2922no—
466C2×Q8—5841no—
467C2×D4—2922no—
468C2×D4—2922no—
469C2×D4—732no—
470C2×D4—1464no—
471C2×D4—1461no—
472C2×D4—1461no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4—1461no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—2922no—
481C22×C4—2922no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—5841no—
487C22×C4—5841no—
488C22×C4—5841no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4C2×D29211yescharacteristic
497C22×C4C2×D29211yescharacteristic
498C22×C4C2×D29211yescharacteristic
499C22×C4C2×D29211yescharacteristic
500C22×C4C2×D29211yescharacteristic
501C22×C4Q8×D7312yes—
502C22×C4#1192402712yes—
503C22×C4C23×D7311yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—44no—
517C4⋊C4—2922no—
518C4⋊C4—2922no—
519C4⋊C4—2922no—
520C4⋊C4—2922no—
521C4⋊C4—2922no—
522C4⋊C4—2922no—
523C4⋊C4—2922no—
524C4⋊C4—5841no—
525C4⋊C4—732no—
526C4⋊C4—732no—
527C4⋊C4—732no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C22⋊C4—22no—
540C22⋊C4—2922no—
541C22⋊C4—2922no—
542C22⋊C4—2922no—
543C22⋊C4—5841no—
544C22⋊C4—1464no—
545C22⋊C4D58414yes—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—2922no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—2921no—
556C23—2921no—
557C23—2921no—
558C23—2921no—
559C23—2921no—
560C23—2921no—
561C23—1462no—
562C23#1455657112yes—
563C23C2×D58412yes—
564C23#1455662412yes—
565C23C22×D29211yescharacteristic
566Q8—5844no—
567D4—2928no—
568D4—2921no—
569D4—2921no—
570D4—1462no—
571D4—1462no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—5844no—
607C2×C4—5844no—
608C2×C4—5844no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4#1455657112yes—
613C2×C4C2×D58412yes—
614C2×C4C2×D58412yes—
615C2×C4C2×Q8×D7311yescharacteristic
616C2×C4#41×D7311yescharacteristic
617C2×C4#41×D7311yescharacteristic
618C2×C4C2×#1192402711yescharacteristic
619C2×C4#1455663611yescharacteristic
620C2×C4#1455663611yescharacteristic
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—2922no—
632C22—2922no—
633C22—2922no—
634C22—5841no—
635C22—5841no—
636C22—5841no—
637C22—5841no—
638C22#1798886814yes—
639C22C2×#1455657111yescharacteristic
640C22#1798923211yescharacteristic
641C22#1798923211yescharacteristic
642C22C22×D58411yescharacteristic
643C22C2×#1455662411yescharacteristic
644C22#1798939911yescharacteristic
645C22#1798939911yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—2922no—
652C4—2922no—
653C4—2922no—
654C4—5841no—
655C4—5841no—
656C4—5841no—
657C4—5841no—
658C4#1798911411yescharacteristic
659C4#1798917211yescharacteristic
660C4#1798929111yescharacteristic
661C4#1798929111yescharacteristic
662C2—24no—
663C2—5842no—
664C2#2261375311yescharacteristic
665C2C2×#1798886811yescharacteristic
666C2#2261979511yescharacteristic
667C2#2262174511yescharacteristic
668C2#2262174511yescharacteristic
669C2#2262204411yescharacteristic
670C2#2262210411yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1850C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenter#1850D73
hypocenterC73#26408
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #22613633C2semidirect
3. #22611490C2semidirect
4. #22611419C2semidirect
5. #22611318C2semidirect
6. #22611068C2semidirect
7. #22610882C2semidirect
8. #22610503C2semidirect
11. #22610181C2semidirect
12. #22610121C2semidirect
13. #22609932C2semidirect
14. C2×#17988350C2semidirect
15. C73×#1850C2semidirect
69. #14556565Q8semidirect
348. #1850D73semidirect
351. C73#26408semidirect

Maximal subgroups

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

subgroupindex
#226136332
#226129452
#226114902
#226114192
#226113182
#226110682
#226108822
#226105032
#226103592
#226102462
#226101812
#226101212
#226099322
C2×#179883502
C73×#18502
#2640873