Finite Group Databasebeta

← #33096617

Subgroups — #33096617

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618566C211yesmaximal, characteristic
2C2×#17988645C211yesmaximal, characteristic
3#22618494C211yesmaximal, characteristic
4#22610705C211yesmaximal, characteristic, complemented
5#22610705C211yesmaximal, characteristic, complemented
6#22610703C211yesmaximal, characteristic, complemented
7#22610703C211yesmaximal, characteristic, complemented
8#22610138C211yesmaximal, characteristic
9#22610136C211yesmaximal, characteristic
10#22610135C211yesmaximal, characteristic
11#22610104C211yesmaximal, characteristic, complemented
12#22610101C211yesmaximal, characteristic, complemented
13#22610070C211yesmaximal, characteristic, complemented
14#22610067C211yesmaximal, characteristic, complemented
15C73×#1855C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18#17988645—21no—
19#17988645—21no—
20#17988645—21no—
21#17988645—21no—
22#17988645C412yes—
23#17988645C412yes—
24#17988645C412yes—
25#17988645C412yes—
26C2×#14556527C2211yescharacteristic
27C2×#14556527C2211yescharacteristic
28C2×#14556526C2211yescharacteristic
29C2×#14556526C2211yescharacteristic
30C2×#14556526C2211yescharacteristic
31C2×#14556526C2211yescharacteristic
32#17988615—21no—
33#17988615—21no—
34#17988613—21no—
35#17988613—21no—
36#17988610C2211yescharacteristic
37#17988610C2211yescharacteristic
38#17988610C2211yescharacteristic
39#17988610C2211yescharacteristic
40#17988610C2211yescharacteristic
41#17988610C2211yescharacteristic
42C2×#14556520C2211yescharacteristic
43C2×#14556520C2211yescharacteristic
44#17988551C2211yescharacteristic
45C2×C22⋊#10596831C2211yescharacteristic
46M16×Dic73C2211yescharacteristic
47M16×Dic73C2211yescharacteristic
48C2×#14556501C2211yescharacteristic
49C2×C8×Dic73C2211yescharacteristic
50#17988352C2211yescharacteristic
51#17988352C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988351C2211yescharacteristic
55#17988351C2211yescharacteristic
56C22×M16×C73C2211yescharacteristic
57C2×#108×C73C2211yescharacteristic
58C2×#108×C73C2211yescharacteristic
59#407×C73C2211yescharacteristic
60C2×C22⋊C8×C73C2211yescharacteristic
61#341×C73C2211yescharacteristic
62#341×C73C2211yescharacteristic
63C2×#11924012—21no—
64C2×#11924012—41no—
65C2×#11924012—41no—
66C2×#11924012C2311yescharacteristic
67C2×#11924012C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70C22×#10596831—21no—
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73#14556550C2×C411yescharacteristic
74#14556550C2×C411yescharacteristic
75#14556550C2×C411yescharacteristic
76#14556550C2×C411yescharacteristic
77#14556550D411yescharacteristic
78#14556550D411yescharacteristic
79#14556550D411yescharacteristic
80#14556550D411yescharacteristic
81C2×C4⋊Dic73Q811yescharacteristic
82C2×C4⋊Dic73Q811yescharacteristic
83C2×C22⋊Dic73Q811yescharacteristic
84C2×C22⋊Dic73Q811yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87#14556527—41no—
88#14556527—41no—
89#14556526—41no—
90#14556526—41no—
91#14556526—41no—
92#14556526—41no—
93#14556520—21no—
94#14556520—21no—
95#14556520—21no—
96#14556520—21no—
97#14556520—21no—
98#14556520—21no—
99#14556520—21no—
100#14556520—21no—
101#14556520—21no—
102#14556520—21no—
103#14556520—21no—
104#14556520—21no—
105#14556520C2×C412yes—
106#14556520C2×C412yes—
107#14556520C2×C412yes—
108#14556520C2×C412yes—
109C22⋊#10596831—21no—
110C22⋊#10596831—21no—
111C22⋊#10596831C2×C412yes—
112C22⋊#10596831C2×C412yes—
113#14556501—21no—
114#14556501—21no—
115#14556501C2×C412yes—
116#14556501C2×C412yes—
117C8×Dic73—21no—
118C8×Dic73—21no—
119C8×Dic73C2×C412yes—
120C8×Dic73C2×C412yes—
121C23×C4×C73C2311yescharacteristic
122C2×M16×C73—21no—
123C2×M16×C73—41no—
124C2×M16×C73—41no—
125C2×M16×C73C2311yescharacteristic
126C2×M16×C73C2311yescharacteristic
127C2×M16×C73C2311yescharacteristic
128C2×M16×C73C2311yescharacteristic
129C22×C8×C73—21no—
130C22×C8×C73C2311yescharacteristic
131C22×C8×C73C2311yescharacteristic
132#108×C73—41no—
133#108×C73—41no—
134C22⋊C8×C73—21no—
135C22⋊C8×C73—21no—
136C22⋊C8×C73C2×C412yes—
137C22⋊C8×C73C2×C412yes—
138C22×Dic73C2×Q811yescharacteristic
139C22×Dic73C2×Q811yescharacteristic
140C22×Dic73C2×Q811yescharacteristic
141C22×Dic73C2×Q811yescharacteristic
142#11924012—21no—
143#11924012—21no—
144#11924012—21no—
145#11924012—21no—
146#11924012—21no—
147#11924012—21no—
148#11924012—21no—
149#11924012—21no—
150#11924012—42no—
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×#10596831—42no—
160C2×#10596831C22×C412yes—
161C2×#10596831C22×C412yes—
162C4⋊Dic73C4⋊C411yescharacteristic
163C4⋊Dic73C4⋊C411yescharacteristic
164C4⋊Dic73C4⋊C411yescharacteristic
165C4⋊Dic73C4⋊C411yescharacteristic
166C4⋊Dic73C4⋊C411yescharacteristic
167C4⋊Dic73C4⋊C411yescharacteristic
168C4⋊Dic73C4⋊C411yescharacteristic
169C4⋊Dic73C4⋊C411yescharacteristic
170C22⋊Dic73C4⋊C411yescharacteristic
171C22⋊Dic73C4⋊C411yescharacteristic
172C22⋊Dic73C4⋊C411yescharacteristic
173C22⋊Dic73C4⋊C411yescharacteristic
174C22⋊Dic73C4⋊C411yescharacteristic
175C22⋊Dic73C4⋊C411yescharacteristic
176C22⋊Dic73C4⋊C411yescharacteristic
177C22⋊Dic73C4⋊C411yescharacteristic
178C4×Dic73C22×C411yescharacteristic
179C4×Dic73C22×C411yescharacteristic
180C4×Dic73C22×C411yescharacteristic
181C4×Dic73C22×C411yescharacteristic
182C4×Dic73C2×D411yescharacteristic
183C4×Dic73C2×D411yescharacteristic
184C4×Dic73C2×D411yescharacteristic
185C4×Dic73C2×D411yescharacteristic
186C24×C73C2×Q811yescharacteristic
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×Q811yescharacteristic
196C22×C4×C73C2411yescharacteristic
197M16×C73—21no—
198M16×C73—21no—
199M16×C73—21no—
200M16×C73—21no—
201M16×C73—21no—
202M16×C73—21no—
203M16×C73—21no—
204M16×C73—21no—
205M16×C73—42no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—42no—
215C2×C8×C73C22×C412yes—
216C2×C8×C73C22×C412yes—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73C2×C4⋊C411yescharacteristic
222C23×C73C2×C4⋊C411yescharacteristic
223C23×C73C22×Q811yescharacteristic
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73C2×C4⋊C411yescharacteristic
235C2×C4×C73C2×C4⋊C411yescharacteristic
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C22×D411yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×C4×C732−1+411yescharacteristic
241C2×C4×C732−1+411yescharacteristic
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73C2×C4⋊C411yescharacteristic
247C2×Dic73C2×C4⋊C411yescharacteristic
248C2×Dic73C2×C4⋊C411yescharacteristic
249C2×Dic73C2×C4⋊C411yescharacteristic
250C2×Dic73C2×C4⋊C411yescharacteristic
251C2×Dic73C2×C4⋊C411yescharacteristic
252C2×Dic73C2×C4⋊C411yescharacteristic
253C2×Dic73C2×C4⋊C411yescharacteristic
254C8×C73—22no—
255C8×C73—22no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73—41no—
260#10596831—22no—
261#10596831—22no—
262#10596831—41no—
263#10596831—41no—
264#10596831—41no—
265#10596831—41no—
266C22×C73—21no—
267C22×C73—21no—
268C22×C73—21no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73—21no—
273C22×C73—21no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73C22×C4⋊C411yescharacteristic
277C22×C73#54411yescharacteristic
278C22×C73#54411yescharacteristic
279C4×C73—21no—
280C4×C73—21no—
281C4×C73#52011yescharacteristic
282C4×C73#52011yescharacteristic
283C4×C73#53511yescharacteristic
284C4×C73#53611yescharacteristic
285Dic73—21no—
286Dic73—21no—
287Dic73—21no—
288Dic73—21no—
289Dic73—21no—
290Dic73—21no—
291Dic73—21no—
292Dic73—21no—
293#21995—731nomaximal, Sylow
294C2×C73—21no—
295C2×C73—21no—
296C2×C73—41no—
297C2×C73—41no—
298C2×C73#231711yescharacteristic
299C2×C73#291611yescharacteristic
300C2×C73#291611yescharacteristic
301#2921—731no—
302C2×#433—731no—
303#2905—731no—
304#1950—731no—
305#1950—731no—
306#1950—731no—
307#1950—731no—
308#1855—731no—
309#1855—731no—
310#1855—731no—
311#1855D7311yescharacteristic
312#1849—731no—
313#1848—731no—
314#1838—731no—
315#1837—731no—
316C73#2199511yescharacteristic, complemented, Sylow
317C22×M16—731no—
318C22×M16C2×D7311yescharacteristic
319C2×#117—731no—
320#433—732no—
321#433—732no—
322#433—732no—
323#433—732no—
324#433—1461no—
325#433—1461no—
326#433—1461no—
327#433—1461no—
328C2×#108—731no—
329C2×#108—731no—
330C2×#108—731no—
331C2×#108—731no—
332C2×#108—731no—
333C2×#108—731no—
334C2×#108C2×D7311yescharacteristic
335C2×#108C2×D7311yescharacteristic
336#424—1461no—
337#424—1461no—
338#424—1461no—
339#424—1461no—
340#423—731no—
341#423—731no—
342#423—731no—
343#423—731no—
344#423—731no—
345#423—731no—
346C2×C4⋊C8—731no—
347C2×C4⋊C8—731no—
348#407—731no—
349#407C2×D7311yescharacteristic
350C2×C22⋊C8—731no—
351C2×C22⋊C8C2×D7311yescharacteristic
352C4×M16—731no—
353C4×M16—731no—
354C2×#97—731no—
355C2×C4×C8—731no—
356#341—731no—
357#341—731no—
358#341—731no—
359#341—731no—
360#341—731no—
361#341—731no—
362#341C2×D7311yescharacteristic
363#341C2×D7311yescharacteristic
364C23×C4C22×D7311yescharacteristic
365C2×M16—21no—
366C2×M16—41no—
367C2×M16—41no—
368C2×M16—2921no—
369C2×M16—2921no—
370C2×M16—731no—
371C2×M16—731no—
372C2×M16—731no—
373C2×M16—731no—
374C2×M16—1461no—
375C2×M16C22×D7311yescharacteristic
376C2×M16C22×D7311yescharacteristic
377C2×M16C22×D7311yescharacteristic
378C2×M16C22×D7311yescharacteristic
379C22×C8—21no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—1461no—
383C22×C8C22×D7311yescharacteristic
384C22×C8C22×D7311yescharacteristic
385#117—731no—
386#117—731no—
387#117—731no—
388#117—731no—
389#117—731no—
390#117—731no—
391#117—731no—
392#117—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C22⋊C4—731no—
396C2×C22⋊C4—731no—
397C2×C42—731no—
398C2×C42—731no—
399#108—41no—
400#108—41no—
401#108—2921no—
402#108—2921no—
403#108—2921no—
404#108—2921no—
405#108—2921no—
406#108—2921no—
407C4⋊C8—732no—
408C4⋊C8—732no—
409C4⋊C8—732no—
410C4⋊C8—732no—
411C4⋊C8—1461no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8—1461no—
418C4⋊C8—1461no—
419C4⋊C8—1461no—
420C4⋊C8—1461no—
421C4⋊C8—1461no—
422C4⋊C8—1461no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—732no—
426C22⋊C8—732no—
427C22⋊C8—1461no—
428C22⋊C8—1461no—
429C22⋊C8C4×D7312yes—
430C22⋊C8C4×D7312yes—
431#97—732no—
432#97—732no—
433#97—1461no—
434#97—1461no—
435C4×C8—732no—
436C4×C8—732no—
437C4×C8—1461no—
438C4×C8—1461no—
439C24Q8×D7311yescharacteristic
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4C2×C4×D7311yescharacteristic
449C22×C4C2×C4×D7311yescharacteristic
450C22×C4D4×D7311yescharacteristic
451C22×C4D4×D7311yescharacteristic
452C22×C4Q8×D7311yescharacteristic
453C22×C4C23×D7311yescharacteristic
454M16—21no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—42no—
463M16—2922no—
464M16—1461no—
465M16—1461no—
466M16—1461no—
467M16—1461no—
468M16—1461no—
469M16—1461no—
470M16—1461no—
471M16—1461no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—42no—
481C2×C8—2922no—
482C2×C8—732no—
483C2×C8—732no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8C2×C4×D7312yes—
493C2×C8C2×C4×D7312yes—
494C4⋊C4—731no—
495C4⋊C4—731no—
496C4⋊C4—731no—
497C4⋊C4—731no—
498C4⋊C4—731no—
499C4⋊C4—731no—
500C4⋊C4—731no—
501C4⋊C4—731no—
502C22⋊C4—731no—
503C22⋊C4—731no—
504C22⋊C4—731no—
505C22⋊C4—731no—
506C22⋊C4—731no—
507C22⋊C4—731no—
508C22⋊C4—731no—
509C22⋊C4—731no—
510C42—731no—
511C42—731no—
512C42—731no—
513C42—731no—
514C42—731no—
515C42—731no—
516C42—731no—
517C42—731no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—41no—
522C23C4⋊C4×D7311yescharacteristic
523C23C4⋊C4×D7311yescharacteristic
524C23C2×Q8×D7311yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—731no—
540C2×C4—731no—
541C2×C4—731no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4C4⋊C4×D7311yescharacteristic
548C2×C4C4⋊C4×D7311yescharacteristic
549C2×C4C22×C4×D7311yescharacteristic
550C2×C4C2×D4×D7311yescharacteristic
551C2×C4#1455663611yescharacteristic
552C2×C4#1455663611yescharacteristic
553C2×C4#1455663811yescharacteristic
554C2×C4#1455663811yescharacteristic
555C8—22no—
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—2921no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—1462no—
566C8—1462no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—81no—
576C22—81no—
577C22C2×C4⋊C4×D7311yescharacteristic
578C22#1798921611yescharacteristic
579C22#1798921611yescharacteristic
580C4—21no—
581C4—21no—
582C4—1461no—
583C4—1461no—
584C4—1461no—
585C4—1461no—
586C4—1461no—
587C4—1461no—
588C4—1461no—
589C4—1461no—
590C4#1798906611yescharacteristic
591C4#1798906611yescharacteristic
592C4#1798915211yescharacteristic
593C4#1798915911yescharacteristic
594C2—21no—
595C2—21no—
596C2—41no—
597C2—41no—
598C2#2261255011yescharacteristic
599C2#2261853911yescharacteristic
600C2#2261854111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1855C2
socleC22×C73C22×C4⋊C4
prime radicalC24×C73C2×Q8
cosocleC22×C4×C73C24
hypercenter#1855D73
hypocenterC73#21995
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
4. #22610705C2semidirect
5. #22610705C2semidirect
6. #22610703C2semidirect
7. #22610703C2semidirect
11. #22610104C2semidirect
12. #22610101C2semidirect
13. #22610070C2semidirect
14. #22610067C2semidirect
316. C73#21995semidirect

Maximal subgroups

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

subgroupindex
#226185662
C2×#179886452
#226184942
#226107052
#226107032
#226101382
#226101362
#226101352
#226101042
#226101012
#226100702
#226100672
C73×#18552
#2199573