Finite Group Databasebeta

← #33098248

Subgroups — #33098248

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988868C211yesmaximal, characteristic
2#22612679C211yesmaximal, characteristic, complemented
3#22612575C211yesmaximal, characteristic
4#22610878C211yesmaximal, characteristic, complemented
5#22610701C211yesmaximal, characteristic, complemented
6#22610669C211yesmaximal, characteristic, complemented
7#22610664C211yesmaximal, characteristic, complemented
8#22610628C211yesmaximal, characteristic
9#22610614C211yesmaximal, characteristic
10#22610559C211yesmaximal, characteristic
11#22610552C211yesmaximal, characteristic
12#22610524C211yesmaximal, characteristic, complemented
13#22610503C211yesmaximal, characteristic, complemented
14#22609942C211yesmaximal, characteristic
15C73×#1768C211yesmaximal, characteristic, complemented
16C2×#14556591C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18#17988868C418yescomplemented
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556513C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27#17988491—21no—
28#17988491—21no—
29#17988489C2211yescharacteristic
30#17988489C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988476—21no—
36#17988476—21no—
37#17988470—21no—
38#17988470—21no—
39#17988453C2211yescharacteristic
40#17988449—21no—
41#17988446—21no—
42C4×C4⋊Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57#376×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63#14556591—22no—
64#14556591C2×C414yes—
65C4×D292—22no—
66C4×D292C2×C414yes—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87#14556525—41no—
88#14556525C2×C414yes—
89#14556525C2×C414yes—
90#14556523—41no—
91#14556513—41no—
92#14556513—41no—
93#14556513C2×C414yes—
94#14556513C2×C414yes—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C4⋊C8×C73—22no—
112C4⋊C8×C73C2×C414yes—
113#97×C73—41no—
114C4×C8×C73—41no—
115#95×C73D411yescharacteristic
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C23×D73—21no—
120C2×Dic146—24no—
121C2×Dic146—81no—
122C2×Dic146—81no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292C22×C412yes—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C2×C4×D73—24no—
137C2×C4×D73—81no—
138C2×C4×D73—81no—
139C4⋊Dic73—24no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73C22×C412yes—
152C4⋊Dic73C22×C412yes—
153C4⋊Dic73C22×C412yes—
154#11924007—81no—
155#11924007—81no—
156#11924007—81no—
157#11924007—81no—
158#11924005—24no—
159#11924005—81no—
160#11924005—81no—
161#11924005—81no—
162#11924005—81no—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73—81no—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73C22×C412yes—
184C2×C8×C73C22×C412yes—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73C22×C412yes—
190C42×C73#4112yes—
191C23×C73C22×D411yescharacteristic
192C22×D73—22no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73C4×D412yes—
216C2×C4×C73C4×D412yes—
217C2×C4×C73C23×C411yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73H(5,𝔽2)11yescharacteristic
220C2×C4×C732−1+411yescharacteristic
221C2×C4×C732−1+411yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—41no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244D292—22no—
245D292—22no—
246D292—41no—
247D292—41no—
248C4×D73—84no—
249Dic146—84no—
250C8×C73—22no—
251C8×C73—22no—
252C8×C73—41no—
253C8×C73—41no—
254C22×C73C2×C4×D411yescharacteristic
255C22×C73#53611yescharacteristic
256C22×C73#53711yescharacteristic
257C22×C73#57611yescharacteristic
258C22×C73#57611yescharacteristic
259C22×C73#57711yescharacteristic
260C22×C73#57811yescharacteristic
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C2×D73—81no—
268C4×C73—22no—
269C4×C73—22no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73#51911yescharacteristic
275C4×C73#52011yescharacteristic
276C4×C73#54111yescharacteristic
277C4×C73#54211yescharacteristic
278Dic73—42no—
279Dic73—42no—
280Dic73—42no—
281Dic73—81no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285#22127—731nomaximal, Sylow
286C2×C73#232311yescharacteristic
287C2×C73#294711yescharacteristic
288C2×C73#294811yescharacteristic
289C2×C73#324111yescharacteristic
290C2×C73#324311yescharacteristic
291C2×C73#325711yescharacteristic
292C2×C73#326211yescharacteristic
293D73—82no—
294C2×#474—731no—
295#2331—731no—
296#2319—731no—
297#1987—731no—
298#1949—731no—
299#1943—731no—
300#1942—731no—
301#1936—731no—
302#1934—731no—
303#1926—731no—
304#1925—731no—
305#1921—731no—
306#1918—731no—
307#1804—731no—
308#1768D7311yescharacteristic, complemented
309C73#2212711yescharacteristic, complemented, Sylow
310C2×C4⋊Q8—731no—
311C2×C4×D4—731no—
312#474—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107—731no—
316C2×#106—731no—
317C2×C4⋊C8C2×D7311yescharacteristic
318C2×C4⋊C8C2×D7311yescharacteristic
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#102—731no—
323C2×#97C2×D7311yescharacteristic
324C2×C4×C8C2×D7311yescharacteristic
325#391—1461no—
326#391—1461no—
327#390—731no—
328#390—731no—
329#389—731no—
330#389—731no—
331#389—731no—
332#389—731no—
333#388—1461no—
334#388—1461no—
335#387—1461no—
336#387—1461no—
337#382—731no—
338#381—1461no—
339#380—1461no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343#376C2×D7311yescharacteristic
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#336C2×D7311yescharacteristic
353#336C2×D7311yescharacteristic
354C22×Q8—1461no—
355C22×D4—731no—
356C23×C4—1461no—
357C22×C8C22×D7311yescharacteristic
358C22×C8C22×D7311yescharacteristic
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C4⋊Q8—734no—
362C4⋊Q8—1462no—
363C4×D4—734no—
364C4×D4—1462no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C42—1461no—
383C2×C42—1461no—
384C2×C42—1461no—
385C2×C42C22×D7311yescharacteristic
386C2×C42C22×D7311yescharacteristic
387C2×C42C22×D7311yescharacteristic
388#107—2921no—
389#107—734no—
390#107—734no—
391#106—2921no—
392C4⋊C8—22no—
393C4⋊C8C4×D7314yes—
394#102—2921no—
395#102—2921no—
396#102—734no—
397#102—734no—
398#97—41no—
399C4×C8—41no—
400#95—1461no—
401#95—1461no—
402#95—1461no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95D29211yescharacteristic
410#95D29211yescharacteristic
411#95D29211yescharacteristic
412#95D29211yescharacteristic
413C24—1461no—
414C2×Q8—5841no—
415C2×Q8—5841no—
416C2×Q8—1464no—
417C2×D4—732no—
418C2×D4—1461no—
419C2×D4—1461no—
420C2×D4—1461no—
421C2×D4—1461no—
422C2×D4—1461no—
423C22×C4—5841no—
424C22×C4—5841no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1464no—
433C22×C4C2×D29211yescharacteristic
434C22×C4C2×D29211yescharacteristic
435C22×C4C2×D29211yescharacteristic
436C22×C4C2×D29211yescharacteristic
437C22×C4C2×D29211yescharacteristic
438C22×C4C2×D29211yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D7312yes—
451C2×C8C2×C4×D7312yes—
452C4⋊C4—5841no—
453C4⋊C4—5841no—
454C4⋊C4—5841no—
455C4⋊C4—5841no—
456C4⋊C4—732no—
457C4⋊C4—732no—
458C4⋊C4—732no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1464no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C22⋊C4—5841no—
472C22⋊C4—5841no—
473C22⋊C4—5841no—
474C22⋊C4—5841no—
475C22⋊C4—1464no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—5841no—
481C42—5841no—
482C42—5841no—
483C42C2×C4×D7312yes—
484C42#1192402812yes—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—2921no—
491C23—1462no—
492C23C22×D29211yescharacteristic
493Q8—5844no—
494D4—2921no—
495D4—2921no—
496D4—1462no—
497D4—1462no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—5844no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4C4×D29212yes—
537C2×C4C4×D29212yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4C2×#1192402811yescharacteristic
540C2×C4#1455663611yescharacteristic
541C2×C4#1455663811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663911yescharacteristic
544C8—22no—
545C8—22no—
546C8—41no—
547C8—41no—
548C22—2922no—
549C22—2922no—
550C22—2922no—
551C22—5841no—
552C22—5841no—
553C22—5841no—
554C22—5841no—
555C22C2×C4×D29211yescharacteristic
556C22#1798916311yescharacteristic
557C22#1798916811yescharacteristic
558C22#1798939211yescharacteristic
559C22#1798939211yescharacteristic
560C22#1798939911yescharacteristic
561C22#1798940411yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—2922no—
569C4—2922no—
570C4—2922no—
571C4—5841no—
572C4—5841no—
573C4—5841no—
574C4—5841no—
575C4#1798906311yescharacteristic
576C4#1798906611yescharacteristic
577C4#1798919511yescharacteristic
578C4#1798920211yescharacteristic
579C2—5842no—
580C2#2261261411yescharacteristic
581C2#2261877611yescharacteristic
582C2#2261878411yescharacteristic
583C2#2262118611yescharacteristic
584C2#2262121611yescharacteristic
585C2#2262134411yescharacteristic
586C2#2262139211yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1768C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1768D73
hypocenterC73#22127
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
2. #22612679C2semidirect
4. #22610878C2semidirect
5. #22610701C2semidirect
6. #22610669C2semidirect
7. #22610664C2semidirect
12. #22610524C2semidirect
13. #22610503C2semidirect
15. C73×#1768C2semidirect
18. #17988868C4semidirect
308. #1768D73semidirect
309. C73#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#179888682
#226126792
#226125752
#226108782
#226107012
#226106692
#226106642
#226106282
#226106142
#226105592
#226105522
#226105242
#226105032
#226099422
C73×#17682
#2212773