Finite Group Databasebeta

← #34018548

Subgroups — #34018548

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626852C211yesmaximal, characteristic, complemented
2#22626850C211yesmaximal, characteristic, complemented
3#22625575C212yesmaximal, complemented
4#22625572C212yesmaximal, complemented
5#22625569C212yesmaximal, complemented
6#22625562C212yesmaximal, complemented
7C2×#17989451C211yesmaximal, characteristic, complemented
8#22625228C211yesmaximal, characteristic, complemented
9#22625226C211yesmaximal, characteristic, complemented
10#22625134C211yesmaximal, characteristic, complemented
11#22618699C211yesmaximal, characteristic
12#17989638C2212yes—
13#17989637C2212yes—
14#17989622C2212yes—
15#17989621C2212yes—
16#17989618—22no—
17#17989616—22no—
18#17989609—21no—
19#17989608—21no—
20#17989603C2211yescharacteristic
21#17989603C2211yescharacteristic
22#17989591—21no—
23#17989590—21no—
24C2×C4⋊#10596833C2211yescharacteristic
25C2×C4⋊#10596833C2211yescharacteristic
26#17989451C2212yes—
27#17989451C2212yes—
28#17989437C2212yes—
29Q8⋊#10596833C2212yes—
30#17989435C2212yes—
31D4⋊#10596833C2212yes—
32#17989428C2212yes—
33C8×#10596833C2212yes—
34C2×#14556611C411yescharacteristic
35#41×Dic73C2211yescharacteristic
36#17989055C2211yescharacteristic
37#17988679C411yescharacteristic
38#17988679C2211yescharacteristic
39#17988678C411yescharacteristic
40#17988678C2211yescharacteristic
41#17988672C411yescharacteristic
42#17988672C2211yescharacteristic
43#17988667C411yescharacteristic
44#17988667C2211yescharacteristic
45#17988623C411yescharacteristic
46#17988584C411yescharacteristic
47C2×C8×Dic73C2211yescharacteristic
48#416×C73C411yescharacteristic
49C2×#11924038—22no—
50C2×#11924037—22no—
51C22×#10596833—21no—
52C22×#10596833—21no—
53#14556658—22no—
54C4⋊#10596833—22no—
55C4⋊#10596833—21no—
56C4⋊#10596833—21no—
57C4⋊#10596833—21no—
58C4⋊#10596833D411yescharacteristic
59C4⋊#10596833D411yescharacteristic
60C4⋊#10596833C2312yes—
61C4⋊#10596833C2312yes—
62#14556648—22no—
63C22⋊#10596833—22no—
64C22⋊#10596833—21no—
65C22⋊#10596833D412yes—
66#14556646—22no—
67#14556646—22no—
68C4×#10596833C2312yes—
69C4×#10596833C2312yes—
70C2×#11924028C2×C411yescharacteristic
71#14556611—24no—
72C2×#11924018C2×C411yescharacteristic
73C2×#11924016C2×C411yescharacteristic
74C2×#11924015C2×C411yescharacteristic
75C2×#11924013C2×C411yescharacteristic
76C22×#10596831C2×C411yescharacteristic
77Q8×Dic73C2×C411yescharacteristic
78Q8×Dic73C2311yescharacteristic
79#14556556C2×C411yescharacteristic
80#14556556C2311yescharacteristic
81D4×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73C2×C411yescharacteristic
84D4×Dic73C2311yescharacteristic
85#14556553C2×C411yescharacteristic
86#14556553C2311yescharacteristic
87#14556551—21no—
88#14556551—21no—
89#14556550—21no—
90#14556550C2×C411yescharacteristic
91#14556548—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C42×D73—21no—
95#14556524C2×C411yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99#14556516C2×C411yescharacteristic
100#14556516C2×C411yescharacteristic
101#14556513C2×C411yescharacteristic
102#14556513C2×C411yescharacteristic
103C8×Dic73C2×C411yescharacteristic
104C8×Dic73C2×C411yescharacteristic
105C8×Dic73C2311yescharacteristic
106C8×Dic73C2311yescharacteristic
107C2×#41×C73C2×C411yescharacteristic
108C22×C8×C73C2×C411yescharacteristic
109#117×C73C2×C411yescharacteristic
110#103×C73C2×C411yescharacteristic
111#103×C73C2×C411yescharacteristic
112#102×C73C2×C411yescharacteristic
113#102×C73C2×C411yescharacteristic
114#11924038—82no—
115#11924037—82no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—22no—
119C2×#10596833—81no—
120C2×#10596833—81no—
121C2×#10596833C2×D412yes—
122C2×#10596833C2×D412yes—
123#11924028—22no—
124#11924028—22no—
125C2×Dic146C22×C411yescharacteristic
126C2×D292C22×C411yescharacteristic
127C2×C22⋊D73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73#4111yescharacteristic
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135#11924018—22no—
136#11924016—22no—
137#11924015—22no—
138#11924013—22no—
139C2×#10596831M1614yescomplemented
140C2×#10596831C22×C411yescharacteristic
141C2×#10596831C22×C411yescharacteristic
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73C22×C411yescharacteristic
145C4⋊Dic73C22×C411yescharacteristic
146#11924007—21no—
147#11924007—21no—
148C22⋊Dic73—21no—
149C22⋊Dic73—21no—
150C22⋊Dic73—21no—
151#11924005—21no—
152#11924005—21no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73C22×C411yescharacteristic
159C4×Dic73C2×D411yescharacteristic
160C4×Dic73#4111yescharacteristic
161C4×Dic73C2411yescharacteristic
162#41×C73—22no—
163#41×C73—22no—
164C2×Q8×C73C22×C411yescharacteristic
165C2×D4×C73—21no—
166C2×D4×C73C22×C411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73C22×C411yescharacteristic
169C2×C8×C73—22no—
170C2×C8×C73C22×C411yescharacteristic
171C2×C8×C73C22×C411yescharacteristic
172C4⋊C4×C73C22×C411yescharacteristic
173C4⋊C4×C73C22×C411yescharacteristic
174C22⋊C4×C73—21no—
175C42×C73—21no—
176C23×C73—21no—
177C23×C73C4×D411yescharacteristic
178C22×D73—21no—
179Q8×C73—21no—
180Q8×C73—21no—
181D4×C73—21no—
182D4×C73—21no—
183D4×C73—42no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—42no—
188C2×C4×C73C4×D411yescharacteristic
189C2×C4×C73C2×M1612yes—
190C2×C4×C73#13112yes—
191C2×C4×C73C23×C411yescharacteristic
192C22⋊D73—42no—
193C2×Dic73—21no—
194C2×Dic73—21no—
195C2×Dic73—21no—
196C2×Dic73—21no—
197C2×Dic73—21no—
198C2×Dic73—41no—
199C2×Dic73—41no—
200C2×Dic73—41no—
201C2×Dic73—81no—
202C2×Dic73C4×D411yescharacteristic
203C2×Dic73C4×D411yescharacteristic
204C2×Dic73C22×D411yescharacteristic
205C2×Dic73C2×#4111yescharacteristic
206D292—21no—
207D292—21no—
208C4×D73—42no—
209C4×D73—41no—
210C4×D73—41no—
211C4×D73—81no—
212Dic146—21no—
213Dic146—21no—
214#10596833—22no—
215#10596833—22no—
216#10596833—42no—
217#10596833—42no—
218#10596833—42no—
219C8×C73—21no—
220C8×C73—21no—
221#10596831C2×M1612yes—
222#10596831C2×M1612yes—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—41no—
226C22×C73—41no—
227C22×C73—41no—
228C22×C73#43612yes—
229C22×C73C2×C4×D411yescharacteristic
230C2×D73—41no—
231C2×D73—41no—
232C2×D73—41no—
233C4×C73—41no—
234C4×C73—41no—
235C4×C73—41no—
236C4×C73#43612yes—
237C4×C73C22×M1611yescharacteristic
238C4×C73C2×#13111yescharacteristic
239Dic73—21no—
240Dic73—21no—
241Dic73—21no—
242Dic73—41no—
243Dic73—41no—
244Dic73—41no—
245Dic73#57611yescharacteristic
246Dic73#57811yescharacteristic
247#17898—731nomaximal, Sylow
248C2×C73—22no—
249C2×C73—81no—
250C2×C73C2×#43611yescharacteristic
251C2×C73#295111yescharacteristic
252C2×C73#295711yescharacteristic
253D73—81no—
254#2989—731no—
255#2989—731no—
256C4×#135—731no—
257#1592—732no—
258#1592—732no—
259#1591—732no—
260#1589—732no—
261C2×#335—731no—
262#1477—731no—
263#1477—731no—
264#1450—731no—
265C73#1789811yescharacteristic, complemented, Sylow
266C2×#135—731no—
267C4×#41—731no—
268C4×#41—731no—
269#446—732no—
270#446—732no—
271C4×Q16—731no—
272C4×Q16—731no—
273C4×SD16—731no—
274C4×SD16—731no—
275C4×SD16—731no—
276C4×SD16—731no—
277C4×D8—731no—
278C4×D8—731no—
279#436—732no—
280#436—732no—
281#435—1462no—
282#435—1462no—
283#432—1461no—
284#432—1461no—
285#431—731no—
286#431—731no—
287#427—731no—
288#423—1461no—
289#423—1461no—
290C2×C4⋊C8—731no—
291C2×C4⋊C8—731no—
292#416—731no—
293#416#6440811yescharacteristic
294C2×C4×C8—731no—
295#335—732no—
296#335—732no—
297Q8⋊C8—732no—
298Q8⋊C8—732no—
299D4⋊C8—732no—
300D4⋊C8—732no—
301#322—732no—
302C82—732no—
303C2×#41—731no—
304C2×#41C2×#6440811yescharacteristic
305#135—1464no—
306C2×Q16—731no—
307C2×SD16—731no—
308C2×SD16—731no—
309C2×D8—731no—
310C2×M16—1462no—
311C2×M16—1462no—
312C22×C8—731no—
313C22×C8—1461no—
314C22×C8—1461no—
315C22×C8C2×#6440811yescharacteristic
316C4×Q8—731no—
317C4×Q8—731no—
318C4×Q8—731no—
319C4×Q8—731no—
320C4×D4—731no—
321C4×D4—731no—
322C4×D4—731no—
323C4×D4—731no—
324C4×D4—1461no—
325C4×D4—1461no—
326C4×D4—1461no—
327C4×D4—1461no—
328#117—731no—
329#117—1461no—
330#117—1461no—
331#117C2×#6440811yescharacteristic
332C2×C42—731no—
333C2×C42—1461no—
334C2×C42—1461no—
335#107—731no—
336#107—731no—
337#106—731no—
338#106—731no—
339C4⋊C8—731no—
340C4⋊C8—731no—
341C4⋊C8—732no—
342C4⋊C8—732no—
343C4⋊C8—1461no—
344C4⋊C8—1461no—
345C4⋊C8—1461no—
346C4⋊C8—1462no—
347C4⋊C8—1462no—
348#103—731no—
349#103—731no—
350#103C2×#6440811yescharacteristic
351#103C2×#6440811yescharacteristic
352#102—731no—
353#102—731no—
354#102C2×#6440811yescharacteristic
355#102C2×#6440811yescharacteristic
356C22⋊C8—732no—
357C22⋊C8—1462no—
358C22⋊C8—1462no—
359C22⋊C8—1461no—
360#97—1462no—
361#97—1462no—
362C4×C8—732no—
363C4×C8—731no—
364C4×C8—731no—
365C4×C8—731no—
366C4×C8—731no—
367C4×C8—732no—
368#41—22no—
369#41—22no—
370#41—1462no—
371#41—1462no—
372C2×Q8—731no—
373C2×Q8C22×#6440811yescharacteristic
374C2×D4—21no—
375C2×D4—731no—
376C2×D4—1461no—
377C2×D4C22×#6440811yescharacteristic
378C22×C4—21no—
379C22×C4—731no—
380C22×C4—731no—
381C22×C4—1461no—
382C22×C4—1461no—
383C22×C4—1461no—
384C22×C4—1461no—
385C22×C4—1461no—
386C22×C4C22×#6440811yescharacteristic
387Q16—1462no—
388SD16—1462no—
389SD16—1462no—
390D8—1462no—
391M16—5842no—
392M16—5842no—
393C2×C8—22no—
394C2×C8—5841no—
395C2×C8—5841no—
396C2×C8—734no—
397C2×C8—731no—
398C2×C8—731no—
399C2×C8—732no—
400C2×C8—732no—
401C2×C8—1462no—
402C2×C8—1462no—
403C2×C8—1462no—
404C2×C8C22×#6440811yescharacteristic
405C2×C8C22×#6440811yescharacteristic
406C4⋊C4—731no—
407C4⋊C4—731no—
408C4⋊C4—1461no—
409C4⋊C4—1461no—
410C4⋊C4—1461no—
411C4⋊C4—1461no—
412C4⋊C4C22×#6440811yescharacteristic
413C4⋊C4C22×#6440811yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—1461no—
416C22⋊C4—1461no—
417C22⋊C4—1461no—
418C22⋊C4—1461no—
419C22⋊C4—1461no—
420C42—21no—
421C42—731no—
422C42—731no—
423C42—731no—
424C42—731no—
425C42—1461no—
426C42—1461no—
427C42—1461no—
428C42—1461no—
429C42—1461no—
430C23—21no—
431C23—1461no—
432C23D4×#6440811yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—1461no—
436Q8—1461no—
437D4—21no—
438D4—21no—
439D4—42no—
440D4—2922no—
441D4—1461no—
442D4—1461no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—2921no—
448C2×C4—2921no—
449C2×C4—2921no—
450C2×C4—2921no—
451C2×C4—2922no—
452C2×C4—2921no—
453C2×C4—5841no—
454C2×C4—5841no—
455C2×C4—731no—
456C2×C4—731no—
457C2×C4—731no—
458C2×C4—731no—
459C2×C4—1461no—
460C2×C4—1461no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4—1461no—
464C2×C4D4×#6440811yescharacteristic
465C2×C4D73⋊M1612yes—
466C2×C4#1455667812yes—
467C2×C4C23×#6440811yescharacteristic
468C8—21no—
469C8—21no—
470C8—2922no—
471C8—2922no—
472C8—2922no—
473C8—732no—
474C8—732no—
475C8—1462no—
476C8—1462no—
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—2921no—
483C22—2921no—
484C22—2921no—
485C22#1798962012yes—
486C22C2×D4×#6440811yescharacteristic
487C4—41no—
488C4—41no—
489C4—41no—
490C4—2921no—
491C4—2921no—
492C4—2921no—
493C4—731no—
494C4—731no—
495C4—1461no—
496C4—1461no—
497C4—1461no—
498C4#1798962012yes—
499C4C2×D73⋊M1611yescharacteristic
500C4C2×#1455667811yescharacteristic
501C2—22no—
502C2—81no—
503C2—5841no—
504C2C2×#1798962011yescharacteristic
505C2#2262669311yescharacteristic
506C2#2262671011yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#416×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C4
cosocleC4×Dic73C24
hypercenter#416#64408
hypocenterC73#17898
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22626852C2semidirect
2. #22626850C2semidirect
3. #22625575C2semidirect
4. #22625572C2semidirect
5. #22625569C2semidirect
6. #22625562C2semidirect
7. C2×#17989451C2semidirect
8. #22625228C2semidirect
9. #22625226C2semidirect
10. #22625134C2semidirect
139. C2×#10596831M16semidirect
265. C73#17898semidirect

Maximal subgroups

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

subgroupindex
#226268522
#226268502
#226255752
#226255722
#226255692
#226255622
C2×#179894512
#226252282
#226252262
#226251342
#226186992
#1789873