Finite Group Databasebeta

← #33084859

Subgroups — #33084859

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988647C211yesmaximal, characteristic
2C2×#17988612C211yesmaximal, characteristic
3#22612428C211yesmaximal, characteristic
4#22609807C211yesmaximal, characteristic, complemented
5#22609804C211yesmaximal, characteristic, complemented
6#22609799C211yesmaximal, characteristic, complemented
7#22609797C211yesmaximal, characteristic, complemented
8#22609790C211yesmaximal, characteristic, complemented
9#22609790C211yesmaximal, characteristic, complemented
10#22609784C211yesmaximal, characteristic, complemented
11#22609716C211yesmaximal, characteristic
12#22609713C211yesmaximal, characteristic
13#22609708C211yesmaximal, characteristic
14#22609706C211yesmaximal, characteristic
15C73×#1767C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556548C2211yescharacteristic
19#17988647C418yescomplemented
20#17988612C418yescomplemented
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35#17988478C412yes—
36#17988478C412yes—
37#17988471C412yes—
38#17988471C412yes—
39C4×#11924007C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42C4×#11924005C2211yescharacteristic
43#17988433C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50C2×C4⋊C8×C73C2211yescharacteristic
51C2×C4⋊C8×C73C2211yescharacteristic
52C2×#97×C73C2211yescharacteristic
53C2×#97×C73C2211yescharacteristic
54C43×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57C22×C4×D73C2311yescharacteristic
58C2×#11924011—41no—
59C2×#11924011—41no—
60C2×#11924011C2×C414yes—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C2×C8×D73—41no—
66C2×C8×D73—41no—
67C2×C8×D73C2×C414yes—
68#14556548C2×C414yes—
69#14556548C2×C414yes—
70C2×#11924007C2×C412yes—
71C2×#11924007C2×C412yes—
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2×C411yescharacteristic
74C2×#11924005C2×C412yes—
75C2×#11924005C2×C412yes—
76C2×#11924005C2×C411yescharacteristic
77C2×#11924005C2×C411yescharacteristic
78C2×C4×Dic73C2311yescharacteristic
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81C4⋊#10596831C2×C414yes—
82C4⋊#10596831C2×C414yes—
83#14556520C2×C414yes—
84#14556520C2×C414yes—
85#14556503C2×C414yes—
86#14556503C2×C414yes—
87#14556502—41no—
88#14556502—41no—
89#14556501—41no—
90#14556501—41no—
91#14556501C2×C414yes—
92C8×Dic73—41no—
93C8×Dic73—41no—
94C8×Dic73C2×C414yes—
95#14556498C2×C412yes—
96#14556498C2×C412yes—
97#14556498C2×C412yes—
98#14556498C2×C412yes—
99#14556498C2×C412yes—
100#14556498C2×C412yes—
101C22×C8×C73C2311yescharacteristic
102C22×C8×C73C2311yescharacteristic
103C22×C8×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105C2×C42×C73C2×C412yes—
106C2×C42×C73C2×C412yes—
107C2×C42×C73C2311yescharacteristic
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C4⋊C8×C73C2×C414yes—
111C4⋊C8×C73C2×C414yes—
112#97×C73—41no—
113#97×C73—41no—
114C23×D73C22×C411yescharacteristic
115C22×Dic73C22×C412yes—
116C22×Dic73C22×C412yes—
117C22×Dic73C22×C411yescharacteristic
118C22×Dic73C22×C411yescharacteristic
119C22×Dic73C22×C411yescharacteristic
120C2×C4×D73—21no—
121C2×C4×D73—21no—
122C2×C4×D73—21no—
123C2×C4×D73—21no—
124C2×C4×D73C22×C412yes—
125C2×C4×D73C22×C412yes—
126#11924011—44no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831C22×C412yes—
136C2×#10596831C22×C412yes—
137C2×#10596831C22×C412yes—
138C2×#10596831C22×C412yes—
139C8×D73—44no—
140#11924007—42no—
141#11924007—42no—
142#11924007C4214yes—
143#11924007C4214yes—
144#11924005—42no—
145#11924005—42no—
146#11924005C4214yes—
147#11924005C4214yes—
148C4×Dic73—21no—
149C4×Dic73—21no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73C22×C412yes—
153C4×Dic73C22×C412yes—
154C22×C4×C73C22×C412yes—
155C22×C4×C73C22×C412yes—
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C2411yescharacteristic
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73—21no—
164C2×C8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73C22×C412yes—
168C2×C8×C73C22×C412yes—
169C2×C8×C73C22×C412yes—
170C2×C8×C73C22×C412yes—
171C42×C73—21no—
172C42×C73—21no—
173C42×C73—21no—
174C42×C73—21no—
175C42×C73—42no—
176C42×C73—42no—
177C42×C73C22×C412yes—
178C42×C73C22×C412yes—
179C23×C73C23×C411yescharacteristic
180C22×D73—21no—
181C22×D73—21no—
182C22×D73—22no—
183C22×D73—22no—
184C22×D73C2×C4212yes—
185C2×C4×C73—22no—
186C2×C4×C73—22no—
187C2×C4×C73—22no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73C2×C4212yes—
194C2×C4×C73C2×C4212yes—
195C2×C4×C73C23×C411yescharacteristic
196C2×C4×C73C23×C411yescharacteristic
197C2×C4×C73H(5,𝔽2)11yescharacteristic
198C2×C4×C73H(5,𝔽2)11yescharacteristic
199C2×C4×C732−1+411yescharacteristic
200C2×C4×C732−1+411yescharacteristic
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—22no—
204C2×Dic73—22no—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—22no—
208C2×Dic73—22no—
209C2×Dic73—22no—
210C2×Dic73—22no—
211C2×Dic73C2×C4212yes—
212C2×Dic73C2×C4212yes—
213C2×Dic73C2×C4212yes—
214C4×D73—22no—
215C4×D73—22no—
216C4×D73—22no—
217C4×D73—22no—
218C8×C73—22no—
219C8×C73—22no—
220C8×C73—22no—
221C8×C73—22no—
222#10596831—22no—
223#10596831—22no—
224#10596831—22no—
225#10596831—22no—
226C22×C73C22×C4211yescharacteristic
227C22×C73#52011yescharacteristic
228C22×C73#52011yescharacteristic
229C22×C73#56812yes—
230C22×C73#56812yes—
231C2×D73—22no—
232C2×D73—22no—
233C2×D73—22no—
234C2×D73—42no—
235C2×D73—42no—
236C4×C73—22no—
237C4×C73—22no—
238C4×C73—42no—
239C4×C73—42no—
240C4×C73#51811yescharacteristic
241C4×C73#51911yescharacteristic
242C4×C73#52011yescharacteristic
243C4×C73#52011yescharacteristic
244Dic73—22no—
245Dic73—22no—
246Dic73—22no—
247Dic73—42no—
248Dic73—42no—
249#20554—731nomaximal, Sylow
250C2×C73#227411yescharacteristic
251C2×C73#287511yescharacteristic
252C2×C73#287511yescharacteristic
253C2×C73#299512yes—
254C2×C73#299512yes—
255D73—42no—
256C2×#433—731no—
257C2×#424—731no—
258#2298—731no—
259#1771—731no—
260#1771—731no—
261#1770—731no—
262#1770—731no—
263#1768—731no—
264#1768—731no—
265#1767—731no—
266#1767D7311yescharacteristic, complemented
267#1753—731no—
268#1753—731no—
269#1752—731no—
270#1752—731no—
271C73#2055411yescharacteristic, complemented, Sylow
272C22×M16—731no—
273C23×C8—731no—
274C2×#117—731no—
275#433—738no—
276#424—738no—
277C2×C4⋊C8—731no—
278C2×C4⋊C8—731no—
279C2×C4⋊C8—731no—
280C2×C4⋊C8—731no—
281C2×C4⋊C8C2×D7311yescharacteristic
282C2×C4⋊C8C2×D7311yescharacteristic
283C2×C22⋊C8—731no—
284C2×C22⋊C8—731no—
285C2×#97—731no—
286C2×#97—731no—
287C2×#97—731no—
288C2×#97—731no—
289C2×#97—731no—
290C2×#97C2×D7311yescharacteristic
291C2×#97C2×D7311yescharacteristic
292C2×C4×C8—731no—
293C2×C4×C8—731no—
294C2×C4×C8—731no—
295#388—732no—
296#388—732no—
297#387—732no—
298#387—732no—
299C4×C4⋊C4—731no—
300C4×C4⋊C4—731no—
301C4×C22⋊C4—731no—
302C4×C22⋊C4—731no—
303#376—731no—
304C43C2×D7311yescharacteristic
305#336—731no—
306#336—731no—
307#336—731no—
308#336—731no—
309#336—731no—
310#336—731no—
311#336C2×D7311yescharacteristic
312#336C2×D7311yescharacteristic
313C23×C4—731no—
314C2×M16—2921no—
315C2×M16—2921no—
316C2×M16—734no—
317C22×C8—2921no—
318C22×C8—2921no—
319C22×C8—734no—
320C22×C8—731no—
321C22×C8—731no—
322C22×C8—731no—
323C22×C8—731no—
324C22×C8C22×D7311yescharacteristic
325C22×C8C22×D7311yescharacteristic
326C22×C8C22×D7311yescharacteristic
327C22×C8C22×D7311yescharacteristic
328#117—734no—
329#117—734no—
330C2×C4⋊C4—732no—
331C2×C4⋊C4—732no—
332C2×C4⋊C4—731no—
333C2×C4⋊C4—731no—
334C2×C22⋊C4—732no—
335C2×C22⋊C4—732no—
336C2×C22⋊C4—731no—
337C2×C22⋊C4—731no—
338C2×C42—731no—
339C2×C42—731no—
340C2×C42—731no—
341C2×C42C4×D7312yes—
342C2×C42C4×D7312yes—
343C2×C42C22×D7311yescharacteristic
344C2×C42C22×D7311yescharacteristic
345C2×C42C22×D7311yescharacteristic
346C4⋊C8—734no—
347C4⋊C8—734no—
348C4⋊C8—734no—
349C4⋊C8—734no—
350C4⋊C8C4×D7314yes—
351C4⋊C8C4×D7314yes—
352C22⋊C8—734no—
353C22⋊C8—734no—
354#97—41no—
355#97—41no—
356#97—2921no—
357#97—2921no—
358#97—2921no—
359#97—2921no—
360#97—734no—
361C4×C8—2921no—
362C4×C8—2921no—
363C4×C8—734no—
364#95—732no—
365#95—732no—
366#95—732no—
367#95—732no—
368#95—732no—
369#95—732no—
370C24—731no—
371C22×C4—731no—
372C22×C4—731no—
373C22×C4—731no—
374C22×C4—732no—
375C22×C4—732no—
376C22×C4—732no—
377C22×C4—732no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4—1461no—
381C22×C4—1461no—
382C22×C4C2×C4×D7312yes—
383C22×C4C2×C4×D7312yes—
384C22×C4C2×C4×D7311yescharacteristic
385C22×C4C2×C4×D7311yescharacteristic
386C22×C4C23×D7311yescharacteristic
387M16—2924no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—2924no—
397C2×C8—732no—
398C2×C8—732no—
399C2×C8—732no—
400C2×C8—732no—
401C2×C8—1461no—
402C2×C8—1461no—
403C2×C8—1461no—
404C2×C8—1461no—
405C2×C8—1461no—
406C2×C8—1461no—
407C2×C8—1461no—
408C2×C8—1461no—
409C2×C8C2×C4×D7312yes—
410C2×C8C2×C4×D7312yes—
411C2×C8C2×C4×D7312yes—
412C2×C8C2×C4×D7312yes—
413C4⋊C4—2922no—
414C4⋊C4—2922no—
415C4⋊C4—734no—
416C4⋊C4—734no—
417C22⋊C4—2922no—
418C22⋊C4—2922no—
419C22⋊C4—734no—
420C22⋊C4—734no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42—42no—
427C42—732no—
428C42—732no—
429C42—1461no—
430C42—1461no—
431C42—1461no—
432C42—1461no—
433C42C2×C4×D7312yes—
434C42C2×C4×D7312yes—
435C23—732no—
436C23—1461no—
437C23—1461no—
438C23—1462no—
439C23—1462no—
440C23C22×C4×D7311yescharacteristic
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—732no—
450C2×C4—732no—
451C2×C4—732no—
452C2×C4—1462no—
453C2×C4—1462no—
454C2×C4—1462no—
455C2×C4—1462no—
456C2×C4—1462no—
457C2×C4—1462no—
458C2×C4—1461no—
459C2×C4—1461no—
460C2×C4—1462no—
461C2×C4—1462no—
462C2×C4—1462no—
463C2×C4—1462no—
464C2×C4—1462no—
465C2×C4—1462no—
466C2×C4C42×D7312yes—
467C2×C4C42×D7312yes—
468C2×C4C22×C4×D7311yescharacteristic
469C2×C4C22×C4×D7311yescharacteristic
470C2×C4#1455663611yescharacteristic
471C2×C4#1455663711yescharacteristic
472C2×C4#1455663811yescharacteristic
473C2×C4#1455663911yescharacteristic
474C8—22no—
475C8—22no—
476C8—22no—
477C8—22no—
478C8—1462no—
479C8—1462no—
480C8—1462no—
481C8—1462no—
482C22—2922no—
483C22—2922no—
484C22—1462no—
485C22—1462no—
486C22—1462no—
487C22C2×C42×D7311yescharacteristic
488C22#1798906711yescharacteristic
489C22#1798906811yescharacteristic
490C22#1798935312yes—
491C22#1798935312yes—
492C4—22no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—2922no—
497C4—2922no—
498C4—1462no—
499C4—1462no—
500C4—1462no—
501C4#1798905911yescharacteristic
502C4#1798906311yescharacteristic
503C4#1798906611yescharacteristic
504C4#1798906911yescharacteristic
505C2—2922no—
506C2#2261228511yescharacteristic
507C2#2261835611yescharacteristic
508C2#2261836011yescharacteristic
509C2#2261909812yes—
510C2#2261910312yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC73×#1767C2
socleC23×C73C23×C4
prime radicalC23×D73C22×C4
cosocleC22×C4×C73C24
hypercenter#1767D73
hypocenterC73#20554
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1767

Complements

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

NH ≅ G/Ntype
4. #22609807C2semidirect
5. #22609804C2semidirect
6. #22609799C2semidirect
7. #22609797C2semidirect
8. #22609790C2semidirect
9. #22609790C2semidirect
10. #22609784C2semidirect
15. C73×#1767C2semidirect
19. #17988647C4semidirect
20. #17988612C4semidirect
266. #1767D73semidirect
271. C73#20554semidirect

Maximal subgroups

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

subgroupindex
C2×#179886472
C2×#179886122
#226124282
#226098072
#226098042
#226097992
#226097972
#226097902
#226097842
#226097162
#226097132
#226097082
#226097062
C73×#17672
#2055473