Finite Group Databasebeta

← #33108067

Subgroups — #33108067

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619713C211yesmaximal, characteristic
2#22618880C211yesmaximal, characteristic, complemented
3#22618500C211yesmaximal, characteristic
4#22610704C211yesmaximal, characteristic, complemented
5#22610683C212yesmaximal, complemented
6#22610528C212yesmaximal, complemented
7#22610207C212yesmaximal
8#22609784C211yesmaximal, characteristic, complemented
9#22609772C212yesmaximal
10C4×#14556519C211yesmaximal, characteristic
11C73×#1994C211yesmaximal, characteristic, complemented
12C2×#14556599C2212yes—
13C4×#11924028C2211yescharacteristic
14#17988891C412yescomplemented
15#17988884C412yescomplemented
16#17988879C411yescharacteristic, complemented
17#17988874C411yescharacteristic, complemented
18#17988869C411yescharacteristic, complemented
19#17988864C411yescharacteristic, complemented
20#17988718—21no—
21#17988718—22no—
22Q8×#10596831—21no—
23C2×#14556526C2212yes—
24#17988623C2212yes—
25#17988615C2212yes—
26#17988614C2212yes—
27#17988612—22no—
28#17988609—21no—
29C2×C4⋊#10596831C2211yescharacteristic
30C2×C4⋊#10596831C2211yescharacteristic
31C4⋊C8×D73—21no—
32C2×#14556519C2211yescharacteristic
33C2×#14556519C2211yescharacteristic
34#17988581C2212yes—
35#17988543C2212yes—
36C4×#11924012C2211yescharacteristic
37C2×#14556502C2212yes—
38#17988351C2212yes—
39#17988347C2211yescharacteristic
40#17988347C2211yescharacteristic
41#17988342C2212yes—
42#533×C73C2211yescharacteristic
43#423×C73C2211yescharacteristic
44C43×C73C2211yescharacteristic
45#339×C73C2212yes—
46#339×C73C2212yes—
47C2×#11924028C2311yescharacteristic
48#14556599—24no—
49#14556599—42no—
50C2×#11924012C2311yescharacteristic
51C2×#11924012C2311yescharacteristic
52C2×#11924011—22no—
53C22×#10596831C2312yes—
54C22×#10596831C2312yes—
55C2×C8×D73—22no—
56C4×Dic146C2×C411yescharacteristic
57C4×Dic146C2×C411yescharacteristic
58C4×D292C2×C411yescharacteristic
59C4×D292C2×C411yescharacteristic
60C4×C22⋊D73—21no—
61C4×C22⋊D73—21no—
62#14556550C2311yescharacteristic
63#14556548—21no—
64C42×D73—21no—
65#14556526—42no—
66#14556524C2×C412yes—
67#14556524C2×C412yes—
68#14556522C2×C412yes—
69#14556522C2×C412yes—
70C4⋊#10596831—22no—
71C4⋊#10596831—22no—
72C4⋊#10596831C2×C411yescharacteristic
73C4⋊#10596831C2×C411yescharacteristic
74C4⋊#10596831C2×C412yes—
75C4⋊#10596831D411yescharacteristic
76C4⋊#10596831D412yes—
77C4⋊#10596831D411yescharacteristic
78#14556520—22no—
79#14556520—22no—
80#14556519—22no—
81#14556519Q814yescomplemented
82#14556514C2×C412yes—
83#14556514C2×C412yes—
84#14556511C2×C412yes—
85#14556511C2×C412yes—
86#14556502—22no—
87#14556502—21no—
88#14556502—42no—
89C4×#10596831—22no—
90C4×#10596831—21no—
91C2×M16×C73C2312yes—
92C4⋊Q8×C73C2×C411yescharacteristic
93C4⋊Q8×C73C2×C411yescharacteristic
94#125×C73C2×C412yes—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73—21no—
99#117×C73C2312yes—
100C2×C42×C73—21no—
101C2×C42×C73—21no—
102C2×C42×C73C2311yescharacteristic
103C2×C42×C73C2311yescharacteristic
104C2×C42×C73C2311yescharacteristic
105C4⋊C8×C73—21no—
106C4⋊C8×C73—21no—
107#11924028—21no—
108#11924028—21no—
109#11924028C2×Q812yes—
110#11924028#4112yes—
111C2×Dic146C22×C411yescharacteristic
112C2×D292C22×C411yescharacteristic
113C2×C22⋊D73—21no—
114C2×C4×D73—21no—
115C2×C4×D73—21no—
116C2×C4×D73—21no—
117#11924012—21no—
118#11924012—21no—
119#11924012—21no—
120#11924012C2×Q812yes—
121#11924011—44no—
122C2×#10596831—21no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—21no—
128C2×#10596831C22×C412yes—
129C2×#10596831C22×C412yes—
130C2×#10596831C2×D412yes—
131C2×#10596831C2×D412yes—
132C8×D73—44no—
133C4⋊Dic73C22×C411yescharacteristic
134C4⋊Dic73C22×C411yescharacteristic
135#11924007—21no—
136#11924007—21no—
137C22⋊Dic73—21no—
138#11924005—21no—
139#11924005—21no—
140C4×Dic73—21no—
141C4×Dic73—21no—
142C4×Dic73—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73#4111yescharacteristic
148C22×C4×C73#4111yescharacteristic
149C22×C4×C73C2411yescharacteristic
150M16×C73—22no—
151M16×C73—22no—
152C2×C8×C73—22no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73C22×C412yes—
158C4⋊C4×C73C22×C412yes—
159C22⋊C4×C73—22no—
160C42×C73—22no—
161C42×C73—21no—
162C42×C73—21no—
163C42×C73—21no—
164C42×C73—42no—
165C42×C73—42no—
166C42×C73C22×C411yescharacteristic
167C42×C73C22×C411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73C2×D411yescharacteristic
170C42×C73C2×Q812yes—
171C23×C73C2×#4111yescharacteristic
172C22×D73—21no—
173Q8×C73—42no—
174Q8×C73—81no—
175C2×C4×C73—22no—
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73C4×D411yescharacteristic
186C2×C4×C73C4×D411yescharacteristic
187C2×C4×C73C4×D411yescharacteristic
188C2×C4×C73C4×D411yescharacteristic
189C2×C4×C73C23×C411yescharacteristic
190C2×C4×C73C22×D411yescharacteristic
191C2×C4×C73C22×Q811yescharacteristic
192C2×C4×C73C2×#4111yescharacteristic
193C2×C4×C73H(5,𝔽2)11yescharacteristic
194C2×C4×C732−1+411yescharacteristic
195C22⋊D73—22no—
196C22⋊D73—22no—
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—21no—
200D292—21no—
201D292C4×Q812yes—
202C4×D73—22no—
203C4×D73—22no—
204C4×D73—42no—
205C4×D73—81no—
206Dic146—21no—
207Dic146C4×Q812yes—
208C8×C73—42no—
209#10596831—22no—
210#10596831—22no—
211#10596831—22no—
212#10596831—22no—
213#10596831—22no—
214#10596831—41no—
215C22×C73—21no—
216C22×C73—21no—
217C22×C73C2×C4×D411yescharacteristic
218C22×C73#55211yescharacteristic
219C22×C73#55411yescharacteristic
220C2×D73—22no—
221C2×D73—41no—
222C2×D73—41no—
223C4×C73—21no—
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—42no—
227C4×C73—42no—
228C4×C73—42no—
229C4×C73—42no—
230C4×C73#44412yes—
231C4×C73C2×C4×Q811yescharacteristic
232C4×C73#52011yescharacteristic
233C4×C73#54611yescharacteristic
234C4×C73D4×Q811yescharacteristic
235Dic73—22no—
236Dic73—41no—
237Dic73—41no—
238#22983—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73#236011yescharacteristic
242C2×C73C2×#44411yescharacteristic
243C2×C73#295611yescharacteristic
244D73—42no—
245#3076—731no—
246#2962—731no—
247#2906—731no—
248#1994D7311yescharacteristic, complemented
249#1950—731no—
250#1946—732no—
251#1922—732no—
252#1866—732no—
253#1767—731no—
254#1765—732no—
255C4×#104—731no—
256C73#2298311yescharacteristic, complemented, Sylow
257C2×#131—732no—
258#533C2×D7311yescharacteristic
259C4×#41—731no—
260#479—732no—
261#478—732no—
262#477—731no—
263#476—731no—
264#475—731no—
265#474—731no—
266#446—1462no—
267#446—1461no—
268C8×Q8—1461no—
269C2×#108—732no—
270#427—732no—
271#424—732no—
272#424—732no—
273#424—1462no—
274#423—1461no—
275#423C2×D7311yescharacteristic
276C2×C4⋊C8—731no—
277C2×C4⋊C8—731no—
278C2×C4⋊C8—1461no—
279C2×#104—731no—
280C2×#104—731no—
281#416—732no—
282#405—732no—
283C4×M16—731no—
284C2×#97—732no—
285C43C2×D7311yescharacteristic
286#341—732no—
287#339—731no—
288#339—731no—
289#339C2×D7312yes—
290#339C2×D7312yes—
291#336—732no—
292C2×#41—731no—
293#131—2922no—
294#131—1464no—
295C2×M16—731no—
296C2×M16—731no—
297C2×M16—1462no—
298C2×M16C22×D7312yes—
299C22×C8—732no—
300C22×C8—732no—
301C22×C8—1462no—
302C4⋊Q8C4×D7311yescharacteristic
303C4⋊Q8C4×D7311yescharacteristic
304#125C4×D7312yes—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—731no—
310C4×Q8—731no—
311C4×D4—731no—
312C4×D4—731no—
313C4×D4—1461no—
314C4×D4—1461no—
315#117—731no—
316#117—1461no—
317#117C22×D7312yes—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—1461no—
321C2×C42C22×D7311yescharacteristic
322C2×C42C22×D7311yescharacteristic
323C2×C42C22×D7311yescharacteristic
324#108—2922no—
325#107—732no—
326#107—732no—
327#106—732no—
328#106—732no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—731no—
332C4⋊C8—731no—
333C4⋊C8—732no—
334C4⋊C8—731no—
335C4⋊C8—731no—
336C4⋊C8—732no—
337C4⋊C8—1462no—
338C4⋊C8—1462no—
339C4⋊C8—1462no—
340C4⋊C8—1462no—
341#104—734no—
342#104—1462no—
343#103—732no—
344#103—732no—
345#102—732no—
346#102—732no—
347#97—2922no—
348#97—1462no—
349#97—1461no—
350C4×C8—1462no—
351C4×C8—1461no—
352#41—732no—
353#41—732no—
354#41—1461no—
355#41—1461no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—731no—
359C2×D4—731no—
360C2×D4—1461no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—1461no—
364C22×C4—1461no—
365C22×C4—1461no—
366C22×C4#1192402711yescharacteristic
367C22×C4#1192402711yescharacteristic
368C22×C4C23×D7311yescharacteristic
369M16—22no—
370M16—22no—
371M16—2924no—
372M16—732no—
373M16—1461no—
374M16—1461no—
375M16—1461no—
376C2×C8—22no—
377C2×C8—2924no—
378C2×C8—732no—
379C2×C8—732no—
380C2×C8—732no—
381C2×C8—732no—
382C2×C8—1461no—
383C2×C8—1461no—
384C2×C8—1462no—
385C2×C8—1462no—
386C2×C8—1462no—
387C2×C8—1462no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—731no—
393C4⋊C4—731no—
394C4⋊C4—1461no—
395C4⋊C4—1461no—
396C4⋊C4C2×C4×D7312yes—
397C4⋊C4C2×C4×D7312yes—
398C22⋊C4—22no—
399C22⋊C4—1461no—
400C22⋊C4—1461no—
401C22⋊C4—1461no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—42no—
407C42—42no—
408C42—1461no—
409C42—1461no—
410C42—1461no—
411C42C2×C4×D7311yescharacteristic
412C42C2×C4×D7311yescharacteristic
413C42D4×D7311yescharacteristic
414C42D4×D7311yescharacteristic
415C42Q8×D7312yes—
416C23—1461no—
417C23C2×#1192402711yescharacteristic
418Q8—42no—
419Q8—81no—
420Q8—732no—
421Q8—1461no—
422D4—732no—
423D4—1462no—
424D4—1462no—
425D4—1461no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—2922no—
437C2×C4—5841no—
438C2×C4—1461no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1462no—
442C2×C4—1462no—
443C2×C4#1455655111yescharacteristic
444C2×C4#1455655111yescharacteristic
445C2×C4#1455655111yescharacteristic
446C2×C4#1455655111yescharacteristic
447C2×C4C22×C4×D7311yescharacteristic
448C2×C4C2×D4×D7311yescharacteristic
449C2×C4C2×Q8×D7311yescharacteristic
450C2×C4#41×D7311yescharacteristic
451C2×C4#1455663711yescharacteristic
452C2×C4#1455663911yescharacteristic
453C8—42no—
454C8—2921no—
455C8—1462no—
456C8—1462no—
457C8—1462no—
458C8—1462no—
459C8—1462no—
460C22—21no—
461C22—21no—
462C22—2921no—
463C22—2921no—
464C22—1462no—
465C22C2×#1455655111yescharacteristic
466C22#1798927111yescharacteristic
467C22#1798929511yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—2921no—
476C4—2921no—
477C4—1462no—
478C4#1798870612yes—
479C4C4×Q8×D7311yescharacteristic
480C4#1798906911yescharacteristic
481C4#1798923011yescharacteristic
482C4#1798925211yescharacteristic
483C2—21no—
484C2—21no—
485C2—2922no—
486C2#2261294111yescharacteristic
487C2C2×#1798870611yescharacteristic
488C2#2261883711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1994C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1994D73
hypocenterC73#22983
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #22618880C2semidirect
4. #22610704C2semidirect
5. #22610683C2semidirect
6. #22610528C2semidirect
8. #22609784C2semidirect
11. C73×#1994C2semidirect
14. #17988891C4semidirect
15. #17988884C4semidirect
16. #17988879C4semidirect
17. #17988874C4semidirect
18. #17988869C4semidirect
19. #17988864C4semidirect
81. #14556519Q8semidirect
248. #1994D73semidirect
256. C73#22983semidirect

Maximal subgroups

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

subgroupindex
#226197132
#226188802
#226185002
#226107042
#226106832
#226105282
#226102072
#226097842
#226097722
C4×#145565192
C73×#19942
#2298373