Finite Group Databasebeta

← #33240638

Subgroups — #33240638

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620443C211yesmaximal, characteristic, complemented
2C2×#17988669C211yesmaximal, characteristic
3#22612091C212yesmaximal, complemented
4#22612087C212yesmaximal, complemented
5#22612077C212yesmaximal, complemented
6#22612067C212yesmaximal, complemented
7#22611717C211yesmaximal, characteristic, complemented
8#22611686C211yesmaximal, characteristic, complemented
9#22611648C211yesmaximal, characteristic
10#22611646C211yesmaximal, characteristic
11C73×#2159C211yesmaximal, characteristic, complemented
12C22×D584C2211yescharacteristic
13#17989140—21no—
14C2×#14556553—21no—
15C2×#14556531C2211yescharacteristic
16#17988991C2211yescharacteristic, complemented
17#17988991C2211yescharacteristic, complemented
18#17988988C2211yescharacteristic, complemented
19#17988988C2211yescharacteristic, complemented
20#17988987—21no—
21#17988984—21no—
22#17988669—21no—
23#17988669—21no—
24#17988669C2212yes—
25#17988669C2212yes—
26#17988625C2211yescharacteristic
27#17988623—21no—
28C2×#14556511—21no—
29C2×C8×Dic73C2211yescharacteristic
30#17988414C2212yescomplemented
31#17988412C2212yescomplemented
32#17988405C2212yescomplemented
33#17988405C2212yescomplemented
34#17988389C2212yes—
35#17988389C2212yes—
36#17988387C2212yes—
37#17988387C2212yes—
38C2×M32×C73C2211yescharacteristic
39#427×C73C2211yescharacteristic
40C2×#107×C73C2211yescharacteristic
41#367×C73C2212yescomplemented
42#366×C73C2212yescomplemented
43C22×D292—21no—
44C22×C4×D73—41no—
45C2×D584—21no—
46C2×D584—22no—
47C2×D584Q814yescomplemented
48C2×D584C2312yes—
49C22×#10596831D411yescharacteristic
50#14556590—21no—
51#14556590—21no—
52#14556582—21no—
53#14556582—21no—
54#14556568—41no—
55#14556568—41no—
56C4×Dic146—41no—
57C4×Dic146—41no—
58#14556553—22no—
59#14556553—22no—
60#14556553—41no—
61#14556553—41no—
62#14556550—21no—
63C2×#11924005—41no—
64C2×C4×Dic73D411yescharacteristic
65#14556531—22no—
66C2×#11924003C2312yes—
67#14556525C2311yescharacteristic
68#14556525C2311yescharacteristic
69#14556524—21no—
70#14556524—21no—
71#14556523—21no—
72#14556522—21no—
73#14556522—21no—
74#14556511—22no—
75#14556511—22no—
76C8×Dic73D411yescharacteristic
77C8×Dic73D411yescharacteristic
78C8×Dic73C2311yescharacteristic
79C8×Dic73C2311yescharacteristic
80C22×C8×C73C2311yescharacteristic
81#117×C73—21no—
82C2×C4⋊C4×C73—21no—
83M32×C73—22no—
84C2×C16×C73C2312yes—
85#107×C73—21no—
86#107×C73C2311yescharacteristic
87#107×C73C2311yescharacteristic
88#107×C73C2312yes—
89#106×C73—21no—
90C23×D73—41no—
91C2×Dic146—41no—
92C2×Dic146—41no—
93C2×D292—24no—
94C2×D292—22no—
95C2×D292—41no—
96C2×D292—42no—
97C22×Dic73D811yescharacteristic
98C22×Dic73D811yescharacteristic
99C2×C4×D73—42no—
100C2×C4×D73—42no—
101C2×C4×D73—84no—
102D584—24no—
103D584—22no—
104D584C2×Q814yes—
105C2×#10596831—22no—
106C2×#10596831C2×D411yescharacteristic
107C2×#10596831C2×D411yescharacteristic
108C4⋊Dic73—21no—
109C4⋊Dic73—21no—
110C4⋊Dic73—41no—
111C4⋊Dic73—41no—
112#11924007—41no—
113#11924007—41no—
114#11924007—41no—
115#11924007—41no—
116C22⋊Dic73—41no—
117#11924005—42no—
118#11924005—42no—
119C4×Dic73—41no—
120C4×Dic73D811yescharacteristic
121C4×Dic73D811yescharacteristic
122C4×Dic73C2×D411yescharacteristic
123C4×Dic73C2×D411yescharacteristic
124#11924003—22no—
125C22×C4×C73—41no—
126C22×C4×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C2×C8×C73C2×Q812yes—
129C2×C8×C73#4112yes—
130C2×C8×C73C2411yescharacteristic
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—41no—
135C22⋊C4×C73—41no—
136C42×C73—41no—
137C16×C73—22no—
138C23×C73C2×D811yescharacteristic
139C22×D73—42no—
140C22×D73—44no—
141C22×D73—82no—
142C22×D73—82no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—82no—
147C2×C4×C73C22⋊Q812yes—
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73#13712yes—
150C2×C4×C73C22×D411yescharacteristic
151C2×Dic73—22no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73C2×D811yescharacteristic
156C2×Dic73C2×D811yescharacteristic
157C2×Dic73C2×D811yescharacteristic
158C2×Dic73C2×D811yescharacteristic
159D292—24no—
160D292—42no—
161D292—44no—
162C4×D73—84no—
163C4×D73—84no—
164Dic146—82no—
165Dic146—82no—
166C8×C73C22⋊Q812yes—
167C8×C73C22×Q811yescharacteristic
168C8×C73C2×#4111yescharacteristic
169#10596831—21no—
170#10596831—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73#47412yes—
174C22×C73C22×D811yescharacteristic
175C2×D73—44no—
176C2×D73—88no—
177C2×D73—82no—
178C2×D73—82no—
179C4×C73—81no—
180C4×C73—82no—
181C4×C73—81no—
182C4×C73#47412yes—
183C4×C73C2×C22⋊Q811yescharacteristic
184C4×C73C2×#13711yescharacteristic
185Dic73—21no—
186Dic73—21no—
187Dic73—21no—
188Dic73—81no—
189Dic73—81no—
190Dic73#50912yes—
191#33627—731nomaximal, Sylow
192C2×C73—22no—
193C2×C73C2×#47411yescharacteristic
194C2×C73C2×#50911yescharacteristic
195C2×C73#341411yescharacteristic
196D73—84no—
197#3160—731no—
198C2×C4×D8—731no—
199#2228—732no—
200#2228—732no—
201#2226—732no—
202#2224—732no—
203#2159—731no—
204#2159D7311yescharacteristic, complemented
205#2150—731no—
206#2140—731no—
207#2140—731no—
208C73#3362711yescharacteristic, complemented, Sylow
209C22×D8—731no—
210#533—1461no—
211C2×C4×D4—1461no—
212C2×M32—731no—
213C2×M32C2×D7311yescharacteristic
214#500—731no—
215#500—731no—
216#499—731no—
217#499—731no—
218#499—1461no—
219#498—1461no—
220C4×D8—732no—
221C4×D8—732no—
222C4×D8—1461no—
223C4×D8—1461no—
224#427—731no—
225#427—1461no—
226#427C2×D7311yescharacteristic
227C2×#107C2×D7311yescharacteristic
228C2×#102—1461no—
229C2×C4×C8—731no—
230#367—732no—
231#367C2×D7312yescomplemented
232#366—732no—
233#366C2×D7312yescomplemented
234C4⋊C16—732no—
235C4⋊C16—732no—
236#357—732no—
237#357—732no—
238#357—732no—
239#357—732no—
240C22×D4—1461no—
241C23×C4—2921no—
242C2×D8—734no—
243C2×D8—732no—
244C2×D8—1461no—
245C2×D8—1462no—
246C22×C8—731no—
247C22×C8C22×D7311yescharacteristic
248C4⋊Q8—1461no—
249C4⋊Q8—1461no—
250#125—1461no—
251#125—1461no—
252C22⋊Q8—2921no—
253C22⋊Q8—2921no—
254C4×Q8—2921no—
255C4×Q8—2921no—
256C4×D4—2921no—
257C4×D4—2921no—
258C4×D4—1462no—
259C4×D4—1462no—
260#117—21no—
261#117—1461no—
262C2×C4⋊C4—21no—
263C2×C22⋊C4—2921no—
264C2×C42—731no—
265M32—22no—
266M32—1462no—
267C2×C16—732no—
268C2×C16C22×D7312yes—
269#107—21no—
270#107—731no—
271#107—731no—
272#107—1461no—
273#107—1461no—
274#107C22×D7311yescharacteristic
275#107C22×D7311yescharacteristic
276#107C22×D7312yes—
277#106—21no—
278#106—1461no—
279#106—1461no—
280#106—1461no—
281#102—1462no—
282#102—1462no—
283C4×C8—731no—
284C4×C8—731no—
285C4×C8—731no—
286C4×C8—731no—
287C24—2921no—
288C2×Q8—2921no—
289C2×Q8—2921no—
290C2×D4—2921no—
291C2×D4—2922no—
292C2×D4—1464no—
293C2×D4—1462no—
294C22×C4—41no—
295C22×C4—2922no—
296C22×C4—2922no—
297C22×C4—5844no—
298C22×C4—731no—
299C22×C4—731no—
300C22×C4D4×D7311yescharacteristic
301D8—734no—
302D8—1464no—
303D8—1462no—
304C2×C8—731no—
305C2×C8—731no—
306C2×C8—1462no—
307C2×C8D4×D7311yescharacteristic
308C2×C8Q8×D7312yes—
309C2×C8#1192402812yes—
310C2×C8C23×D7311yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—2921no—
318C4⋊C4—2921no—
319C4⋊C4—2921no—
320C4⋊C4—2921no—
321C4⋊C4—1461no—
322C4⋊C4—1461no—
323C22⋊C4—41no—
324C22⋊C4—2922no—
325C22⋊C4—2922no—
326C22⋊C4—2921no—
327C42—41no—
328C42—2921no—
329C42—731no—
330C42—731no—
331C42—731no—
332C42—731no—
333C16—22no—
334C16—1462no—
335C23—2924no—
336C23—2922no—
337C23—5842no—
338C23—5842no—
339C23D8×D7311yescharacteristic
340Q8—5842no—
341Q8—5842no—
342D4—2924no—
343D4—2922no—
344D4—1464no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—2921no—
352C2×C4—5844no—
353C2×C4—5844no—
354C2×C4—731no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—731no—
358C2×C4—1462no—
359C2×C4#1455656912yes—
360C2×C4D8×D7311yescharacteristic
361C2×C4#1455662312yes—
362C2×C4C2×D4×D7311yescharacteristic
363C8—1461no—
364C8—1461no—
365C8#1455656912yes—
366C8C2×Q8×D7311yescharacteristic
367C8C2×#1192402811yescharacteristic
368C22—21no—
369C22—21no—
370C22—2924no—
371C22—5848no—
372C22—5842no—
373C22—5842no—
374C22#1798886612yes—
375C22C2×D8×D7311yescharacteristic
376C4—82no—
377C4—81no—
378C4—81no—
379C4—5841no—
380C4—5841no—
381C4—732no—
382C4—1461no—
383C4—1461no—
384C4—1461no—
385C4#1798886612yes—
386C4C2×#1455656911yescharacteristic
387C4C2×#1455662311yescharacteristic
388C2—22no—
389C2—5844no—
390C2C2×#1798886611yescharacteristic
391C2#509×D7311yescharacteristic
392C2#2262306611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC22×D584C22
hypercenter#2159D73
hypocenterC73#33627
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. #22620443C2semidirect
3. #22612091C2semidirect
4. #22612087C2semidirect
5. #22612077C2semidirect
6. #22612067C2semidirect
7. #22611717C2semidirect
8. #22611686C2semidirect
11. C73×#2159C2semidirect
16. #17988991C22semidirect
17. #17988991C22semidirect
18. #17988988C22semidirect
19. #17988988C22semidirect
30. #17988414C22semidirect
31. #17988412C22semidirect
32. #17988405C22semidirect
33. #17988405C22semidirect
41. #367×C73C22semidirect
42. #366×C73C22semidirect
47. C2×D584Q8semidirect
204. #2159D73semidirect
208. C73#33627semidirect
231. #367C2×D73semidirect
233. #366C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204432
C2×#179886692
#226120912
#226120872
#226120772
#226120672
#226117172
#226116862
#226116482
#226116462
C73×#21592
#3362773