Finite Group Databasebeta

← #33240724

Subgroups — #33240724

Order 18688.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620443C211yesmaximal, characteristic, complemented
2C2×#17988704C211yesmaximal, characteristic
3#22612105C214yesmaximal, complemented
4#22612100C214yesmaximal, complemented
5#22611704C211yesmaximal, characteristic, complemented
6#22611703C211yesmaximal, characteristic, complemented
7C2×#17988407C211yesmaximal, characteristic, complemented
8#22611667C211yesmaximal, characteristic
9C73×#2145C211yesmaximal, characteristic
10C2×#14556598—21no—
11#17989140—21no—
12C2×#14556531C2211yescharacteristic
13#17988991—21no—
14#17988988—21no—
15#17988987C2212yescomplemented
16#17988984C2212yescomplemented
17#17988704—21no—
18#17988704—21no—
19#17988704C2212yes—
20#17988704C2212yes—
21C2×#14556527C2211yescharacteristic
22#17988625C2211yescharacteristic
23#17988623—21no—
24C2×#14556518—21no—
25C2×C8×Dic73C2211yescharacteristic
26#17988410C2214yescomplemented
27#17988409C2214yescomplemented
28#17988407C2212yescomplemented
29#17988407C2212yescomplemented
30#17988397C2214yes—
31C2×#135×C73C2211yescharacteristic
32C2×M32×C73C2211yescharacteristic
33#427×C73C2211yescharacteristic
34#360×C73C2214yes—
35#14556598—22no—
36#14556598—22no—
37#14556598—41no—
38#14556598—41no—
39C2×#11924012—21no—
40C2×#11924012—41no—
41C22×#10596831—41no—
42C22×#10596831D411yescharacteristic
43#14556590—22no—
44#14556582—22no—
45#14556568—42no—
46C4×Dic146—42no—
47#14556550—21no—
48C2×C4×Dic73D411yescharacteristic
49#14556531—21no—
50#14556531C2312yes—
51C2×#11924003—21no—
52#14556527—21no—
53#14556527C2312yes—
54#14556525—21no—
55#14556524—22no—
56#14556523C2312yes—
57#14556522—22no—
58#14556518—22no—
59#14556518—22no—
60C8×Dic73D411yescharacteristic
61C8×Dic73D411yescharacteristic
62C8×Dic73C2311yescharacteristic
63C8×Dic73C2311yescharacteristic
64C2×#41×C73—21no—
65#135×C73—21no—
66#135×C73—21no—
67#135×C73Q812yescomplemented
68#135×C73C2312yes—
69C2×Q16×C73Q811yescharacteristic, complemented
70C2×SD16×C73—21no—
71C2×D8×C73Q811yescharacteristic, complemented
72C22×C8×C73C2311yescharacteristic
73#117×C73—21no—
74M32×C73—21no—
75M32×C73C2312yes—
76C2×C16×C73—21no—
77#107×C73—21no—
78#106×C73C2312yes—
79C2×Dic146—42no—
80C22×Dic73—21no—
81#11924012—22no—
82#11924012—41no—
83#11924012—42no—
84#11924012—42no—
85C2×#10596831—41no—
86C2×#10596831—42no—
87C2×#10596831—42no—
88C2×#10596831SD1611yescharacteristic
89C2×#10596831SD1611yescharacteristic
90C2×#10596831SD1611yescharacteristic
91C2×#10596831SD1611yescharacteristic
92C2×#10596831C2×D411yescharacteristic
93C2×#10596831C2×D411yescharacteristic
94C4⋊Dic73—22no—
95C4⋊Dic73—42no—
96#11924007—42no—
97#11924007—42no—
98C22⋊Dic73—41no—
99C4×Dic73—21no—
100C4×Dic73—41no—
101C4×Dic73C2×D411yescharacteristic
102C4×Dic73C2×D411yescharacteristic
103#11924003—22no—
104#41×C73—22no—
105#41×C73—22no—
106#41×C73—41no—
107#41×C73—41no—
108C2×Q8×C73—21no—
109C2×D4×C73—21no—
110C2×D4×C73—41no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113Q16×C73—21no—
114Q16×C73C2×Q812yes—
115SD16×C73—22no—
116SD16×C73—22no—
117D8×C73—21no—
118D8×C73C2×Q812yes—
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×Q811yescharacteristic
121C2×C8×C73C2×Q811yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C2×C8×C73#4111yescharacteristic
124C2×C8×C73C2411yescharacteristic
125C4⋊C4×C73—22no—
126C22⋊C4×C73—41no—
127C42×C73—41no—
128C16×C73—22no—
129C23×C73—41no—
130C23×C73AGL(1,ℤ/8)11yescharacteristic
131Q8×C73—22no—
132Q8×C73—41no—
133D4×C73—22no—
134D4×C73—42no—
135D4×C73—42no—
136D4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73—42no—
141C2×C4×C73C22⋊Q811yescharacteristic
142C2×C4×C73C22⋊Q811yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73C2×SD1611yescharacteristic
145C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C73C22×D411yescharacteristic
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—42no—
150C2×Dic73—42no—
151Dic146—84no—
152C8×C73C22⋊Q811yescharacteristic
153C8×C73C22⋊Q811yescharacteristic
154C8×C73C22×Q811yescharacteristic
155C8×C73C2×#4111yescharacteristic
156#10596831—42no—
157#10596831C2×SD1611yescharacteristic
158#10596831C2×SD1611yescharacteristic
159#10596831C2×SD1611yescharacteristic
160#10596831C2×SD1611yescharacteristic
161C22×C73—41no—
162C22×C73—42no—
163C22×C73—82no—
164C22×C73#47611yescharacteristic
165C22×C73#47611yescharacteristic
166C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
167C4×C73—42no—
168C4×C73—82no—
169C4×C73#47611yescharacteristic
170C4×C73#47611yescharacteristic
171C4×C73C2×C22⋊Q811yescharacteristic
172C4×C73C22×SD1611yescharacteristic
173Dic73—22no—
174Dic73—41no—
175Dic73—82no—
176#33637—731nomaximal, Sylow
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—21no—
180C2×C73—82no—
181C2×C73C2×#47611yescharacteristic
182#3160—731no—
183C2×#443—731no—
184#2231—734no—
185#2230—734no—
186#2155—731no—
187#2155—731no—
188C2×#364—731no—
189#2145—731no—
190#2145D7311yescharacteristic
191C73#3363711yescharacteristic, complemented, Sylow
192C2×#135C2×D7311yescharacteristic
193C2×#131—1461no—
194#533—1461no—
195C2×M32—731no—
196C2×M32C2×D7311yescharacteristic
197#500—1461no—
198#499—732no—
199#499—1461no—
200#498—732no—
201#443—732no—
202#443—732no—
203#443—1461no—
204#443—1461no—
205C2×#108—731no—
206#427—731no—
207#427—1461no—
208#427C2×D7311yescharacteristic
209C2×#104—1461no—
210C2×C4×C8—731no—
211#365—734no—
212#365—734no—
213#364—732no—
214#364—732no—
215#360—734no—
216#360C2×D7314yes—
217C2×#41—21no—
218#135—21no—
219#135—21no—
220#135Dic14612yescomplemented
221#135C22×D7312yes—
222C2×Q16Dic14611yescharacteristic, complemented
223C2×SD16—21no—
224C2×D8Dic14611yescharacteristic, complemented
225#131—2921no—
226#131—2921no—
227#131—1462no—
228#131—1462no—
229C2×M16—2921no—
230C2×M16—1461no—
231C22×C8—2921no—
232C22×C8—731no—
233C22×C8C22×D7311yescharacteristic
234C4⋊Q8—1462no—
235#125—1462no—
236C22⋊Q8—2922no—
237C4×Q8—2922no—
238#117—21no—
239#117—1461no—
240C2×C42—731no—
241M32—21no—
242M32—732no—
243M32—1461no—
244M32C22×D7312yes—
245C2×C16—21no—
246C2×C16—1461no—
247#108—732no—
248#108—1461no—
249#107—21no—
250#107—1462no—
251#107—1461no—
252#106—732no—
253#106—1462no—
254#106C22×D7312yes—
255#104—1462no—
256#104—1462no—
257C4×C8—731no—
258C4×C8—731no—
259C4×C8—731no—
260C4×C8—731no—
261#41—22no—
262#41—22no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—2922no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—1461no—
271C22×C4D4×D7311yescharacteristic
272Q16—21no—
273Q16C2×Dic14612yes—
274SD16—22no—
275SD16—22no—
276D8—21no—
277D8C2×Dic14612yes—
278M16—2921no—
279M16—2922no—
280M16—2922no—
281M16—1462no—
282C2×C8—2921no—
283C2×C8—2922no—
284C2×C8—2922no—
285C2×C8—731no—
286C2×C8—731no—
287C2×C8—731no—
288C2×C8—731no—
289C2×C8—731no—
290C2×C8—731no—
291C2×C8D4×D7311yescharacteristic
292C2×C8C2×Dic14611yescharacteristic
293C2×C8C2×Dic14611yescharacteristic
294C2×C8#1192402711yescharacteristic
295C2×C8#1192402711yescharacteristic
296C2×C8C23×D7311yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—2922no—
299C4⋊C4—2922no—
300C4⋊C4—2922no—
301C4⋊C4—1462no—
302C22⋊C4—41no—
303C22⋊C4—2921no—
304C42—41no—
305C42—2921no—
306C42—731no—
307C42—731no—
308C42—1461no—
309C16—22no—
310C16—1462no—
311C23—41no—
312C23#1455661511yescharacteristic
313Q8—22no—
314Q8—41no—
315Q8—5844no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—1461no—
327C2×C4—1461no—
328C2×C4#1455656811yescharacteristic
329C2×C4#1455656811yescharacteristic
330C2×C4SD16×D7311yescharacteristic
331C2×C4SD16×D7311yescharacteristic
332C2×C4#1455661511yescharacteristic
333C2×C4C2×D4×D7311yescharacteristic
334C8—2922no—
335C8—731no—
336C8—731no—
337C8—731no—
338C8—731no—
339C8#1455656811yescharacteristic
340C8#1455656811yescharacteristic
341C8C22×Dic14611yescharacteristic
342C8C2×#1192402711yescharacteristic
343C22—41no—
344C22—42no—
345C22—82no—
346C22#1798887511yescharacteristic
347C22#1798887511yescharacteristic
348C22C2×#1455661511yescharacteristic
349C4—42no—
350C4—82no—
351C4—2921no—
352C4—5842no—
353C4—1462no—
354C4#1798887511yescharacteristic
355C4#1798887511yescharacteristic
356C4C2×#1455656811yescharacteristic
357C4C2×SD16×D7311yescharacteristic
358C2—21no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2C2×#1798887511yescharacteristic
subgroupG/H
centerC4#17988875
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2145C2
socleC2×C73C2×#476
prime radicalC2×#135×C73C22
hypercenter#2145D73
hypocenterC73#33637
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
1. #22620443C2semidirect
3. #22612105C2semidirect
4. #22612100C2semidirect
5. #22611704C2semidirect
6. #22611703C2semidirect
7. C2×#17988407C2semidirect
15. #17988987C22semidirect
16. #17988984C22semidirect
26. #17988410C22semidirect
27. #17988409C22semidirect
28. #17988407C22semidirect
29. #17988407C22semidirect
67. #135×C73Q8semidirect
69. C2×Q16×C73Q8semidirect
71. C2×D8×C73Q8semidirect
191. C73#33637semidirect
220. #135Dic146semidirect
222. C2×Q16Dic146semidirect
224. C2×D8Dic146semidirect

Maximal subgroups

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

subgroupindex
#226204432
C2×#179887042
#226121052
#226121002
#226117042
#226117032
C2×#179884072
#226116672
C73×#21452
#3363773