Finite Group Databasebeta

← #33244126

Subgroups — #33244126

Order 18688.

466 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989016C211yesmaximal, characteristic, complemented
2#508×D73C211yesmaximal, characteristic, complemented
3#22612251C211yesmaximal, characteristic, complemented
4#22612248C211yesmaximal, characteristic, complemented
5#22612247C212yesmaximal, complemented
6#22612244C211yesmaximal, characteristic, complemented
7#22612242C212yesmaximal, complemented
8#22612239C212yesmaximal, complemented
9#22612237C212yesmaximal, complemented
10M64×D73C211yesmaximal, characteristic, complemented
11C73×#2263C211yesmaximal, characteristic, complemented
12C2×#14556610—21no—
13#135×D73—21no—
14#17989020C2211yescharacteristic, complemented
15#17989019C2211yescharacteristic
16#17989018—21no—
17#17989017—21no—
18#17989016—21no—
19#17989016—21no—
20#17989016C2212yes—
21#17989016C2211yescharacteristic, complemented
22#17989016C2212yescomplemented
23C2×Dic584C2211yescharacteristic, complemented
24Q32×D73C2211yescharacteristic, complemented
25C2×#14556535—21no—
26SD32×D73—21no—
27D16×D73C2211yescharacteristic, complemented
28C2×C16×D73C2211yescharacteristic
29Dic1168C2212yescomplemented
30#17988427C2212yescomplemented
31#17988426C2212yescomplemented
32#17988425C2212yescomplemented
33#17988425C2212yescomplemented
34#17988422C2212yescomplemented
35#17988421C2211yescharacteristic, complemented
36#17988420C2212yescomplemented
37C32×D73C2212yescomplemented
38#508×C73C2211yescharacteristic
39C2×D16×C73C2211yescharacteristic, complemented
40SD64×C73C2212yescomplemented
41D32×C73C2212yescomplemented
42M64×C73C2211yescharacteristic, complemented
43C2×#11924027—41no—
44#41×D73—41no—
45#14556614—21no—
46#14556613—21no—
47#14556612—41no—
48#14556611—41no—
49#14556610—22no—
50#14556610—22no—
51#14556610—21no—
52#14556610—41no—
53#14556610—41no—
54C2×Dic292—21no—
55Q16×D73—21no—
56C2×#11924016—41no—
57SD16×D73—41no—
58D8×D73—21no—
59C2×C8×D73D411yescharacteristic
60Dic584—21no—
61Dic584C2311yescharacteristic
62Dic584C2312yes—
63#14556537—21no—
64#14556536—21no—
65#14556535—21no—
66#14556535—22no—
67#14556535—22no—
68#14556534—21no—
69D1168C2311yescharacteristic
70#14556532—21no—
71C2×#11924003D411yescharacteristic
72C16×D73D411yescharacteristic
73C16×D73D411yescharacteristic
74C16×D73C2311yescharacteristic
75C16×D73C2311yescharacteristic
76#14556497C2312yes—
77#135×C73—21no—
78C2×D8×C73—21no—
79Q32×C73C2311yescharacteristic
80SD32×C73—21no—
81D16×C73—21no—
82D16×C73C2312yes—
83D16×C73C2311yescharacteristic
84C2×C16×C73C2311yescharacteristic
85C32×C73C2312yes—
86#11924029—41no—
87#11924028—41no—
88#11924028—81no—
89#11924027—42no—
90#11924027—42no—
91#11924027—41no—
92#11924027—81no—
93#11924027—81no—
94#11924027—81no—
95C2×Dic146—41no—
96Q8×D73—41no—
97C2×C22⋊D73—81no—
98D4×D73—41no—
99D4×D73—81no—
100C22×Dic73—81no—
101C2×C4×D73—81no—
102C2×C4×D73D811yescharacteristic
103Dic292—22no—
104Dic292—21no—
105Dic292—41no—
106#11924018—41no—
107#11924017—41no—
108#11924016—41no—
109#11924016—42no—
110#11924016—42no—
111#11924015—41no—
112D584—21no—
113#11924013—41no—
114C2×#10596831D811yescharacteristic
115C8×D73D811yescharacteristic
116C8×D73D811yescharacteristic
117C8×D73C2×D411yescharacteristic
118C8×D73C2×D411yescharacteristic
119#11924003C2×D411yescharacteristic
120#11924003C2×D411yescharacteristic
121#41×C73—41no—
122C2×D4×C73—41no—
123Q16×C73—21no—
124SD16×C73—41no—
125D8×C73—22no—
126D8×C73—21no—
127D8×C73—41no—
128C2×C8×C73C2×D411yescharacteristic
129C16×C73C2×D411yescharacteristic
130C16×C73C2411yescharacteristic
131C23×C73—81no—
132C22×D73—81no—
133C22×D73D1611yescharacteristic
134Q8×C73—41no—
135D4×C73—42no—
136D4×C73—41no—
137D4×C73—81no—
138D4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73C2×D811yescharacteristic
141C22⋊D73—81no—
142C22⋊D73—81no—
143C22⋊D73—82no—
144C22⋊D73—82no—
145C22⋊D73—81no—
146C2×Dic73—81no—
147C2×Dic73—81no—
148C2×Dic73—82no—
149C2×Dic73—82no—
150C2×Dic73D1611yescharacteristic
151D292—41no—
152D292—81no—
153C4×D73—81no—
154C4×D73—81no—
155C4×D73—81no—
156C4×D73D1611yescharacteristic
157C4×D73D1611yescharacteristic
158C4×D73C2×D811yescharacteristic
159C4×D73C2×D811yescharacteristic
160Dic146—42no—
161Dic146—41no—
162Dic146—81no—
163Dic146—81no—
164C8×C73C2×D811yescharacteristic
165C8×C73C22×D411yescharacteristic
166#10596831C2×D811yescharacteristic
167#10596831C2×D811yescharacteristic
168C22×C73—82no—
169C22×C73—81no—
170C22×C73—162no—
171C22×C73C2×D1611yescharacteristic
172C2×D73—41no—
173C2×D73—81no—
174C2×D73—162no—
175C2×D73C2×D1611yescharacteristic
176C2×D73C2×D1611yescharacteristic
177C4×C73—81no—
178C4×C73C2×D1611yescharacteristic
179C4×C73C22×D811yescharacteristic
180Dic73—81no—
181Dic73—82no—
182Dic73C2×D1611yescharacteristic
183Dic73C2×D1611yescharacteristic
184#33981—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—161no—
187C2×C73—162no—
188C2×C73C22×D1611yescharacteristic
189D73—21no—
190D73—21no—
191D73—161no—
192C2×#508—731no—
193C2×#508—731no—
194#2264—731no—
195#2264—732no—
196#2264—731no—
197#2263—731no—
198#2263—732no—
199#2263D7311yescharacteristic, complemented
200#2262—732no—
201#2262—732no—
202C2×M64—731no—
203C73#3398111yescharacteristic, complemented, Sylow
204C2×#135—1461no—
205C2×#135—1461no—
206#508—731no—
207#508—732no—
208#508—732no—
209#508—731no—
210#508—731no—
211#508—1461no—
212#508—1461no—
213#508—1461no—
214#508—1461no—
215#508C2×D7311yescharacteristic
216C2×Q32—731no—
217C2×Q32—731no—
218C2×SD32—1461no—
219C2×SD32—1461no—
220C2×D16—731no—
221C2×D16C2×D7311yescharacteristic, complemented
222C22×C16—731no—
223Q64—732no—
224Q64—732no—
225SD64—732no—
226SD64—732no—
227SD64—732no—
228SD64C2×D7312yescomplemented
229D32—732no—
230D32C2×D7312yescomplemented
231M64—731no—
232M64—732no—
233M64C2×D7311yescharacteristic, complemented
234C2×C32—732no—
235C2×#41—2921no—
236C2×#41—2921no—
237#135—21no—
238#135—2921no—
239#135—2921no—
240#135—2921no—
241#135—2921no—
242#135—1462no—
243#135—1462no—
244#135—1461no—
245#135—1461no—
246#135—1461no—
247C2×Q16—1461no—
248C2×Q16—1461no—
249C2×SD16—2921no—
250C2×SD16—2921no—
251C2×D8—21no—
252C2×D8—1461no—
253C22×C8—731no—
254Q32—731no—
255Q32—732no—
256Q32—1461no—
257Q32—1461no—
258Q32C22×D7311yescharacteristic
259SD32—21no—
260SD32—1462no—
261SD32—1462no—
262SD32—1461no—
263SD32—1461no—
264SD32—1461no—
265D16—21no—
266D16—731no—
267D16—1461no—
268D16C22×D7312yes—
269D16C22×D7311yescharacteristic
270C2×C16—731no—
271C2×C16—731no—
272C2×C16—731no—
273C2×C16—731no—
274C2×C16—731no—
275C2×C16C22×D7311yescharacteristic
276C32—732no—
277C32C22×D7312yes—
278#41—41no—
279#41—2922no—
280#41—2922no—
281#41—2921no—
282#41—2921no—
283#41—2921no—
284#41—5841no—
285#41—5841no—
286#41—5841no—
287#41—5841no—
288C2×Q8—2921no—
289C2×Q8—2921no—
290C2×D4—41no—
291C2×D4—2921no—
292C2×D4—5841no—
293C2×D4—5841no—
294C22×C4—5841no—
295C22×C4—5841no—
296C22×C4—731no—
297Q16—21no—
298Q16—2921no—
299Q16—2921no—
300Q16—1462no—
301Q16—1461no—
302SD16—41no—
303SD16—2922no—
304SD16—2922no—
305SD16—2921no—
306SD16—2921no—
307SD16—2921no—
308D8—22no—
309D8—21no—
310D8—41no—
311D8—2921no—
312D8—1461no—
313C2×C8—731no—
314C2×C8—731no—
315C2×C8—731no—
316C2×C8—731no—
317C2×C8—731no—
318C2×C8D4×D7311yescharacteristic
319C16—731no—
320C16—731no—
321C16D4×D7311yescharacteristic
322C16C23×D7311yescharacteristic
323C23—81no—
324C23—5841no—
325C23—731no—
326Q8—41no—
327Q8—2922no—
328Q8—2921no—
329Q8—5841no—
330Q8—5841no—
331D4—42no—
332D4—41no—
333D4—81no—
334D4—81no—
335D4—2921no—
336D4—5841no—
337D4—5841no—
338D4—5842no—
339D4—5842no—
340D4—5841no—
341D4—5841no—
342C2×C4—81no—
343C2×C4—5841no—
344C2×C4—5841no—
345C2×C4—5841no—
346C2×C4—5841no—
347C2×C4—5842no—
348C2×C4—5842no—
349C2×C4—5841no—
350C2×C4—731no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—731no—
354C2×C4—731no—
355C2×C4D8×D7311yescharacteristic
356C8—731no—
357C8—731no—
358C8D8×D7311yescharacteristic
359C8C2×D4×D7311yescharacteristic
360C22—81no—
361C22—82no—
362C22—162no—
363C22—2921no—
364C22—5841no—
365C22—731no—
366C22—731no—
367C22—11682no—
368C22D16×D7311yescharacteristic
369C4—81no—
370C4—5841no—
371C4—5842no—
372C4—731no—
373C4—731no—
374C4D16×D7311yescharacteristic
375C4C2×D8×D7311yescharacteristic
376C2—21no—
377C2—162no—
378C2—161no—
379C2—11681no—
380C2—1461no—
381C2—1461no—
382C2C2×D16×D7311yescharacteristic
subgroupG/H
centerC2C2×D16×D73
commutator, cosocleC16×C73C24
FrattiniC16C23×D73
FittingC73×#2263C2
socleC2×C73C22×D16
hypercenter#2263D73
hypocenterC73#33981
LCSG ▷ C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. C2×#17989016C2semidirect
2. #508×D73C2semidirect
3. #22612251C2semidirect
4. #22612248C2semidirect
5. #22612247C2semidirect
6. #22612244C2semidirect
7. #22612242C2semidirect
8. #22612239C2semidirect
9. #22612237C2semidirect
10. M64×D73C2semidirect
11. C73×#2263C2semidirect
14. #17989020C22semidirect
21. #17989016C22semidirect
22. #17989016C22semidirect
23. C2×Dic584C22semidirect
24. Q32×D73C22semidirect
27. D16×D73C22semidirect
29. Dic1168C22semidirect
30. #17988427C22semidirect
31. #17988426C22semidirect
32. #17988425C22semidirect
33. #17988425C22semidirect
34. #17988422C22semidirect
35. #17988421C22semidirect
36. #17988420C22semidirect
37. C32×D73C22semidirect
39. C2×D16×C73C22semidirect
40. SD64×C73C22semidirect
41. D32×C73C22semidirect
42. M64×C73C22semidirect
199. #2263D73semidirect
203. C73#33981semidirect
221. C2×D16C2×D73semidirect
228. SD64C2×D73semidirect
230. D32C2×D73semidirect
233. M64C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890162
#508×D732
#226122512
#226122482
#226122472
#226122442
#226122422
#226122392
#226122372
M64×D732
C73×#22632
#3398173