Finite Group Databasebeta

← #33244137

Subgroups — #33244137

Order 18688.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623081C211yesmaximal, characteristic, complemented
2#22623069C211yesmaximal, characteristic, complemented
3#22612250C212yesmaximal, complemented
4#22612248C211yesmaximal, characteristic, complemented
5#22612243C211yesmaximal, characteristic, complemented
6#22612242C212yesmaximal, complemented
7#22612239C212yesmaximal, complemented
8C2×#17988426C211yesmaximal, characteristic, complemented
9SD64×D73C212yesmaximal, complemented
10#22612224C211yesmaximal, characteristic, complemented
11C73×#2262C211yesmaximal, characteristic, complemented
12#17989401—21no—
13#17989389—21no—
14#17989027—21no—
15#17989026—22no—
16#17989024—22no—
17#17989022—21no—
18#17989020C2212yes—
19#17989016C2212yescomplemented
20C2×Dic584C2211yescharacteristic, complemented
21Q32×D73C2212yescomplemented
22C2×D1168C2211yescharacteristic
23D16×D73C2212yes—
24#17989003C2211yescharacteristic
25#17988427C2212yescomplemented
26#17988426C2212yescomplemented
27#17988426C2212yescomplemented
28#17988425C2212yescomplemented
29#17988424C2212yescomplemented
30#17988422C2212yescomplemented
31#17988421C2211yescharacteristic, complemented
32#17988420C2212yescomplemented
33C32×D73C2212yescomplemented
34#508×C73C2211yescharacteristic
35#508×C73C2211yescharacteristic, complemented
36Q64×C73C2211yescharacteristic, complemented
37SD64×C73C2211yescharacteristic, complemented
38SD64×C73C2211yescharacteristic, complemented
39D32×C73C2211yescharacteristic, complemented
40C2×C32×C73C2211yescharacteristic, complemented
41#14556638—41no—
42#14556637—41no—
43#14556621—41no—
44#14556620—42no—
45#14556618—42no—
46#14556616—41no—
47#14556614—22no—
48#14556610—22no—
49C2×Dic292—21no—
50Q16×D73—22no—
51C2×D584—21no—
52D8×D73—22no—
53#14556597D411yescharacteristic
54Dic584—21no—
55Dic584C2312yes—
56#14556537—22no—
57#14556535—22no—
58#14556534—22no—
59D1168—21no—
60D1168C2312yes—
61#14556532—22no—
62#14556531D411yescharacteristic
63#14556530D412yes—
64C16×D73C2312yes—
65#14556497C2312yes—
66#135×C73—21no—
67#135×C73—21no—
68Q32×C73C2311yescharacteristic
69Q32×C73C2311yescharacteristic
70SD32×C73—21no—
71SD32×C73—21no—
72D16×C73C2311yescharacteristic
73D16×C73C2311yescharacteristic
74C2×C16×C73C2311yescharacteristic
75C32×C73C2311yescharacteristic
76C32×C73C2311yescharacteristic
77#11924029—42no—
78#11924028—81no—
79#11924028—81no—
80#11924028D811yescharacteristic
81#11924027—42no—
82#11924027—82no—
83C2×Dic146—41no—
84C2×Dic146—81no—
85Q8×D73—42no—
86C2×D292—41no—
87C2×D292—81no—
88D4×D73—42no—
89D4×D73—82no—
90Dic292—22no—
91Dic292—41no—
92#11924018—42no—
93#11924016—42no—
94#11924015—42no—
95D584—22no—
96D584—41no—
97#11924013—42no—
98#11924012D811yescharacteristic
99#11924011D812yes—
100C8×D73C2×D412yes—
101#11924003C2×D412yes—
102#41×C73—41no—
103#41×C73—41no—
104Q16×C73—21no—
105Q16×C73—21no—
106SD16×C73—41no—
107SD16×C73—41no—
108D8×C73—21no—
109D8×C73—21no—
110C2×C8×C73C2×D411yescharacteristic
111C16×C73C2×D411yescharacteristic
112C16×C73C2411yescharacteristic
113C22×D73—81no—
114C22×D73—82no—
115Q8×C73—41no—
116Q8×C73—41no—
117D4×C73—41no—
118D4×C73—41no—
119D4×C73—81no—
120D4×C73—81no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73C2×D811yescharacteristic
124C22⋊D73—82no—
125C22⋊D73—82no—
126C22⋊D73D1612yes—
127C2×Dic73—81no—
128C2×Dic73—82no—
129D292—42no—
130D292—82no—
131D292—81no—
132D292D1611yescharacteristic
133C4×D73—82no—
134C4×D73—82no—
135C4×D73C2×D812yes—
136Dic146—42no—
137Dic146—81no—
138Dic146—82no—
139Dic146D1611yescharacteristic
140C8×C73C2×D811yescharacteristic
141C8×C73C22×D411yescharacteristic
142#10596831C2×D812yes—
143C22×C73—81no—
144C22×C73—81no—
145C22×C73C2×D1611yescharacteristic
146C2×D73—82no—
147C2×D73—322no—
148C2×D73—321no—
149C2×D73C2×D1612yes—
150C4×C73—81no—
151C4×C73—81no—
152C4×C73C2×D1611yescharacteristic
153C4×C73C22×D811yescharacteristic
154Dic73—82no—
155Dic73C2×D1612yes—
156#33983—731nomaximal, Sylow
157C2×C73—21no—
158C2×C73—161no—
159C2×C73—161no—
160C2×C73C22×D1611yescharacteristic
161D73—22no—
162D73—162no—
163#3417—731no—
164#3415—731no—
165#2264—731no—
166#2264—732no—
167#2263—731no—
168#2263—732no—
169#2262—732no—
170#2262D7311yescharacteristic, complemented
171C2×SD64—732no—
172C2×SD64—731no—
173#2258—731no—
174C73#3398311yescharacteristic, complemented, Sylow
175#578—1461no—
176#576—1461no—
177#510—1462no—
178#510—1461no—
179#509—1462no—
180#509—1461no—
181#508—732no—
182#508—732no—
183#508C2×D7311yescharacteristic
184#508C2×D7311yescharacteristic, complemented
185C2×Q32—732no—
186C2×Q32—731no—
187C2×D16—732no—
188C2×D16—731no—
189#504—731no—
190Q64—732no—
191Q64C2×D7311yescharacteristic, complemented
192SD64—732no—
193SD64—732no—
194SD64—732no—
195SD64—732no—
196SD64C2×D7311yescharacteristic, complemented
197SD64C2×D7311yescharacteristic, complemented
198D32—732no—
199D32C2×D7311yescharacteristic, complemented
200M64—731no—
201M64—732no—
202C2×C32—732no—
203C2×C32C2×D7311yescharacteristic, complemented
2042−1+4—2921no—
205H(5,𝔽2)—2921no—
206#137—2922no—
207#137—2921no—
208AGL(1,ℤ/8)—2922no—
209AGL(1,ℤ/8)—2921no—
210#135—21no—
211#135—21no—
212#135—1462no—
213#135—1462no—
214C2×Q16—1462no—
215C2×Q16—1461no—
216C2×D8—1462no—
217C2×D8—1461no—
218#131—731no—
219Q32—732no—
220Q32—1462no—
221Q32—1461no—
222Q32C22×D7311yescharacteristic
223Q32C22×D7311yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—1462no—
227SD32—1462no—
228D16—732no—
229D16—1462no—
230D16—1461no—
231D16C22×D7311yescharacteristic
232D16C22×D7311yescharacteristic
233M32—731no—
234M32—732no—
235C2×C16—732no—
236C2×C16C22×D7311yescharacteristic
237C32—732no—
238C32C22×D7311yescharacteristic
239C32C22×D7311yescharacteristic
240#41—41no—
241#41—41no—
242#41—2922no—
243#41—2922no—
244#41—5841no—
245#41—5841no—
246#41—5842no—
247#41—731no—
248C2×Q8—2922no—
249C2×Q8—2921no—
250C2×Q8—5841no—
251C2×D4—2922no—
252C2×D4—2921no—
253C2×D4—5841no—
254C2×D4—5842no—
255Q16—21no—
256Q16—21no—
257Q16—2922no—
258Q16—2921no—
259Q16—1462no—
260SD16—41no—
261SD16—41no—
262SD16—2922no—
263SD16—2922no—
264D8—21no—
265D8—21no—
266D8—2922no—
267D8—2921no—
268D8—1462no—
269M16—731no—
270M16—732no—
271C2×C8—732no—
272C2×C8D4×D7311yescharacteristic
273C16—732no—
274C16D4×D7311yescharacteristic
275C16C23×D7311yescharacteristic
276C23—5842no—
277C23—5841no—
278Q8—41no—
279Q8—41no—
280Q8—2922no—
281Q8—5842no—
282Q8—5841no—
283Q8—731no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—2922no—
289D4—5842no—
290D4—5842no—
291D4—5842no—
292D4—5841no—
293D4—732no—
294D4—731no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—5842no—
298C2×C4—5842no—
299C2×C4—5841no—
300C2×C4—5842no—
301C2×C4—732no—
302C2×C4D8×D7311yescharacteristic
303C8—732no—
304C8D8×D7311yescharacteristic
305C8C2×D4×D7311yescharacteristic
306C22—81no—
307C22—81no—
308C22—23361no—
309C22—23362no—
310C22—5842no—
311C22—732no—
312C22D16×D7311yescharacteristic
313C4—81no—
314C4—81no—
315C4—5842no—
316C4—732no—
317C4D16×D7311yescharacteristic
318C4C2×D8×D7311yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—11682no—
323C2—1462no—
324C2C2×D16×D7311yescharacteristic
subgroupG/H
centerC2C2×D16×D73
commutator, cosocleC16×C73C24
FrattiniC16C23×D73
FittingC73×#2262C2
socleC2×C73C22×D16
hypercenter#2262D73
hypocenterC73#33983
LCSG ▷ C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #22623081C2semidirect
2. #22623069C2semidirect
3. #22612250C2semidirect
4. #22612248C2semidirect
5. #22612243C2semidirect
6. #22612242C2semidirect
7. #22612239C2semidirect
8. C2×#17988426C2semidirect
9. SD64×D73C2semidirect
10. #22612224C2semidirect
11. C73×#2262C2semidirect
19. #17989016C22semidirect
20. C2×Dic584C22semidirect
21. Q32×D73C22semidirect
25. #17988427C22semidirect
26. #17988426C22semidirect
27. #17988426C22semidirect
28. #17988425C22semidirect
29. #17988424C22semidirect
30. #17988422C22semidirect
31. #17988421C22semidirect
32. #17988420C22semidirect
33. C32×D73C22semidirect
35. #508×C73C22semidirect
36. Q64×C73C22semidirect
37. SD64×C73C22semidirect
38. SD64×C73C22semidirect
39. D32×C73C22semidirect
40. C2×C32×C73C22semidirect
170. #2262D73semidirect
174. C73#33983semidirect
184. #508C2×D73semidirect
191. Q64C2×D73semidirect
196. SD64C2×D73semidirect
197. SD64C2×D73semidirect
199. D32C2×D73semidirect
203. C2×C32C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230812
#226230692
#226122502
#226122482
#226122432
#226122422
#226122392
C2×#179884262
SD64×D732
#226122242
C73×#22622
#3398373