Finite Group Databasebeta

← #33244134

Subgroups — #33244134

Order 18688.

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

normal subgroups only

This group has 472 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#22623069C211yesmaximal, characteristic, complemented
2#22623068C211yesmaximal, characteristic, complemented
3#22612246C211yesmaximal, characteristic, complemented
4#22612245C212yesmaximal, complemented
5#22612242C212yesmaximal, complemented
6#22612239C212yesmaximal, complemented
7#22612238C211yesmaximal, characteristic, complemented
8C2×#17988422C211yesmaximal, characteristic, complemented
9D32×D73C212yesmaximal, complemented
10#22612226C211yesmaximal, characteristic, complemented
11C73×#2263C211yesmaximal, characteristic, complemented
12#17989389—21no—
13#17989388—21no—
14#17989024—22no—
15#17989023—21no—
16#17989022—21no—
17#17989021—22no—
18#17989020C2212yescomplemented
19#17989019C2211yescharacteristic, complemented
20#17989016C2212yescomplemented
21C2×D1168C2211yescharacteristic, complemented
22D16×D73C2212yescomplemented
23D16×D73C2212yescomplemented
24#17989003C2211yescharacteristic
25#17988427C2211yescharacteristic, complemented
26#17988426C2212yescomplemented
27#17988425C2211yescharacteristic, complemented
28#17988424C2211yescharacteristic, complemented
29D2336C2212yescomplemented
30#17988422C2211yescharacteristic, complemented
31#17988422C2212yescomplemented
32#17988422C2212yescomplemented
33#17988420C2212yescomplemented
34C2×#14556497C2211yescharacteristic, complemented
35C32×D73C2212yescomplemented
36#508×C73C2211yescharacteristic, complemented
37C2×D16×C73C2211yescharacteristic, complemented
38SD64×C73C2212yescomplemented
39D32×C73C2212yescomplemented
40M64×C73C2211yescharacteristic, complemented
41#14556637—41no—
42#14556636—41no—
43#14556618—42no—
44#14556617—41no—
45#14556616—41no—
46#14556615—42no—
47#14556614—22no—
48#14556613—21no—
49#14556610—22no—
50C2×D584—21no—
51D8×D73—22no—
52D8×D73—22no—
53#14556597D411yescharacteristic
54Dic584C2311yescharacteristic
55#14556536—21no—
56#14556535—22no—
57#14556534—22no—
58D1168—21no—
59D1168C2312yes—
60D1168C2311yescharacteristic
61#14556532—22no—
62#14556532—22no—
63#14556531D411yescharacteristic
64#14556530D412yes—
65C16×D73C2312yes—
66#14556497C2311yescharacteristic
67#14556497C2311yescharacteristic
68#135×C73—21no—
69C2×D8×C73—21no—
70Q32×C73C2311yescharacteristic
71SD32×C73—21no—
72D16×C73—21no—
73D16×C73C2312yes—
74D16×C73C2311yescharacteristic
75C2×C16×C73C2311yescharacteristic
76C32×C73C2312yes—
77#11924029—42no—
78#11924028—41no—
79#11924028—81no—
80#11924028D811yescharacteristic
81#11924027—42no—
82#11924027—81no—
83C2×D292—41no—
84C2×D292—81no—
85C2×C22⋊D73—82no—
86D4×D73—42no—
87D4×D73—42no—
88D4×D73—82no—
89D4×D73—81no—
90Dic292—21no—
91#11924017—41no—
92#11924016—42no—
93#11924015—42no—
94D584—21no—
95D584—22no—
96D584—41no—
97#11924013—42no—
98#11924013—42no—
99#11924012D811yescharacteristic
100#11924011D812yes—
101C8×D73C2×D412yes—
102#11924003C2×D412yes—
103#41×C73—41no—
104C2×D4×C73—41no—
105Q16×C73—21no—
106SD16×C73—41no—
107D8×C73—22no—
108D8×C73—21no—
109D8×C73—41no—
110C2×C8×C73C2×D411yescharacteristic
111C16×C73C2×D411yescharacteristic
112C16×C73C2411yescharacteristic
113C23×C73—81no—
114C22×D73—81no—
115C22×D73—82no—
116C22×D73—82no—
117Q8×C73—41no—
118D4×C73—42no—
119D4×C73—41no—
120D4×C73—81no—
121D4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73C2×D811yescharacteristic
124C22⋊D73—81no—
125C22⋊D73—82no—
126C22⋊D73—82no—
127C22⋊D73—82no—
128C22⋊D73D1612yes—
129C2×Dic73—82no—
130D292—41no—
131D292—42no—
132D292—82no—
133D292—81no—
134D292D1611yescharacteristic
135C4×D73—81no—
136C4×D73—82no—
137C4×D73C2×D812yes—
138Dic146—41no—
139Dic146D1611yescharacteristic
140C8×C73C2×D811yescharacteristic
141C8×C73C22×D411yescharacteristic
142#10596831C2×D812yes—
143C22×C73—82no—
144C22×C73—81no—
145C22×C73—321no—
146C22×C73C2×D1611yescharacteristic
147C2×D73—82no—
148C2×D73—81no—
149C2×D73—322no—
150C2×D73—322no—
151C2×D73—321no—
152C2×D73C2×D1612yes—
153C4×C73—81no—
154C4×C73C2×D1611yescharacteristic
155C4×C73C22×D811yescharacteristic
156Dic73—81no—
157Dic73C2×D1612yes—
158#33982—731nomaximal, Sylow
159C2×C73—21no—
160C2×C73—161no—
161C2×C73—162no—
162C2×C73C22×D1611yescharacteristic
163D73—22no—
164D73—162no—
165D73—161no—
166#3415—731no—
167#3415—731no—
168#2263—732no—
169#2263—732no—
170#2263—731no—
171#2263D7311yescharacteristic, complemented
172#2262—731no—
173#2262—732no—
174C2×D32—731no—
175C2×D32—732no—
176#2258—731no—
177C73#3398211yescharacteristic, complemented, Sylow
178#576—1461no—
179#576—1461no—
180#509—1462no—
181#509—1462no—
182#509—1461no—
183#509—1461no—
184#508—732no—
185#508—732no—
186#508—731no—
187#508C2×D7311yescharacteristic, complemented
188C2×D16—732no—
189C2×D16—732no—
190C2×D16—731no—
191C2×D16C2×D7311yescharacteristic, complemented
192#504—731no—
193Q64—731no—
194SD64—731no—
195SD64—731no—
196SD64—732no—
197SD64C2×D7312yescomplemented
198D32—732no—
199D32—731no—
200D32—732no—
201D32—732no—
202D32C2×D7312yescomplemented
203M64—732no—
204M64C2×D7311yescharacteristic, complemented
205C2×C32—731no—
206C2×C32—732no—
207H(5,𝔽2)—2921no—
208H(5,𝔽2)—2921no—
209AGL(1,ℤ/8)—2922no—
210AGL(1,ℤ/8)—2922no—
211AGL(1,ℤ/8)—2921no—
212AGL(1,ℤ/8)—2921no—
213#135—21no—
214#135—1462no—
215#135—1462no—
216#135—1461no—
217C2×D8—21no—
218C2×D8—1462no—
219C2×D8—1462no—
220C2×D8—1461no—
221#131—731no—
222Q32—731no—
223Q32C22×D7311yescharacteristic
224SD32—21no—
225SD32—1462no—
226SD32—1462no—
227SD32—1461no—
228D16—21no—
229D16—732no—
230D16—731no—
231D16—1462no—
232D16—1462no—
233D16—1461no—
234D16C22×D7312yes—
235D16C22×D7311yescharacteristic
236M32—731no—
237M32—732no—
238C2×C16—732no—
239C2×C16C22×D7311yescharacteristic
240C32—731no—
241C32—731no—
242C32C22×D7312yes—
243#41—41no—
244#41—2922no—
245#41—2922no—
246#41—2921no—
247#41—5841no—
248#41—5841no—
249#41—731no—
250C2×D4—41no—
251C2×D4—2922no—
252C2×D4—2922no—
253C2×D4—2921no—
254C2×D4—5841no—
255C2×D4—5842no—
256C2×D4—5841no—
257C2×D4—5842no—
258Q16—21no—
259Q16—1461no—
260SD16—41no—
261SD16—2922no—
262SD16—2922no—
263SD16—2921no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—2922no—
268D8—2922no—
269D8—2921no—
270D8—1461no—
271D8—1462no—
272M16—731no—
273M16—732no—
274C2×C8—732no—
275C2×C8D4×D7311yescharacteristic
276C16—732no—
277C16D4×D7311yescharacteristic
278C16C23×D7311yescharacteristic
279C23—81no—
280C23—5842no—
281C23—5842no—
282C23—5841no—
283Q8—41no—
284Q8—2921no—
285Q8—731no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—2921no—
291D4—2922no—
292D4—5842no—
293D4—5842no—
294D4—5841no—
295D4—5842no—
296D4—5842no—
297D4—5841no—
298D4—732no—
299D4—731no—
300C2×C4—81no—
301C2×C4—5842no—
302C2×C4—5842no—
303C2×C4—5841no—
304C2×C4—732no—
305C2×C4D8×D7311yescharacteristic
306C8—732no—
307C8D8×D7311yescharacteristic
308C8C2×D4×D7311yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—23361no—
313C22—23362no—
314C22—23362no—
315C22—5842no—
316C22—5841no—
317C22—732no—
318C22D16×D7311yescharacteristic
319C4—81no—
320C4—5841no—
321C4—732no—
322C4D16×D7311yescharacteristic
323C4C2×D8×D7311yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—11682no—
328C2—11681no—
329C2—1462no—
330C2C2×D16×D7311yescharacteristic
subgroupG/H
centerC2C2×D16×D73
commutator, cosocleC16×C73C24
FrattiniC16C23×D73
FittingC73×#2263C2
socleC2×C73C22×D16
hypercenter#2263D73
hypocenterC73#33982
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. #22623069C2semidirect
2. #22623068C2semidirect
3. #22612246C2semidirect
4. #22612245C2semidirect
5. #22612242C2semidirect
6. #22612239C2semidirect
7. #22612238C2semidirect
8. C2×#17988422C2semidirect
9. D32×D73C2semidirect
10. #22612226C2semidirect
11. C73×#2263C2semidirect
18. #17989020C22semidirect
19. #17989019C22semidirect
20. #17989016C22semidirect
21. C2×D1168C22semidirect
22. D16×D73C22semidirect
23. D16×D73C22semidirect
25. #17988427C22semidirect
26. #17988426C22semidirect
27. #17988425C22semidirect
28. #17988424C22semidirect
29. D2336C22semidirect
30. #17988422C22semidirect
31. #17988422C22semidirect
32. #17988422C22semidirect
33. #17988420C22semidirect
34. C2×#14556497C22semidirect
35. C32×D73C22semidirect
36. #508×C73C22semidirect
37. C2×D16×C73C22semidirect
38. SD64×C73C22semidirect
39. D32×C73C22semidirect
40. M64×C73C22semidirect
171. #2263D73semidirect
177. C73#33982semidirect
187. #508C2×D73semidirect
191. C2×D16C2×D73semidirect
197. SD64C2×D73semidirect
202. D32C2×D73semidirect
204. M64C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230692
#226230682
#226122462
#226122452
#226122422
#226122392
#226122382
C2×#179884222
D32×D732
#226122262
C73×#22632
#3398273