Finite Group Databasebeta

← #34078501

Subgroups — #34078501

Order 19328.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#23753048C211yesmaximal, characteristic, complemented
2#23752723C212yesmaximal, complemented
3#23752401C214yesmaximal, complemented
4#23752398C214yesmaximal, complemented
5#23752186C211yesmaximal, characteristic, complemented
6#23752185C212yesmaximal, complemented
7#525×C151C211yesmaximal, characteristic, complemented
8#18012787—21no—
9#18012786—22no—
10#18012785—22no—
11#18012784—21no—
12#18012784C2214yes—
13C2×#14559004—22no—
14C2×#14559004C2212yes—
15#41×D151—24no—
16#18012722C2214yescomplemented
17#18012718—22no—
18#18012713—22no—
19#18012708C2214yescomplemented
20C4×C22⋊D151—24no—
21#18012698C2212yescomplemented
22#18012654C2214yescomplemented
23#18012653C2218yescomplemented
24C2×#41×C151C2211yescharacteristic
25C23×C4×C151C2211yescharacteristic, complemented
26#123×C151C2212yescomplemented
27C22⋊Q8×C151—21no—
28#121×C151—21no—
29#120×C151C2212yescomplemented
30C4×D4×C151—22no—
31#117×C151C2211yescharacteristic, complemented
32#14559005—24no—
33#14559004—24no—
34#14559004—24no—
35#14559004—22no—
36#14559004—22no—
37#14559004—24no—
38#14559004—22no—
39#14559004—22no—
40#14559003—24no—
41#14559003—28no—
42C2×Dic302—22no—
43C2×Dic302D412yescomplemented
44Q8×D151—24no—
45C2×D604—22no—
46C2×D604D412yescomplemented
47C2×C22⋊D151—24no—
48C2×C22⋊D151—24no—
49C2×C22⋊D151C2314yes—
50D4×D151—24no—
51D4×D151—28no—
52C2×C4×D151—24no—
53C2×C4×D151D414yescomplemented
54C4⋊Dic151—22no—
55Dic151⋊C4—24no—
56C22⋊Dic151—24no—
57C22⋊Dic151C2314yes—
58(C2×D151)⋊C4—24no—
59C4×Dic151—22no—
60C24×C151C2311yescharacteristic
61#41×C151—22no—
62#41×C151—22no—
63C2×Q8×C151D411yescharacteristic, complemented
64C2×D4×C151—22no—
65C2×D4×C151D411yescharacteristic, complemented
66C2×D4×C151C2312yes—
67C22×C4×C151—22no—
68C22×C4×C151—22no—
69C22×C4×C151—21no—
70C22×C4×C151—21no—
71C22×C4×C151D412yescomplemented
72C22×C4×C151C2311yescharacteristic
73C22×C4×C151C2311yescharacteristic
74C4⋊C4×C151—21no—
75C4⋊C4×C151—22no—
76C22⋊C4×C151—22no—
77C22⋊C4×C151—22no—
78C22⋊C4×C151C2312yes—
79C42×C151—21no—
80C22×D151—24no—
81C22×D151C2×D414yes—
82Dic302—24no—
83Dic302—22no—
84Dic302—22no—
85D604—24no—
86D604—22no—
87D604—22no—
88C22⋊D151—28no—
89C22⋊D151—24no—
90C22⋊D151—24no—
91C22⋊D151—48no—
92C2×Dic151—24no—
93C2×Dic151—24no—
94C2×Dic151C2×D414yes—
95C4×D151—28no—
96C4×D151—24no—
97C4×D151—24no—
98C23×C151—22no—
99C23×C151—21no—
100C23×C151—42no—
101C23×C151C2×D412yes—
102C23×C151C2411yescharacteristic
103Q8×C151—22no—
104D4×C151—24no—
105D4×C151—22no—
106D4×C151—44no—
107C2×C4×C151—22no—
108C2×C4×C151—22no—
109C2×C4×C151—22no—
110C2×C4×C151—22no—
111C2×C4×C151—21no—
112C2×C4×C151—41no—
113C2×C4×C151—42no—
114C2×C4×C151—41no—
115C2×C4×C151C2×D412yes—
116C2×C4×C151C2×D412yes—
117C2×C4×C151C2×D412yes—
118C2×C4×C151C2×D411yescharacteristic
119C2×C4×C151C2×D411yescharacteristic
120C22×C151—22no—
121C22×C151—24no—
122C22×C151—22no—
123C22×C151—41no—
124C22×C151—42no—
125C22×C151—42no—
126C22×C151—41no—
127C22×C151—41no—
128C22×C151C22×D412yes—
129C22×C151C22×D411yescharacteristic
130C2×D151—24no—
131C2×D151—44no—
132C2×D151—84no—
133C4×C151—22no—
134C4×C151—42no—
135C4×C151—41no—
136C4×C151#12011yescharacteristic
137C4×C151#12012yes—
138C4×C151#12011yescharacteristic
139Dic151—24no—
140Dic151—44no—
141C2×C151—22no—
142C2×C151—21no—
143C2×C151—42no—
144C2×C151—42no—
145C2×C151C2×#12011yescharacteristic
146D151—44no—
147C151#302511yescharacteristic, complemented, Sylow
148#3025—1511nomaximal, Sylow
149#585—1511no—
150#525—1512no—
151#525D15111yescharacteristic, complemented
152#458—1514no—
153B(4,𝔽2)—1514no—
154#410—1511no—
155#410—1512no—
1562−1+4—3022no—
1572−1+4—3021no—
158H(5,𝔽2)—3022no—
159H(5,𝔽2)—3021no—
160H(5,𝔽2)—1514no—
161C2×#41—3024no—
162C2×#41—3022no—
163C2×#41—1512no—
164C2×#41C2×D15111yescharacteristic
165C23×C4C2×D15111yescharacteristic, complemented
166#123—1514no—
167#123C2×D15112yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—3022no—
170#121—21no—
171#121—3022no—
172#120—1514no—
173#120C2×D15112yescomplemented
174C4×D4—22no—
175C4×D4—3024no—
176#117—1512no—
177#117C2×D15111yescharacteristic, complemented
178#99—1514no—
179#99—1518no—
180C24C22×D15111yescharacteristic
181#41—22no—
182#41—22no—
183#41—3024no—
184#41—3024no—
185#41—3024no—
186#41—3024no—
187#41—3028no—
188#41—3022no—
189#41—3022no—
190#41—3022no—
191#41—3024no—
192#41—3022no—
193C2×Q8—3022no—
194C2×Q8—3024no—
195C2×Q8—1512no—
196C2×Q8C22⋊D15111yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—3022no—
199C2×D4—3024no—
200C2×D4—3028no—
201C2×D4—3024no—
202C2×D4—3024no—
203C2×D4—1514no—
204C2×D4—1512no—
205C2×D4C22⋊D15111yescharacteristic, complemented
206C2×D4C22×D15112yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—3024no—
212C22×C4—1514no—
213C22×C4C22⋊D15112yescomplemented
214C22×C4C22×D15111yescharacteristic
215C22×C4C22×D15111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—3024no—
219C4⋊C4—3022no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—3024no—
223C22⋊C4—3024no—
224C22⋊C4—1514no—
225C22⋊C4C22×D15112yes—
226C42—21no—
227C42—3022no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3024no—
232C23—1514no—
233C23C2×C22⋊D15112yes—
234C23C23×D15111yescharacteristic
235Q8—22no—
236Q8—3024no—
237Q8—3022no—
238Q8—3022no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—3024no—
243D4—3028no—
244D4—3024no—
245D4—3024no—
246D4—3022no—
247D4—3022no—
248D4—6048no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—3028no—
258C2×C4—3024no—
259C2×C4—3024no—
260C2×C4—3024no—
261C2×C4—3024no—
262C2×C4—1514no—
263C2×C4D4×D15112yes—
264C2×C4D4×D15112yes—
265C2×C4C2×C22⋊D15112yes—
266C2×C4C2×C22⋊D15111yescharacteristic
267C2×C4C2×C22⋊D15111yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—3024no—
277C22—6044no—
278C22—12084no—
279C22C2×D4×D15112yes—
280C22C22×C22⋊D15111yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—3024no—
285C4—6044no—
286C4#1801270711yescharacteristic
287C4#1801270712yes—
288C4#1801270711yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—6044no—
294C2C2×#1801270711yescharacteristic
subgroupG/H
centerC4#18012707
commutator, cosocleC23×C151C24
FrattiniC23C23×D151
Fitting#525×C151C2
socleC2×C151C2×#120
hypercenter#525D151
hypocenterC151#3025
LCSG ▷ C23×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #23753048C2semidirect
2. #23752723C2semidirect
3. #23752401C2semidirect
4. #23752398C2semidirect
5. #23752186C2semidirect
6. #23752185C2semidirect
7. #525×C151C2semidirect
16. #18012722C22semidirect
19. #18012708C22semidirect
21. #18012698C22semidirect
22. #18012654C22semidirect
23. #18012653C22semidirect
25. C23×C4×C151C22semidirect
26. #123×C151C22semidirect
29. #120×C151C22semidirect
31. #117×C151C22semidirect
43. C2×Dic302D4semidirect
46. C2×D604D4semidirect
53. C2×C4×D151D4semidirect
63. C2×Q8×C151D4semidirect
65. C2×D4×C151D4semidirect
71. C22×C4×C151D4semidirect
147. C151#3025semidirect
151. #525D151semidirect
165. C23×C4C2×D151semidirect
167. #123C2×D151semidirect
173. #120C2×D151semidirect
177. #117C2×D151semidirect
196. C2×Q8C22⋊D151semidirect
205. C2×D4C22⋊D151semidirect
213. C22×C4C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
#237530482
#237527232
#237524012
#237523982
#237521862
#237521852
#525×C1512
#3025151