Finite Group Databasebeta

← #34078350

Subgroups — #34078350

Order 19328.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14559004C211yesmaximal, characteristic, complemented
2C2×#18012769C211yesmaximal, characteristic, complemented
3C2×#18012764C211yesmaximal, characteristic, complemented
4#23752372C214yesmaximal, complemented
5#23752355C214yesmaximal, complemented
6#23752220C211yesmaximal, characteristic, complemented
7#23752216C211yesmaximal, characteristic, complemented
8#23752176C211yesmaximal, characteristic, complemented
9#525×C151C211yesmaximal, characteristic, complemented
10C2×#14559004—22no—
11C2×#14559004—24no—
12C2×#14559004—24no—
13C2×#14559004C2214yes—
14C22×Dic302C2211yescharacteristic
15C22×D604C2211yescharacteristic
16C22×C22⋊D151—21no—
17C22×C4×D151—21no—
18#18012769—24no—
19#18012764—24no—
20C2×#14558994C2212yescomplemented
21C2×#14558992C2212yescomplemented
22C2×#14558991C2212yescomplemented
23C2×#14558989C2212yescomplemented
24C2×#14558988C2211yescharacteristic
25C2×#14558988C2211yescharacteristic
26#18012662C2214yescomplemented
27#18012659C2214yescomplemented
28C22⋊(C151⋊C8)C2214yescomplemented
29C2×#41×C151C2211yescharacteristic
30C23×C4×C151C2211yescharacteristic
31#123×C151—21no—
32C22⋊Q8×C151C2212yescomplemented
33#121×C151C2212yescomplemented
34#120×C151—21no—
35C4×D4×C151—22no—
36#117×C151C2211yescharacteristic
37C23×D151—21no—
38#14559004—216no—
39#14559004—28no—
40#14559004—28no—
41C2×Dic302—22no—
42C2×Dic302—21no—
43C2×Dic302D414yescomplemented
44C2×Dic302C2312yes—
45C2×D604—22no—
46C2×D604—21no—
47C2×D604D414yescomplemented
48C2×D604C2312yes—
49C2×C22⋊D151—28no—
50C2×C22⋊D151—24no—
51C22×Dic151—21no—
52C2×C4×D151—28no—
53C2×C4×D151—24no—
54#14558994—24no—
55#14558992—24no—
56#14558991—24no—
57#14558989—24no—
58#14558988—22no—
59#14558988—22no—
60C2×C151⋊C8C2312yes—
61C2×C151⋊C8C2312yes—
62C24×C151D411yescharacteristic
63#41×C151—22no—
64#41×C151—22no—
65C2×Q8×C151C2311yescharacteristic
66C2×D4×C151—22no—
67C2×D4×C151—21no—
68C2×D4×C151C2311yescharacteristic
69C22×C4×C151—22no—
70C22×C4×C151—22no—
71C22×C4×C151—21no—
72C22×C4×C151D411yescharacteristic
73C22×C4×C151D412yes—
74C22×C4×C151C2312yes—
75C22×C4×C151C2311yescharacteristic
76C4⋊C4×C151—22no—
77C4⋊C4×C151C2312yes—
78C22⋊C4×C151—21no—
79C22⋊C4×C151—22no—
80C22⋊C4×C151—22no—
81C42×C151—21no—
82C22×D151—24no—
83C22×D151—22no—
84C22×D151—44no—
85Dic302—24no—
86Dic302—22no—
87Dic302C2×D414yes—
88D604—24no—
89D604—22no—
90D604C2×D414yes—
91C22⋊D151—216no—
92C2×Dic151—24no—
93C2×Dic151—22no—
94C4×D151—216no—
95C151⋊C8—22no—
96C151⋊C8—22no—
97C23×C151—22no—
98C23×C151—21no—
99C23×C151—42no—
100C23×C151C2×D412yes—
101C23×C151C2×D411yescharacteristic
102Q8×C151—21no—
103Q8×C151—21no—
104D4×C151—21no—
105D4×C151—21no—
106D4×C151—44no—
107D4×C151—42no—
108C2×C4×C151—22no—
109C2×C4×C151—24no—
110C2×C4×C151—22no—
111C2×C4×C151—22no—
112C2×C4×C151—21no—
113C2×C4×C151—22no—
114C2×C4×C151—42no—
115C2×C4×C151C2×D412yes—
116C2×C4×C151C2×D412yes—
117C2×C4×C151C2×D412yes—
118C2×C4×C151C2×D411yescharacteristic
119C2×C4×C151C2411yescharacteristic
120C22×C151—22no—
121C22×C151—22no—
122C22×C151—22no—
123C22×C151—22no—
124C22×C151—44no—
125C22×C151—41no—
126C22×C151—42no—
127C22×C151#12012yes—
128C22×C151C22×D411yescharacteristic
129C2×D151—24no—
130C2×D151—48no—
131C2×D151—44no—
132C4×C151—22no—
133C4×C151—41no—
134C4×C151—42no—
135C4×C151#12012yes—
136C4×C151C22×D411yescharacteristic
137C4×C151C22×D411yescharacteristic
138Dic151—24no—
139C2×C151—22no—
140C2×C151—42no—
141C2×C151—81no—
142C2×C151C2×#12011yescharacteristic
143C2×C151#57512yes—
144D151—44no—
145C151#300511yescharacteristic, complemented, Sylow
146#3005—1511nomaximal, Sylow
147C22×#41—1511no—
148C2×#137—1511no—
149C2×AGL(1,ℤ/8)—1511no—
150#525D15111yescharacteristic, complemented
151#452—1514no—
152#449—1514no—
153#419—1511no—
154#418—1511no—
155#407—1511no—
156C2×#41—3024no—
157C2×#41—3024no—
158C2×#41—3022no—
159C2×#41—1514no—
160C2×#41C2×D15111yescharacteristic
161C22×Q8—1511no—
162C22×D4—3021no—
163C22×D4—1511no—
164C23×C4—3021no—
165C23×C4C2×D15111yescharacteristic
166#137—3024no—
167AGL(1,ℤ/8)—3024no—
168C2×Q16—1512no—
169C2×SD16—1512no—
170C2×SD16—1512no—
171C2×D8—1512no—
172C2×M16—1511no—
173C2×M16—1511no—
174#123—21no—
175C22⋊Q8C2×D15112yescomplemented
176#121C2×D15112yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D15111yescharacteristic
180#103—1514no—
181#102—1514no—
182C22⋊C8—1514no—
183C24—3021no—
184C24C22⋊D15111yescharacteristic
185#41—22no—
186#41—22no—
187#41—30216no—
188#41—3028no—
189#41—3028no—
190C2×Q8—3022no—
191C2×Q8—3021no—
192C2×Q8—1514no—
193C2×Q8—1512no—
194C2×Q8C22×D15111yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3028no—
198C2×D4—3024no—
199C2×D4—3022no—
200C2×D4—3021no—
201C2×D4—1514no—
202C2×D4—1512no—
203C2×D4C22×D15111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—3028no—
208C22×C4—3021no—
209C22×C4—3024no—
210C22×C4C22⋊D15112yes—
211C22×C4C22⋊D15111yescharacteristic
212C22×C4C22×D15112yes—
213C22×C4C22×D15111yescharacteristic
214Q16—3024no—
215SD16—3024no—
216SD16—3024no—
217D8—3024no—
218M16—3022no—
219M16—3022no—
220C2×C8—1512no—
221C2×C8—1512no—
222C4⋊C4—22no—
223C4⋊C4C22×D15112yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3024no—
232C23—3022no—
233C23—6044no—
234C23C2×C22⋊D15112yes—
235C23C2×C22⋊D15111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3024no—
239Q8—3022no—
240Q8—1514no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—30216no—
246D4—3024no—
247D4—3022no—
248D4—1514no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—30216no—
257C2×C4—3024no—
258C2×C4—3022no—
259C2×C4D4×D15112yes—
260C2×C4D4×D15112yes—
261C2×C4C2×C22⋊D15112yes—
262C2×C4C2×C22⋊D15111yescharacteristic
263C2×C4C23×D15111yescharacteristic
264C8—3022no—
265C8—3022no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—3024no—
274C22—6048no—
275C22—6044no—
276C22#1801270712yes—
277C22C22×C22⋊D15111yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—3024no—
282C4#1801270712yes—
283C4C2×D4×D15111yescharacteristic
284C4C2×D4×D15111yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—6044no—
289C2C2×#1801270711yescharacteristic
290C2#2375301012yes—
subgroupG/H
centerC22C22×C22⋊D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#525×C151C2
socleC22×C151C22×D4
hypercenter#525D151
hypocenterC151#3005
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#14559004C2semidirect
2. C2×#18012769C2semidirect
3. C2×#18012764C2semidirect
4. #23752372C2semidirect
5. #23752355C2semidirect
6. #23752220C2semidirect
7. #23752216C2semidirect
8. #23752176C2semidirect
9. #525×C151C2semidirect
20. C2×#14558994C22semidirect
21. C2×#14558992C22semidirect
22. C2×#14558991C22semidirect
23. C2×#14558989C22semidirect
26. #18012662C22semidirect
27. #18012659C22semidirect
28. C22⋊(C151⋊C8)C22semidirect
32. C22⋊Q8×C151C22semidirect
33. #121×C151C22semidirect
43. C2×Dic302D4semidirect
47. C2×D604D4semidirect
145. C151#3005semidirect
150. #525D151semidirect
175. C22⋊Q8C2×D151semidirect
176. #121C2×D151semidirect

Maximal subgroups

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

subgroupindex
C22×#145590042
C2×#180127692
C2×#180127642
#237523722
#237523552
#237522202
#237522162
#237521762
#525×C1512
#3005151