Finite Group Databasebeta

← #451×D151 #34078301

Subgroups — #451×D151 #34078301

Order 19328.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×D151C211yesmaximal, characteristic
2C2×Q16×D151C212yesmaximal, complemented
3C22⋊Q8×D151C211yesmaximal, characteristic
4#23752370C211yesmaximal, characteristic, complemented
5#23752369C212yesmaximal, complemented
6#23752368C211yesmaximal, characteristic, complemented
7#23752367C211yesmaximal, characteristic, complemented
8#23752366C212yesmaximal, complemented
9#103×D151C212yesmaximal, complemented
10C22⋊C8×D151C211yesmaximal, characteristic
11#451×C151C211yesmaximal, characteristic, complemented
12C22×Dic302—21no—
13C22×Dic302C2211yescharacteristic
14C2×Q8×D151—28no—
15C2×Q8×D151—21no—
16C2×Q8×D151—21no—
17C2×Q8×D151C2211yescharacteristic
18C2×Q8×D151C2212yes—
19C2×Q8×D151C2212yes—
20C22×C4×D151—21no—
21C22×C4×D151C2211yescharacteristic
22C2×Dic604C2212yescomplemented
23C2×#14558994C2212yescomplemented
24C2×#14558994C2212yescomplemented
25Q16×D151—28no—
26C2×C8×D151C2212yes—
27#18012719C2211yescharacteristic
28#18012718C2211yescharacteristic
29#18012717—21no—
30#18012716—21no—
31#18012715C2211yescharacteristic
32C4⋊C4×D151—21no—
33C4⋊C4×D151C2211yescharacteristic
34C22⋊C4×D151—21no—
35#18012664C2212yescomplemented
36#18012663C2212yescomplemented
37#18012662C2212yescomplemented
38C22⋊(C151⋊C8)C2211yescharacteristic
39(C2×D151)⋊C8C2212yes—
40C22×Q8×C151C2211yescharacteristic
41C2×Q16×C151C2212yescomplemented
42C22⋊Q8×C151C2211yescharacteristic
43#103×C151C2212yescomplemented
44C22⋊C8×C151C2211yescharacteristic
45C23×D151D411yescharacteristic
46C2×Dic302—28no—
47C2×Dic302—24no—
48C2×Dic302—22no—
49C2×Dic302—22no—
50C2×Dic302—21no—
51C2×Dic302—21no—
52C2×Dic302C2311yescharacteristic
53C2×Dic302C2312yes—
54Q8×D151—28no—
55Q8×D151—28no—
56Q8×D151—24no—
57Q8×D151—24no—
58Q8×D151—24no—
59Q8×D151—24no—
60Q8×D151D418yes—
61C22×Dic151—21no—
62C22×Dic151D411yescharacteristic
63C2×C4×D151—28no—
64C2×C4×D151—22no—
65C2×C4×D151—22no—
66C2×C4×D151—24no—
67C2×C4×D151—21no—
68C2×C4×D151—21no—
69C2×C4×D151D411yescharacteristic
70C2×C4×D151D411yescharacteristic
71C2×C4×D151C2311yescharacteristic
72C2×C4×D151C2311yescharacteristic
73Dic604—24no—
74#14558994—24no—
75#14558994—24no—
76C2×C151⋊C8C2312yes—
77C8×D151—24no—
78C4⋊Dic151—21no—
79C4⋊Dic151C2311yescharacteristic
80Dic151⋊C4—21no—
81Dic151⋊C4—21no—
82Dic151⋊C4—21no—
83C22⋊Dic151—21no—
84(C2×D151)⋊C4—21no—
85(C2×D151)⋊C4—21no—
86C2×Q8×C151—24no—
87C2×Q8×C151—21no—
88C2×Q8×C151C2311yescharacteristic
89C2×Q8×C151C2312yes—
90C22×C4×C151—21no—
91C22×C4×C151C2311yescharacteristic
92Q16×C151—24no—
93C2×C8×C151C2312yes—
94C4⋊C4×C151—21no—
95C4⋊C4×C151C2311yescharacteristic
96C22⋊C4×C151—21no—
97C22×D151—22no—
98C22×D151—22no—
99C22×D151Q1614yescomplemented
100C22×D151C2×D411yescharacteristic
101C22×D151C2×D411yescharacteristic
102Dic302—24no—
103Dic302—28no—
104Dic302—22no—
105Dic302—28no—
106Dic302—22no—
107Dic302—42no—
108Dic302C2×D414yes—
109C2×Dic151—24no—
110C2×Dic151—22no—
111C2×Dic151—22no—
112C2×Dic151—21no—
113C2×Dic151—21no—
114C2×Dic151C2×D411yescharacteristic
115C2×Dic151C2×D411yescharacteristic
116C4×D151—28no—
117C4×D151—28no—
118C4×D151—22no—
119C4×D151—22no—
120C4×D151—22no—
121C4×D151—42no—
122C4×D151—42no—
123C4×D151C2×D414yes—
124C151⋊C8—22no—
125C23×C151C2×D411yescharacteristic
126Q8×C151—24no—
127Q8×C151—22no—
128Q8×C151—22no—
129Q8×C151C2×D414yes—
130C2×C4×C151—24no—
131C2×C4×C151—22no—
132C2×C4×C151—22no—
133C2×C4×C151—21no—
134C2×C4×C151—21no—
135C2×C4×C151C2×D411yescharacteristic
136C2×C4×C151C2411yescharacteristic
137C8×C151—22no—
138C22×C151—21no—
139C22×C151—21no—
140C22×C151C2×Q1612yescomplemented
141C22×C151C22×D411yescharacteristic
142C2×D151—21no—
143C2×D151—21no—
144C2×D151—28no—
145C2×D151#12012yes—
146C2×D151C2×Q1612yes—
147C2×D151C2×Q1612yescomplemented
148C2×D151#13712yes—
149C4×C151—24no—
150C4×C151—21no—
151C4×C151—41no—
152C4×C151—41no—
153C4×C151C22×D412yes—
154Dic151—24no—
155Dic151—21no—
156Dic151—41no—
157Dic151—41no—
158Dic151#12012yes—
159C2×C151—22no—
160C2×C151C2×#12011yescharacteristic
161C2×C151C22×Q1611yescharacteristic
162C2×C151C2×#13711yescharacteristic
163D151—22no—
164D151#45114yesdirect product factor
165C151C2×#45111yescharacteristic, complemented, Sylow
166C2×#451—1511nomaximal, Sylow
167C23×Q8—1511no—
168C22×Q16—1512no—
169C2×C22⋊Q8—1511no—
170#451—1511no—
171#451—1512no—
172#451—1511no—
173#451—1512no—
174#451—1511no—
175#451D15111yescharacteristic, direct product factor
176C2×#103—1512no—
177C2×C22⋊C8—1511no—
178C22×Q8—3021no—
179C22×Q8—3028no—
180C22×Q8—3021no—
181C22×Q8—3021no—
182C22×Q8—1511no—
183C22×Q8—1512no—
184C22×Q8—1512no—
185C22×Q8—1511no—
186C22×Q8C2×D15111yescharacteristic
187C23×C4—3021no—
188C23×C4—1511no—
189C2×Q16—3028no—
190C2×Q16—1512no—
191C2×Q16—1512no—
192C2×Q16—1512no—
193C2×Q16C2×D15112yescomplemented
194C22×C8—1512no—
195C22⋊Q8—3021no—
196C22⋊Q8—3021no—
197C22⋊Q8—1511no—
198C22⋊Q8—1511no—
199C22⋊Q8—1511no—
200C22⋊Q8C2×D15111yescharacteristic
201C2×C4⋊C4—3021no—
202C2×C4⋊C4—1511no—
203C2×C22⋊C4—3021no—
204#103—1512no—
205#103—1512no—
206#103—1512no—
207#103C2×D15112yescomplemented
208C22⋊C8—1511no—
209C22⋊C8—1512no—
210C22⋊C8C2×D15111yescharacteristic
211C24—1511no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—3028no—
215C2×Q8—3024no—
216C2×Q8—3022no—
217C2×Q8—3022no—
218C2×Q8—3021no—
219C2×Q8—3028no—
220C2×Q8—3028no—
221C2×Q8—3024no—
222C2×Q8—3024no—
223C2×Q8—3024no—
224C2×Q8—3021no—
225C2×Q8—3024no—
226C2×Q8—1518no—
227C2×Q8—1511no—
228C2×Q8—1512no—
229C2×Q8C22×D15111yescharacteristic
230C2×Q8C22×D15112yes—
231C22×C4—21no—
232C22×C4—3028no—
233C22×C4—3022no—
234C22×C4—3022no—
235C22×C4—3024no—
236C22×C4—3021no—
237C22×C4—3021no—
238C22×C4—3021no—
239C22×C4—1511no—
240C22×C4—1511no—
241C22×C4—1511no—
242C22×C4—1511no—
243C22×C4—1511no—
244C22×C4C22×D15111yescharacteristic
245Q16—24no—
246Q16—3024no—
247Q16—3024no—
248Q16—3024no—
249C2×C8—3024no—
250C2×C8—1512no—
251C2×C8C22×D15112yes—
252C4⋊C4—21no—
253C4⋊C4—3021no—
254C4⋊C4—3021no—
255C4⋊C4—3021no—
256C4⋊C4—3021no—
257C4⋊C4—1511no—
258C4⋊C4C22×D15111yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—3021no—
261C22⋊C4—3021no—
262C22⋊C4—3021no—
263C23—3022no—
264C23—3022no—
265C23—1514no—
266C23—1511no—
267C23—1511no—
268C23D4×D15111yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—3028no—
273Q8—3024no—
274Q8—3028no—
275Q8—3022no—
276Q8—3022no—
277Q8—6042no—
278Q8—1514no—
279Q8D4×D15114yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3028no—
286C2×C4—3028no—
287C2×C4—3022no—
288C2×C4—3022no—
289C2×C4—3024no—
290C2×C4—3022no—
291C2×C4—3022no—
292C2×C4—3022no—
293C2×C4—3021no—
294C2×C4—3021no—
295C2×C4—6042no—
296C2×C4—6042no—
297C2×C4—1511no—
298C2×C4—1511no—
299C2×C4—1514no—
300C2×C4D4×D15111yescharacteristic
301C2×C4C23×D15111yescharacteristic
302C8—22no—
303C8—3022no—
304C22—21no—
305C22—21no—
306C22—3028no—
307C22—3021no—
308C22—3021no—
309C22—1512no—
310C22—1512no—
311C22—1512no—
312C22—1512no—
313C22Q16×D15112yescomplemented
314C22C2×D4×D15111yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—3024no—
320C4—3021no—
321C4—6041no—
322C4—6041no—
323C4—1512no—
324C4C2×D4×D15112yes—
325C2—22no—
326C2—3022no—
327C2—1514no—
328C2#120×D15111yescharacteristic
329C2C2×Q16×D15111yescharacteristic
330C2#137×D15111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#451×C151C2
socleC22×C151C22×D4
prime radicalC23×D151D4
hypercenter#451D151
hypocenterC151C2×#451
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D151C2semidirect
4. #23752370C2semidirect
5. #23752369C2semidirect
6. #23752368C2semidirect
7. #23752367C2semidirect
8. #23752366C2semidirect
9. #103×D151C2semidirect
11. #451×C151C2semidirect
22. C2×Dic604C22semidirect
23. C2×#14558994C22semidirect
24. C2×#14558994C22semidirect
35. #18012664C22semidirect
36. #18012663C22semidirect
37. #18012662C22semidirect
41. C2×Q16×C151C22semidirect
43. #103×C151C22semidirect
99. C22×D151Q16semidirect
140. C22×C151C2×Q16semidirect
147. C2×D151C2×Q16semidirect
164. D151#451direct (N × H)
165. C151C2×#451semidirect
175. #451D151direct (N × H)
193. C2×Q16C2×D151semidirect
207. #103C2×D151semidirect
313. C22Q16×D151semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1512
C2×Q16×D1512
C22⋊Q8×D1512
#237523702
#237523692
#237523682
#237523672
#237523662
#103×D1512
C22⋊C8×D1512
#451×C1512
C2×#451151