Finite Group Databasebeta

← #34078369

Subgroups — #34078369

Order 19328.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14559005C211yesmaximal, characteristic
2C2×#18012771C212yesmaximal, complemented
3C22⋊Q8×D151C211yesmaximal, characteristic, complemented
4#23752377C212yesmaximal, complemented
5#23752373C212yesmaximal, complemented
6#23752368C211yesmaximal, characteristic, complemented
7#23752365C211yesmaximal, characteristic, complemented
8#23752362C211yesmaximal, characteristic, complemented
9#23752212C212yesmaximal, complemented
10#23752174C211yesmaximal, characteristic, complemented
11#451×C151C211yesmaximal, characteristic, complemented
12C2×#14559005—28no—
13C2×#14559005—21no—
14C2×#14559005—21no—
15C2×#14559005C2212yes—
16C2×#14559005C2212yes—
17C2×Q8×D151C2211yescharacteristic, complemented
18C22×D604—21no—
19C22×D604C2211yescharacteristic
20C22×C4×D151—21no—
21C22×C4×D151C2211yescharacteristic
22#18012771—28no—
23C2×#14558994C2212yescomplemented
24C2×#14558993C2212yescomplemented
25C2×#14558991C2212yescomplemented
26C2×#14558987C2212yescomplemented
27#18012719C2211yescharacteristic, complemented
28#18012718C2211yescharacteristic
29#18012717—21no—
30#18012716—21no—
31#18012715C2211yescharacteristic, complemented
32C4⋊C4×D151—21no—
33C4⋊C4×D151C2211yescharacteristic, complemented
34C22⋊C4×D151—21no—
35#18012663C2212yescomplemented
36#18012661C2212yescomplemented
37#18012659C2212yescomplemented
38C22⋊(C151⋊C8)C2211yescharacteristic
39(C2×D151)⋊C8C2212yescomplemented
40C22×Q8×C151C2211yescharacteristic
41C2×Q16×C151C2212yescomplemented
42C22⋊Q8×C151C2211yescharacteristic
43#103×C151C2212yescomplemented
44C22⋊C8×C151C2211yescharacteristic
45C23×D151—21no—
46C23×D151D411yescharacteristic
47#14559005—28no—
48#14559005—28no—
49#14559005—24no—
50#14559005—24no—
51#14559005—24no—
52#14559005D418yes—
53C2×Dic302—21no—
54C2×Dic302C2311yescharacteristic
55Q8×D151—24no—
56C2×D604—28no—
57C2×D604—22no—
58C2×D604—22no—
59C2×D604—24no—
60C2×D604—21no—
61C2×D604C2312yes—
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
73#14558994—24no—
74#14558993—24no—
75#14558991—24no—
76#14558987—24no—
77C2×C151⋊C8C2312yes—
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—24no—
98C22×D151—22no—
99C22×D151—22no—
100C22×D151—22no—
101C22×D151—41no—
102C22×D151—42no—
103C22×D151—42no—
104C22×D151C2×D411yescharacteristic
105C22×D151C2×D411yescharacteristic
106Dic302—22no—
107Dic302—42no—
108D604—28no—
109D604—24no—
110D604—28no—
111D604—22no—
112D604C2×D414yes—
113C2×Dic151—22no—
114C2×Dic151—21no—
115C2×Dic151—21no—
116C2×Dic151C2×D411yescharacteristic
117C2×Dic151C2×D411yescharacteristic
118C4×D151—28no—
119C4×D151—22no—
120C4×D151—22no—
121C4×D151—28no—
122C4×D151—22no—
123C4×D151—42no—
124C4×D151—42no—
125C4×D151C2×D414yes—
126C151⋊C8—22no—
127C23×C151C2×D411yescharacteristic
128Q8×C151—24no—
129Q8×C151—22no—
130Q8×C151—22no—
131Q8×C151C2×D414yes—
132C2×C4×C151—24no—
133C2×C4×C151—22no—
134C2×C4×C151—22no—
135C2×C4×C151—21no—
136C2×C4×C151—21no—
137C2×C4×C151C2×D411yescharacteristic
138C2×C4×C151C2411yescharacteristic
139C8×C151—22no—
140C22×C151—21no—
141C22×C151—21no—
142C22×C151#13712yescomplemented
143C22×C151C22×D411yescharacteristic
144C2×D151—21no—
145C2×D151—21no—
146C2×D151—21no—
147C2×D151—24no—
148C2×D151—21no—
149C2×D151—21no—
150C2×D151—44no—
151C2×D151—48no—
152C2×D151—42no—
153C2×D151—42no—
154C2×D151#12012yes—
155C4×C151—24no—
156C4×C151—21no—
157C4×C151—41no—
158C4×C151—41no—
159C4×C151C22×D412yes—
160Dic151—21no—
161Dic151—41no—
162Dic151—41no—
163Dic151#12012yes—
164C2×C151—22no—
165C2×C151C2×#12011yescharacteristic
166C2×C151C2×#13711yescharacteristic
167C2×C151#57511yescharacteristic
168D151—22no—
169D151—41no—
170D151—44no—
171C151#300711yescharacteristic, complemented, Sylow
172#3007—1511nomaximal, Sylow
173C22×#41—1511no—
174C2×#137—1512no—
175C2×C22⋊Q8—1511no—
176#452—1512no—
177#452—1512no—
178#451—1511no—
179#451D15111yescharacteristic, complemented
180#450—1511no—
181#450—1511no—
182#417—1512no—
183#407—1511no—
184C2×#41—3028no—
185C2×#41—3021no—
186C2×#41—3021no—
187C2×#41—1512no—
188C2×#41—1512no—
189C22×Q8—1511no—
190C22×Q8C2×D15111yescharacteristic
191C22×D4—3021no—
192C22×D4—1511no—
193C23×C4—3021no—
194C23×C4—1511no—
195#137—3028no—
196C2×Q16—1512no—
197C2×Q16C2×D15112yescomplemented
198C2×SD16—1512no—
199C2×SD16—1512no—
200C2×M16—1512no—
201C22⋊Q8—3021no—
202C22⋊Q8—3021no—
203C22⋊Q8—1511no—
204C22⋊Q8—1511no—
205C22⋊Q8—1511no—
206C22⋊Q8C2×D15111yescharacteristic
207C2×C4⋊C4—3021no—
208C2×C4⋊C4—1511no—
209C2×C22⋊C4—3021no—
210#103—1512no—
211#103C2×D15112yescomplemented
212#102—1512no—
213#102—1512no—
214C22⋊C8—1511no—
215C22⋊C8—1512no—
216C22⋊C8C2×D15111yescharacteristic
217C24—3021no—
218C24—1511no—
219#41—3028no—
220#41—3028no—
221#41—3024no—
222#41—3024no—
223#41—3024no—
224#41—1518no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—3021no—
228C2×Q8—3024no—
229C2×Q8—1511no—
230C2×Q8C22×D15111yescharacteristic
231C2×Q8C22×D15112yes—
232C2×D4—3028no—
233C2×D4—3022no—
234C2×D4—3022no—
235C2×D4—3024no—
236C2×D4—3021no—
237C2×D4—1512no—
238C22×C4—21no—
239C22×C4—3028no—
240C22×C4—3022no—
241C22×C4—3022no—
242C22×C4—3024no—
243C22×C4—3021no—
244C22×C4—3021no—
245C22×C4—1511no—
246C22×C4—1511no—
247C22×C4—1511no—
248C22×C4—1511no—
249C22×C4—1511no—
250C22×C4C22×D15111yescharacteristic
251Q16—24no—
252Q16—3024no—
253SD16—3024no—
254SD16—3024no—
255M16—3024no—
256C2×C8—1512no—
257C2×C8C22×D15112yes—
258C4⋊C4—21no—
259C4⋊C4—3021no—
260C4⋊C4—3021no—
261C4⋊C4—3021no—
262C4⋊C4—3021no—
263C4⋊C4—1511no—
264C4⋊C4C22×D15111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3021no—
267C22⋊C4—3021no—
268C22⋊C4—3021no—
269C23—3024no—
270C23—3022no—
271C23—3022no—
272C23—3022no—
273C23—6041no—
274C23—6042no—
275C23—6042no—
276C23—1511no—
277C23—1511no—
278C23D4×D15111yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—3022no—
283Q8—6042no—
284Q8D4×D15114yes—
285D4—3028no—
286D4—3024no—
287D4—3028no—
288D4—3022no—
289D4—1514no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3028no—
296C2×C4—3022no—
297C2×C4—3028no—
298C2×C4—3022no—
299C2×C4—3022no—
300C2×C4—3022no—
301C2×C4—3021no—
302C2×C4—3021no—
303C2×C4—6042no—
304C2×C4—6042no—
305C2×C4—1511no—
306C2×C4—1511no—
307C2×C4—1514no—
308C2×C4D4×D15111yescharacteristic
309C2×C4C23×D15111yescharacteristic
310C8—22no—
311C8—3022no—
312C22—21no—
313C22—21no—
314C22—3021no—
315C22—3021no—
316C22—3021no—
317C22—3021no—
318C22—3021no—
319C22—3024no—
320C22—6044no—
321C22—6048no—
322C22—6042no—
323C22—6042no—
324C22—1512no—
325C22#1801277112yescomplemented
326C22C2×D4×D15111yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—3021no—
332C4—6041no—
333C4—6041no—
334C4—1512no—
335C4C2×D4×D15112yes—
336C2—22no—
337C2—3022no—
338C2—6041no—
339C2—6044no—
340C2#120×D15111yescharacteristic
341C2C2×#1801277111yescharacteristic
342C2#2375301111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#451×C151C2
socleC22×C151C22×D4
prime radicalC22×#14559005C2
hypercenter#451D151
hypocenterC151#3007
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×#18012771C2semidirect
3. C22⋊Q8×D151C2semidirect
4. #23752377C2semidirect
5. #23752373C2semidirect
6. #23752368C2semidirect
7. #23752365C2semidirect
8. #23752362C2semidirect
9. #23752212C2semidirect
10. #23752174C2semidirect
11. #451×C151C2semidirect
17. C2×Q8×D151C22semidirect
23. C2×#14558994C22semidirect
24. C2×#14558993C22semidirect
25. C2×#14558991C22semidirect
26. C2×#14558987C22semidirect
27. #18012719C22semidirect
31. #18012715C22semidirect
33. C4⋊C4×D151C22semidirect
35. #18012663C22semidirect
36. #18012661C22semidirect
37. #18012659C22semidirect
39. (C2×D151)⋊C8C22semidirect
41. C2×Q16×C151C22semidirect
43. #103×C151C22semidirect
142. C22×C151#137semidirect
171. C151#3007semidirect
179. #451D151semidirect
197. C2×Q16C2×D151semidirect
211. #103C2×D151semidirect
325. C22#18012771semidirect

Maximal subgroups

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

subgroupindex
C22×#145590052
C2×#180127712
C22⋊Q8×D1512
#237523772
#237523732
#237523682
#237523652
#237523622
#237522122
#237521742
#451×C1512
#3007151