Finite Group Databasebeta

← #34078329

Subgroups — #34078329

Order 19328.

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

normal subgroups only

This group has 764 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×#14559003C211yesmaximal, characteristic, complemented
2C2×#18012760C212yesmaximal, complemented
3#23752725C211yesmaximal, characteristic, complemented
4#23752371C212yesmaximal, complemented
5#23752367C211yesmaximal, characteristic, complemented
6#23752358C212yesmaximal, complemented
7#23752353C211yesmaximal, characteristic, complemented
8#23752346C211yesmaximal, characteristic, complemented
9#23752205C212yesmaximal, complemented
10C22⋊C8×D151C211yesmaximal, characteristic, complemented
11#450×C151C211yesmaximal, characteristic, complemented
12C2×#14559005C2211yescharacteristic, complemented
13C2×#14559003—28no—
14C2×#14559003—21no—
15C2×#14559003—21no—
16C2×#14559003C2212yes—
17C2×#14559003C2212yes—
18C22×Dic302C2211yescharacteristic
19C22×C22⋊D151—21no—
20C23×Dic151—21no—
21C22×C4×D151C2211yescharacteristic
22#18012760—28no—
23C2×#14558994C2212yescomplemented
24C2×#14558993C2212yescomplemented
25C2×#14558989C2212yescomplemented
26C2×C8×D151C2212yescomplemented
27#18012724—21no—
28#18012715C2211yescharacteristic, complemented
29#18012714C2211yescharacteristic, complemented
30#18012713C2211yescharacteristic
31#18012709—21no—
32#18012701—21no—
33(C22⋊D151)⋊C4—21no—
34#18012697C2211yescharacteristic, complemented
35#18012664C2212yescomplemented
36#18012662C2212yescomplemented
37#18012660C2212yescomplemented
38C22⋊(C151⋊C8)C2211yescharacteristic
39(C2×D151)⋊C8C2212yescomplemented
40C22×D4×C151C2211yescharacteristic
41C2×SD16×C151C2212yescomplemented
42C22⋊Q8×C151C2211yescharacteristic
43#102×C151C2212yescomplemented
44C22⋊C8×C151C2211yescharacteristic
45C23×D151D411yescharacteristic
46#14559005—24no—
47#14559003—28no—
48#14559003—28no—
49#14559003—24no—
50#14559003—24no—
51#14559003—24no—
52#14559003D418yescomplemented
53C2×Dic302—24no—
54C2×Dic302—21no—
55C2×Dic302C2312yes—
56C2×D604—21no—
57C2×D604—21no—
58C2×D604C2311yescharacteristic
59C2×C22⋊D151—28no—
60C2×C22⋊D151—22no—
61C2×C22⋊D151—22no—
62C2×C22⋊D151—21no—
63C22×Dic151—28no—
64C22×Dic151—22no—
65C22×Dic151—22no—
66C22×Dic151—21no—
67C22×Dic151D411yescharacteristic
68C2×C4×D151—24no—
69C2×C4×D151—21no—
70C2×C4×D151D411yescharacteristic
71C2×C4×D151D411yescharacteristic
72C2×C4×D151C2311yescharacteristic
73C2×C4×D151C2311yescharacteristic
74#14558994—24no—
75#14558993—24no—
76#14558989—24no—
77C2×C151⋊C8C2312yes—
78C8×D151—24no—
79C4⋊Dic151C2311yescharacteristic
80Dic151⋊C4—21no—
81(C2×D151)⋊C4—21no—
82(C2×D151)⋊C4—21no—
83(C2×D151)⋊C4—21no—
84(C2×D151)⋊C4—21no—
85C4×Dic151—21no—
86C24×C151—21no—
87C2×Q8×C151C2311yescharacteristic
88C2×D4×C151—24no—
89C2×D4×C151—21no—
90C2×D4×C151C2312yes—
91C22×C4×C151C2311yescharacteristic
92SD16×C151—24no—
93C2×C8×C151C2312yes—
94C4⋊C4×C151—21no—
95C4⋊C4×C151C2311yescharacteristic
96C22⋊C4×C151—21no—
97C22×D151—22no—
98C22×D151—22no—
99C22×D151—21no—
100C22×D151—41no—
101C22×D151C2×D411yescharacteristic
102C22×D151C2×D411yescharacteristic
103Dic302—24no—
104Dic302—22no—
105Dic302C2×D414yes—
106D604—22no—
107D604—42no—
108D604—42no—
109C22⋊D151—28no—
110C22⋊D151—28no—
111C22⋊D151—42no—
112C2×Dic151—24no—
113C2×Dic151—28no—
114C2×Dic151—22no—
115C2×Dic151—22no—
116C2×Dic151—28no—
117C2×Dic151—21no—
118C2×Dic151C2×D411yescharacteristic
119C2×Dic151C2×D411yescharacteristic
120C4×D151—22no—
121C4×D151—22no—
122C4×D151—22no—
123C4×D151—42no—
124C4×D151C2×D414yes—
125C151⋊C8—22no—
126C23×C151—24no—
127C23×C151—22no—
128C23×C151—42no—
129C23×C151—42no—
130C23×C151C2×D411yescharacteristic
131Q8×C151—22no—
132D4×C151—24no—
133D4×C151—22no—
134D4×C151C2×D414yes—
135C2×C4×C151—22no—
136C2×C4×C151—21no—
137C2×C4×C151—21no—
138C2×C4×C151C2×D411yescharacteristic
139C2×C4×C151C2411yescharacteristic
140C8×C151—22no—
141C22×C151—21no—
142C22×C151—21no—
143C22×C151—24no—
144C22×C151—48no—
145C22×C151—42no—
146C22×C151—42no—
147C22×C151#13512yescomplemented
148C22×C151C22×D411yescharacteristic
149C2×D151—21no—
150C2×D151—21no—
151C2×D151—21no—
152C2×D151—21no—
153C2×D151—21no—
154C2×D151—44no—
155C2×D151—41no—
156C2×D151—41no—
157C2×D151—41no—
158C2×D151#12012yes—
159C4×C151—21no—
160C4×C151—41no—
161C4×C151—41no—
162C4×C151C22×D412yes—
163Dic151—21no—
164Dic151—24no—
165Dic151—41no—
166Dic151#12012yes—
167C2×C151—22no—
168C2×C151—44no—
169C2×C151C2×#12011yescharacteristic
170C2×C151C2×#13511yescharacteristic
171C2×C151#57511yescharacteristic
172D151—22no—
173D151—41no—
174D151—81no—
175C151#300211yescharacteristic, complemented, Sylow
176#3002—1511nomaximal, Sylow
177C22×#41—1511no—
178C2×#135—1512no—
179#525—1511no—
180#452—1512no—
181#451—1511no—
182#450D15111yescharacteristic, complemented
183#449—1512no—
184#448—1511no—
185#447—1511no—
186#416—1512no—
187C2×C22⋊C8—1511no—
188C2×#41—3028no—
189C2×#41—3021no—
190C2×#41—3021no—
191C2×#41—1511no—
192C2×#41—1512no—
193C2×#41—1512no—
194C22×Q8—1511no—
195C22×D4—3021no—
196C22×D4C2×D15111yescharacteristic
197C23×C4—3021no—
198C23×C4—1511no—
199#135—3028no—
200C2×Q16—1512no—
201C2×SD16—1512no—
202C2×SD16C2×D15112yescomplemented
203C2×D8—1512no—
204C22×C8—1512no—
205#123—3021no—
206C22⋊Q8—1511no—
207C22⋊Q8C2×D15111yescharacteristic
208#121—1511no—
209#121—1511no—
210#120—3021no—
211C4×D4—3021no—
212C4×D4—3021no—
213#117—1511no—
214#103—1512no—
215#103—1512no—
216#102—1512no—
217#102C2×D15112yescomplemented
218C22⋊C8—1511no—
219C22⋊C8—1512no—
220C22⋊C8C2×D15111yescharacteristic
221C24—21no—
222C24—1511no—
223#41—3028no—
224#41—3028no—
225#41—3024no—
226#41—3024no—
227#41—3024no—
228#41—3024no—
229#41—1518no—
230C2×Q8—3024no—
231C2×Q8—3021no—
232C2×Q8—1512no—
233C2×Q8C22×D15111yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—3021no—
237C2×D4—3021no—
238C2×D4—3028no—
239C2×D4—3022no—
240C2×D4—3022no—
241C2×D4—3021no—
242C2×D4—1511no—
243C2×D4C22×D15112yes—
244C22×C4—3024no—
245C22×C4—3021no—
246C22×C4—3028no—
247C22×C4—3021no—
248C22×C4—3022no—
249C22×C4—3022no—
250C22×C4—1511no—
251C22×C4—1511no—
252C22×C4—1511no—
253C22×C4—1511no—
254C22×C4—1511no—
255C22×C4C22×D15111yescharacteristic
256Q16—3024no—
257SD16—24no—
258SD16—3024no—
259D8—3024no—
260C2×C8—3024no—
261C2×C8—1512no—
262C2×C8C22×D15112yes—
263C4⋊C4—21no—
264C4⋊C4—3021no—
265C4⋊C4—1511no—
266C4⋊C4C22×D15111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3021no—
269C22⋊C4—3021no—
270C22⋊C4—3021no—
271C22⋊C4—3021no—
272C42—3021no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—3022no—
278C23—3022no—
279C23—3021no—
280C23—6041no—
281C23—1511no—
282C23—1511no—
283C23D4×D15111yescharacteristic
284Q8—22no—
285Q8—3024no—
286Q8—3022no—
287Q8—1514no—
288D4—24no—
289D4—22no—
290D4—3028no—
291D4—3028no—
292D4—3022no—
293D4—6042no—
294D4—6042no—
295D4—6042no—
296D4D4×D15114yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3022no—
301C2×C4—3022no—
302C2×C4—3024no—
303C2×C4—3028no—
304C2×C4—3022no—
305C2×C4—3022no—
306C2×C4—3028no—
307C2×C4—3022no—
308C2×C4—3021no—
309C2×C4—6042no—
310C2×C4—1511no—
311C2×C4—1511no—
312C2×C4—1514no—
313C2×C4D4×D15111yescharacteristic
314C2×C4C23×D15111yescharacteristic
315C8—22no—
316C8—3022no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—3021no—
324C22—3021no—
325C22—3021no—
326C22—3021no—
327C22—3021no—
328C22—6044no—
329C22—6041no—
330C22—6041no—
331C22—6041no—
332C22—1512no—
333C22#1801276012yescomplemented
334C22C2×D4×D15111yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—3021no—
339C4—3024no—
340C4—6041no—
341C4—1512no—
342C4C2×D4×D15112yes—
343C2—22no—
344C2—44no—
345C2—3022no—
346C2—6041no—
347C2—12081no—
348C2#120×D15111yescharacteristic
349C2C2×#1801276011yescharacteristic
350C2#2375300811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
hypercenter#450D151
hypocenterC151#3002
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14559003C2semidirect
2. C2×#18012760C2semidirect
3. #23752725C2semidirect
4. #23752371C2semidirect
5. #23752367C2semidirect
6. #23752358C2semidirect
7. #23752353C2semidirect
8. #23752346C2semidirect
9. #23752205C2semidirect
10. C22⋊C8×D151C2semidirect
11. #450×C151C2semidirect
12. C2×#14559005C22semidirect
23. C2×#14558994C22semidirect
24. C2×#14558993C22semidirect
25. C2×#14558989C22semidirect
26. C2×C8×D151C22semidirect
28. #18012715C22semidirect
29. #18012714C22semidirect
34. #18012697C22semidirect
35. #18012664C22semidirect
36. #18012662C22semidirect
37. #18012660C22semidirect
39. (C2×D151)⋊C8C22semidirect
41. C2×SD16×C151C22semidirect
43. #102×C151C22semidirect
52. #14559003D4semidirect
147. C22×C151#135semidirect
175. C151#3002semidirect
182. #450D151semidirect
202. C2×SD16C2×D151semidirect
217. #102C2×D151semidirect
333. C22#18012760semidirect

Maximal subgroups

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

subgroupindex
C22×#145590032
C2×#180127602
#237527252
#237523712
#237523672
#237523582
#237523532
#237523462
#237522052
C22⋊C8×D1512
#450×C1512
#3002151