Finite Group Databasebeta

← #34079913

Subgroups — #34079913

Order 19328.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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, complemented
2C2×#18012710C211yesmaximal, characteristic
3#23752548C211yesmaximal, characteristic, complemented
4#23752545C211yesmaximal, characteristic, complemented
5#23752524C212yesmaximal, complemented
6#23752457C212yesmaximal, complemented
7#23752455C212yesmaximal, complemented
8#23752363C211yesmaximal, characteristic, complemented
9#23752349C212yesmaximal, complemented
10#23752174C211yesmaximal, characteristic, complemented
11#450×C151C211yesmaximal, characteristic, complemented
12C2×Q8×D151C2211yescharacteristic, complemented
13C22×C22⋊D151—21no—
14C22×C4×D151C2211yescharacteristic
15C2×#14558992C2212yescomplemented
16C2×#14558987C2212yescomplemented
17#18012719C2211yescharacteristic, complemented
18#18012718C2211yescharacteristic
19#18012717—21no—
20#18012716—21no—
21#18012715C2211yescharacteristic, complemented
22#18012710—21no—
23#18012710—21no—
24#18012710C2212yes—
25#18012710C2212yes—
26C2×C4⋊Dic151C2211yescharacteristic
27C4⋊C4×D151—21no—
28C4⋊C4×D151C2211yescharacteristic, complemented
29C2×C22⋊Dic151—21no—
30C22⋊C4×D151—21no—
31#18012673C2212yescomplemented
32#18012672C2212yescomplemented
33#18012664C2212yescomplemented
34#18012663C2212yescomplemented
35#18012660C2212yescomplemented
36C22⋊(C151⋊C8)C2211yescharacteristic
37(C2×D151)⋊C8C2212yescomplemented
38C22×D4×C151C2211yescharacteristic
39C2×SD16×C151C2212yescomplemented
40C22⋊Q8×C151C2211yescharacteristic
41#102×C151C2212yescomplemented
42C22⋊C8×C151C2211yescharacteristic
43C23×D151D411yescharacteristic
44C2×Dic302—21no—
45C2×Dic302C2311yescharacteristic
46Q8×D151—42no—
47C2×C22⋊D151—22no—
48C2×C22⋊D151—22no—
49C2×C22⋊D151—44no—
50C22×Dic151—21no—
51C22×Dic151D411yescharacteristic
52C2×C4×D151—21no—
53C2×C4×D151—21no—
54C2×C4×D151—42no—
55C2×C4×D151D411yescharacteristic
56C2×C4×D151D411yescharacteristic
57C2×C4×D151C2311yescharacteristic
58C2×C4×D151C2311yescharacteristic
59#14558992—42no—
60#14558987—42no—
61C2×C151⋊C8C2312yes—
62C4⋊Dic151—21no—
63C4⋊Dic151—21no—
64C4⋊Dic151C2312yes—
65C4⋊Dic151C2311yescharacteristic
66Dic151⋊C4—21no—
67Dic151⋊C4—21no—
68Dic151⋊C4—21no—
69C22⋊Dic151—22no—
70C22⋊Dic151—22no—
71C22⋊Dic151—21no—
72(C2×D151)⋊C4—21no—
73(C2×D151)⋊C4—21no—
74C24×C151—21no—
75C2×Q8×C151C2311yescharacteristic
76C2×D4×C151—21no—
77C2×D4×C151—42no—
78C2×D4×C151C2312yes—
79C22×C4×C151C2311yescharacteristic
80SD16×C151—42no—
81C2×C8×C151C2312yes—
82C4⋊C4×C151—21no—
83C4⋊C4×C151C2311yescharacteristic
84C22⋊C4×C151—21no—
85C22×D151—22no—
86C22×D151—22no—
87C22×D151—41no—
88C22×D151C2×D411yescharacteristic
89C22×D151C2×D411yescharacteristic
90Dic302—22no—
91Dic302—42no—
92C22⋊D151—44no—
93C22⋊D151—44no—
94C2×Dic151—21no—
95C2×Dic151—21no—
96C2×Dic151—22no—
97C2×Dic151—41no—
98C2×Dic151—42no—
99C2×Dic151C2×D411yescharacteristic
100C2×Dic151C2×D411yescharacteristic
101C4×D151—22no—
102C4×D151—22no—
103C4×D151—41no—
104C4×D151—41no—
105C4×D151—42no—
106C4×D151—42no—
107C151⋊C8—22no—
108C23×C151—22no—
109C23×C151—42no—
110C23×C151—42no—
111C23×C151—42no—
112C23×C151C2×D411yescharacteristic
113Q8×C151—22no—
114D4×C151—22no—
115D4×C151—22no—
116D4×C151—42no—
117C2×C4×C151—22no—
118C2×C4×C151—21no—
119C2×C4×C151—21no—
120C2×C4×C151C2×D411yescharacteristic
121C2×C4×C151C2411yescharacteristic
122C8×C151—22no—
123C22×C151—21no—
124C22×C151—21no—
125C22×C151—42no—
126C22×C151—42no—
127C22×C151—42no—
128C22×C151—84no—
129C22×C151#13712yes—
130C22×C151C22×D411yescharacteristic
131C2×D151—21no—
132C2×D151—21no—
133C2×D151—21no—
134C2×D151—21no—
135C2×D151—21no—
136C2×D151—44no—
137C2×D151AGL(1,ℤ/8)12yes—
138C4×C151—21no—
139C4×C151—41no—
140C4×C151—41no—
141C4×C151H(5,𝔽2)12yes—
142Dic151—21no—
143Dic151—21no—
144Dic151—41no—
145Dic151—41no—
146Dic151—42no—
147C2×C151—22no—
148C2×C151—82no—
149C2×C151#53411yescharacteristic
150C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C151C2×#13711yescharacteristic
152D151—22no—
153D151—41no—
154C151#319311yescharacteristic, complemented, Sylow
155#3193—1511nomaximal, Sylow
156C2×C22⋊Q8—1511no—
157C2×#121—1511no—
158#484—1511no—
159#484—1511no—
160#480—1512no—
161#468—1512no—
162#468—1512no—
163#450—1511no—
164#450D15111yescharacteristic, complemented
165#448—1512no—
166#407—1511no—
167C22×Q8—1511no—
168C22×D4—3021no—
169C22×D4C2×D15111yescharacteristic
170C23×C4—1511no—
171C2×SD16—1512no—
172C2×SD16C2×D15112yescomplemented
173C2×M16—1512no—
174C22⋊Q8—3021no—
175C22⋊Q8—3021no—
176C22⋊Q8—1511no—
177C22⋊Q8—1511no—
178C22⋊Q8—1511no—
179C22⋊Q8C2×D15111yescharacteristic
180#121—3021no—
181#121—3021no—
182#121—1512no—
183#121—1512no—
184C2×C4⋊C4—3021no—
185C2×C4⋊C4—1511no—
186C2×C4⋊C4—1511no—
187C2×C22⋊C4—3021no—
188C2×C22⋊C4—3021no—
189#107—1512no—
190#107—1512no—
191#103—1512no—
192#103—1512no—
193#102—1512no—
194#102C2×D15112yescomplemented
195C22⋊C8—1511no—
196C22⋊C8—1512no—
197C22⋊C8C2×D15111yescharacteristic
198C24—21no—
199C24—1511no—
200C2×Q8—3021no—
201C2×Q8—6042no—
202C2×Q8—1511no—
203C2×Q8C22×D15111yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—3022no—
207C2×D4—3022no—
208C2×D4—6044no—
209C2×D4C22×D15112yes—
210C22×C4—3021no—
211C22×C4—3021no—
212C22×C4—3021no—
213C22×C4—6042no—
214C22×C4—1511no—
215C22×C4—1511no—
216C22×C4—1511no—
217C22×C4—1511no—
218C22×C4—1511no—
219C22×C4C22×D15111yescharacteristic
220SD16—42no—
221SD16—6042no—
222M16—6042no—
223C2×C8—1512no—
224C2×C8C22×D15112yes—
225C4⋊C4—21no—
226C4⋊C4—3021no—
227C4⋊C4—3021no—
228C4⋊C4—3021no—
229C4⋊C4—3021no—
230C4⋊C4—3021no—
231C4⋊C4—1511no—
232C4⋊C4—1512no—
233C4⋊C4C22×D15111yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—3021no—
236C22⋊C4—3021no—
237C22⋊C4—3021no—
238C22⋊C4—3022no—
239C22⋊C4—3022no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—3022no—
245C23—3022no—
246C23—6041no—
247C23—1511no—
248C23—1511no—
249C23D4×D15111yescharacteristic
250Q8—22no—
251Q8—3022no—
252Q8—6042no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—6044no—
257D4—6044no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—3022no—
262C2×C4—3022no—
263C2×C4—3021no—
264C2×C4—3021no—
265C2×C4—3022no—
266C2×C4—6041no—
267C2×C4—6041no—
268C2×C4—6042no—
269C2×C4—6042no—
270C2×C4—6041no—
271C2×C4—6042no—
272C2×C4—1511no—
273C2×C4—1511no—
274C2×C4D4×D15111yescharacteristic
275C2×C4C23×D15111yescharacteristic
276C8—22no—
277C8—3022no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—3021no—
285C22—3021no—
286C22—3021no—
287C22—3021no—
288C22—3021no—
289C22—6044no—
290C22—1512no—
291C22#1801276812yes—
292C22C2×D4×D15111yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—3021no—
297C4—3021no—
298C4—6042no—
299C4—6041no—
300C4—6041no—
301C4#1801278412yes—
302C2—22no—
303C2—82no—
304C2—3022no—
305C2—6041no—
306C2#2375277211yescharacteristic
307C2AGL(1,ℤ/8)×D15111yescharacteristic
308C2C2×#1801276811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
prime radicalC2×#18012710C2
hypercenter#450D151
hypocenterC151#3193
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⋊Q8×D151C2semidirect
3. #23752548C2semidirect
4. #23752545C2semidirect
5. #23752524C2semidirect
6. #23752457C2semidirect
7. #23752455C2semidirect
8. #23752363C2semidirect
9. #23752349C2semidirect
10. #23752174C2semidirect
11. #450×C151C2semidirect
12. C2×Q8×D151C22semidirect
15. C2×#14558992C22semidirect
16. C2×#14558987C22semidirect
17. #18012719C22semidirect
21. #18012715C22semidirect
28. C4⋊C4×D151C22semidirect
31. #18012673C22semidirect
32. #18012672C22semidirect
33. #18012664C22semidirect
34. #18012663C22semidirect
35. #18012660C22semidirect
37. (C2×D151)⋊C8C22semidirect
39. C2×SD16×C151C22semidirect
41. #102×C151C22semidirect
154. C151#3193semidirect
164. #450D151semidirect
172. C2×SD16C2×D151semidirect
194. #102C2×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×#180127102
#237525482
#237525452
#237525242
#237524572
#237524552
#237523632
#237523492
#237521742
#450×C1512
#3193151