Finite Group Databasebeta

← #34079573

Subgroups — #34079573

Order 19328.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#18012771C212yesmaximal, complemented
2C2×#18012766C212yesmaximal, complemented
3#23752763C211yesmaximal, characteristic, complemented
4#124×D151C211yesmaximal, characteristic, complemented
5#23752604C211yesmaximal, characteristic, complemented
6#23752602C211yesmaximal, characteristic, complemented
7#23752601C211yesmaximal, characteristic, complemented
8#23752597C212yesmaximal, complemented
9#23752593C212yesmaximal, complemented
10#23752164C211yesmaximal, characteristic, complemented
11#497×C151C211yesmaximal, characteristic, complemented
12C2×#14559005C2212yes—
13C2×Q8×D151C2211yescharacteristic
14C2×D4×D151C2211yescharacteristic
15#18012771—28no—
16#18012766—28no—
17C2×#14558994C2212yescomplemented
18C2×#14558993C2212yescomplemented
19C2×#14558991C2212yes—
20C2×#14558991C2212yescomplemented
21C2×D1208C2212yes—
22C2×#14558989C2212yes—
23C2×#14558987C2212yes—
24#18012736C2211yescharacteristic, complemented
25#18012728C2211yescharacteristic
26#18012728C2212yescomplemented
27#18012727C2211yescharacteristic
28#18012726C2211yescharacteristic
29#18012725—22no—
30#18012714—22no—
31#18012701—22no—
32C22⋊C4×D151—22no—
33C42×D151C2211yescharacteristic
34#18012650C2211yescharacteristic, complemented
35C8×Dic151C2212yescomplemented
36C2×Q16×C151C2212yescomplemented
37C2×SD16×C151C2212yes—
38C4⋊Q8×C151C2211yescharacteristic, complemented
39#124×C151C2211yescharacteristic
40#97×C151C2211yescharacteristic, complemented
41C23×D151—21no—
42#14559005—24no—
43#14559005—24no—
44C2×Dic302—21no—
45C2×Dic302C2311yescharacteristic
46Q8×D151—24no—
47C2×D604—22no—
48C2×D604—22no—
49C2×D604C2311yescharacteristic
50C2×D604C2312yes—
51C2×C22⋊D151—21no—
52D4×D151—24no—
53C2×C4×D151—21no—
54C2×C4×D151—22no—
55C2×C4×D151D412yes—
56C2×C4×D151C2311yescharacteristic
57#14558994—24no—
58#14558993—24no—
59#14558991—24no—
60#14558991—24no—
61D1208—24no—
62#14558989—24no—
63#14558987D418yescomplemented
64C2×C151⋊C8C2312yes—
65C22⋊Dic151—22no—
66(C2×D151)⋊C4—22no—
67(C2×D151)⋊C4—24no—
68(C2×D151)⋊C4—22no—
69C4×Dic151D412yes—
70C4×Dic151C2311yescharacteristic
71C2×Q8×C151C2311yescharacteristic
72C2×Q8×C151C2312yes—
73C2×D4×C151C2311yescharacteristic
74Q16×C151—24no—
75SD16×C151—24no—
76C2×C8×C151C2312yes—
77C4⋊C4×C151—22no—
78C22⋊C4×C151—22no—
79C42×C151C2311yescharacteristic
80C22×D151—21no—
81C22×D151—22no—
82C22×D151—44no—
83C22×D151—42no—
84C22×D151C2×D411yescharacteristic
85Dic302—22no—
86Dic302—42no—
87D604—22no—
88D604—22no—
89D604—22no—
90D604—44no—
91D604—44no—
92C22⋊D151—42no—
93C2×Dic151—21no—
94C2×Dic151C2×D412yes—
95C2×Dic151C2×D411yescharacteristic
96C4×D151—42no—
97C4×D151—42no—
98C4×D151—44no—
99C4×D151C2×D414yes—
100C151⋊C8C2×D414yes—
101C23×C151—21no—
102Q8×C151—22no—
103Q8×C151—22no—
104Q8×C151—22no—
105D4×C151—22no—
106C2×C4×C151—22no—
107C2×C4×C151—21no—
108C2×C4×C151C2×D412yes—
109C2×C4×C151C2411yescharacteristic
110C8×C151C2×D414yes—
111C22×C151—41no—
112C22×C151—42no—
113C22×C151C22×D411yescharacteristic
114C2×D151—22no—
115C2×D151—42no—
116C2×D151—41no—
117C2×D151—44no—
118C2×D151—42no—
119C2×D151—84no—
120C2×D151#12712yes—
121C4×C151—22no—
122C4×C151—42no—
123C4×C151—41no—
124C4×C151C22×D412yes—
125Dic151—22no—
126Dic151—41no—
127Dic151#12712yes—
128C2×C151—81no—
129C2×C151C2×#12711yescharacteristic
130C2×C151#57512yes—
131D151—22no—
132D151—82no—
133D151—81no—
134C151#315311yescharacteristic, complemented, Sylow
135#3153—1511nomaximal, Sylow
136C2×#137—1512no—
137C2×AGL(1,ℤ/8)—1512no—
138#532—1511no—
139C2×#124—1511no—
140#497—1511no—
141#497D15111yescharacteristic, complemented
142#496—1511no—
143#496—1511no—
144#495—1512no—
145#495—1512no—
146C4×M16—1511no—
147C2×#41—1512no—
148C22×Q8—1511no—
149C22×D4—1511no—
150#137—3028no—
151AGL(1,ℤ/8)—3028no—
152C2×Q16—1512no—
153C2×Q16C2×D15112yescomplemented
154C2×SD16—1512no—
155C2×SD16—1512no—
156C2×SD16—1512no—
157C2×SD16C2×D15112yes—
158C2×D8—1512no—
159C2×D8—1512no—
160C2×M16—1512no—
161C4⋊Q8C2×D15111yescharacteristic, complemented
162#127—1511no—
163#124—3022no—
164#124—1511no—
165#124—1512no—
166#124—1511no—
167#124—1511no—
168#124C2×D15111yescharacteristic
169#121—3022no—
170C4×D4—3022no—
171C2×C22⋊C4—3022no—
172C2×C42—1511no—
173#97—1511no—
174#97C2×D15111yescharacteristic, complemented
175C4×C8—1512no—
176C24—3021no—
177#41—3024no—
178#41—3024no—
179C2×Q8—3021no—
180C2×Q8—3024no—
181C2×Q8—1511no—
182C2×Q8C22×D15111yescharacteristic
183C2×Q8C22×D15112yes—
184C2×D4—3022no—
185C2×D4—3024no—
186C2×D4—3021no—
187C2×D4—3022no—
188C2×D4—1511no—
189C2×D4—1512no—
190C2×D4C22×D15111yescharacteristic
191C22×C4—3021no—
192C22×C4—3022no—
193C22×C4—1512no—
194C22×C4—1511no—
195Q16—24no—
196Q16—3024no—
197SD16—24no—
198SD16—3024no—
199SD16—3024no—
200SD16—3024no—
201D8—3024no—
202D8—3024no—
203M16—1518no—
204C2×C8—1512no—
205C2×C8C22×D15112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—3022no—
209C22⋊C4—3024no—
210C22⋊C4—3022no—
211C22⋊C4—3022no—
212C42—1512no—
213C42—1511no—
214C42C22×D15111yescharacteristic
215C23—21no—
216C23—3021no—
217C23—3022no—
218C23—6044no—
219C23—6042no—
220C23—1511no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—3022no—
225Q8—6042no—
226D4—22no—
227D4—3022no—
228D4—3022no—
229D4—3022no—
230D4—6044no—
231D4—6044no—
232D4—6042no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—3021no—
236C2×C4—6042no—
237C2×C4—6042no—
238C2×C4—6044no—
239C2×C4—1512no—
240C2×C4—1511no—
241C2×C4—1514no—
242C2×C4D4×D15112yes—
243C2×C4C23×D15111yescharacteristic
244C8—1514no—
245C8D4×D15114yes—
246C22—41no—
247C22—42no—
248C22—3022no—
249C22—6042no—
250C22—6041no—
251C22—6044no—
252C22—6042no—
253C22—1512no—
254C22—12084no—
255C22C2×D4×D15111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—3022no—
260C4—6041no—
261C4—1512no—
262C4C2×D4×D15112yes—
263C2—81no—
264C2—3022no—
265C2—12081no—
266C2—12082no—
267C2#127×D15111yescharacteristic
268C2#2375301112yes—
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#497×C151C2
socleC22×C151C22×D4
hypercenter#497D151
hypocenterC151#3153
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#18012771C2semidirect
2. C2×#18012766C2semidirect
3. #23752763C2semidirect
4. #124×D151C2semidirect
5. #23752604C2semidirect
6. #23752602C2semidirect
7. #23752601C2semidirect
8. #23752597C2semidirect
9. #23752593C2semidirect
10. #23752164C2semidirect
11. #497×C151C2semidirect
17. C2×#14558994C22semidirect
18. C2×#14558993C22semidirect
20. C2×#14558991C22semidirect
24. #18012736C22semidirect
26. #18012728C22semidirect
34. #18012650C22semidirect
35. C8×Dic151C22semidirect
36. C2×Q16×C151C22semidirect
38. C4⋊Q8×C151C22semidirect
40. #97×C151C22semidirect
63. #14558987D4semidirect
134. C151#3153semidirect
141. #497D151semidirect
153. C2×Q16C2×D151semidirect
161. C4⋊Q8C2×D151semidirect
174. #97C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127712
C2×#180127662
#237527632
#124×D1512
#237526042
#237526022
#237526012
#237525972
#237525932
#237521642
#497×C1512
#3153151