Finite Group Databasebeta

← #34078904

Subgroups — #34078904

Order 19328.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#23752765C211yesmaximal, characteristic, complemented
2C2×D4×Dic151C211yesmaximal, characteristic
3#23752514C212yesmaximal, complemented
4#23752510C212yesmaximal, complemented
5#23752500C212yesmaximal, complemented
6#23752490C212yesmaximal, complemented
7#23752255C211yesmaximal, characteristic, complemented
8#23752254C211yesmaximal, characteristic, complemented
9#23752235C211yesmaximal, characteristic, complemented
10#23752217C211yesmaximal, characteristic
11#418×C151C211yesmaximal, characteristic
12C23×Dic151—21no—
13C2×#14558988C2211yescharacteristic
14#18012738C2211yescharacteristic, complemented
15#18012738C2211yescharacteristic, complemented
16#18012730C2211yescharacteristic, complemented
17#18012730C2211yescharacteristic, complemented
18#18012716—21no—
19#18012716—21no—
20C4×Dic302—21no—
21C4×Dic302—21no—
22D4×Dic151—21no—
23D4×Dic151—21no—
24D4×Dic151C2212yes—
25D4×Dic151C2212yes—
26#18012698C2211yescharacteristic
27C2×C4⋊Dic151C2211yescharacteristic
28C2×C22⋊Dic151—21no—
29C2×C4×Dic151C2211yescharacteristic
30#18012673C2212yescomplemented
31#18012672C2212yescomplemented
32#18012671C2212yescomplemented
33#18012670C2212yescomplemented
34Dic151⋊C8C2212yescomplemented
35Dic151⋊C8C2212yescomplemented
36#18012660C2212yes—
37#18012660C2212yes—
38C22×D4×C151C2211yescharacteristic
39C2×M16×C151C2211yescharacteristic
40#117×C151C2211yescharacteristic
41#102×C151C2212yes—
42#102×C151C2212yes—
43C2×Dic302—21no—
44C2×Dic302—21no—
45C22×Dic151—22no—
46C22×Dic151—22no—
47C22×Dic151—21no—
48C22×Dic151—44no—
49C22×Dic151D411yescharacteristic
50C22×Dic151D411yescharacteristic
51#14558988—22no—
52C2×C151⋊C8C2312yes—
53C4⋊Dic151—21no—
54C4⋊Dic151—21no—
55C4⋊Dic151—21no—
56C4⋊Dic151C2312yes—
57C4⋊Dic151C2311yescharacteristic
58C4⋊Dic151C2311yescharacteristic
59Dic151⋊C4—21no—
60Dic151⋊C4—21no—
61Dic151⋊C4—21no—
62Dic151⋊C4—21no—
63C22⋊Dic151—22no—
64C22⋊Dic151—22no—
65C22⋊Dic151—21no—
66C4×Dic151—21no—
67C4×Dic151D411yescharacteristic
68C4×Dic151D411yescharacteristic
69C4×Dic151C2311yescharacteristic
70C4×Dic151C2311yescharacteristic
71C24×C151—21no—
72C2×D4×C151—21no—
73C2×D4×C151—22no—
74C2×D4×C151Q814yescomplemented
75C2×D4×C151C2312yes—
76C22×C4×C151C2311yescharacteristic
77M16×C151—22no—
78C2×C8×C151C2312yes—
79C4⋊C4×C151C2311yescharacteristic
80C4⋊C4×C151C2311yescharacteristic
81C22⋊C4×C151—21no—
82C42×C151—21no—
83Dic302—42no—
84Dic302—42no—
85C2×Dic151—22no—
86C2×Dic151—21no—
87C2×Dic151—21no—
88C2×Dic151—22no—
89C2×Dic151—41no—
90C2×Dic151—44no—
91C2×Dic151—42no—
92C2×Dic151—44no—
93C2×Dic151C2×D411yescharacteristic
94C2×Dic151C2×D411yescharacteristic
95C2×Dic151C2×D411yescharacteristic
96C2×Dic151C2×D411yescharacteristic
97C151⋊C8—22no—
98C23×C151—24no—
99C23×C151—22no—
100C23×C151—42no—
101C23×C151—42no—
102C23×C151C2×D411yescharacteristic
103D4×C151—24no—
104D4×C151—22no—
105D4×C151C2×Q814yes—
106C2×C4×C151—21no—
107C2×C4×C151—21no—
108C2×C4×C151C2×D411yescharacteristic
109C2×C4×C151C2×Q812yes—
110C2×C4×C151#4112yes—
111C2×C4×C151C2411yescharacteristic
112C8×C151—22no—
113C22×C151—21no—
114C22×C151—21no—
115C22×C151—24no—
116C22×C151—48no—
117C22×C151—42no—
118C22×C151—42no—
119C22×C151C22⋊Q812yes—
120C22×C151C22×D411yescharacteristic
121C4×C151—41no—
122C4×C151—41no—
123C4×C151C22⋊Q812yes—
124C4×C151C22×Q811yescharacteristic
125C4×C151C2×#4111yescharacteristic
126Dic151—21no—
127Dic151—21no—
128Dic151—21no—
129Dic151—42no—
130Dic151—41no—
131Dic151—41no—
132Dic151AGL(1,ℤ/8)12yes—
133C2×C151—22no—
134C2×C151—44no—
135C2×C151C2×C22⋊Q811yescharacteristic
136C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C151#57511yescharacteristic
138C151#307711yescharacteristic, complemented, Sylow
139#3077—1511nomaximal, Sylow
140#533—1511no—
141C2×C4×D4—1511no—
142#478—1512no—
143#478—1512no—
144#476—1512no—
145#474—1512no—
146#428—1511no—
147#428—1511no—
148#423—1511no—
149#418—1511no—
150#418D15111yescharacteristic
151C22×D4C2×D15111yescharacteristic
152C23×C4—3021no—
153C2×M16—1511no—
154C2×M16C2×D15111yescharacteristic
155C4⋊Q8—1511no—
156C4⋊Q8—1511no—
157#125—1511no—
158#125—1511no—
159C22⋊Q8—3021no—
160C22⋊Q8—3021no—
161C4×Q8—3021no—
162C4×Q8—3021no—
163C4×D4—3021no—
164C4×D4—3021no—
165C4×D4—1512no—
166C4×D4—1512no—
167#117—1511no—
168#117C2×D15111yescharacteristic
169C2×C4⋊C4—1511no—
170C2×C22⋊C4—3021no—
171C2×C42—1511no—
172#107—1512no—
173#107—1512no—
174#106—1512no—
175#106—1512no—
176C4⋊C8—1512no—
177C4⋊C8—1512no—
178#102—1512no—
179#102—1512no—
180#102C2×D15112yes—
181#102C2×D15112yes—
182C24—21no—
183C2×Q8—3021no—
184C2×Q8—3021no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic30214yescomplemented
188C2×D4C22×D15112yes—
189C22×C4—3021no—
190C22×C4—3022no—
191C22×C4—3022no—
192C22×C4—6044no—
193C22×C4—1511no—
194C22×C4—1511no—
195C22×C4C22×D15111yescharacteristic
196M16—22no—
197M16—3022no—
198C2×C8—1512no—
199C2×C8C22×D15112yes—
200C4⋊C4—3021no—
201C4⋊C4—3021no—
202C4⋊C4—3021no—
203C4⋊C4—3021no—
204C4⋊C4—3021no—
205C4⋊C4—3021no—
206C4⋊C4—3021no—
207C4⋊C4—1511no—
208C4⋊C4—1511no—
209C4⋊C4—1512no—
210C4⋊C4C22×D15111yescharacteristic
211C4⋊C4C22×D15111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—3021no—
214C22⋊C4—3022no—
215C22⋊C4—3022no—
216C42—21no—
217C42—3021no—
218C42—1511no—
219C42—1511no—
220C42—1511no—
221C42—1511no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D15111yescharacteristic
227Q8—6042no—
228Q8—6042no—
229D4—24no—
230D4—22no—
231D4C2×Dic30214yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—3022no—
235C2×C4—3021no—
236C2×C4—3021no—
237C2×C4—3022no—
238C2×C4—6041no—
239C2×C4—6044no—
240C2×C4—6042no—
241C2×C4—6044no—
242C2×C4—1511no—
243C2×C4—1511no—
244C2×C4—1511no—
245C2×C4—1511no—
246C2×C4D4×D15111yescharacteristic
247C2×C4C2×Dic30212yes—
248C2×C4#1455900312yes—
249C2×C4C23×D15111yescharacteristic
250C8—22no—
251C8—3022no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1801271612yes—
259C22C2×D4×D15111yescharacteristic
260C4—41no—
261C4—41no—
262C4—3021no—
263C4—3021no—
264C4—3021no—
265C4—6042no—
266C4—6041no—
267C4—6041no—
268C4—1512no—
269C4#1801271612yes—
270C4C22×Dic30211yescharacteristic
271C4C2×#1455900311yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1801271611yescharacteristic
275C2AGL(1,ℤ/8)×D15111yescharacteristic
276C2#2375300811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#418×C151C2
socleC22×C151C22×D4
prime radicalC22×D4×C151C22
hypercenter#418D151
hypocenterC151#3077
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #23752765C2semidirect
3. #23752514C2semidirect
4. #23752510C2semidirect
5. #23752500C2semidirect
6. #23752490C2semidirect
7. #23752255C2semidirect
8. #23752254C2semidirect
9. #23752235C2semidirect
14. #18012738C22semidirect
15. #18012738C22semidirect
16. #18012730C22semidirect
17. #18012730C22semidirect
30. #18012673C22semidirect
31. #18012672C22semidirect
32. #18012671C22semidirect
33. #18012670C22semidirect
34. Dic151⋊C8C22semidirect
35. Dic151⋊C8C22semidirect
74. C2×D4×C151Q8semidirect
138. C151#3077semidirect
187. C2×D4Dic302semidirect

Maximal subgroups

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

subgroupindex
#237527652
C2×D4×Dic1512
#237525142
#237525102
#237525002
#237524902
#237522552
#237522542
#237522352
#237522172
#418×C1512
#3077151