Finite Group Databasebeta

← #34078906

Subgroups — #34078906

Order 19328.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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×#18012701C211yesmaximal, characteristic
3#23752515C212yesmaximal, complemented
4#23752511C212yesmaximal, complemented
5#23752501C212yesmaximal, complemented
6#23752491C212yesmaximal, complemented
7#23752252C211yesmaximal, characteristic, complemented
8#23752235C211yesmaximal, characteristic, complemented
9#23752218C211yesmaximal, characteristic
10#23752216C211yesmaximal, characteristic
11#428×C151C211yesmaximal, characteristic, complemented
12C22×D604C2211yescharacteristic
13C22×C4×D151—21no—
14C2×#14558988C2211yescharacteristic
15#18012738C2211yescharacteristic, complemented
16#18012738C2211yescharacteristic, complemented
17#18012730C2211yescharacteristic, complemented
18#18012730C2211yescharacteristic, complemented
19#18012716—21no—
20#18012716—21no—
21C4×Dic302—21no—
22C4×Dic302—21no—
23#18012701—21no—
24#18012701—21no—
25#18012701C2212yes—
26#18012701C2212yes—
27#18012698C2211yescharacteristic
28C2×(C2×D151)⋊C4—21no—
29C2×C4×Dic151C2211yescharacteristic
30#18012672C2212yescomplemented
31#18012670C2212yescomplemented
32Dic151⋊C8C2212yescomplemented
33Dic151⋊C8C2212yescomplemented
34#18012661C2212yes—
35#18012661C2212yes—
36#18012659C2212yes—
37#18012659C2212yes—
38C2×M16×C151C2211yescharacteristic
39#117×C151C2211yescharacteristic
40C2×C4⋊C4×C151C2211yescharacteristic
41#107×C151C2212yescomplemented
42#106×C151C2212yescomplemented
43C23×D151—21no—
44C2×Dic302—21no—
45C2×Dic302—21no—
46C2×D604—21no—
47C2×D604—22no—
48C2×D604Q814yescomplemented
49C2×D604C2312yes—
50C22×Dic151D411yescharacteristic
51C22×Dic151D411yescharacteristic
52C2×C4×D151—22no—
53C2×C4×D151—22no—
54C2×C4×D151—44no—
55#14558988—22no—
56C2×C151⋊C8C2312yes—
57C4⋊Dic151—21no—
58C4⋊Dic151—21no—
59C4⋊Dic151C2311yescharacteristic
60C4⋊Dic151C2311yescharacteristic
61Dic151⋊C4—21no—
62Dic151⋊C4—21no—
63Dic151⋊C4—21no—
64Dic151⋊C4—21no—
65C22⋊Dic151—21no—
66(C2×D151)⋊C4—22no—
67(C2×D151)⋊C4—22no—
68C4×Dic151—21no—
69C4×Dic151D411yescharacteristic
70C4×Dic151D411yescharacteristic
71C4×Dic151C2311yescharacteristic
72C4×Dic151C2311yescharacteristic
73C22×C4×C151—21no—
74C22×C4×C151C2311yescharacteristic
75M16×C151—22no—
76C2×C8×C151C2312yes—
77C4⋊C4×C151—21no—
78C4⋊C4×C151C2312yes—
79C4⋊C4×C151C2311yescharacteristic
80C4⋊C4×C151C2311yescharacteristic
81C22⋊C4×C151—21no—
82C42×C151—21no—
83C22×D151—24no—
84C22×D151—22no—
85C22×D151—42no—
86C22×D151—42no—
87Dic302—42no—
88Dic302—42no—
89D604—24no—
90D604—22no—
91D604C2×Q814yes—
92C2×Dic151—22no—
93C2×Dic151—21no—
94C2×Dic151—21no—
95C2×Dic151—41no—
96C2×Dic151C2×D411yescharacteristic
97C2×Dic151C2×D411yescharacteristic
98C2×Dic151C2×D411yescharacteristic
99C2×Dic151C2×D411yescharacteristic
100C4×D151—44no—
101C4×D151—44no—
102C151⋊C8—22no—
103C23×C151C2×D411yescharacteristic
104C2×C4×C151—22no—
105C2×C4×C151—21no—
106C2×C4×C151—21no—
107C2×C4×C151—42no—
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×C151C22⋊Q812yes—
116C22×C151C22×D411yescharacteristic
117C2×D151—24no—
118C2×D151—48no—
119C2×D151—42no—
120C2×D151—42no—
121C4×C151—42no—
122C4×C151—41no—
123C4×C151—41no—
124C4×C151C22⋊Q812yes—
125C4×C151C22×Q811yescharacteristic
126C4×C151C2×#4111yescharacteristic
127Dic151—21no—
128Dic151—21no—
129Dic151—21no—
130Dic151—41no—
131Dic151—41no—
132Dic151AGL(1,ℤ/8)12yes—
133C2×C151—22no—
134C2×C151C2×C22⋊Q811yescharacteristic
135C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C151#57511yescharacteristic
137D151—44no—
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#428D15111yescharacteristic, complemented
148#423—1511no—
149#418—1511no—
150#418—1511no—
151C22×D4—1511no—
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⋊C4C2×D15111yescharacteristic
170C2×C22⋊C4—3021no—
171C2×C42—1511no—
172#107—1512no—
173#107C2×D15112yescomplemented
174#106—1512no—
175#106C2×D15112yescomplemented
176C4⋊C8—1512no—
177C4⋊C8—1512no—
178#102—1512no—
179#102—1512no—
180#102—1512no—
181#102—1512no—
182C24—3021no—
183C2×Q8—3021no—
184C2×Q8—3021no—
185C2×D4—3021no—
186C2×D4—3022no—
187C2×D4—1514no—
188C2×D4—1512no—
189C22×C4—21no—
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—21no—
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⋊C4C22×D15111yescharacteristic
210C4⋊C4C22×D15111yescharacteristic
211C4⋊C4C22×D15112yes—
212C22⋊C4—21no—
213C22⋊C4—3022no—
214C22⋊C4—3022no—
215C22⋊C4—3021no—
216C42—21no—
217C42—3021no—
218C42—1511no—
219C42—1511no—
220C42—1511no—
221C42—1511no—
222C23—3024no—
223C23—3022no—
224C23—6042no—
225C23—6042no—
226C23D4×D15111yescharacteristic
227Q8—6042no—
228Q8—6042no—
229D4—3024no—
230D4—3022no—
231D4—1514no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—3022no—
237C2×C4—3021no—
238C2×C4—3021no—
239C2×C4—6044no—
240C2×C4—6044no—
241C2×C4—6041no—
242C2×C4—1511no—
243C2×C4—1511no—
244C2×C4—1511no—
245C2×C4—1511no—
246C2×C4D4×D15111yescharacteristic
247C2×C4Q8×D15112yes—
248C2×C4#1455900412yes—
249C2×C4C23×D15111yescharacteristic
250C8—22no—
251C8—3022no—
252C22—21no—
253C22—21no—
254C22—3024no—
255C22—6048no—
256C22—6042no—
257C22—6042no—
258C22#1801271712yes—
259C22C2×D4×D15111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—3021no—
264C4—3021no—
265C4—3021no—
266C4—6041no—
267C4—6041no—
268C4—1512no—
269C4#1801271712yes—
270C4C2×Q8×D15111yescharacteristic
271C4C2×#1455900411yescharacteristic
272C2—22no—
273C2—6044no—
274C2C2×#1801271711yescharacteristic
275C2AGL(1,ℤ/8)×D15111yescharacteristic
276C2#2375301111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#428×C151C2
socleC22×C151C22×D4
prime radicalC22×D604C22
hypercenter#428D151
hypocenterC151#3077
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #23752765C2semidirect
3. #23752515C2semidirect
4. #23752511C2semidirect
5. #23752501C2semidirect
6. #23752491C2semidirect
7. #23752252C2semidirect
8. #23752235C2semidirect
11. #428×C151C2semidirect
15. #18012738C22semidirect
16. #18012738C22semidirect
17. #18012730C22semidirect
18. #18012730C22semidirect
30. #18012672C22semidirect
31. #18012670C22semidirect
32. Dic151⋊C8C22semidirect
33. Dic151⋊C8C22semidirect
41. #107×C151C22semidirect
42. #106×C151C22semidirect
48. C2×D604Q8semidirect
138. C151#3077semidirect
147. #428D151semidirect
173. #107C2×D151semidirect
175. #106C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237527652
C2×#180127012
#237525152
#237525112
#237525012
#237524912
#237522522
#237522352
#237522182
#237522162
#428×C1512
#3077151