Finite Group Databasebeta

← #34078730

Subgroups — #34078730

Order 19328.

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

normal subgroups only

This group has 548 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
1C2×#18012767C211yesmaximal, characteristic, complemented
2C2×M16×D151C211yesmaximal, characteristic, complemented
3#23752724C211yesmaximal, characteristic, complemented
4C2×#18012714C211yesmaximal, characteristic, complemented
5#23752466C212yesmaximal, complemented
6#23752459C212yesmaximal, complemented
7#23752449C212yesmaximal, complemented
8#23752442C212yesmaximal, complemented
9#23752216C211yesmaximal, characteristic, complemented
10#23752211C211yesmaximal, characteristic, complemented
11#428×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13C22×D604—21no—
14C22×D604C2211yescharacteristic
15C22×C4×D151C2211yescharacteristic
16#18012767—24no—
17C2×#14558993C2212yescomplemented
18C2×D1208C2212yescomplemented
19C2×#14558988C2211yescharacteristic
20C2×#14558987C2212yescomplemented
21M16×D151—24no—
22C2×C8×D151C2212yescomplemented
23#18012723—21no—
24#18012719C2211yescharacteristic, complemented
25#18012719C2211yescharacteristic, complemented
26#18012714—21no—
27#18012714—21no—
28#18012714C2212yes—
29#18012714C2212yes—
30#18012714C2211yescharacteristic, complemented
31#18012714C2211yescharacteristic, complemented
32#18012709—21no—
33C4×D604—21no—
34C4×D604—21no—
35C2×(C2×D151)⋊C4—21no—
36#18012662C2212yescomplemented
37#18012659C2212yescomplemented
38#18012659C2212yescomplemented
39#18012659C2212yescomplemented
40C2×M16×C151C2211yescharacteristic
41#117×C151C2211yescharacteristic
42C2×C4⋊C4×C151C2211yescharacteristic
43#107×C151C2212yescomplemented
44#106×C151C2212yescomplemented
45C23×D151—21no—
46C23×D151D411yescharacteristic
47#14559004—22no—
48#14559004—22no—
49C2×Dic302C2311yescharacteristic
50C2×D604—21no—
51C2×D604—21no—
52C2×D604—22no—
53C2×D604—22no—
54C2×D604—22no—
55C2×D604—21no—
56C2×D604—22no—
57C2×D604—44no—
58C2×D604C2312yes—
59C2×D604C2311yescharacteristic
60C2×C22⋊D151—21no—
61C22×Dic151D411yescharacteristic
62C2×C4×D151—22no—
63C2×C4×D151—21no—
64C2×C4×D151—22no—
65C2×C4×D151D411yescharacteristic
66C2×C4×D151D411yescharacteristic
67C2×C4×D151C2311yescharacteristic
68C2×C4×D151C2311yescharacteristic
69#14558993—24no—
70D1208—24no—
71#14558988—22no—
72#14558987—24no—
73C2×C151⋊C8C2312yes—
74C8×D151—24no—
75C4⋊Dic151—21no—
76C4⋊Dic151—21no—
77(C2×D151)⋊C4—21no—
78(C2×D151)⋊C4—21no—
79(C2×D151)⋊C4—22no—
80(C2×D151)⋊C4—22no—
81(C2×D151)⋊C4—21no—
82(C2×D151)⋊C4—21no—
83C22×C4×C151—21no—
84C22×C4×C151C2311yescharacteristic
85M16×C151D414yescomplemented
86C2×C8×C151C2312yes—
87C4⋊C4×C151—21no—
88C4⋊C4×C151C2312yes—
89C4⋊C4×C151C2311yescharacteristic
90C4⋊C4×C151C2311yescharacteristic
91C22⋊C4×C151—21no—
92C42×C151—21no—
93C22×D151—22no—
94C22×D151—22no—
95C22×D151—21no—
96C22×D151—22no—
97C22×D151—41no—
98C22×D151—42no—
99C22×D151—42no—
100C22×D151—42no—
101C22×D151C2×D411yescharacteristic
102C22×D151C2×D411yescharacteristic
103Dic302—21no—
104Dic302—21no—
105D604—21no—
106D604—22no—
107D604—22no—
108D604—21no—
109D604—42no—
110D604—44no—
111D604—42no—
112D604—44no—
113D604—42no—
114C22⋊D151—42no—
115C2×Dic151—22no—
116C2×Dic151—21no—
117C2×Dic151C2×D411yescharacteristic
118C2×Dic151C2×D411yescharacteristic
119C4×D151—22no—
120C4×D151—22no—
121C4×D151—21no—
122C4×D151—21no—
123C4×D151—21no—
124C4×D151—21no—
125C4×D151—42no—
126C151⋊C8—22no—
127C23×C151C2×D411yescharacteristic
128C2×C4×C151—21no—
129C2×C4×C151—21no—
130C2×C4×C151—22no—
131C2×C4×C151—42no—
132C2×C4×C151C2×D411yescharacteristic
133C2×C4×C151C2×D412yes—
134C2×C4×C151#4112yes—
135C2×C4×C151C2411yescharacteristic
136C8×C151C2×D414yescomplemented
137C22×C151—21no—
138C22×C151—21no—
139C22×C151#12112yes—
140C22×C151C22×D411yescharacteristic
141C2×D151—21no—
142C2×D151—21no—
143C2×D151—21no—
144C2×D151—21no—
145C2×D151—21no—
146C2×D151—44no—
147C2×D151—41no—
148C2×D151—42no—
149C2×D151—42no—
150C2×D151—41no—
151C2×D151—41no—
152C2×D151—42no—
153C2×D151—84no—
154C2×D151AGL(1,ℤ/8)12yescomplemented
155C4×C151—42no—
156C4×C151—41no—
157C4×C151—41no—
158C4×C151#12112yes—
159C4×C151C22×D411yescharacteristic
160C4×C151C2×#4111yescharacteristic
161Dic151—21no—
162Dic151—21no—
163Dic151—41no—
164C2×C151—22no—
165C2×C151C2×#12111yescharacteristic
166C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C151#57511yescharacteristic
168D151—22no—
169D151—41no—
170D151—81no—
171D151—82no—
172C151#305511yescharacteristic, complemented, Sylow
173#3055—1511nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1511no—
175C22×M16—1511no—
176#525—1511no—
177C2×#121—1511no—
178#469—1512no—
179#468—1512no—
180#466—1512no—
181#465—1512no—
182#428D15111yescharacteristic, complemented
183#418—1511no—
184#417—1511no—
185C2×#41—1511no—
186C22×D4—3021no—
187C22×D4—1511no—
188C23×C4—1511no—
189AGL(1,ℤ/8)—3024no—
190C2×SD16—1512no—
191C2×D8—1512no—
192C2×M16—3024no—
193C2×M16—1511no—
194C2×M16—1512no—
195C2×M16C2×D15111yescharacteristic
196C22×C8—1512no—
197#123—3021no—
198C22⋊Q8—1511no—
199C22⋊Q8—1511no—
200#121—3021no—
201#121—3021no—
202#121—1512no—
203#121—1512no—
204#121—1511no—
205#121—1511no—
206#120—3021no—
207C4×D4—3021no—
208C4×D4—3021no—
209#117C2×D15111yescharacteristic
210C2×C4⋊C4C2×D15111yescharacteristic
211C2×C22⋊C4—3021no—
212#107C2×D15112yescomplemented
213#106C2×D15112yescomplemented
214#103—1512no—
215#102—1512no—
216#102—1512no—
217#102—1512no—
218C24—3021no—
219C24—1511no—
220#41—3022no—
221#41—3022no—
222C2×Q8—1511no—
223C2×D4—3021no—
224C2×D4—3021no—
225C2×D4—3022no—
226C2×D4—3022no—
227C2×D4—3021no—
228C2×D4—3021no—
229C2×D4—3022no—
230C2×D4—3022no—
231C2×D4—6044no—
232C2×D4—1511no—
233C2×D4—1512no—
234C22×C4—21no—
235C22×C4—3022no—
236C22×C4—3022no—
237C22×C4—3021no—
238C22×C4—1511no—
239C22×C4—1511no—
240C22×C4—1511no—
241C22×C4—1511no—
242C22×C4—1511no—
243C22×C4C22×D15111yescharacteristic
244SD16—3024no—
245D8—3024no—
246M16—3022no—
247M16—3024no—
248M16D60414yescomplemented
249C2×C8—3024no—
250C2×C8—1512no—
251C2×C8C22×D15112yes—
252C4⋊C4—21no—
253C4⋊C4—3021no—
254C4⋊C4—3021no—
255C4⋊C4C22×D15111yescharacteristic
256C4⋊C4C22×D15111yescharacteristic
257C4⋊C4C22×D15112yes—
258C22⋊C4—21no—
259C22⋊C4—3021no—
260C22⋊C4—3021no—
261C22⋊C4—3022no—
262C22⋊C4—3022no—
263C22⋊C4—3021no—
264C22⋊C4—3021no—
265C42—21no—
266C23—3022no—
267C23—3022no—
268C23—3021no—
269C23—3022no—
270C23—6041no—
271C23—6042no—
272C23—6042no—
273C23—6042no—
274C23—1511no—
275C23—1511no—
276C23D4×D15111yescharacteristic
277Q8—3021no—
278Q8—3021no—
279D4—3021no—
280D4—3022no—
281D4—3022no—
282D4—3021no—
283D4—6044no—
284D4—6042no—
285D4—6044no—
286D4—6042no—
287D4—6042no—
288D4—6042no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3022no—
294C2×C4—3022no—
295C2×C4—3022no—
296C2×C4—3021no—
297C2×C4—3021no—
298C2×C4—3021no—
299C2×C4—3021no—
300C2×C4—3021no—
301C2×C4—6042no—
302C2×C4—1511no—
303C2×C4—1511no—
304C2×C4D4×D15111yescharacteristic
305C2×C4C2×D60412yes—
306C2×C4#1455900512yes—
307C2×C4C23×D15111yescharacteristic
308C8—3022no—
309C8C2×D60414yescomplemented
310C22—21no—
311C22—21no—
312C22—3021no—
313C22—3021no—
314C22—3021no—
315C22—3021no—
316C22—3021no—
317C22—6044no—
318C22—6041no—
319C22—6042no—
320C22—6042no—
321C22—6041no—
322C22—6041no—
323C22—6042no—
324C22—1512no—
325C22—12084no—
326C22#1801271412yes—
327C22C2×D4×D15111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3021no—
332C4—3021no—
333C4—6041no—
334C4#1801271412yes—
335C4C22×D60411yescharacteristic
336C4C2×#1455900511yescharacteristic
337C2—22no—
338C2—3022no—
339C2—6041no—
340C2—12082no—
341C2—12081no—
342C2C2×#1801271411yescharacteristic
343C2AGL(1,ℤ/8)×D15111yescharacteristic
344C2#2375301111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#428×C151C2
socleC22×C151C22×D4
hypercenter#428D151
hypocenterC151#3055
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. C2×#18012767C2semidirect
2. C2×M16×D151C2semidirect
3. #23752724C2semidirect
4. C2×#18012714C2semidirect
5. #23752466C2semidirect
6. #23752459C2semidirect
7. #23752449C2semidirect
8. #23752442C2semidirect
9. #23752216C2semidirect
10. #23752211C2semidirect
11. #428×C151C2semidirect
17. C2×#14558993C22semidirect
18. C2×D1208C22semidirect
20. C2×#14558987C22semidirect
22. C2×C8×D151C22semidirect
24. #18012719C22semidirect
25. #18012719C22semidirect
30. #18012714C22semidirect
31. #18012714C22semidirect
36. #18012662C22semidirect
37. #18012659C22semidirect
38. #18012659C22semidirect
39. #18012659C22semidirect
43. #107×C151C22semidirect
44. #106×C151C22semidirect
85. M16×C151D4semidirect
136. C8×C151C2×D4semidirect
154. C2×D151AGL(1,ℤ/8)semidirect
172. C151#3055semidirect
182. #428D151semidirect
212. #107C2×D151semidirect
213. #106C2×D151semidirect
248. M16D604semidirect
309. C8C2×D604semidirect

Maximal subgroups

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

subgroupindex
C2×#180127672
C2×M16×D1512
#237527242
C2×#180127142
#237524662
#237524592
#237524492
#237524422
#237522162
#237522112
#428×C1512
#3055151