Finite Group Databasebeta

← #34078727

Subgroups — #34078727

Order 19328.

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

normal subgroups only

This group has 548 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
1C2×#18012765C211yesmaximal, characteristic, complemented
2C2×M16×D151C211yesmaximal, characteristic, complemented
3#23752724C211yesmaximal, characteristic, complemented
4C2×#18012710C211yesmaximal, characteristic, complemented
5#23752464C212yesmaximal, complemented
6#23752457C212yesmaximal, complemented
7#23752447C212yesmaximal, complemented
8#23752440C212yesmaximal, complemented
9#23752255C211yesmaximal, characteristic, complemented
10#23752214C211yesmaximal, characteristic, complemented
11#418×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13C22×C22⋊D151—21no—
14C22×C4×D151C2211yescharacteristic
15#18012765—24no—
16C2×#14558992C2212yescomplemented
17C2×#14558989C2212yescomplemented
18C2×#14558988C2211yescharacteristic
19C2×#14558987C2212yescomplemented
20M16×D151—24no—
21C2×C8×D151C2212yescomplemented
22#18012723—21no—
23#18012719C2211yescharacteristic, complemented
24#18012719C2211yescharacteristic, complemented
25#18012714C2211yescharacteristic, complemented
26#18012714C2211yescharacteristic, complemented
27#18012710—21no—
28#18012710—21no—
29#18012710C2212yes—
30#18012710C2212yes—
31#18012709—21no—
32C4×D604—21no—
33C4×D604—21no—
34C2×C4⋊Dic151C2211yescharacteristic
35C2×C22⋊Dic151—21no—
36#18012672C2212yescomplemented
37#18012670C2212yescomplemented
38#18012664C2212yescomplemented
39#18012661C2212yescomplemented
40C22×D4×C151C2211yescharacteristic
41C2×M16×C151C2211yescharacteristic
42#117×C151C2211yescharacteristic
43#102×C151C2212yescomplemented
44#102×C151C2212yescomplemented
45C23×D151D411yescharacteristic
46#14559004—22no—
47#14559004—22no—
48C2×Dic302C2311yescharacteristic
49C2×D604—21no—
50C2×D604—21no—
51C2×D604C2311yescharacteristic
52C2×C22⋊D151—22no—
53C2×C22⋊D151—22no—
54C2×C22⋊D151—21no—
55C2×C22⋊D151—44no—
56C22×Dic151—21no—
57C22×Dic151D411yescharacteristic
58C2×C4×D151—22no—
59C2×C4×D151—21no—
60C2×C4×D151—22no—
61C2×C4×D151D411yescharacteristic
62C2×C4×D151D411yescharacteristic
63C2×C4×D151C2311yescharacteristic
64C2×C4×D151C2311yescharacteristic
65#14558992—24no—
66#14558989—24no—
67#14558988D414yescomplemented
68#14558987—24no—
69C2×C151⋊C8C2312yes—
70C8×D151—24no—
71C4⋊Dic151—21no—
72C4⋊Dic151—21no—
73C4⋊Dic151—21no—
74C4⋊Dic151C2312yes—
75C22⋊Dic151—22no—
76C22⋊Dic151—22no—
77(C2×D151)⋊C4—21no—
78(C2×D151)⋊C4—21no—
79(C2×D151)⋊C4—21no—
80(C2×D151)⋊C4—21no—
81C24×C151—21no—
82C2×D4×C151—22no—
83C2×D4×C151—21no—
84C2×D4×C151—22no—
85C2×D4×C151C2312yes—
86C22×C4×C151C2311yescharacteristic
87M16×C151—22no—
88C2×C8×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—41no—
97C22×D151C2×D411yescharacteristic
98C22×D151C2×D411yescharacteristic
99Dic302—21no—
100Dic302—21no—
101D604—21no—
102D604—21no—
103D604—42no—
104D604—42no—
105C22⋊D151—44no—
106C22⋊D151—42no—
107C22⋊D151—44no—
108C2×Dic151—22no—
109C2×Dic151—21no—
110C2×Dic151—22no—
111C2×Dic151—42no—
112C2×Dic151C2×D411yescharacteristic
113C2×Dic151C2×D411yescharacteristic
114C4×D151—22no—
115C4×D151—22no—
116C4×D151—21no—
117C4×D151—21no—
118C4×D151—21no—
119C4×D151—21no—
120C4×D151—42no—
121C151⋊C8C2×D414yescomplemented
122C23×C151—22no—
123C23×C151—42no—
124C23×C151—42no—
125C23×C151—42no—
126C23×C151C2×D411yescharacteristic
127D4×C151—22no—
128D4×C151—22no—
129D4×C151—42no—
130C2×C4×C151—21no—
131C2×C4×C151—21no—
132C2×C4×C151C2×D411yescharacteristic
133C2×C4×C151C2×D412yes—
134C2×C4×C151#4112yes—
135C2×C4×C151C2411yescharacteristic
136C8×C151—22no—
137C22×C151—21no—
138C22×C151—21no—
139C22×C151—42no—
140C22×C151—42no—
141C22×C151—42no—
142C22×C151—84no—
143C22×C151#12112yes—
144C22×C151C22×D411yescharacteristic
145C2×D151—21no—
146C2×D151—21no—
147C2×D151—21no—
148C2×D151—21no—
149C2×D151—21no—
150C2×D151—44no—
151C2×D151—41no—
152C2×D151—41no—
153C2×D151—41no—
154C2×D151AGL(1,ℤ/8)12yescomplemented
155C4×C151—41no—
156C4×C151—41no—
157C4×C151#12112yes—
158C4×C151C22×D411yescharacteristic
159C4×C151C2×#4111yescharacteristic
160Dic151—21no—
161Dic151—21no—
162Dic151—41no—
163Dic151—42no—
164C2×C151—22no—
165C2×C151—82no—
166C2×C151C2×#12111yescharacteristic
167C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C151#57511yescharacteristic
169D151—22no—
170D151—41no—
171D151—81no—
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#428—1511no—
183#418D15111yescharacteristic, complemented
184#417—1511no—
185C2×#41—1511no—
186C22×D4—3021no—
187C22×D4C2×D15111yescharacteristic
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⋊C4—1511no—
211C2×C22⋊C4—3021no—
212#107—1512no—
213#106—1512no—
214#103—1512no—
215#102—1512no—
216#102C2×D15112yescomplemented
217#102C2×D15112yescomplemented
218C24—21no—
219C24—1511no—
220#41—3022no—
221#41—3022no—
222C2×Q8—1511no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3021no—
227C2×D4—3021no—
228C2×D4—3022no—
229C2×D4—3022no—
230C2×D4—3021no—
231C2×D4—6044no—
232C2×D4—1511no—
233C2×D4C22×D15112yes—
234C22×C4—3022no—
235C22×C4—3022no—
236C22×C4—3021no—
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—22no—
247M16—3024no—
248M16—1514no—
249C2×C8—3024no—
250C2×C8—1512no—
251C2×C8C22×D15112yes—
252C4⋊C4—3021no—
253C4⋊C4—3021no—
254C4⋊C4—3021no—
255C4⋊C4—1512no—
256C4⋊C4C22×D15111yescharacteristic
257C4⋊C4C22×D15111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3021no—
260C22⋊C4—3021no—
261C22⋊C4—3021no—
262C22⋊C4—3021no—
263C22⋊C4—3022no—
264C22⋊C4—3022no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3022no—
271C23—3022no—
272C23—3021no—
273C23—6041no—
274C23—1511no—
275C23—1511no—
276C23D4×D15111yescharacteristic
277Q8—3021no—
278Q8—3021no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3021no—
283D4—3021no—
284D4—6042no—
285D4—6042no—
286D4—6044no—
287D4—6044no—
288D4—6042no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3022no—
292C2×C4—3022no—
293C2×C4—3022no—
294C2×C4—3021no—
295C2×C4—3021no—
296C2×C4—3021no—
297C2×C4—3021no—
298C2×C4—3021no—
299C2×C4—3022no—
300C2×C4—6042no—
301C2×C4—6042no—
302C2×C4—1511no—
303C2×C4—1511no—
304C2×C4D4×D15111yescharacteristic
305C2×C4D4×D15112yes—
306C2×C4#1455900412yes—
307C2×C4C23×D15111yescharacteristic
308C8—22no—
309C8—1514no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3021no—
317C22—3021no—
318C22—3021no—
319C22—3021no—
320C22—3021no—
321C22—6044no—
322C22—6041no—
323C22—6041no—
324C22—6041no—
325C22—1512no—
326C22#1801271212yes—
327C22C2×D4×D15111yescharacteristic
328C4—41no—
329C4—41no—
330C4—3021no—
331C4—3021no—
332C4—6042no—
333C4—6041no—
334C4#1801271212yes—
335C4C2×D4×D15111yescharacteristic
336C4C2×#1455900411yescharacteristic
337C2—22no—
338C2—82no—
339C2—3022no—
340C2—6041no—
341C2—12081no—
342C2C2×#1801271211yescharacteristic
343C2AGL(1,ℤ/8)×D15111yescharacteristic
344C2#2375300811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#418×C151C2
socleC22×C151C22×D4
hypercenter#418D151
hypocenterC151#3055
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. C2×#18012765C2semidirect
2. C2×M16×D151C2semidirect
3. #23752724C2semidirect
4. C2×#18012710C2semidirect
5. #23752464C2semidirect
6. #23752457C2semidirect
7. #23752447C2semidirect
8. #23752440C2semidirect
9. #23752255C2semidirect
10. #23752214C2semidirect
11. #418×C151C2semidirect
16. C2×#14558992C22semidirect
17. C2×#14558989C22semidirect
19. C2×#14558987C22semidirect
21. C2×C8×D151C22semidirect
23. #18012719C22semidirect
24. #18012719C22semidirect
25. #18012714C22semidirect
26. #18012714C22semidirect
36. #18012672C22semidirect
37. #18012670C22semidirect
38. #18012664C22semidirect
39. #18012661C22semidirect
43. #102×C151C22semidirect
44. #102×C151C22semidirect
67. #14558988D4semidirect
121. C151⋊C8C2×D4semidirect
154. C2×D151AGL(1,ℤ/8)semidirect
172. C151#3055semidirect
183. #418D151semidirect
216. #102C2×D151semidirect
217. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127652
C2×M16×D1512
#237527242
C2×#180127102
#237524642
#237524572
#237524472
#237524402
#237522552
#237522142
#418×C1512
#3055151