Finite Group Databasebeta

← #34078732

Subgroups — #34078732

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×#18012764C211yesmaximal, characteristic, complemented
2C2×M16×D151C211yesmaximal, characteristic, complemented
3#23752719C211yesmaximal, characteristic, complemented
4C2×#18012714C211yesmaximal, characteristic, complemented
5#23752464C212yesmaximal, complemented
6#23752461C212yesmaximal, complemented
7#23752447C212yesmaximal, complemented
8#23752444C212yesmaximal, complemented
9#23752252C211yesmaximal, characteristic, complemented
10#23752218C211yesmaximal, characteristic, complemented
11#417×C151C211yesmaximal, characteristic, complemented
12C22×D604—21no—
13C22×D604C2211yescharacteristic
14C22×C4×D151C2211yescharacteristic
15#18012764—24no—
16C2×#14558991C2212yescomplemented
17C2×#14558989C2212yescomplemented
18C2×#14558988C2211yescharacteristic
19C2×#14558987C2212yescomplemented
20M16×D151—24no—
21C2×C8×D151C2212yescomplemented
22#18012720—21no—
23#18012715C2211yescharacteristic, complemented
24#18012715C2211yescharacteristic, complemented
25#18012714—21no—
26#18012714—21no—
27#18012714C2212yes—
28#18012714C2212yes—
29#18012710C2211yescharacteristic, complemented
30#18012710C2211yescharacteristic, complemented
31#18012707—21no—
32C4×C22⋊D151—21no—
33C4×C22⋊D151—21no—
34#18012698C2211yescharacteristic
35C2×(C2×D151)⋊C4—21no—
36#18012672C2212yescomplemented
37#18012670C2212yescomplemented
38#18012661C2212yescomplemented
39#18012661C2212yescomplemented
40C2×#41×C151C2211yescharacteristic
41C2×M16×C151C2211yescharacteristic
42C2×C4⋊C4×C151C2211yescharacteristic
43#103×C151C2212yescomplemented
44#102×C151C2212yescomplemented
45C23×D151—21no—
46C23×D151D411yescharacteristic
47C2×D604—22no—
48C2×D604—22no—
49C2×D604—22no—
50C2×D604—21no—
51C2×D604—22no—
52C2×D604—44no—
53C2×D604C2312yes—
54C2×C22⋊D151—21no—
55C2×C22⋊D151—21no—
56C22×Dic151D411yescharacteristic
57C2×C4×D151—22no—
58C2×C4×D151—22no—
59C2×C4×D151D411yescharacteristic
60C2×C4×D151D411yescharacteristic
61C2×C4×D151C2311yescharacteristic
62C2×C4×D151C2311yescharacteristic
63#14558991—24no—
64#14558989—24no—
65#14558988D414yescomplemented
66#14558987—24no—
67C2×C151⋊C8C2312yes—
68C8×D151—24no—
69C4⋊Dic151C2311yescharacteristic
70C4⋊Dic151C2311yescharacteristic
71Dic151⋊C4—21no—
72Dic151⋊C4—21no—
73C22⋊Dic151—21no—
74C22⋊Dic151—21no—
75C22⋊Dic151—21no—
76(C2×D151)⋊C4—21no—
77(C2×D151)⋊C4—21no—
78(C2×D151)⋊C4—22no—
79(C2×D151)⋊C4—22no—
80C4×Dic151—21no—
81#41×C151—22no—
82#41×C151—22no—
83C2×Q8×C151C2311yescharacteristic
84C2×D4×C151—21no—
85C2×D4×C151C2311yescharacteristic
86C22×C4×C151—21no—
87C22×C4×C151—21no—
88C22×C4×C151C2311yescharacteristic
89M16×C151—22no—
90C2×C8×C151C2312yes—
91C4⋊C4×C151—21no—
92C4⋊C4×C151C2312yes—
93C22×D151—22no—
94C22×D151—22no—
95C22×D151—22no—
96C22×D151—41no—
97C22×D151—42no—
98C22×D151—42no—
99C22×D151—42no—
100C22×D151C2×D411yescharacteristic
101C22×D151C2×D411yescharacteristic
102D604—22no—
103D604—22no—
104D604—44no—
105D604—44no—
106D604—42no—
107C22⋊D151—42no—
108C22⋊D151—42no—
109C2×Dic151—22no—
110C2×Dic151—21no—
111C2×Dic151—21no—
112C2×Dic151C2×D411yescharacteristic
113C2×Dic151C2×D411yescharacteristic
114C4×D151—22no—
115C4×D151—22no—
116C4×D151—21no—
117C4×D151—21no—
118C4×D151—21no—
119C4×D151—21no—
120C151⋊C8C2×D414yescomplemented
121C23×C151—21no—
122C23×C151C2×D411yescharacteristic
123Q8×C151—21no—
124Q8×C151—21no—
125D4×C151—21no—
126D4×C151—21no—
127D4×C151—42no—
128C2×C4×C151—21no—
129C2×C4×C151—22no—
130C2×C4×C151—42no—
131C2×C4×C151—42no—
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—41no—
140C22×C151—41no—
141C22×C151—41no—
142C22×C151#12112yes—
143C22×C151C22×D411yescharacteristic
144C2×D151—21no—
145C2×D151—21no—
146C2×D151—21no—
147C2×D151—21no—
148C2×D151—21no—
149C2×D151—44no—
150C2×D151—42no—
151C2×D151—42no—
152C2×D151—42no—
153C2×D151—84no—
154C2×D151AGL(1,ℤ/8)12yescomplemented
155C4×C151—41no—
156C4×C151—42no—
157C4×C151#12112yes—
158C4×C151C22×D411yescharacteristic
159C4×C151C2×#4111yescharacteristic
160Dic151—21no—
161Dic151—21no—
162Dic151—41no—
163Dic151—41no—
164C2×C151—22no—
165C2×C151—81no—
166C2×C151C2×#12111yescharacteristic
167C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C151#57511yescharacteristic
169D151—22no—
170D151—41no—
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#428—1511no—
183#418—1511no—
184#417D15111yescharacteristic, complemented
185C2×#41C2×D15111yescharacteristic
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—1511no—
203#121—1511no—
204#121—1512no—
205#121—1512no—
206#120—3021no—
207C4×D4—3021no—
208C4×D4—3021no—
209#117—1511no—
210C2×C4⋊C4C2×D15111yescharacteristic
211C2×C22⋊C4—3021no—
212#107—1512no—
213#106—1512no—
214#103C2×D15112yescomplemented
215#102—1512no—
216#102—1512no—
217#102C2×D15112yescomplemented
218C24—3021no—
219C24—1511no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D15111yescharacteristic
223C2×D4—21no—
224C2×D4—3022no—
225C2×D4—3022no—
226C2×D4—3021no—
227C2×D4—3021no—
228C2×D4—3022no—
229C2×D4—3021no—
230C2×D4—3022no—
231C2×D4—6044no—
232C2×D4—1512no—
233C2×D4C22×D15111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—3022no—
237C22×C4—3022no—
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—21no—
253C4⋊C4—3021no—
254C4⋊C4—3021no—
255C4⋊C4—1511no—
256C4⋊C4—1511no—
257C4⋊C4C22×D15112yes—
258C22⋊C4—3021no—
259C22⋊C4—3021no—
260C22⋊C4—3022no—
261C22⋊C4—3022no—
262C22⋊C4—3021no—
263C22⋊C4—3021no—
264C22⋊C4—3021no—
265C42—3021no—
266C23—21no—
267C23—3022no—
268C23—3022no—
269C23—3022no—
270C23—6041no—
271C23—6042no—
272C23—6042no—
273C23—6042no—
274C23—1511no—
275C23—1511no—
276C23D4×D15111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—3022no—
283D4—3022no—
284D4—6044no—
285D4—6044no—
286D4—6042no—
287D4—6042no—
288D4—6042no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
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—3021no—
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—41no—
313C22—41no—
314C22—41no—
315C22—3021no—
316C22—3021no—
317C22—3021no—
318C22—3021no—
319C22—3021no—
320C22—6044no—
321C22—6042no—
322C22—6042no—
323C22—6042no—
324C22—1512no—
325C22—12084no—
326C22#1801271212yes—
327C22C2×D4×D15111yescharacteristic
328C4—42no—
329C4—41no—
330C4—3021no—
331C4—3021no—
332C4—6041no—
333C4—6041no—
334C4#1801271212yes—
335C4C2×D4×D15111yescharacteristic
336C4C2×#1455900411yescharacteristic
337C2—22no—
338C2—81no—
339C2—3022no—
340C2—6041no—
341C2—12082no—
342C2C2×#1801271211yescharacteristic
343C2AGL(1,ℤ/8)×D15111yescharacteristic
344C2#2375301111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#417×C151C2
socleC22×C151C22×D4
hypercenter#417D151
hypocenterC151#3055
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#18012764C2semidirect
2. C2×M16×D151C2semidirect
3. #23752719C2semidirect
4. C2×#18012714C2semidirect
5. #23752464C2semidirect
6. #23752461C2semidirect
7. #23752447C2semidirect
8. #23752444C2semidirect
9. #23752252C2semidirect
10. #23752218C2semidirect
11. #417×C151C2semidirect
16. C2×#14558991C22semidirect
17. C2×#14558989C22semidirect
19. C2×#14558987C22semidirect
21. C2×C8×D151C22semidirect
23. #18012715C22semidirect
24. #18012715C22semidirect
29. #18012710C22semidirect
30. #18012710C22semidirect
36. #18012672C22semidirect
37. #18012670C22semidirect
38. #18012661C22semidirect
39. #18012661C22semidirect
43. #103×C151C22semidirect
44. #102×C151C22semidirect
65. #14558988D4semidirect
120. C151⋊C8C2×D4semidirect
154. C2×D151AGL(1,ℤ/8)semidirect
172. C151#3055semidirect
184. #417D151semidirect
214. #103C2×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×#180127642
C2×M16×D1512
#237527192
C2×#180127142
#237524642
#237524612
#237524472
#237524442
#237522522
#237522182
#417×C1512
#3055151