Finite Group Databasebeta

← #34078458

Subgroups — #34078458

Order 19328.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#23753048C211yesmaximal, characteristic, complemented
2#23753010C212yesmaximal, complemented
3#23752394C212yesmaximal, complemented
4#23752389C212yesmaximal, complemented
5#23752384C212yesmaximal, complemented
6#23752379C212yesmaximal, complemented
7C2×#18012667C212yesmaximal, complemented
8#23752196C211yesmaximal, characteristic, complemented
9#532×C151C211yesmaximal, characteristic, complemented
10#18012787C2212yes—
11#18012786—22no—
12#18012785—22no—
13#18012784—22no—
14#18012784C2212yes—
15C2×#14559004—22no—
16C2×#14559004C2212yes—
17#41×D151—24no—
18#18012770C2214yescomplemented
19#18012769—22no—
20#18012765C2214yescomplemented
21#18012764—22no—
22#18012759—24no—
23C2×#14558988C2212yescomplemented
24#18012667C2214yescomplemented
25#18012667C2214yescomplemented
26#18012658C2212yescomplemented
27#18012656C2212yescomplemented
28C2×#41×C151C2211yescharacteristic
29C2×#41×C151C2211yescharacteristic, complemented
30C4⋊Q8×C151C2211yescharacteristic, complemented
31#127×C151C2211yescharacteristic, complemented
32#124×C151C2211yescharacteristic, complemented
33#124×C151C2211yescharacteristic, complemented
34#121×C151—22no—
35C4×D4×C151—22no—
36C2×C42×C151C2211yescharacteristic, complemented
37#14559005—24no—
38#14559004—24no—
39#14559004—24no—
40#14559004—22no—
41#14559004—24no—
42#14559004—22no—
43#14559004D414yescomplemented
44#14559004C2314yes—
45#14559003—28no—
46#14559003—24no—
47C2×Dic302—22no—
48C2×Dic302D412yescomplemented
49Q8×D151—24no—
50C2×D604—22no—
51C2×D604D412yescomplemented
52C2×C22⋊D151—24no—
53C2×C22⋊D151—22no—
54D4×D151—28no—
55D4×D151—24no—
56C2×C4×D151—24no—
57C2×C4×D151—22no—
58#14558994—24no—
59#14558992—24no—
60#14558991—24no—
61#14558989—24no—
62#14558988—22no—
63#14558988C2314yes—
64C2×C151⋊C8—22no—
65#41×C151—22no—
66#41×C151—22no—
67#41×C151—22no—
68#41×C151—22no—
69C2×Q8×C151C2311yescharacteristic
70C2×Q8×C151C2311yescharacteristic
71C2×D4×C151—22no—
72C2×D4×C151—21no—
73C2×D4×C151D412yes—
74C2×D4×C151C2311yescharacteristic
75C2×D4×C151C2311yescharacteristic
76C22×C4×C151—21no—
77C22×C4×C151—21no—
78C22×C4×C151D412yes—
79C22×C4×C151C2311yescharacteristic
80C4⋊C4×C151—22no—
81C22⋊C4×C151—22no—
82C22⋊C4×C151—22no—
83C42×C151—21no—
84C42×C151C2311yescharacteristic
85C42×C151C2311yescharacteristic
86C22×D151—24no—
87C22×D151—22no—
88Dic302—24no—
89Dic302—22no—
90Dic302C2×D414yes—
91D604—24no—
92D604—22no—
93D604C2×D414yes—
94C22⋊D151—28no—
95C22⋊D151—24no—
96C22⋊D151—24no—
97C2×Dic151—24no—
98C2×Dic151—22no—
99C4×D151—28no—
100C4×D151—24no—
101C4×D151—24no—
102C151⋊C8—24no—
103C23×C151—21no—
104C23×C151C2×D412yes—
105C23×C151C2×D411yescharacteristic
106Q8×C151—22no—
107Q8×C151—22no—
108D4×C151—24no—
109D4×C151—22no—
110D4×C151—22no—
111D4×C151—44no—
112D4×C151—42no—
113C2×C4×C151—21no—
114C2×C4×C151—22no—
115C2×C4×C151—22no—
116C2×C4×C151—21no—
117C2×C4×C151—21no—
118C2×C4×C151—42no—
119C2×C4×C151—41no—
120C2×C4×C151—41no—
121C2×C4×C151C2×D412yes—
122C2×C4×C151C2×D412yes—
123C2×C4×C151C2×D412yes—
124C2×C4×C151C2×D411yescharacteristic
125C2×C4×C151C2411yescharacteristic
126C22×C151—22no—
127C22×C151—41no—
128C22×C151—42no—
129C22×C151—42no—
130C22×C151—41no—
131C22×C151#12012yes—
132C22×C151C22×D411yescharacteristic
133C2×D151—24no—
134C2×D151—82no—
135C2×D151—84no—
136C4×C151—22no—
137C4×C151—22no—
138C4×C151—41no—
139C4×C151—41no—
140C4×C151#12011yescharacteristic
141C4×C151#12011yescharacteristic
142C4×C151C22×D412yes—
143Dic151—24no—
144C2×C151—21no—
145C2×C151—22no—
146C2×C151—42no—
147C2×C151—81no—
148C2×C151C2×#12011yescharacteristic
149D151—44no—
150C151#301811yescharacteristic, complemented, Sylow
151#3018—1511nomaximal, Sylow
152#585—1511no—
153#575—1512no—
154#532D15111yescharacteristic, complemented
155#456—1512no—
156#455—1512no—
157#454—1512no—
158#453—1512no—
159C2×#104—1512no—
160#413—1511no—
1612−1+4—3022no—
1622−1+4—1512no—
163H(5,𝔽2)—3022no—
164H(5,𝔽2)—3022no—
165H(5,𝔽2)—1512no—
166C2×#41—3024no—
167C2×#41—3022no—
168C2×#41—1512no—
169C2×#41C2×D15111yescharacteristic
170C2×#41C2×D15111yescharacteristic, complemented
171#137—3022no—
172#137—1514no—
173AGL(1,ℤ/8)—3022no—
174AGL(1,ℤ/8)—1514no—
175#135—3024no—
176C2×M16—1512no—
177C4⋊Q8C2×D15111yescharacteristic, complemented
178#127C2×D15111yescharacteristic, complemented
179#124C2×D15111yescharacteristic, complemented
180#124C2×D15111yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D15111yescharacteristic, complemented
184#104—1514no—
185#104—1514no—
186#101—1512no—
187#100—1512no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—3024no—
193#41—3024no—
194#41—3028no—
195#41—3024no—
196#41—3024no—
197#41—3022no—
198#41—3022no—
199#41—3024no—
200#41—1514no—
201#41—1514no—
202C2×Q8—3022no—
203C2×Q8—3024no—
204C2×Q8—1512no—
205C2×Q8C22×D15111yescharacteristic
206C2×Q8C22×D15111yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—3022no—
210C2×D4—3028no—
211C2×D4—3024no—
212C2×D4—3024no—
213C2×D4—3022no—
214C2×D4—1512no—
215C2×D4C22⋊D15112yes—
216C2×D4C22×D15111yescharacteristic
217C2×D4C22×D15111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—3024no—
221C22×C4—3022no—
222C22×C4C22⋊D15112yes—
223C22×C4C22×D15111yescharacteristic
224Q16—3024no—
225SD16—3024no—
226SD16—3024no—
227D8—3024no—
228M16—3022no—
229M16—1514no—
230C2×C8—3022no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D15111yescharacteristic
236C42C22×D15111yescharacteristic
237C23—21no—
238C23—3024no—
239C23—3022no—
240C23C2×C22⋊D15112yes—
241C23C2×C22⋊D15111yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—3024no—
245Q8—3022no—
246Q8—1514no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—3024no—
253D4—3028no—
254D4—3024no—
255D4—3024no—
256D4—3022no—
257D4—1514no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3028no—
267C2×C4—3024no—
268C2×C4—3022no—
269C2×C4—3024no—
270C2×C4—3024no—
271C2×C4D4×D15112yes—
272C2×C4D4×D15112yes—
273C2×C4C2×C22⋊D15112yes—
274C2×C4C2×C22⋊D15111yescharacteristic
275C2×C4C23×D15111yescharacteristic
276C8—3024no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—3024no—
283C22—12082no—
284C22—12084no—
285C22#1801270712yes—
286C22C22×C22⋊D15111yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3024no—
292C4#1801270711yescharacteristic
293C4#1801270711yescharacteristic
294C4C2×D4×D15112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—6044no—
300C2C2×#1801270711yescharacteristic
subgroupG/H
centerC4#18012707
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#532×C151C2
socleC2×C151C2×#120
hypercenter#532D151
hypocenterC151#3018
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #23753048C2semidirect
2. #23753010C2semidirect
3. #23752394C2semidirect
4. #23752389C2semidirect
5. #23752384C2semidirect
6. #23752379C2semidirect
7. C2×#18012667C2semidirect
8. #23752196C2semidirect
9. #532×C151C2semidirect
18. #18012770C22semidirect
20. #18012765C22semidirect
23. C2×#14558988C22semidirect
24. #18012667C22semidirect
25. #18012667C22semidirect
26. #18012658C22semidirect
27. #18012656C22semidirect
29. C2×#41×C151C22semidirect
30. C4⋊Q8×C151C22semidirect
31. #127×C151C22semidirect
32. #124×C151C22semidirect
33. #124×C151C22semidirect
36. C2×C42×C151C22semidirect
43. #14559004D4semidirect
48. C2×Dic302D4semidirect
51. C2×D604D4semidirect
150. C151#3018semidirect
154. #532D151semidirect
170. C2×#41C2×D151semidirect
177. C4⋊Q8C2×D151semidirect
178. #127C2×D151semidirect
179. #124C2×D151semidirect
180. #124C2×D151semidirect
183. C2×C42C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530482
#237530102
#237523942
#237523892
#237523842
#237523792
C2×#180126672
#237521962
#532×C1512
#3018151