Finite Group Databasebeta

← #34079404

Subgroups — #34079404

Order 19328.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752822C211yesmaximal, characteristic
2#23752690C211yesmaximal, characteristic
3#23752554C211yesmaximal, characteristic
4#23752550C211yesmaximal, characteristic
5#23752521C211yesmaximal, characteristic, complemented
6#23752517C211yesmaximal, characteristic, complemented
7#23752420C211yesmaximal, characteristic, complemented
8#23752420C211yesmaximal, characteristic, complemented
9#23752368C211yesmaximal, characteristic
10#23752319C211yesmaximal, characteristic, complemented
11#23752316C211yesmaximal, characteristic, complemented
12Q16×Dic151C211yesmaximal, characteristic, complemented
13#23752304C211yesmaximal, characteristic, complemented
14#23752270C211yesmaximal, characteristic
15#451×C151C211yesmaximal, characteristic
16C2×#14558994C2211yescharacteristic
17C2×#14558994C2211yescharacteristic
18#18012737C2211yescharacteristic
19#18012732—21no—
20#18012732—21no—
21#18012730—21no—
22#18012730C2211yescharacteristic
23#18012729—21no—
24#18012718C2211yescharacteristic
25Q8×Dic151—21no—
26Q8×Dic151—21no—
27Q8×Dic151C2211yescharacteristic
28Q8×Dic151C2211yescharacteristic
29Q8×Dic151C2211yescharacteristic
30Q8×Dic151C2211yescharacteristic
31Q8×Dic151C2211yescharacteristic
32#18012704C2211yescharacteristic
33#18012698—21no—
34#18012698—21no—
35#18012698C2211yescharacteristic
36#18012698C2211yescharacteristic
37#18012673C2211yescharacteristic
38#18012672C2211yescharacteristic
39#18012671C2211yescharacteristic
40#18012670C2211yescharacteristic
41Dic151⋊C8C2211yescharacteristic
42Dic151⋊C8C2211yescharacteristic
43#18012664C2211yescharacteristic
44#18012664C2211yescharacteristic
45#18012663C2211yescharacteristic
46#18012663C2211yescharacteristic
47#18012663C2211yescharacteristic
48#18012663C2211yescharacteristic
49C22⋊(C151⋊C8)C2211yescharacteristic
50#18012649C2211yescharacteristic
51C8×Dic151C2211yescharacteristic
52C22×Q8×C151C2211yescharacteristic
53C2×Q16×C151C2211yescharacteristic
54C2×Q16×C151C2211yescharacteristic
55C22⋊Q8×C151C2211yescharacteristic
56#103×C151C2211yescharacteristic
57#103×C151C2211yescharacteristic
58C22⋊C8×C151C2211yescharacteristic
59C2×Dic302C2311yescharacteristic
60#14558994—22no—
61#14558994—22no—
62C2×C151⋊C8C2311yescharacteristic
63C2×C151⋊C8C2311yescharacteristic
64C4⋊Dic151—21no—
65C4⋊Dic151—21no—
66C4⋊Dic151—21no—
67C4⋊Dic151—21no—
68C4⋊Dic151—21no—
69C4⋊Dic151—21no—
70C4⋊Dic151D411yescharacteristic
71C4⋊Dic151D411yescharacteristic
72C4⋊Dic151C2311yescharacteristic
73C4⋊Dic151C2311yescharacteristic
74C4⋊Dic151C2311yescharacteristic
75Dic151⋊C4—21no—
76Dic151⋊C4—21no—
77Dic151⋊C4—21no—
78Dic151⋊C4—21no—
79Dic151⋊C4—21no—
80C22⋊Dic151—21no—
81C22⋊Dic151—21no—
82C22⋊Dic151D411yescharacteristic
83C22⋊Dic151D411yescharacteristic
84C4×Dic151—21no—
85C4×Dic151—21no—
86C4×Dic151—21no—
87C4×Dic151—21no—
88C4×Dic151—21no—
89C4×Dic151—21no—
90C4×Dic151—21no—
91C4×Dic151C2311yescharacteristic
92C4×Dic151C2311yescharacteristic
93C2×Q8×C151—22no—
94C2×Q8×C151—22no—
95C2×Q8×C151—21no—
96C2×Q8×C151C2311yescharacteristic
97C2×Q8×C151C2311yescharacteristic
98C2×Q8×C151C2311yescharacteristic
99C22×C4×C151—21no—
100C22×C4×C151C2311yescharacteristic
101Q16×C151—22no—
102Q16×C151—22no—
103C2×C8×C151C2311yescharacteristic
104C2×C8×C151C2311yescharacteristic
105C4⋊C4×C151—21no—
106C4⋊C4×C151C2311yescharacteristic
107C22⋊C4×C151—21no—
108Dic302—21no—
109Dic302—21no—
110C2×Dic151—21no—
111C2×Dic151—21no—
112C2×Dic151—21no—
113C2×Dic151—21no—
114C2×Dic151C2×D411yescharacteristic
115C2×Dic151C2×D411yescharacteristic
116C2×Dic151C2×D411yescharacteristic
117C2×Dic151C2×D411yescharacteristic
118C151⋊C8—21no—
119C151⋊C8—21no—
120C23×C151C2×D411yescharacteristic
121Q8×C151—24no—
122Q8×C151—21no—
123Q8×C151—21no—
124Q8×C151—21no—
125Q8×C151—21no—
126Q8×C151#4112yes—
127Q8×C151#4112yes—
128C2×C4×C151—24no—
129C2×C4×C151—21no—
130C2×C4×C151—21no—
131C2×C4×C151—21no—
132C2×C4×C151—21no—
133C2×C4×C151—21no—
134C2×C4×C151—21no—
135C2×C4×C151C2×D411yescharacteristic
136C2×C4×C151C2411yescharacteristic
137C8×C151—21no—
138C8×C151—21no—
139C22×C151—21no—
140C22×C151—21no—
141C22×C151—21no—
142C22×C151C22×D411yescharacteristic
143C4×C151—22no—
144C4×C151—22no—
145C4×C151—21no—
146C4×C151—41no—
147C4×C151—41no—
148C4×C151C2×#4111yescharacteristic
149C4×C151C2×#4111yescharacteristic
150Dic151—21no—
151Dic151—21no—
152Dic151—21no—
153Dic151—21no—
154Dic151—41no—
155Dic151—41no—
156Dic151—41no—
157Dic151—41no—
158C2×C151—41no—
159C2×C151#52511yescharacteristic
160C2×C151#57811yescharacteristic
161C2×C151#57811yescharacteristic
162C151#312711yescharacteristic, complemented, Sylow
163#3127—1511nomaximal, Sylow
164#541—1511no—
165#519—1511no—
166#485—1511no—
167#485—1511no—
168#479—1511no—
169#479—1511no—
170#462—1511no—
171#462—1511no—
172#451—1511no—
173#451D15111yescharacteristic
174#441—1511no—
175#441—1511no—
176C4×Q16—1511no—
177C4×Q16—1511no—
178#433—1511no—
179C22×Q8C2×D15111yescharacteristic
180C2×Q16—1511no—
181C2×Q16—1511no—
182C2×Q16C2×D15111yescharacteristic
183C2×Q16C2×D15111yescharacteristic
184C4⋊Q8—1511no—
185#126—3021no—
186#126—3021no—
187#125—3021no—
188#125—3021no—
189#125—1511no—
190C22⋊Q8—1511no—
191C22⋊Q8C2×D15111yescharacteristic
192C4×Q8—3021no—
193C4×Q8—3021no—
194C4×Q8—1511no—
195C4×Q8—1511no—
196C4×Q8—1511no—
197C4×Q8—1511no—
198C4×Q8—1511no—
199C4×Q8—1511no—
200#117—3021no—
201#117—3021no—
202#117—1511no—
203#117—1511no—
204#107—1511no—
205#107—1511no—
206#106—1511no—
207#106—1511no—
208C4⋊C8—1511no—
209C4⋊C8—1511no—
210#103—1511no—
211#103—1511no—
212#103—1511no—
213#103—1511no—
214#103—1511no—
215#103—1511no—
216#103C2×D15111yescharacteristic
217#103C2×D15111yescharacteristic
218C22⋊C8—1511no—
219C22⋊C8C2×D15111yescharacteristic
220#97—1511no—
221C4×C8—1511no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—1511no—
226C2×Q8C22×D15111yescharacteristic
227C2×Q8C22×D15111yescharacteristic
228C2×Q8C22×D15111yescharacteristic
229C22×C4—21no—
230C22×C4C22×D15111yescharacteristic
231Q16—22no—
232Q16—22no—
233Q16—3022no—
234Q16—3022no—
235C2×C8—1511no—
236C2×C8—1511no—
237C2×C8C22×D15111yescharacteristic
238C2×C8C22×D15111yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—3021no—
241C4⋊C4—3021no—
242C4⋊C4—3021no—
243C4⋊C4—3021no—
244C4⋊C4—3021no—
245C4⋊C4—3021no—
246C4⋊C4—3021no—
247C4⋊C4—3021no—
248C4⋊C4—3021no—
249C4⋊C4—3021no—
250C4⋊C4—3021no—
251C4⋊C4—1511no—
252C4⋊C4—1511no—
253C4⋊C4—1511no—
254C4⋊C4—1511no—
255C4⋊C4—1511no—
256C4⋊C4C22×D15111yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—3021no—
259C22⋊C4—3021no—
260C22⋊C4—1511no—
261C22⋊C4—1511no—
262C42—3021no—
263C42—3021no—
264C42—3021no—
265C42—3021no—
266C42—3021no—
267C42—3021no—
268C42—3021no—
269C42—1511no—
270C42—1511no—
271C23D4×D15111yescharacteristic
272Q8—24no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—3021no—
278Q8—3021no—
279Q8#1455900312yes—
280Q8#1455900312yes—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—3021no—
289C2×C4—3021no—
290C2×C4—3021no—
291C2×C4—3021no—
292C2×C4—1511no—
293C2×C4—1511no—
294C2×C4—1511no—
295C2×C4—1511no—
296C2×C4D4×D15111yescharacteristic
297C2×C4C23×D15111yescharacteristic
298C8—21no—
299C8—21no—
300C8—3021no—
301C8—3021no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D15111yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—3021no—
312C4—3021no—
313C4—3021no—
314C4—3021no—
315C4—6041no—
316C4—6041no—
317C4—6041no—
318C4—6041no—
319C4C2×#1455900311yescharacteristic
320C4C2×#1455900311yescharacteristic
321C2—41no—
322C2#2375271811yescharacteristic
323C2#2375302511yescharacteristic
324C2#2375302811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#451×C151C2
socleC22×C151C22×D4
prime radicalC23×C151C2×D4
hypercenter#451D151
hypocenterC151#3127
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #23752521C2semidirect
6. #23752517C2semidirect
7. #23752420C2semidirect
8. #23752420C2semidirect
10. #23752319C2semidirect
11. #23752316C2semidirect
12. Q16×Dic151C2semidirect
13. #23752304C2semidirect
162. C151#3127semidirect

Maximal subgroups

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

subgroupindex
#237528222
#237526902
#237525542
#237525502
#237525212
#237525172
#237524202
#237523682
#237523192
#237523162
Q16×Dic1512
#237523042
#237522702
#451×C1512
#3127151