Finite Group Databasebeta

← #30676223

Subgroups — #30676223

Order 16768.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22553933C211yesmaximal, characteristic, complemented
2#22553647C211yesmaximal, characteristic, complemented
3#22553625C211yesmaximal, characteristic, complemented
4#22553607C211yesmaximal, characteristic, complemented
5#22553592C211yesmaximal, characteristic, complemented
6#22553534C211yesmaximal, characteristic, complemented
7#22553527C211yesmaximal, characteristic, complemented
8#22553474C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic
10#22553412C211yesmaximal, characteristic, complemented
11#22553410C211yesmaximal, characteristic, complemented
12C4×#14523503C211yesmaximal, characteristic, complemented
13C4×#14523498C211yesmaximal, characteristic, complemented
14#22553373C211yesmaximal, characteristic, complemented
15#520×C131C211yesmaximal, characteristic, complemented
16C2×#14523503C2211yescharacteristic
17C2×#14523501C2211yescharacteristic
18C2×#14523500C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20#16862246C2211yescharacteristic
21#16862241—21no—
22#16862234C2211yescharacteristic
23#16862232—21no—
24#16862228—21no—
25#16862225C2211yescharacteristic
26#16862224—21no—
27#16862220C2211yescharacteristic
28C4×Dic262C2211yescharacteristic
29C4×D524C2211yescharacteristic
30#16862179C2211yescharacteristic
31#16862179C2211yescharacteristic
32#16862179C2211yescharacteristic
33#16862179C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic
35C4⋊(C131⋊C8)C2211yescharacteristic
36#16862170C2211yescharacteristic
37#16862170C2211yescharacteristic
38#16862169C2211yescharacteristic
39#16862169C2211yescharacteristic
40#16862167C2211yescharacteristic
41#16862167C2211yescharacteristic
42#16862166C2211yescharacteristic
43#16862166C2211yescharacteristic
44C22⋊(C131⋊C8)C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46#16862157C2211yescharacteristic
47C4×C131⋊C8C2211yescharacteristic
48C2×#41×C131C2211yescharacteristic
49C4×Q8×C131C2211yescharacteristic
50C4×Q8×C131C2211yescharacteristic
51C4×D4×C131—21no—
52C4×D4×C131—21no—
53C4×D4×C131C2211yescharacteristic
54C4×D4×C131C2211yescharacteristic
55#117×C131—21no—
56#117×C131C2211yescharacteristic
57C2×C4⋊C4×C131—21no—
58C2×C4⋊C4×C131C2211yescharacteristic
59C2×Dic262—21no—
60C2×Dic262C2311yescharacteristic
61C2×D524C2311yescharacteristic
62C2×C22⋊D131—21no—
63C2×C4×D131—21no—
64#14523503—22no—
65#14523501—22no—
66#14523500—22no—
67#14523498—22no—
68C2×C131⋊C8C2311yescharacteristic
69C2×C131⋊C8C2311yescharacteristic
70C2×C131⋊C8C2311yescharacteristic
71C2×C131⋊C8C2311yescharacteristic
72C4⋊Dic131—21no—
73C4⋊Dic131C2311yescharacteristic
74C4⋊Dic131C2311yescharacteristic
75Dic131⋊C4—21no—
76Dic131⋊C4—21no—
77C22⋊Dic131—21no—
78(C2×D131)⋊C4—21no—
79(C2×D131)⋊C4—21no—
80(C2×D131)⋊C4—21no—
81(C2×D131)⋊C4—21no—
82C4×Dic131—21no—
83#41×C131—22no—
84#41×C131—22no—
85C2×Q8×C131C2311yescharacteristic
86C2×D4×C131—21no—
87C2×D4×C131C2311yescharacteristic
88C22×C4×C131—21no—
89C22×C4×C131—21no—
90C22×C4×C131—21no—
91C22×C4×C131—21no—
92C22×C4×C131C2311yescharacteristic
93C4⋊C4×C131—21no—
94C4⋊C4×C131—21no—
95C4⋊C4×C131—21no—
96C4⋊C4×C131D411yescharacteristic
97C4⋊C4×C131D411yescharacteristic
98C4⋊C4×C131C2311yescharacteristic
99C4⋊C4×C131C2311yescharacteristic
100C22⋊C4×C131—21no—
101C22⋊C4×C131—21no—
102C22⋊C4×C131D411yescharacteristic
103C22⋊C4×C131D411yescharacteristic
104C42×C131—21no—
105C42×C131—21no—
106C42×C131C2311yescharacteristic
107C42×C131C2311yescharacteristic
108C22×D131—21no—
109Dic262—21no—
110Dic262—21no—
111Dic262—81no—
112D524—21no—
113D524—21no—
114C22⋊D131—81no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C4×D131—81no—
119C131⋊C8—21no—
120C131⋊C8—21no—
121C131⋊C8—21no—
122C131⋊C8—21no—
123C23×C131—21no—
124C23×C131C2×D411yescharacteristic
125Q8×C131—21no—
126Q8×C131#4112yes—
127D4×C131—24no—
128D4×C131—21no—
129D4×C131#4112yes—
130C2×C4×C131—24no—
131C2×C4×C131—21no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—21no—
135C2×C4×C131—21no—
136C2×C4×C131—81no—
137C2×C4×C131—81no—
138C2×C4×C131—81no—
139C2×C4×C131C2×D411yescharacteristic
140C2×C4×C131C2×D411yescharacteristic
141C2×C4×C131C2×D411yescharacteristic
142C2×C4×C131C2×D411yescharacteristic
143C2×C4×C131C2×D411yescharacteristic
144C2×C4×C131C2411yescharacteristic
145C22×C131—21no—
146C22×C131—22no—
147C22×C131—21no—
148C22×C131—21no—
149C22×C131—41no—
150C22×C131—41no—
151C22×C131C22×D411yescharacteristic
152C2×D131—41no—
153C2×D131—41no—
154C2×D131—41no—
155C4×C131—21no—
156C4×C131—22no—
157C4×C131—21no—
158C4×C131—21no—
159C4×C131—21no—
160C4×C131—21no—
161C4×C131—41no—
162C4×C131—41no—
163C4×C131C2×#4111yescharacteristic
164C4×C131C2×#4111yescharacteristic
165Dic131—41no—
166Dic131—41no—
167Dic131—41no—
168C2×C131—41no—
169C2×C131—42no—
170C2×C131#52511yescharacteristic
171C2×C131#57611yescharacteristic
172C2×C131#57811yescharacteristic
173D131—81no—
174C131#312411yescharacteristic, complemented, Sylow
175#3124—1311nomaximal, Sylow
176#542—1311no—
177#520D13111yescharacteristic, complemented
178#484—1311no—
179#480—1311no—
180#477—1311no—
181#474—1311no—
182#464—1311no—
183#463—1311no—
184#452—1311no—
185#449—1311no—
186#440—1311no—
187#440—1311no—
188C4×Q16—1311no—
189C4×D8—1311no—
190#433—1311no—
191C2×#41C2×D13111yescharacteristic
192C2×Q16—1311no—
193C2×SD16—1311no—
194C2×SD16—1311no—
195C2×D8—1311no—
196C4⋊Q8—1311no—
197#126—2621no—
198#124—2621no—
199#124—1311no—
200#123—2621no—
201C22⋊Q8—2621no—
202C22⋊Q8—1311no—
203#121—1311no—
204C4×Q8—1311no—
205C4×Q8C2×D13111yescharacteristic
206C4×Q8C2×D13111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1311no—
210C4×D4C2×D13111yescharacteristic
211C4×D4C2×D13111yescharacteristic
212#117—21no—
213#117C2×D13111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D13111yescharacteristic
216#107—1311no—
217#107—1311no—
218#107—1311no—
219#107—1311no—
220C4⋊C8—1311no—
221C4⋊C8—1311no—
222#103—1311no—
223#103—1311no—
224#103—1311no—
225#103—1311no—
226#102—1311no—
227#102—1311no—
228#102—1311no—
229#102—1311no—
230C22⋊C8—1311no—
231C22⋊C8—1311no—
232#97—1311no—
233C4×C8—1311no—
234#41—22no—
235#41—22no—
236C2×Q8—2621no—
237C2×Q8—1311no—
238C2×Q8C22×D13111yescharacteristic
239C2×D4—21no—
240C2×D4—2621no—
241C2×D4—1311no—
242C2×D4C22×D13111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2621no—
248C22×C4C22×D13111yescharacteristic
249Q16—2622no—
250SD16—2622no—
251SD16—2622no—
252D8—2622no—
253C2×C8—1311no—
254C2×C8—1311no—
255C2×C8—1311no—
256C2×C8—1311no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—2621no—
261C4⋊C4—2621no—
262C4⋊C4—2621no—
263C4⋊C4—1311no—
264C4⋊C4—1311no—
265C4⋊C4C22⋊D13111yescharacteristic
266C4⋊C4C22⋊D13111yescharacteristic
267C4⋊C4C22×D13111yescharacteristic
268C4⋊C4C22×D13111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2621no—
272C22⋊C4—2621no—
273C22⋊C4—2621no—
274C22⋊C4—2621no—
275C22⋊C4—2621no—
276C22⋊C4C22⋊D13111yescharacteristic
277C22⋊C4C22⋊D13111yescharacteristic
278C42—21no—
279C42—21no—
280C42—2621no—
281C42C22×D13111yescharacteristic
282C42C22×D13111yescharacteristic
283C23—21no—
284C23—2621no—
285C23C2×C22⋊D13111yescharacteristic
286Q8—21no—
287Q8—2621no—
288Q8—2621no—
289Q8—10481no—
290Q8#1452351312yes—
291D4—24no—
292D4—21no—
293D4—2621no—
294D4—2621no—
295D4—10481no—
296D4#1452351312yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2621no—
304C2×C4—2621no—
305C2×C4—2621no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—10481no—
310C2×C4C2×C22⋊D13111yescharacteristic
311C2×C4C2×C22⋊D13111yescharacteristic
312C2×C4C2×C22⋊D13111yescharacteristic
313C2×C4C2×C22⋊D13111yescharacteristic
314C2×C4C2×C22⋊D13111yescharacteristic
315C2×C4C23×D13111yescharacteristic
316C8—2621no—
317C8—2621no—
318C8—2621no—
319C8—2621no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—5241no—
327C22—5241no—
328C22—5241no—
329C22C22×C22⋊D13111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—5241no—
339C4—5241no—
340C4—5241no—
341C4C2×#1452351311yescharacteristic
342C4C2×#1452351311yescharacteristic
343C2—41no—
344C2—42no—
345C2—10481no—
346C2#2255382311yescharacteristic
347C2#2255411511yescharacteristic
348C2#2255412711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#520×C131C2
socleC22×C131C22×D4
prime radical#22553457C2
hypercenter#520D131
hypocenterC131#3124
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #22553933C2semidirect
2. #22553647C2semidirect
3. #22553625C2semidirect
4. #22553607C2semidirect
5. #22553592C2semidirect
6. #22553534C2semidirect
7. #22553527C2semidirect
8. #22553474C2semidirect
10. #22553412C2semidirect
11. #22553410C2semidirect
12. C4×#14523503C2semidirect
13. C4×#14523498C2semidirect
14. #22553373C2semidirect
15. #520×C131C2semidirect
174. C131#3124semidirect
177. #520D131semidirect

Maximal subgroups

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

subgroupindex
#225539332
#225536472
#225536252
#225536072
#225535922
#225535342
#225535272
#225534742
#225534572
#225534122
#225534102
C4×#145235032
C4×#145234982
#225533732
#520×C1312
#3124131