Finite Group Databasebeta

← #44553140

Subgroups — #44553140

Order 32128.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548709C211yesmaximal, characteristic
2#30631584C211yesmaximal, characteristic, complemented
3#30631582C211yesmaximal, characteristic, complemented
4#30631579C211yesmaximal, characteristic
5#30631578C211yesmaximal, characteristic
6#30631577C211yesmaximal, characteristic
7#30631574C211yesmaximal, characteristic, complemented
8#30631571C211yesmaximal, characteristic, complemented
9#30631568C211yesmaximal, characteristic, complemented
10#30631561C211yesmaximal, characteristic
11#30631558C211yesmaximal, characteristic
12#30631553C211yesmaximal, characteristic, complemented
13C4×Dic251⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D251)⋊C4C211yesmaximal, characteristic
15C4×C4⋊C4×C251C211yesmaximal, characteristic, complemented
16C22×C4×D251C2211yescharacteristic
17#22548709C418yescomplemented
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C42×C251C2211yescharacteristic
48C2×C42×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C22×Dic251C2311yescharacteristic
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C2×C4×D251—22no—
61C2×C4×D251—41no—
62C2×C4×D251—41no—
63C4⋊Dic251—41no—
64C4⋊Dic251—41no—
65Dic251⋊C4—22no—
66Dic251⋊C4—41no—
67Dic251⋊C4—41no—
68Dic251⋊C4C2×C414yescomplemented
69Dic251⋊C4C2×C414yescomplemented
70Dic251⋊C4C2×C414yescomplemented
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—41no—
73(C2×D251)⋊C4—41no—
74(C2×D251)⋊C4C2×C414yes—
75(C2×D251)⋊C4C2×C414yes—
76(C2×D251)⋊C4C2×C414yes—
77C4×Dic251—22no—
78C4×Dic251—41no—
79C4×Dic251—41no—
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C4⋊C4×C251—22no—
88C4⋊C4×C251—22no—
89C42×C251—41no—
90C42×C251—41no—
91C42×C251C2×C414yescomplemented
92C22×D251—21no—
93C22×D251—21no—
94C22×D251—21no—
95C22×D251—21no—
96C22×D251—21no—
97C22×D251—21no—
98C22×D251C22×C412yes—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251C22×C412yes—
118C2×Dic251C22×C412yes—
119C2×Dic251C22×C412yes—
120C4×D251—44no—
121C23×C251C2411yescharacteristic
122C2×C4×C251—21no—
123C2×C4×C251—21no—
124C2×C4×C251—21no—
125C2×C4×C251—21no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251C22×C412yes—
139C2×C4×C251C22×C412yes—
140C2×C4×C251C22×C412yes—
141C2×C4×C251#4112yes—
142C2×C4×C251#4112yes—
143C22×C251C23×C411yescharacteristic
144C22×C251C2×#4111yescharacteristic
145C22×C251C2×#4111yescharacteristic
146C22×C251H(5,𝔽2)11yescharacteristic
147C22×C251H(5,𝔽2)11yescharacteristic
148C22×C2512−1+411yescharacteristic
149C22×C2512−1+411yescharacteristic
150C2×D251—22no—
151C2×D251—22no—
152C2×D251—22no—
153C2×D251—41no—
154C2×D251—41no—
155C2×D251—41no—
156C2×D251—41no—
157C4×C251—22no—
158C4×C251—22no—
159C4×C251—22no—
160C4×C251—22no—
161C4×C251—22no—
162C4×C251—41no—
163C4×C251—41no—
164Dic251—22no—
165Dic251—22no—
166Dic251—22no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171C2×C251C4×#4111yescharacteristic
172C2×C251#52011yescharacteristic
173C2×C251#52011yescharacteristic
174C2×C251#53811yescharacteristic
175C2×C251#53911yescharacteristic
176C2×C251#54111yescharacteristic
177C2×C251#54211yescharacteristic
178D251—42no—
179C251#233411yescharacteristic, complemented, Sylow
180#2334—2511nomaximal, Sylow
181C2×#126—2511no—
182#389—2511no—
183#389—2511no—
184#388—2511no—
185#388—2511no—
186#388—2511no—
187#387—2511no—
188#387—2511no—
189#387—2511no—
190#385—2511no—
191#385—2511no—
192#382—2511no—
193C4×C4⋊C4—2511no—
194C4×C4⋊C4D25111yescharacteristic, complemented
195C4×C22⋊C4—2511no—
196C23×C4—2511no—
197#126—2518no—
198C2×C4⋊C4—2511no—
199C2×C4⋊C4—2511no—
200C2×C4⋊C4—2511no—
201C2×C4⋊C4—2511no—
202C2×C4⋊C4—2511no—
203C2×C4⋊C4—2511no—
204C2×C4⋊C4—2511no—
205C2×C4⋊C4—2511no—
206C2×C4⋊C4C2×D25111yescharacteristic
207C2×C4⋊C4C2×D25111yescharacteristic
208C2×C22⋊C4—2511no—
209C2×C22⋊C4—2511no—
210C2×C22⋊C4—2511no—
211C2×C22⋊C4—2511no—
212C2×C22⋊C4—2511no—
213C2×C22⋊C4—2511no—
214C2×C42—2511no—
215C2×C42—2511no—
216C2×C42—2511no—
217C2×C42C2×D25111yescharacteristic
218C2×C42C2×D25111yescharacteristic
219C2×C42C2×D25111yescharacteristic
220#95—2511no—
221#95—2511no—
222#95—2511no—
223#95—2511no—
224#95—2511no—
225#95—2511no—
226#95—2511no—
227#95—2511no—
228#95—2511no—
229#95—2511no—
230#95C2×D25111yescharacteristic
231#95C2×D25111yescharacteristic
232C24—2511no—
233C22×C4—10041no—
234C22×C4—10041no—
235C22×C4—5022no—
236C22×C4—2511no—
237C22×C4—2511no—
238C22×C4—2511no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4C22×D25111yescharacteristic
244C22×C4C22×D25111yescharacteristic
245C22×C4C22×D25111yescharacteristic
246C22×C4C22×D25111yescharacteristic
247C22×C4C22×D25111yescharacteristic
248C22×C4C22×D25111yescharacteristic
249C22×C4C22×D25111yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—10041no—
253C4⋊C4—10041no—
254C4⋊C4—10041no—
255C4⋊C4—10041no—
256C4⋊C4—5022no—
257C4⋊C4—2514no—
258C4⋊C4—2514no—
259C4⋊C4—2514no—
260C22⋊C4—10041no—
261C22⋊C4—10041no—
262C22⋊C4—5022no—
263C22⋊C4—2514no—
264C22⋊C4—2514no—
265C22⋊C4—2514no—
266C42—41no—
267C42—41no—
268C42—10041no—
269C42—10041no—
270C42—5022no—
271C42C4×D25114yescomplemented
272C23—5021no—
273C23—5021no—
274C23—5021no—
275C23—5021no—
276C23—5021no—
277C23—5021no—
278C23—2512no—
279C23C23×D25111yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—10044no—
297C2×C4—5021no—
298C2×C4—5021no—
299C2×C4—5021no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—2512no—
316C2×C4—2512no—
317C2×C4—2512no—
318C2×C4C2×C4×D25112yes—
319C2×C4C2×C4×D25112yes—
320C2×C4C2×C4×D25112yes—
321C2×C4#1686081012yes—
322C2×C4#1686081012yes—
323C22—10041no—
324C22—10041no—
325C22—10041no—
326C22—10041no—
327C22—5022no—
328C22—5022no—
329C22—5022no—
330C22C22×C4×D25111yescharacteristic
331C22C2×#1686081011yescharacteristic
332C22C2×#1686081011yescharacteristic
333C22#2254876011yescharacteristic
334C22#2254876111yescharacteristic
335C22#2254876211yescharacteristic
336C22#2254876311yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—10041no—
345C4—10041no—
346C4—10041no—
347C4—10041no—
348C4—5022no—
349C4—5022no—
350C4—5022no—
351C2—10042no—
352C2C4×#1686081011yescharacteristic
353C2#3063216711yescharacteristic
354C2#3063216811yescharacteristic
355C2#3063227011yescharacteristic
356C2#3063227911yescharacteristic
357C2#3063229511yescharacteristic
358C2#3063230911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
FittingC4×C4⋊C4×C251C2
socle, cosocleC23×C251C24
prime radicalC23×D251C23
hypercenterC4×C4⋊C4D251
hypocenterC251#2334
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #30631584C2semidirect
3. #30631582C2semidirect
7. #30631574C2semidirect
8. #30631571C2semidirect
9. #30631568C2semidirect
12. #30631553C2semidirect
13. C4×Dic251⋊C4C2semidirect
15. C4×C4⋊C4×C251C2semidirect
17. #22548709C4semidirect
68. Dic251⋊C4C2×C4semidirect
69. Dic251⋊C4C2×C4semidirect
70. Dic251⋊C4C2×C4semidirect
91. C42×C251C2×C4semidirect
179. C251#2334semidirect
194. C4×C4⋊C4D251semidirect
271. C42C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487092
#306315842
#306315822
#306315792
#306315782
#306315772
#306315742
#306315712
#306315682
#306315612
#306315582
#306315532
C4×Dic251⋊C42
C4×(C2×D251)⋊C42
C4×C4⋊C4×C2512
#2334251