Finite Group Databasebeta

← #44550772

Subgroups — #44550772

Order 32128.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#22548685C211yesmaximal, characteristic
2#30632160C211yesmaximal, characteristic, complemented
3C2×#22548629C211yesmaximal, characteristic, complemented
4#30631542C211yesmaximal, characteristic, complemented
5#30631542C211yesmaximal, characteristic, complemented
6#30631453C211yesmaximal, characteristic, complemented
7#342×C251C211yesmaximal, characteristic, complemented
8C24×D251C2211yescharacteristic
9C22×D1004—21no—
10C22×C22⋊D251C2211yescharacteristic
11#22548685—22no—
12#22548685C414yescomplemented
13C4×C22⋊D251—22no—
14(C22⋊D251)⋊C4—22no—
15#22548671—21no—
16C2×C22⋊Dic251C2211yescharacteristic
17C2×C22⋊Dic251C2211yescharacteristic
18C2×(C2×D251)⋊C4—21no—
19C2×(C2×D251)⋊C4—22no—
20C2×(C2×D251)⋊C4—22no—
21C22⋊C4×D251—21no—
22#22548629—22no—
23C2×C22⋊C4×C251C2211yescharacteristic
24C2×C22⋊C4×C251C2211yescharacteristic
25C2×C22⋊C4×C251C2211yescharacteristic
26C23×D251—21no—
27C23×D251—21no—
28C23×D251—21no—
29C23×D251—21no—
30C23×D251—21no—
31C23×D251—21no—
32C23×D251—44no—
33C23×D251C2×C412yescomplemented
34C23×D251D411yescharacteristic
35C2×D1004—24no—
36C2×D1004—44no—
37C2×C22⋊D251—21no—
38C2×C22⋊D251—21no—
39C2×C22⋊D251—21no—
40C2×C22⋊D251—21no—
41C2×C22⋊D251C2×C412yes—
42C2×C22⋊D251D412yes—
43C22×Dic251—21no—
44C22×Dic251D411yescharacteristic
45C22×Dic251D412yescomplemented
46C2×C4×D251—22no—
47Dic251⋊C4—22no—
48C22⋊Dic251—21no—
49C22⋊Dic251—21no—
50C22⋊Dic251—21no—
51C22⋊Dic251—21no—
52(C2×D251)⋊C4—22no—
53(C2×D251)⋊C4—24no—
54(C2×D251)⋊C4—44no—
55(C2×D251)⋊C4—44no—
56C4×Dic251—22no—
57C24×C251C2311yescharacteristic
58C22×C4×C251—21no—
59C22×C4×C251—21no—
60C22×C4×C251D412yescomplemented
61C22⋊C4×C251—21no—
62C22⋊C4×C251—21no—
63C22⋊C4×C251—21no—
64C22⋊C4×C251—21no—
65C22⋊C4×C251—21no—
66C22⋊C4×C251C2×C412yescomplemented
67C22×D251—24no—
68C22×D251—24no—
69C22×D251—24no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—21no—
74C22×D251—21no—
75C22×D251—44no—
76C22×D251—44no—
77C22×D251—44no—
78C22×D251—44no—
79C22×D251—42no—
80C22×D251—42no—
81C22×D251—42no—
82C22×D251—41no—
83C22×D251—41no—
84C22×D251—41no—
85C22×D251—41no—
86C22×D251—42no—
87C22×D251C22⋊C412yescomplemented
88D1004—48no—
89C22⋊D251—22no—
90C22⋊D251—22no—
91C22⋊D251—22no—
92C22⋊D251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—22no—
96C2×Dic251—42no—
97C2×Dic251—42no—
98C2×Dic251—41no—
99C2×Dic251C22⋊C412yescomplemented
100C4×D251—82no—
101C23×C251—42no—
102C23×C251C22×C411yescharacteristic
103C23×C251C2×D411yescharacteristic
104C23×C251C2×D411yescharacteristic
105C23×C251C2×D411yescharacteristic
106C23×C251C2×D411yescharacteristic
107C23×C251#4111yescharacteristic
108C23×C251#4111yescharacteristic
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251—42no—
112C2×C4×C251—42no—
113C2×C4×C251—42no—
114C2×C4×C251—41no—
115C22×C251—22no—
116C22×C251—22no—
117C22×C251—22no—
118C22×C251—41no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251—41no—
122C22×C251C2×C22⋊C411yescharacteristic
123C22×C251C4×D411yescharacteristic
124C22×C251C4×D411yescharacteristic
125C22×C251#12011yescharacteristic
126C22×C251#12111yescharacteristic
127C22×C251#12111yescharacteristic
128C22×C251#12311yescharacteristic
129C2×D251—24no—
130C2×D251—24no—
131C2×D251—24no—
132C2×D251—22no—
133C2×D251—44no—
134C2×D251—42no—
135C2×D251—44no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—44no—
139C2×D251—42no—
140C2×D251—42no—
141C2×D251—44no—
142C2×D251—42no—
143C2×D251—81no—
144C4×C251—42no—
145C4×C251—42no—
146C4×C251—81no—
147Dic251—22no—
148Dic251—42no—
149Dic251—81no—
150C2×C251—21no—
151C2×C251—21no—
152C2×C251—21no—
153C2×C251—21no—
154C2×C251—21no—
155C2×C251—21no—
156C2×C251#38611yescharacteristic
157C2×C251B(4,𝔽2)12yes—
158D251—44no—
159D251—42no—
160C251#188911yescharacteristic, complemented, Sylow
161#1889—2511nomaximal, Sylow
162C2×#120—2511no—
163#518—2511no—
164C2×#99—2511no—
165#379—2511no—
166#379—2511no—
167#342—2511no—
168#342D25111yescharacteristic, complemented
169C25—2511no—
170C22×D4—5021no—
171C22×D4—2511no—
172#120—5022no—
173#120—2514no—
174C4×D4—5022no—
175C4×D4—5022no—
176#117—5021no—
177C2×C22⋊C4—5021no—
178C2×C22⋊C4—5022no—
179C2×C22⋊C4—5022no—
180C2×C22⋊C4—5021no—
181C2×C22⋊C4—2511no—
182C2×C22⋊C4—2511no—
183C2×C22⋊C4C2×D25111yescharacteristic
184C2×C22⋊C4C2×D25111yescharacteristic
185C2×C22⋊C4C2×D25111yescharacteristic
186#99—5022no—
187C24—10044no—
188C24—5021no—
189C24—5021no—
190C24—5021no—
191C24—5021no—
192C24—5021no—
193C24—5021no—
194C24—2512no—
195C24—2511no—
196C24C22×D25111yescharacteristic
197C2×D4—10044no—
198C2×D4—5024no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—2512no—
204C2×D4—2512no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—5022no—
208C22×C4—5021no—
209C22×C4—2512no—
210C22×C4—2511no—
211C22×C4D100412yescomplemented
212C4⋊C4—5022no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—10044no—
219C22⋊C4—10044no—
220C22⋊C4—5022no—
221C22⋊C4—5024no—
222C22⋊C4—5021no—
223C22⋊C4—5021no—
224C22⋊C4—5021no—
225C22⋊C4—5021no—
226C22⋊C4C4×D25112yescomplemented
227C42—5022no—
228C23—42no—
229C23—10042no—
230C23—10042no—
231C23—10042no—
232C23—10042no—
233C23—10044no—
234C23—10044no—
235C23—10044no—
236C23—10044no—
237C23—10041no—
238C23—10041no—
239C23—10041no—
240C23—10041no—
241C23—5024no—
242C23—5024no—
243C23—5024no—
244C23—5021no—
245C23—5022no—
246C23—5022no—
247C23—5022no—
248C23—5021no—
249C23—2512no—
250C23C2×C4×D25111yescharacteristic
251C23D4×D25111yescharacteristic
252C23D4×D25111yescharacteristic
253C23D4×D25111yescharacteristic
254C23C2×D100411yescharacteristic
255C23#1686081011yescharacteristic
256C23#1686081111yescharacteristic
257D4—10048no—
258D4—5022no—
259D4—5022no—
260D4—5022no—
261D4—5022no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—20082no—
269C2×C4—10042no—
270C2×C4—10041no—
271C2×C4—10042no—
272C2×C4—5022no—
273C2×C4—5021no—
274C2×C4—5021no—
275C2×C4—2512no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—20081no—
284C22—10042no—
285C22—10042no—
286C22—10044no—
287C22—10044no—
288C22—10044no—
289C22—10042no—
290C22—10042no—
291C22—10042no—
292C22—10042no—
293C22—10044no—
294C22—5024no—
295C22—5024no—
296C22—5024no—
297C22—5022no—
298C22C22⋊C4×D25111yescharacteristic
299C22#2254867711yescharacteristic
300C22C4×D100411yescharacteristic
301C22#2254868511yescharacteristic
302C22#2254868811yescharacteristic
303C22#2254869011yescharacteristic
304C22#2254870011yescharacteristic
305C4—42no—
306C4—42no—
307C4—81no—
308C4—20081no—
309C4—10042no—
310C4—5022no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—10042no—
318C2—10044no—
319C2#3063156511yescharacteristic
320C2#3063187512yes—
subgroupG/H
centerC22#22548685
commutatorC23×C251C22×C4
FrattiniC24C22×D251
Fitting#342×C251C2
socleC22×C251#120
prime radicalC2×#22548685C2
cosocleC24×C251C23
hypercenter#342D251
hypocenterC251#1889
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #30632160C2semidirect
3. C2×#22548629C2semidirect
4. #30631542C2semidirect
5. #30631542C2semidirect
6. #30631453C2semidirect
7. #342×C251C2semidirect
12. #22548685C4semidirect
33. C23×D251C2×C4semidirect
45. C22×Dic251D4semidirect
60. C22×C4×C251D4semidirect
66. C22⋊C4×C251C2×C4semidirect
87. C22×D251C22⋊C4semidirect
99. C2×Dic251C22⋊C4semidirect
160. C251#1889semidirect
168. #342D251semidirect
211. C22×C4D1004semidirect
226. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486852
#306321602
C2×#225486292
#306315422
#306314532
#342×C2512
#1889251