Finite Group Databasebeta

← #44550311

Subgroups — #44550311

Order 32128.

670 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22548760C211yesmaximal, characteristic
2C2×#22548629C212yesmaximal, complemented
3#30631567C211yesmaximal, characteristic, complemented
4#30631564C212yesmaximal, complemented
5C2×#99×C251C211yesmaximal, characteristic, complemented
6#22548760—24no—
7#22548760—22no—
8#22548760C414yescomplemented
9C2×#16860810—21no—
10C2×#16860809—22no—
11C22×C22⋊D251—21no—
12C22×C22⋊D251C2212yes—
13C2×D4×D251—22no—
14C23×Dic251C2211yescharacteristic, complemented
15C2×C22⋊Dic251—21no—
16C2×C22⋊Dic251C2211yescharacteristic, complemented
17C2×(C2×D251)⋊C4—22no—
18#22548629—24no—
19C251⋊#95—22no—
20C251⋊#95—21no—
21C22×D4×C251C2211yescharacteristic
22C2×C22⋊C4×C251C2212yescomplemented
23#99×C251—22no—
24C23×D251—21no—
25C23×D251D412yes—
26#16860810—24no—
27#16860810—24no—
28#16860809—28no—
29#16860809—28no—
30C2×Dic502—21no—
31C2×D1004—21no—
32C2×C22⋊D251—22no—
33C2×C22⋊D251—24no—
34C2×C22⋊D251—24no—
35C2×C22⋊D251—22no—
36C2×C22⋊D251—22no—
37C2×C22⋊D251—22no—
38C2×C22⋊D251—22no—
39C2×C22⋊D251C2×C414yes—
40C2×C22⋊D251D414yes—
41D4×D251—28no—
42D4×D251—28no—
43C22×Dic251—21no—
44C22×Dic251—22no—
45C22×Dic251—21no—
46C22×Dic251—21no—
47C22×Dic251—42no—
48C22×Dic251D412yes—
49C2×C4×D251—22no—
50C22⋊Dic251—22no—
51C22⋊Dic251—42no—
52(C2×D251)⋊C4—44no—
53C24×C251D411yescharacteristic
54C24×C251C2311yescharacteristic
55C2×D4×C251—22no—
56C2×D4×C251—22no—
57C2×D4×C251C2×C412yes—
58C2×D4×C251D412yes—
59C22×C4×C251—22no—
60C22×C4×C251D411yescharacteristic
61C22⋊C4×C251—22no—
62C22⋊C4×C251—22no—
63C22×D251—22no—
64C22×D251—24no—
65C22×D251—22no—
66C22×D251—42no—
67C22×D251—42no—
68C22×D251—81no—
69C22×D251—81no—
70C22×D251C22⋊C414yescomplemented
71Dic502—24no—
72D1004—24no—
73C22⋊D251—28no—
74C22⋊D251—24no—
75C22⋊D251—24no—
76C22⋊D251—24no—
77C22⋊D251—24no—
78C2×Dic251—24no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—24no—
82C2×Dic251—22no—
83C2×Dic251—42no—
84C2×Dic251—41no—
85C2×Dic251—41no—
86C2×Dic251—41no—
87C2×Dic251—41no—
88C2×Dic251—41no—
89C2×Dic251C22⋊C414yescomplemented
90C4×D251—28no—
91C23×C251—21no—
92C23×C251—21no—
93C23×C251—41no—
94C23×C251—41no—
95C23×C251—41no—
96C23×C251—41no—
97C23×C251C22⋊C412yescomplemented
98C23×C251C22×C411yescharacteristic
99C23×C251C2×D412yes—
100C23×C251C2×D412yes—
101C23×C251C2×D411yescharacteristic
102C23×C251C2×D411yescharacteristic
103D4×C251—24no—
104D4×C251—24no—
105C2×C4×C251—21no—
106C2×C4×C251—21no—
107C2×C4×C251—42no—
108C2×C4×C251—42no—
109C2×C4×C251—42no—
110C2×C4×C251C22⋊C412yescomplemented
111C22×C251—22no—
112C22×C251—21no—
113C22×C251—22no—
114C22×C251—21no—
115C22×C251—22no—
116C22×C251—41no—
117C22×C251—42no—
118C22×C251—41no—
119C22×C251—42no—
120C22×C251—41no—
121C22×C251—41no—
122C22×C251—81no—
123C22×C251C2×C22⋊C412yes—
124C22×C251C2×C22⋊C411yescharacteristic
125C22×C251#12012yes—
126C22×C251#12011yescharacteristic
127C22×C251#12011yescharacteristic
128C2×D251—24no—
129C2×D251—42no—
130C2×D251—42no—
131C2×D251—44no—
132C2×D251—82no—
133C2×D251—84no—
134C4×C251—22no—
135C4×C251—82no—
136Dic251—24no—
137Dic251—42no—
138Dic251—81no—
139C2×C251—21no—
140C2×C251—22no—
141C2×C251—22no—
142C2×C251—21no—
143C2×C251—42no—
144C2×C251#37911yescharacteristic
145C2×C251B(4,𝔽2)11yescharacteristic
146C2×C251#45811yescharacteristic
147D251—44no—
148C251#179011yescharacteristic, complemented, Sylow
149#1790—2511nomaximal, Sylow
150C2×H(5,𝔽2)—2511no—
151C2×#99—2512no—
152C2×#99D25111yescharacteristic, complemented
153#386—2511no—
154#386—2512no—
155H(5,𝔽2)—5024no—
156H(5,𝔽2)—5022no—
157H(5,𝔽2)—2514no—
158C2×#41—5022no—
159C2×#41—5021no—
160C22×D4—5022no—
161C22×D4—5021no—
162C22×D4—2512no—
163C22×D4C2×D25111yescharacteristic
164C23×C4—2511no—
165C2×C22⋊C4—5022no—
166C2×C22⋊C4—5021no—
167C2×C22⋊C4—2511no—
168C2×C22⋊C4C2×D25112yescomplemented
169#99—22no—
170#99—5024no—
171#95—5022no—
172#95—5021no—
173C24—5021no—
174C24—2512no—
175C24D100411yescharacteristic
176C24C22×D25111yescharacteristic
177#41—5024no—
178#41—5028no—
179#41—5028no—
180#41—5024no—
181C2×Q8—5021no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—5021no—
185C2×D4—5028no—
186C2×D4—5028no—
187C2×D4—5022no—
188C2×D4—5024no—
189C2×D4—5024no—
190C2×D4—5022no—
191C2×D4—5022no—
192C2×D4—5022no—
193C2×D4—5022no—
194C2×D4—2514no—
195C2×D4—2514no—
196C2×D4C4×D25112yes—
197C2×D4C22⋊D25112yes—
198C22×C4—22no—
199C22×C4—10042no—
200C22×C4—5022no—
201C22×C4—5021no—
202C22×C4—5022no—
203C22×C4—5021no—
204C22×C4—5021no—
205C22×C4—2512no—
206C22×C4D100411yescharacteristic
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—10044no—
210C22⋊C4—10042no—
211C22⋊C4—5022no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—20081no—
219C23—20081no—
220C23—10042no—
221C23—10042no—
222C23—5024no—
223C23—5022no—
224C23—5022no—
225C23—2514no—
226C23(C2×D251)⋊C412yescomplemented
227C23C2×C4×D25111yescharacteristic
228C23D4×D25112yes—
229C23D4×D25112yes—
230C23C2×C22⋊D25111yescharacteristic
231C23C2×D100411yescharacteristic
232Q8—5024no—
233D4—24no—
234D4—24no—
235D4—5024no—
236D4—5028no—
237D4—5024no—
238D4—5024no—
239D4—5024no—
240D4—5024no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—10041no—
247C2×C4—10041no—
248C2×C4—10041no—
249C2×C4—10042no—
250C2×C4—10041no—
251C2×C4—10041no—
252C2×C4—5028no—
253C2×C4—5024no—
254C2×C4—5022no—
255C2×C4—5024no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—2514no—
259C2×C4(C2×D251)⋊C412yescomplemented
260C22—22no—
261C22—21no—
262C22—22no—
263C22—21no—
264C22—22no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—20082no—
273C22—20084no—
274C22—10044no—
275C22—10042no—
276C22—10042no—
277C22—5024no—
278C22C22⋊C4×D25112yes—
279C22C2×(C2×D251)⋊C411yescharacteristic
280C22#2254868312yes—
281C22#2254868511yescharacteristic
282C22#2254868511yescharacteristic
283C4—22no—
284C4—82no—
285C4—20081no—
286C4—10042no—
287C4—5024no—
288C2—22no—
289C2—21no—
290C2—21no—
291C2—22no—
292C2—42no—
293C2—10044no—
294C2#3063154211yescharacteristic
295C2#3063187511yescharacteristic
296C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutatorC23×C251C22×C4
FrattiniC24C22×D251
FittingC2×#99×C251C2
socleC22×C251#120
prime radicalC2×#22548760C2
cosocleC24×C251C23
hypercenterC2×#99D251
hypocenterC251#1790
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#22548629C2semidirect
3. #30631567C2semidirect
4. #30631564C2semidirect
5. C2×#99×C251C2semidirect
8. #22548760C4semidirect
14. C23×Dic251C22semidirect
16. C2×C22⋊Dic251C22semidirect
22. C2×C22⋊C4×C251C22semidirect
70. C22×D251C22⋊C4semidirect
89. C2×Dic251C22⋊C4semidirect
97. C23×C251C22⋊C4semidirect
110. C2×C4×C251C22⋊C4semidirect
148. C251#1790semidirect
152. C2×#99D251semidirect
168. C2×C22⋊C4C2×D251semidirect
226. C23(C2×D251)⋊C4semidirect
259. C2×C4(C2×D251)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225486292
#306315672
#306315642
C2×#99×C2512
#1790251