Finite Group Databasebeta

← #44550812

Subgroups — #44550812

Order 32128.

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

normal subgroups only

This group has 358 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
1#30632199C211yesmaximal, characteristic
2C2×#22548672C211yesmaximal, characteristic, complemented
3C2×#22548643C211yesmaximal, characteristic, complemented
4C2×#22548641C211yesmaximal, characteristic, complemented
5#30631650C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#339×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×C4×D251C2211yescharacteristic
10C2×#16860794C2211yescharacteristic
11#22548699—21no—
12#22548695C412yescomplemented
13#22548690C412yescomplemented
14#22548685—21no—
15C4×D1004—22no—
16#22548672—22no—
17#22548672—22no—
18C2×Dic251⋊C4—21no—
19C2×(C2×D251)⋊C4—21no—
20C2×C4×Dic251C2211yescharacteristic
21#22548643—22no—
22#22548641—22no—
23(C2×D251)⋊C8—22no—
24C2×M16×C251C2211yescharacteristic
25#117×C251C2211yescharacteristic
26C2×C42×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28#16860810—21no—
29#16860810—21no—
30#16860810—21no—
31#16860810—21no—
32C2×Dic502C2×C411yescharacteristic
33C2×D1004—22no—
34C2×D1004C2×C411yescharacteristic
35C2×C22⋊D251—21no—
36C22×Dic251—21no—
37C22×Dic251D411yescharacteristic
38C2×C4×D251—21no—
39C2×C4×D251—21no—
40C2×C4×D251—21no—
41C2×C4×D251—21no—
42C2×C4×D251—21no—
43C2×C4×D251C2×C412yes—
44C2×C4×D251D412yes—
45#16860794—21no—
46#16860794—21no—
47C2×C251⋊C8—21no—
48C4⋊Dic251—22no—
49Dic251⋊C4—24no—
50(C2×D251)⋊C4—24no—
51(C2×D251)⋊C4—22no—
52(C2×D251)⋊C4—22no—
53C4×Dic251—21no—
54C4×Dic251D412yescomplemented
55C22×C4×C251—21no—
56C22×C4×C251C2311yescharacteristic
57M16×C251—21no—
58M16×C251—21no—
59C2×C8×C251—21no—
60C4⋊C4×C251C2×C412yes—
61C22⋊C4×C251—21no—
62C42×C251—21no—
63C42×C251—21no—
64C42×C251D412yescomplemented
65C22×D251—21no—
66C22×D251—21no—
67C22×D251—21no—
68C22×D251—42no—
69C22×D251C22⋊C412yes—
70Dic502—21no—
71Dic502—21no—
72D1004—21no—
73D1004—21no—
74D1004—44no—
75C22⋊D251—41no—
76C22⋊D251—41no—
77C2×Dic251—21no—
78C2×Dic251—21no—
79C2×Dic251—21no—
80C2×Dic251—22no—
81C2×Dic251—42no—
82C2×Dic251C22⋊C412yes—
83C4×D251—22no—
84C4×D251—22no—
85C4×D251—22no—
86C4×D251—22no—
87C4×D251—41no—
88C4×D251—41no—
89C251⋊C8—41no—
90C23×C251C2×D411yescharacteristic
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—42no—
94C2×C4×C251C22×C411yescharacteristic
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251#4111yescharacteristic
99C2×C4×C251#4111yescharacteristic
100C8×C251—41no—
101C22×C251—22no—
102C22×C251C2×C22⋊C411yescharacteristic
103C22×C251#12011yescharacteristic
104C22×C251#12311yescharacteristic
105C2×D251—22no—
106C2×D251—22no—
107C2×D251—22no—
108C2×D251—42no—
109C2×D251—42no—
110C2×D251—41no—
111C2×D251—42no—
112C4×C251—42no—
113C4×C251—42no—
114C4×C251C4×D411yescharacteristic
115C4×C251C4×D411yescharacteristic
116C4×C251#12111yescharacteristic
117C4×C251#12111yescharacteristic
118Dic251—22no—
119Dic251—42no—
120Dic251—41no—
121C2×C251—21no—
122C2×C251—21no—
123C2×C251#38611yescharacteristic
124C2×C251#42112yes—
125D251—42no—
126D251—81no—
127C251#189711yescharacteristic, complemented, Sylow
128#1897—2511nomaximal, Sylow
129#525—2511no—
130C2×#117—2511no—
131C2×#104—2511no—
132C2×#104—2511no—
133#407—2511no—
134#339—2511no—
135#339D25111yescharacteristic, complemented
136C2×#41—2511no—
137C23×C4—2511no—
138C2×M16—2511no—
139C2×M16C2×D25111yescharacteristic
140#123—5021no—
141C22⋊Q8—2512no—
142#121—2512no—
143#120—5021no—
144C4×D4—5022no—
145#117—5022no—
146#117—5022no—
147#117C2×D25111yescharacteristic
148C2×C4⋊C4—5021no—
149C2×C22⋊C4—5021no—
150C2×C42—2511no—
151C2×C42C2×D25111yescharacteristic
152#104—5022no—
153#104—5022no—
154C22⋊C8—5022no—
155C24—2511no—
156#41—5021no—
157#41—5021no—
158#41—5021no—
159#41—5021no—
160C2×Q8—2511no—
161C2×D4—5022no—
162C2×D4—5021no—
163C2×D4—2511no—
164C22×C4—21no—
165C22×C4—5021no—
166C22×C4—5021no—
167C22×C4—5021no—
168C22×C4—5021no—
169C22×C4—5021no—
170C22×C4—5021no—
171C22×C4—2511no—
172C22×C4—2512no—
173C22×C4—2512no—
174C22×C4C22×D25111yescharacteristic
175M16—21no—
176M16—21no—
177M16—5021no—
178M16—5021no—
179C2×C8—21no—
180C2×C8—5021no—
181C4⋊C4—5024no—
182C4⋊C4—5022no—
183C4⋊C4C4×D25112yes—
184C22⋊C4—21no—
185C22⋊C4—5024no—
186C22⋊C4—5022no—
187C22⋊C4—5022no—
188C42—21no—
189C42—21no—
190C42—5021no—
191C42—2512no—
192C42D100412yescomplemented
193C23—10042no—
194C23—5021no—
195C23—5021no—
196C23—5021no—
197C23—2512no—
198C23D4×D25111yescharacteristic
199Q8—5021no—
200Q8—5021no—
201D4—10044no—
202D4—10041no—
203D4—10041no—
204D4—5021no—
205D4—5021no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—10042no—
210C2×C4—10041no—
211C2×C4—10041no—
212C2×C4—5021no—
213C2×C4—5022no—
214C2×C4—5021no—
215C2×C4—5021no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—2512no—
221C2×C4C2×C4×D25111yescharacteristic
222C2×C4D4×D25111yescharacteristic
223C2×C4D4×D25111yescharacteristic
224C2×C4C2×D100411yescharacteristic
225C2×C4#1686081011yescharacteristic
226C2×C4#1686081111yescharacteristic
227C8—41no—
228C8—10041no—
229C22—22no—
230C22—10042no—
231C22—10042no—
232C22—10042no—
233C22—10041no—
234C22—5022no—
235C22—5022no—
236C22—5022no—
237C22C22⋊C4×D25111yescharacteristic
238C22#2254868511yescharacteristic
239C22#2254870011yescharacteristic
240C4—42no—
241C4—42no—
242C4—10042no—
243C4—10041no—
244C4—5022no—
245C4#2254867711yescharacteristic
246C4C4×D100411yescharacteristic
247C4#2254868811yescharacteristic
248C4#2254869011yescharacteristic
249C2—21no—
250C2—21no—
251C2—20081no—
252C2—10042no—
253C2#3063156511yescharacteristic
254C2#3063170212yes—
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#339×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632199C2
cosocleC22×C4×C251C23
hypercenter#339D251
hypocenterC251#1897
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#22548672C2semidirect
3. C2×#22548643C2semidirect
4. C2×#22548641C2semidirect
5. #30631650C2semidirect
6. #30631446C2semidirect
7. #339×C251C2semidirect
12. #22548695C4semidirect
13. #22548690C4semidirect
54. C4×Dic251D4semidirect
64. C42×C251D4semidirect
127. C251#1897semidirect
135. #339D251semidirect
192. C42D1004semidirect

Maximal subgroups

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

subgroupindex
#306321992
C2×#225486722
C2×#225486432
C2×#225486412
#306316502
#306314462
#339×C2512
#1897251