Finite Group Databasebeta

← #44550337

Subgroups — #44550337

Order 32128.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22548763C211yesmaximal, characteristic
2#30631705C211yesmaximal, characteristic, complemented
3#30631703C211yesmaximal, characteristic, complemented
4#30631696C211yesmaximal, characteristic, complemented
5#30631669C211yesmaximal, characteristic, complemented
6#30631660C211yesmaximal, characteristic, complemented
7#419×C251C211yesmaximal, characteristic, complemented
8#22548763—22no—
9#22548763—22no—
10#22548763—22no—
11#22548763—22no—
12C2×#16860811—21no—
13C2×#16860811C412yescomplemented
14C2×#16860810—22no—
15C2×#16860810C2211yescharacteristic
16C2×#16860810C2211yescharacteristic
17C2×Q8×D251—21no—
18C2×Q8×D251C412yescomplemented
19C2×#16860794C2211yescharacteristic
20#22548674C2211yescharacteristic
21#22548643—22no—
22#22548641—22no—
23#22548639—22no—
24#22548633—21no—
25#22548631—21no—
26#22548629—22no—
27C22×Q8×C251C2211yescharacteristic
28C2×M16×C251C2211yescharacteristic
29#117×C251C2211yescharacteristic
30#103×C251—22no—
31#16860811—24no—
32#16860811—24no—
33#16860811—24no—
34#16860811—24no—
35#16860810—28no—
36#16860810—28no—
37#16860810—21no—
38#16860810—21no—
39#16860810—21no—
40#16860810—21no—
41#16860810—21no—
42#16860810—21no—
43#16860810D412yes—
44#16860810D412yes—
45C2×Dic502—22no—
46C2×Dic502C2×C411yescharacteristic
47C2×Dic502D411yescharacteristic
48Q8×D251—24no—
49Q8×D251—24no—
50Q8×D251—24no—
51Q8×D251—24no—
52C2×D1004—22no—
53C2×D1004C2×C411yescharacteristic
54C2×D1004D411yescharacteristic
55C2×C22⋊D251—21no—
56C2×C22⋊D251D412yes—
57C2×C4×D251—22no—
58C2×C4×D251—22no—
59C2×C4×D251—21no—
60C2×C4×D251C2×C412yes—
61#16860794—21no—
62#16860794—21no—
63C2×C251⋊C8—21no—
64C4⋊Dic251—21no—
65C22⋊Dic251—21no—
66C4×Dic251—21no—
67C2×Q8×C251—21no—
68C2×Q8×C251—21no—
69C2×Q8×C251—21no—
70C2×Q8×C251C2×C412yes—
71C2×Q8×C251D412yes—
72C2×Q8×C251D412yes—
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75M16×C251—21no—
76M16×C251—21no—
77C2×C8×C251—21no—
78C4⋊C4×C251—21no—
79C22⋊C4×C251—21no—
80C42×C251—21no—
81C22×D251—21no—
82C22×D251C22⋊C412yes—
83Dic502—28no—
84Dic502—21no—
85Dic502—21no—
86Dic502—21no—
87Dic502C22⋊C412yes—
88D1004—28no—
89D1004—21no—
90D1004—21no—
91D1004—21no—
92D1004C22⋊C412yes—
93C22⋊D251—22no—
94C22⋊D251—22no—
95C22⋊D251—22no—
96C22⋊D251—22no—
97C2×Dic251—21no—
98C2×Dic251—41no—
99C2×Dic251C22⋊C412yes—
100C4×D251—28no—
101C4×D251—28no—
102C4×D251—22no—
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—22no—
106C251⋊C8—41no—
107C23×C251C2×D411yescharacteristic
108Q8×C251—22no—
109Q8×C251—22no—
110Q8×C251—22no—
111Q8×C251C22⋊C414yes—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—41no—
116C2×C4×C251C22×C411yescharacteristic
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D411yescharacteristic
122C8×C251—41no—
123C22×C251—41no—
124C22×C251C2×C22⋊C411yescharacteristic
125C22×C251#12011yescharacteristic
126C22×C251#12011yescharacteristic
127C2×D251—22no—
128C2×D251—22no—
129C2×D251—42no—
130C2×D251—42no—
131C4×C251—24no—
132C4×C251—81no—
133C4×C251C2×C22⋊C411yescharacteristic
134C4×C251C2×C22⋊C411yescharacteristic
135C4×C251#12011yescharacteristic
136C4×C251#12011yescharacteristic
137Dic251—22no—
138Dic251—22no—
139Dic251—81no—
140C2×C251—21no—
141C2×C251—21no—
142C2×C251—21no—
143C2×C251#37911yescharacteristic
144D251—42no—
145D251—42no—
146C251#179611yescharacteristic, complemented, Sylow
147#1796—2511nomaximal, Sylow
148C2×2−1+4—2511no—
149#421—2511no—
150#421—2511no—
151#419—2511no—
152#419D25111yescharacteristic, complemented
153#413—2511no—
154#410—2511no—
1552−1+4—5022no—
1562−1+4—5022no—
1572−1+4—5022no—
1582−1+4—5022no—
159C2×#41—5022no—
160C2×#41—5021no—
161C2×#41—2512no—
162C2×#41—2511no—
163C2×#41—2511no—
164C22×Q8—5021no—
165C22×Q8—2512no—
166C22×Q8C2×D25111yescharacteristic
167C2×M16—2511no—
168C2×M16C2×D25111yescharacteristic
169#117—2511no—
170#117C2×D25111yescharacteristic
171#104—5022no—
172#104—5022no—
173#103—22no—
174#103—5022no—
175#101—5021no—
176#100—5021no—
177#99—5022no—
178#41—5028no—
179#41—5028no—
180#41—5024no—
181#41—5024no—
182#41—5024no—
183#41—5024no—
184#41—5021no—
185#41—5021no—
186#41—5021no—
187#41—5021no—
188#41—5021no—
189#41—5021no—
190#41—2512no—
191#41—2512no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—5022no—
196C2×Q8—5024no—
197C2×Q8—5024no—
198C2×Q8—5024no—
199C2×Q8—5024no—
200C2×Q8—2511no—
201C2×Q8—2511no—
202C2×Q8C4×D25112yes—
203C2×Q8C22⋊D25112yes—
204C2×Q8D100412yes—
205C2×D4—5022no—
206C2×D4—5021no—
207C2×D4—2512no—
208C2×D4—2511no—
209C2×D4—2511no—
210C22×C4—21no—
211C22×C4—5022no—
212C22×C4—5022no—
213C22×C4—5021no—
214C22×C4—2512no—
215C22×C4C22×D25111yescharacteristic
216M16—21no—
217M16—21no—
218M16—5021no—
219M16—5021no—
220C2×C8—21no—
221C2×C8—5021no—
222C4⋊C4—21no—
223C4⋊C4—5021no—
224C22⋊C4—21no—
225C22⋊C4—5021no—
226C42—21no—
227C42—5021no—
228C23—5021no—
229C23—2512no—
230C23D4×D25111yescharacteristic
231Q8—22no—
232Q8—22no—
233Q8—22no—
234Q8—5028no—
235Q8—5021no—
236Q8—5021no—
237Q8—5021no—
238Q8—2512no—
239Q8(C2×D251)⋊C414yes—
240D4—5028no—
241D4—5022no—
242D4—5022no—
243D4—5022no—
244D4—5022no—
245D4—5021no—
246D4—5021no—
247D4—5021no—
248D4—2512no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—10041no—
254C2×C4—5028no—
255C2×C4—5028no—
256C2×C4—5021no—
257C2×C4—5022no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—2512no—
262C2×C4C2×C4×D25111yescharacteristic
263C2×C4D4×D25111yescharacteristic
264C2×C4D4×D25111yescharacteristic
265C2×C4D4×D25111yescharacteristic
266C2×C4C2×C22⋊D25111yescharacteristic
267C2×C4C2×D100411yescharacteristic
268C8—41no—
269C8—10041no—
270C22—41no—
271C22—10042no—
272C22—10042no—
273C22—5022no—
274C22—5022no—
275C22C22⋊C4×D25111yescharacteristic
276C22#2254868311yescharacteristic
277C22#2254868511yescharacteristic
278C4—24no—
279C4—81no—
280C4—20081no—
281C4—5022no—
282C4—5022no—
283C4C22⋊C4×D25111yescharacteristic
284C4C2×(C2×D251)⋊C411yescharacteristic
285C4#2254868311yescharacteristic
286C4#2254868511yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—10042no—
291C2—10042no—
292C2#3063154211yescharacteristic
subgroupG/H
centerC2#30631542
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#419×C251C2
socleC2×C251#379
prime radicalC2×#22548763C2
cosocleC22×C4×C251C23
hypercenter#419D251
hypocenterC251#1796
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #30631705C2semidirect
3. #30631703C2semidirect
4. #30631696C2semidirect
5. #30631669C2semidirect
6. #30631660C2semidirect
7. #419×C251C2semidirect
13. C2×#16860811C4semidirect
18. C2×Q8×D251C4semidirect
146. C251#1796semidirect
152. #419D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487632
#306317052
#306317032
#306316962
#306316692
#306316602
#419×C2512
#1796251