Finite Group Databasebeta

← #44551367

Subgroups — #44551367

Order 32128.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#16860797C211yesmaximal, characteristic, complemented
2C2×#22548691C211yesmaximal, characteristic, complemented
3C2×#22548690C211yesmaximal, characteristic
4C2×#22548637C211yesmaximal, characteristic, complemented
5C2×(C2×D251)⋊C8C211yesmaximal, characteristic, complemented
6#30631449C211yesmaximal, characteristic, complemented
7C2×#103×C251C211yesmaximal, characteristic, complemented
8C22×D1004—21no—
9C22×D1004C2211yescharacteristic
10C22×C4×D251C2211yescharacteristic
11C2×#16860797—22no—
12C2×#16860797—22no—
13C2×#16860797—22no—
14C22×C251⋊C8C2211yescharacteristic
15#22548691—22no—
16#22548691—22no—
17#22548690—22no—
18#22548690—22no—
19C2×C4⋊Dic251C2211yescharacteristic
20C2×Dic251⋊C4—21no—
21C2×(C2×D251)⋊C4—21no—
22C2×(C2×D251)⋊C4—21no—
23#22548637—22no—
24(C2×D251)⋊C8—22no—
25C22×Q8×C251C2211yescharacteristic
26C22×C8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#103×C251—22no—
29C23×D251—21no—
30C23×D251D411yescharacteristic
31C2×D1004—24no—
32C2×D1004—21no—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004—21no—
36C2×D1004—21no—
37C2×D1004—42no—
38C2×D1004—42no—
39C2×D1004D412yes—
40C22×Dic251—21no—
41C22×Dic251D411yescharacteristic
42C2×C4×D251—21no—
43C2×C4×D251—21no—
44C2×C4×D251—21no—
45C2×C4×D251—21no—
46C2×C4×D251—21no—
47C2×C4×D251—21no—
48#16860797—28no—
49C2×C251⋊C8—21no—
50C2×C251⋊C8—21no—
51C2×C251⋊C8D412yescomplemented
52C4⋊Dic251—21no—
53C4⋊Dic251D412yescomplemented
54Dic251⋊C4—24no—
55(C2×D251)⋊C4—24no—
56(C2×D251)⋊C4—24no—
57C2×Q8×C251—21no—
58C2×Q8×C251—21no—
59C2×Q8×C251—21no—
60C2×Q8×C251—21no—
61C2×Q8×C251—21no—
62C2×Q8×C251D412yescomplemented
63C22×C4×C251—21no—
64C22×C4×C251—21no—
65C22×C4×C251C2311yescharacteristic
66C2×C8×C251—21no—
67C2×C8×C251—21no—
68C2×C8×C251—21no—
69C4⋊C4×C251—21no—
70C4⋊C4×C251D412yes—
71C22×D251—21no—
72C22×D251—21no—
73C22×D251—21no—
74C22×D251—21no—
75C22×D251—21no—
76C22×D251—21no—
77C22×D251—22no—
78C22×D251—41no—
79C22×D251—41no—
80C22×D251—41no—
81C22×D251—41no—
82C22×D251—41no—
83C22×D251—41no—
84C22×D251SD1612yescomplemented
85D1004—22no—
86D1004—22no—
87D1004—48no—
88D1004—41no—
89D1004—41no—
90C2×Dic251—21no—
91C2×Dic251—22no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—41no—
95C2×Dic251—41no—
96C4×D251—41no—
97C4×D251—41no—
98C4×D251—41no—
99C4×D251—41no—
100C251⋊C8—22no—
101C23×C251C2×D411yescharacteristic
102Q8×C251—22no—
103Q8×C251—22no—
104Q8×C251—41no—
105Q8×C251—41no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—41no—
109C2×C4×C251—41no—
110C2×C4×C251—41no—
111C2×C4×C251—41no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4111yescharacteristic
118C8×C251—41no—
119C22×C251#12011yescharacteristic
120C22×C251#12011yescharacteristic
121C22×C251#12111yescharacteristic
122C22×C251C2×SD1611yescharacteristic
123C22×C251#13511yescharacteristic
124C22×C251AGL(1,ℤ/8)11yescharacteristic
125C22×C251#13711yescharacteristic
126C2×D251—22no—
127C2×D251—22no—
128C2×D251—22no—
129C2×D251—41no—
130C2×D251—41no—
131C2×D251—41no—
132C2×D251—41no—
133C2×D251—42no—
134C2×D251—42no—
135C2×D251—42no—
136C2×D251—81no—
137C2×D251—81no—
138C2×D251—81no—
139C2×D251—81no—
140C4×C251—42no—
141C4×C251—42no—
142C4×C251#12011yescharacteristic
143C4×C251#12111yescharacteristic
144C4×C251#12111yescharacteristic
145C4×C251#12711yescharacteristic
146Dic251—41no—
147Dic251—42no—
148C2×C251#39211yescharacteristic
149C2×C251#44811yescharacteristic
150C2×C251#44911yescharacteristic
151C2×C251#45011yescharacteristic
152C2×C251#45211yescharacteristic
153C2×C251#46511yescharacteristic
154C2×C251#46811yescharacteristic
155D251—42no—
156D251—82no—
157C251#200011yescharacteristic, complemented, Sylow
158#2000—2511nomaximal, Sylow
159C22×SD16—2511no—
160C2×C22⋊Q8—2511no—
161C2×#121—2511no—
162C2×#103D25111yescharacteristic, complemented
163C2×#102—2511no—
164C2×C22⋊C8—2511no—
165#340—2511no—
166C22×Q8C2×D25111yescharacteristic
167C22×D4—5021no—
168C22×D4—2511no—
169C23×C4—2511no—
170C2×SD16—5022no—
171C2×SD16—5022no—
172C2×SD16—5022no—
173C22×C8—2511no—
174C22×C8C2×D25111yescharacteristic
175C22⋊Q8—5022no—
176C22⋊Q8—5022no—
177#121—5022no—
178#121—5022no—
179C2×C4⋊C4—5021no—
180C2×C4⋊C4—2511no—
181C2×C4⋊C4C2×D25111yescharacteristic
182C2×C22⋊C4—5021no—
183C2×C22⋊C4—5021no—
184#103—22no—
185#102—5022no—
186C22⋊C8—5022no—
187C24—5021no—
188C24—2511no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8D100412yescomplemented
195C2×D4—10042no—
196C2×D4—10042no—
197C2×D4—5024no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—2512no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4—5021no—
209C22×C4—5021no—
210C22×C4—5021no—
211C22×C4—5021no—
212C22×C4—5021no—
213C22×C4—2511no—
214C22×C4C22×D25111yescharacteristic
215SD16—5028no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—5021no—
220C2×C8—5021no—
221C2×C8—2512no—
222C4⋊C4—21no—
223C4⋊C4—5024no—
224C4⋊C4—5021no—
225C4⋊C4—2512no—
226C4⋊C4C22⋊D25112yes—
227C22⋊C4—5024no—
228C22⋊C4—5024no—
229C23—10041no—
230C23—10041no—
231C23—10041no—
232C23—10041no—
233C23—10041no—
234C23—10041no—
235C23—5022no—
236C23—5021no—
237C23—5021no—
238C23—5021no—
239C23—5021no—
240C23—5021no—
241C23—5021no—
242C23—2512no—
243C23D4×D25111yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—10048no—
249D4—10041no—
250D4—10041no—
251D4—5022no—
252D4—5022no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—10041no—
260C2×C4—10041no—
261C2×C4—10041no—
262C2×C4—10041no—
263C2×C4—10041no—
264C2×C4—10041no—
265C2×C4—5021no—
266C2×C4—5022no—
267C2×C4—5021no—
268C2×C4—5021no—
269C2×C4D4×D25111yescharacteristic
270C2×C4D4×D25111yescharacteristic
271C2×C4D4×D25111yescharacteristic
272C2×C4C2×C22⋊D25111yescharacteristic
273C2×C4C2×D100411yescharacteristic
274C2×C4#1686081011yescharacteristic
275C8—41no—
276C8—5022no—
277C22—20081no—
278C22—20081no—
279C22—20081no—
280C22—20081no—
281C22—10042no—
282C22—10042no—
283C22—10042no—
284C22—10041no—
285C22—10041no—
286C22—10041no—
287C22—10041no—
288C22—5022no—
289C22—5022no—
290C22—5022no—
291C22#2254868311yescharacteristic
292C22#2254868511yescharacteristic
293C22#2254868811yescharacteristic
294C22SD16×D25111yescharacteristic
295C22#2254873811yescharacteristic
296C22#2254874211yescharacteristic
297C22#2254874711yescharacteristic
298C4—42no—
299C4—42no—
300C4—10042no—
301C4—10041no—
302C4#2254868511yescharacteristic
303C4#2254868811yescharacteristic
304C4#2254868911yescharacteristic
305C4#2254871111yescharacteristic
306C2—20082no—
307C2—10042no—
308C2#3063159611yescharacteristic
309C2#3063182711yescharacteristic
310C2#3063183511yescharacteristic
311C2#3063183611yescharacteristic
312C2#3063184811yescharacteristic
313C2#3063191911yescharacteristic
314C2#3063193611yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
FittingC2×#103×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548690C2
hypercenterC2×#103D251
hypocenterC251#2000
LCSG ▷ C22×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#16860797C2semidirect
2. C2×#22548691C2semidirect
4. C2×#22548637C2semidirect
5. C2×(C2×D251)⋊C8C2semidirect
6. #30631449C2semidirect
7. C2×#103×C251C2semidirect
51. C2×C251⋊C8D4semidirect
53. C4⋊Dic251D4semidirect
62. C2×Q8×C251D4semidirect
84. C22×D251SD16semidirect
157. C251#2000semidirect
162. C2×#103D251semidirect
194. C2×Q8D1004semidirect

Maximal subgroups

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

subgroupindex
C22×#168607972
C2×#225486912
C2×#225486902
C2×#225486372
C2×(C2×D251)⋊C82
#306314492
C2×#103×C2512
#2000251