Finite Group Databasebeta

← #44551378

Subgroups — #44551378

Order 32128.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22548740C211yesmaximal, characteristic, complemented
2C2×#22548711C211yesmaximal, characteristic, complemented
3C2×#22548642C211yesmaximal, characteristic, complemented
4#30631691C211yesmaximal, characteristic, complemented
5C2×#22548632C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#535×C251C211yesmaximal, characteristic, complemented
8C22×D1004C2211yescharacteristic
9C22×C22⋊D251—22no—
10#22548740—22no—
11#22548740—22no—
12C2×#16860797—21no—
13C2×#16860795—21no—
14C2×#16860794C2211yescharacteristic
15C2×#16860794C2211yescharacteristic
16#22548711—22no—
17#22548711—22no—
18C2×C4×Dic251C2211yescharacteristic
19#22548642—22no—
20#22548635—22no—
21#22548632—22no—
22C2×#41×C251C2211yescharacteristic
23C22×D4×C251C2211yescharacteristic
24#127×C251—21no—
25#124×C251—21no—
26#121×C251—22no—
27#120×C251—22no—
28#117×C251C2211yescharacteristic
29C23×D251—21no—
30C2×D1004—21no—
31C2×D1004—21no—
32C2×D1004—21no—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004D412yescomplemented
36C2×C22⋊D251—28no—
37C2×C22⋊D251—48no—
38C22×Dic251—21no—
39#16860797—24no—
40#16860795—24no—
41#16860794—21no—
42#16860794—21no—
43#16860794—21no—
44#16860794D412yescomplemented
45C2×C251⋊C8—21no—
46C2×C251⋊C8—21no—
47C4×Dic251—21no—
48C4×Dic251D412yescomplemented
49C24×C251D412yes—
50#41×C251—21no—
51#41×C251—21no—
52#41×C251—21no—
53#41×C251—21no—
54C2×Q8×C251D411yescharacteristic, complemented
55C2×D4×C251—21no—
56C2×D4×C251—24no—
57C2×D4×C251—21no—
58C2×D4×C251—21no—
59C2×D4×C251—21no—
60C2×D4×C251—21no—
61C2×D4×C251—21no—
62C2×D4×C251—21no—
63C2×D4×C251D411yescharacteristic, complemented
64C22×C4×C251—21no—
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251D412yescomplemented
67C22⋊C4×C251—24no—
68C22⋊C4×C251—21no—
69C42×C251—21no—
70C22×D251—22no—
71C22×D251—44no—
72C22×D251—41no—
73C22×D251—41no—
74D1004—22no—
75D1004—22no—
76D1004—41no—
77D1004—41no—
78C22⋊D251—416no—
79C2×Dic251—24no—
80C2×Dic251—22no—
81C251⋊C8—22no—
82C251⋊C8—41no—
83C23×C251—22no—
84C23×C251—21no—
85C23×C251—22no—
86C23×C251—22no—
87C23×C251—44no—
88C23×C251C2×D411yescharacteristic
89Q8×C251—21no—
90Q8×C251—21no—
91D4×C251—21no—
92D4×C251—21no—
93D4×C251—41no—
94D4×C251—41no—
95D4×C251—41no—
96D4×C251—41no—
97D4×C251—41no—
98D4×C251—41no—
99D4×C251—84no—
100C2×C4×C251—22no—
101C2×C4×C251—21no—
102C2×C4×C251—41no—
103C2×C4×C251—41no—
104C2×C4×C251C2×D411yescharacteristic
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251#4111yescharacteristic
110C22×C251—22no—
111C22×C251—44no—
112C22×C251—41no—
113C22×C251—42no—
114C22×C251—42no—
115C22×C251—44no—
116C22×C251—82no—
117C22×C251#12011yescharacteristic
118C22×C251#12011yescharacteristic
119C22×C251#12111yescharacteristic
120C2×D251—44no—
121C2×D251—42no—
122C2×D251—82no—
123C2×D251—82no—
124C4×C251—41no—
125C4×C251—42no—
126C4×C251#12011yescharacteristic
127C4×C251#12111yescharacteristic
128C4×C251#12111yescharacteristic
129C4×C251#12711yescharacteristic
130Dic251—24no—
131C2×C251—21no—
132C2×C251—21no—
133C2×C251—81no—
134C2×C251—82no—
135C2×C251#39211yescharacteristic
136C2×C251#45312yes—
137D251—82no—
138C251#200211yescharacteristic, complemented, Sylow
139#2002—2511nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2511no—
141#535D25111yescharacteristic, complemented
142C2×#127—2511no—
143C2×#104—2511no—
144#418—2511no—
145C2×#100—2511no—
146#339—2511no—
147C2×#41C2×D25111yescharacteristic
148C22×D4—5022no—
149C22×D4—2511no—
150C22×D4C2×D25111yescharacteristic
151AGL(1,ℤ/8)—5022no—
152AGL(1,ℤ/8)—5022no—
153C2×SD16—5021no—
154C2×D8—5021no—
155C2×M16—2511no—
156C2×M16—2511no—
157#127—21no—
158#127—5022no—
159#127—5022no—
160#124—21no—
161#121—22no—
162#120—22no—
163#117C2×D25111yescharacteristic
164C2×C42—2511no—
165#104—5022no—
166#102—5022no—
167#100—5022no—
168C24—5021no—
169C24C22⋊D25112yes—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D25111yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—24no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—10048no—
184C2×D4—5028no—
185C2×D4—5021no—
186C2×D4—5021no—
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—5021no—
190C2×D4—2512no—
191C2×D4C22⋊D25111yescharacteristic, complemented
192C22×C4—21no—
193C22×C4—5021no—
194C22×C4C22×D25111yescharacteristic
195SD16—5024no—
196D8—5024no—
197M16—5021no—
198M16—5021no—
199M16—5021no—
200M16—2512no—
201C2×C8—5021no—
202C2×C8—5021no—
203C4⋊C4C22⋊D25112yescomplemented
204C22⋊C4—24no—
205C22⋊C4—21no—
206C42—21no—
207C42—5021no—
208C42—2512no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—44no—
214C23—10044no—
215C23—10041no—
216C23—10041no—
217C23—5022no—
218C23C2×C22⋊D25111yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—21no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—84no—
230D4—100416no—
231D4—10041no—
232D4—10041no—
233D4—5022no—
234D4—5022no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—5024no—
240C2×C4—5022no—
241C2×C4D4×D25111yescharacteristic
242C2×C4D4×D25111yescharacteristic
243C2×C4D4×D25111yescharacteristic
244C2×C4C2×C22⋊D25111yescharacteristic
245C2×C4C2×C22⋊D25111yescharacteristic
246C2×C4#1686080911yescharacteristic
247C8—10041no—
248C8—5022no—
249C22—22no—
250C22—44no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—20082no—
257C22—20082no—
258C22—10044no—
259C22—10042no—
260C22#2254868311yescharacteristic
261C22#2254868411yescharacteristic
262C22#2254868711yescharacteristic
263C4—41no—
264C4—42no—
265C4—5024no—
266C4#2254868311yescharacteristic
267C4#2254868611yescharacteristic
268C4#2254868711yescharacteristic
269C4#2254871111yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—20082no—
275C2#3063159511yescharacteristic
276C2#3063185512yes—
subgroupG/H
centerC22#22548684
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#535×C251C2
socleC22×C251#120
hypercenter#535D251
hypocenterC251#2002
LCSG ▷ C22×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C2×#22548711C2semidirect
3. C2×#22548642C2semidirect
4. #30631691C2semidirect
5. C2×#22548632C2semidirect
6. #30631446C2semidirect
7. #535×C251C2semidirect
35. C2×D1004D4semidirect
44. #16860794D4semidirect
48. C4×Dic251D4semidirect
54. C2×Q8×C251D4semidirect
63. C2×D4×C251D4semidirect
66. C4⋊C4×C251D4semidirect
138. C251#2002semidirect
141. #535D251semidirect
174. C2×Q8C22⋊D251semidirect
191. C2×D4C22⋊D251semidirect
203. C4⋊C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C2×#225487112
C2×#225486422
#306316912
C2×#225486322
#306314462
#535×C2512
#2002251