Finite Group Databasebeta

← #44548723

Subgroups — #44548723

Order 32128.

322 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 65 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×#22548688C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic, complemented
3C22⋊C8×D251C211yesmaximal, characteristic, complemented
4#30631411C211yesmaximal, characteristic, complemented
5#30631410C211yesmaximal, characteristic, complemented
6#30631410C211yesmaximal, characteristic, complemented
7#323×C251C211yesmaximal, characteristic, complemented
8C22×D1004C411yescharacteristic
9C22×C22⋊D251C411yescharacteristic
10C22×C22⋊D251C411yescharacteristic
11C22×C4×D251C2211yescharacteristic
12C2×#16860793—21no—
13C2×C8×D251—21no—
14#22548688—21no—
15#22548688—21no—
16#22548688—21no—
17#22548688—21no—
18C2×Dic251⋊C4C411yescharacteristic
19C2×(C2×D251)⋊C4C2211yescharacteristic
20C22⋊(C251⋊C8)C2211yescharacteristic
21C22⋊(C251⋊C8)C2211yescharacteristic
22(C2×D251)⋊C8—21no—
23(C2×D251)⋊C8—21no—
24C2×C22⋊C4×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26C22⋊C8×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic
28C23×D251C2×C411yescharacteristic
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004—42no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34C2×C22⋊D251—21no—
35C2×C22⋊D251—21no—
36C2×C22⋊D251—42no—
37C2×C22⋊D251—42no—
38C22×Dic251C2×C411yescharacteristic
39C22×Dic251C2×C411yescharacteristic
40C2×C4×D251—42no—
41C2×C4×D251D411yescharacteristic
42C2×C4×D251D411yescharacteristic
43C2×C4×D251D411yescharacteristic
44C2×C4×D251D411yescharacteristic
45#16860793—42no—
46C2×C251⋊C8—21no—
47C2×C251⋊C8—21no—
48C8×D251—42no—
49Dic251⋊C4—21no—
50Dic251⋊C4—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53C24×C251C2×C411yescharacteristic
54C22×C4×C251C2×C411yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C2×C8×C251—21no—
57C2×C8×C251—21no—
58C22⋊C4×C251—21no—
59C22⋊C4×C251—21no—
60C22×D251—21no—
61C22×D251—21no—
62C22×D251—22no—
63C22×D251—21no—
64C22×D251—21no—
65C22×D251—21no—
66C22×D251—41no—
67C22×D251—41no—
68C22×D251—41no—
69C22×D251C22⋊C411yescharacteristic
70C22×D251C22⋊C411yescharacteristic
71D1004—42no—
72D1004—42no—
73C22⋊D251—42no—
74C22⋊D251—42no—
75C22⋊D251—42no—
76C22⋊D251—42no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—41no—
80C2×Dic251C22⋊C411yescharacteristic
81C2×Dic251C22⋊C411yescharacteristic
82C4×D251—22no—
83C4×D251—22no—
84C4×D251—41no—
85C4×D251—41no—
86C251⋊C8—41no—
87C251⋊C8—41no—
88C23×C251—21no—
89C23×C251—21no—
90C23×C251—21no—
91C23×C251—41no—
92C23×C251—41no—
93C23×C251C22×C411yescharacteristic
94C2×C4×C251—21no—
95C2×C4×C251—41no—
96C2×C4×C251—41no—
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C8×C251—41no—
100C8×C251—41no—
101C22×C251—21no—
102C22×C251—21no—
103C22×C251—41no—
104C22×C251—41no—
105C22×C251—41no—
106C22×C251—81no—
107C22×C251—81no—
108C22×C251C2×C22⋊C411yescharacteristic
109C22×C251#13111yescharacteristic
110C22×C251#13111yescharacteristic
111C2×D251—21no—
112C2×D251—21no—
113C2×D251—21no—
114C2×D251—21no—
115C2×D251—21no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—41no—
121C2×D251—81no—
122C2×D251—81no—
123C2×D251#10012yes—
124C4×C251—21no—
125C4×C251—21no—
126C4×C251—41no—
127Dic251—21no—
128Dic251—41no—
129Dic251#9912yes—
130C2×C251—21no—
131C2×C251—21no—
132C2×C251—81no—
133C2×C251#40811yescharacteristic
134C2×C251C2×#9911yescharacteristic
135C2×C251C2×#10011yescharacteristic
136D251—22no—
137D251—41no—
138D251—81no—
139C251#147111yescharacteristic, complemented, Sylow
140#1471—2511nomaximal, Sylow
141C2×#121—2511no—
142#407—2511no—
143C2×C22⋊C8—2511no—
144#323—2511no—
145#323—2511no—
146#323—2511no—
147#323D25111yescharacteristic, complemented
148C22×D4—2511no—
149C22×D4—2511no—
150C22×D4—2511no—
151C23×C4—2511no—
152C2×M16—5021no—
153C22×C8—5021no—
154#121—5021no—
155#121—5021no—
156#121—5021no—
157#121—5021no—
158C2×C4⋊C4—2511no—
159C2×C22⋊C4—2511no—
160C2×C22⋊C4C2×D25111yescharacteristic
161C22⋊C8—5021no—
162C22⋊C8—5021no—
163C22⋊C8—2511no—
164C22⋊C8—2511no—
165C22⋊C8C2×D25111yescharacteristic
166C22⋊C8C2×D25111yescharacteristic
167C24—2511no—
168C24—2511no—
169C24C4×D25111yescharacteristic
170C2×D4—10042no—
171C2×D4—10042no—
172C2×D4—10042no—
173C2×D4—5021no—
174C2×D4—5021no—
175C2×D4—5021no—
176C2×D4—5021no—
177C2×D4—5021no—
178C2×D4—5021no—
179C22×C4—10042no—
180C22×C4—2511no—
181C22×C4—2511no—
182C22×C4—2511no—
183C22×C4—2511no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4C4×D25111yescharacteristic
187C22×C4C22×D25111yescharacteristic
188M16—10042no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—10042no—
192C2×C8—5021no—
193C2×C8—5021no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—5021no—
199C22⋊C4—5021no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—10041no—
206C23—10041no—
207C23—10041no—
208C23—5022no—
209C23—5021no—
210C23—5021no—
211C23—5021no—
212C23—5021no—
213C23—5021no—
214C23—2511no—
215C23—2511no—
216C23C2×C4×D25111yescharacteristic
217D4—10042no—
218D4—10042no—
219D4—10042no—
220D4—10042no—
221D4—10042no—
222D4—10042no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—10041no—
227C2×C4—10041no—
228C2×C4—10041no—
229C2×C4—5022no—
230C2×C4—5022no—
231C2×C4—5021no—
232C2×C4—5022no—
233C2×C4—2511no—
234C2×C4—2511no—
235C2×C4D4×D25111yescharacteristic
236C2×C4D4×D25111yescharacteristic
237C8—41no—
238C8—41no—
239C8—10041no—
240C8—10041no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—20081no—
249C22—20081no—
250C22—10042no—
251C22—10042no—
252C22—10041no—
253C22—10041no—
254C22—10041no—
255C22—5021no—
256C22—5021no—
257C22—5021no—
258C22—5021no—
259C22—5021no—
260C22—2512no—
261C22C22⋊C4×D25111yescharacteristic
262C22#2254872111yescharacteristic
263C22#2254872311yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—10041no—
268C4—5021no—
269C4—2512no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—20081no—
274C2—10041no—
275C2—5022no—
276C2#3063165211yescharacteristic
277C2#99×D25111yescharacteristic
278C2#100×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×#22548688C2
cosocleC22×C4×C251C23
hypercenter#323D251
hypocenterC251#1471
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30631649C2semidirect
3. C22⋊C8×D251C2semidirect
4. #30631411C2semidirect
5. #30631410C2semidirect
6. #30631410C2semidirect
7. #323×C251C2semidirect
139. C251#1471semidirect
147. #323D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486882
#306316492
C22⋊C8×D2512
#306314112
#306314102
#323×C2512
#1471251