Finite Group Databasebeta

← #44558390

Subgroups — #44558390

Order 32128.

644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548687C212yesmaximal, complemented
2#30631616C212yesmaximal, complemented
3#30631615C212yesmaximal, complemented
4#30631596C211yesmaximal, characteristic, complemented
5#30631596C212yesmaximal, complemented
6#30631595C212yesmaximal, complemented
7#30631595C212yesmaximal, complemented
8#30631548C211yesmaximal, characteristic, complemented
9#392×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C22×C22⋊D251C2212yes—
13C22×C22⋊D251C2212yes—
14C23×Dic251C2211yescharacteristic
15#22548687—28no—
16C2×Dic251⋊C4C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2211yescharacteristic
19C2×C22⋊Dic251C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2211yescharacteristic
27C22×D4×C251C2212yes—
28C22×D4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2211yescharacteristic
30C2×C22⋊C4×C251C2212yes—
31#95×C251C2211yescharacteristic
32C23×D251C2312yes—
33C2×D1004—41no—
34C2×D1004—42no—
35C2×C22⋊D251—24no—
36C2×C22⋊D251—24no—
37C2×C22⋊D251—42no—
38C2×C22⋊D251—42no—
39C2×C22⋊D251—42no—
40C2×C22⋊D251—42no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C22×Dic251—41no—
45C22×Dic251—41no—
46C22×Dic251D414yescomplemented
47C22×Dic251C2312yes—
48C22×Dic251C2312yes—
49C22×Dic251C2312yes—
50Dic251⋊C4—24no—
51C22⋊Dic251—24no—
52C22⋊Dic251—41no—
53C22⋊Dic251—42no—
54(C2×D251)⋊C4—24no—
55(C2×D251)⋊C4—42no—
56(C2×D251)⋊C4—42no—
57C24×C251C2312yes—
58C24×C251C2311yescharacteristic
59C2×D4×C251—24no—
60C2×D4×C251—42no—
61C2×D4×C251—41no—
62C2×D4×C251—42no—
63C2×D4×C251—42no—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C22×C4×C251C2312yes—
67C22⋊C4×C251—41no—
68C22⋊C4×C251—42no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—22no—
75C22×D251—22no—
76D1004—44no—
77C22⋊D251—48no—
78C22⋊D251—48no—
79C22⋊D251—48no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—44no—
89C2×Dic251C2×D414yes—
90C23×C251—21no—
91C23×C251—22no—
92C23×C251—22no—
93C23×C251—21no—
94C23×C251—22no—
95C23×C251—22no—
96C23×C251—22no—
97C23×C251—22no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—22no—
101C23×C251C2×D412yes—
102C23×C251C2411yescharacteristic
103D4×C251—48no—
104D4×C251—44no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251—22no—
112C22×C251—22no—
113C22×C251—24no—
114C22×C251—41no—
115C22×C251—42no—
116C22×C251—41no—
117C22×C251—42no—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251—42no—
121C22×C251—42no—
122C22×C251—42no—
123C22×C251—42no—
124C22×C251C22×D411yescharacteristic
125C22×C251H(5,𝔽2)11yescharacteristic
126C22×C251H(5,𝔽2)12yes—
127C22×C251H(5,𝔽2)12yes—
128C22×C251H(5,𝔽2)11yescharacteristic
129C2×D251—42no—
130C2×D251—42no—
131C2×D251—42no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—42no—
135C2×D251—42no—
136C4×C251—42no—
137C4×C251—41no—
138C4×C251—41no—
139Dic251—24no—
140Dic251—42no—
141Dic251—42no—
142C2×C251—42no—
143C2×C251—82no—
144C2×C251#53411yescharacteristic
145C2×C251#53512yes—
146C2×C251#56011yescharacteristic
147C2×C251#56012yes—
148C2×C251#56011yescharacteristic
149D251—82no—
150C251#281511yescharacteristic, complemented, Sylow
151#2815—2511nomaximal, Sylow
152C2×#121—2512no—
153#397—2512no—
154#397—2512no—
155#392—2512no—
156#392—2512no—
157#392—2512no—
158#392—2511no—
159#392D25111yescharacteristic, complemented
160#381—2511no—
161C22×D4—2512no—
162C22×D4—2512no—
163C22×D4—2512no—
164C22×D4—2511no—
165C22×D4C2×D25112yes—
166C22×D4C2×D25111yescharacteristic
167C23×C4—2511no—
168#121—5028no—
169C2×C4⋊C4—2512no—
170C2×C22⋊C4—2511no—
171C2×C22⋊C4—2512no—
172C2×C22⋊C4—2512no—
173C2×C22⋊C4—2512no—
174C2×C22⋊C4—2512no—
175C2×C22⋊C4—2512no—
176C2×C22⋊C4C2×D25111yescharacteristic
177C2×C22⋊C4C2×D25112yes—
178#95—2512no—
179#95—2512no—
180#95—2512no—
181#95—2511no—
182#95C2×D25111yescharacteristic
183C24—2512no—
184C24C22×D25112yes—
185C24C22×D25111yescharacteristic
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—10041no—
192C2×D4—10042no—
193C2×D4—10042no—
194C2×D4—10042no—
195C2×D4—10042no—
196C2×D4—10042no—
197C2×D4—10042no—
198C2×D4—10042no—
199C2×D4—10042no—
200C2×D4—5024no—
201C2×D4—5024no—
202C22×C4—10041no—
203C22×C4—10041no—
204C22×C4—2514no—
205C22×C4—2512no—
206C22×C4—2512no—
207C22×C4—2512no—
208C22×C4C22×D25111yescharacteristic
209C22×C4C22×D25111yescharacteristic
210C22×C4C22×D25112yes—
211C4⋊C4—5024no—
212C22⋊C4—41no—
213C22⋊C4—42no—
214C22⋊C4—10042no—
215C22⋊C4—10042no—
216C22⋊C4—10042no—
217C22⋊C4—10041no—
218C22⋊C4—5024no—
219C22⋊C4—5024no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—5022no—
232C23—5022no—
233C23—5022no—
234C23—5022no—
235C23—5022no—
236C23—5022no—
237C23—5022no—
238C23D4×D25112yes—
239C23C23×D25111yescharacteristic
240D4—48no—
241D4—44no—
242D4—10044no—
243D4—10048no—
244D4—10048no—
245D4—10048no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—10044no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—2514no—
263C22—22no—
264C22—24no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—10042no—
276C22—10042no—
277C22—10042no—
278C22—10042no—
279C22—10042no—
280C22—10042no—
281C22—10042no—
282C22C2×D4×D25111yescharacteristic
283C22#2254876011yescharacteristic
284C22#2254876012yes—
285C22#2254876012yes—
286C22#2254876111yescharacteristic
287C4—42no—
288C4—41no—
289C4—41no—
290C4—10042no—
291C4—10042no—
292C4—5024no—
293C2—42no—
294C2—82no—
295C2—20082no—
296C2#3063224511yescharacteristic
297C2#3063225212yes—
298C2#3063242411yescharacteristic
299C2#3063242512yes—
300C2#3063242611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2815
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548687C2semidirect
2. #30631616C2semidirect
3. #30631615C2semidirect
4. #30631596C2semidirect
5. #30631596C2semidirect
6. #30631595C2semidirect
7. #30631595C2semidirect
8. #30631548C2semidirect
9. #392×C251C2semidirect
46. C22×Dic251D4semidirect
150. C251#2815semidirect
159. #392D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486872
#306316162
#306316152
#306315962
#306315952
#306315482
#392×C2512
#2815251