Finite Group Databasebeta

← #44558366

Subgroups — #44558366

Order 32128.

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

normal subgroups only

This group has 578 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×#22548700C211yesmaximal, characteristic, complemented
2C2×#22548686C211yesmaximal, characteristic, complemented
3#30631626C211yesmaximal, characteristic, complemented
4#30631624C211yesmaximal, characteristic, complemented
5#30631610C212yesmaximal, complemented
6#30631603C212yesmaximal, complemented
7#30631602C212yesmaximal, complemented
8#30631596C212yesmaximal, complemented
9#30631595C211yesmaximal, characteristic, complemented
10#30631546C211yesmaximal, characteristic, complemented
11#394×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C22⋊D251C2212yes—
14C22×C22⋊D251C2212yes—
15C22×C4×D251C2211yescharacteristic
16#22548700—24no—
17#22548686—24no—
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2212yes—
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×C22⋊Dic251C2212yes—
23C2×C22⋊Dic251C2212yes—
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2212yes—
28C2×(C2×D251)⋊C4C2212yes—
29C251⋊#95C2211yescharacteristic
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2212yes—
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2311yescharacteristic
38C23×D251C2311yescharacteristic
39C2×D1004—22no—
40C2×D1004—42no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C2×C22⋊D251—42no—
45C2×C22⋊D251—42no—
46C2×C22⋊D251—42no—
47C22×Dic251C2312yes—
48C22×Dic251C2312yes—
49C22×Dic251C2311yescharacteristic
50C22×Dic251C2311yescharacteristic
51C2×C4×D251—42no—
52C2×C4×D251D414yescomplemented
53C4⋊Dic251—22no—
54C4⋊Dic251—42no—
55Dic251⋊C4—22no—
56Dic251⋊C4—42no—
57C22⋊Dic251—24no—
58C22⋊Dic251—42no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—22no—
62(C2×D251)⋊C4—42no—
63(C2×D251)⋊C4—42no—
64C24×C251C2312yes—
65C2×D4×C251—22no—
66C2×D4×C251—42no—
67C22×C4×C251C2311yescharacteristic
68C22×C4×C251C2312yes—
69C22×C4×C251C2311yescharacteristic
70C22×C4×C251C2311yescharacteristic
71C4⋊C4×C251—22no—
72C22⋊C4×C251—42no—
73C22⋊C4×C251—42no—
74C22×D251—21no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—21no—
78C22×D251—21no—
79C22×D251—22no—
80C22×D251—21no—
81C22×D251—22no—
82C22×D251—21no—
83C22×D251C2×D412yes—
84D1004—44no—
85C22⋊D251—48no—
86C22⋊D251—48no—
87C2×Dic251—22no—
88C2×Dic251—21no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—22no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251C2×D412yes—
97C4×D251—44no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—22no—
101C23×C251—22no—
102C23×C251—22no—
103C23×C251—22no—
104C23×C251—22no—
105C23×C251C2411yescharacteristic
106D4×C251—44no—
107C2×C4×C251—22no—
108C2×C4×C251—21no—
109C2×C4×C251—21no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—21no—
116C2×C4×C251C2×D412yes—
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)12yes—
126C22×C251H(5,𝔽2)11yescharacteristic
127C22×C251H(5,𝔽2)12yes—
128C22×C2512−1+411yescharacteristic
129C2×D251—22no—
130C2×D251—22no—
131C2×D251—22no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—41no—
135C2×D251—41no—
136C2×D251—41no—
137C2×D251—42no—
138C2×D251—42no—
139C4×C251—22no—
140C4×C251—41no—
141C4×C251—41no—
142C4×C251—42no—
143Dic251—22no—
144Dic251—41no—
145Dic251—42no—
146Dic251—42no—
147C2×C251—82no—
148C2×C251#53411yescharacteristic
149C2×C251#53511yescharacteristic
150C2×C251#53711yescharacteristic
151C2×C251#56012yes—
152C2×C251#56212yes—
153D251—42no—
154D251—81no—
155C251#281311yescharacteristic, complemented, Sylow
156#2813—2511nomaximal, Sylow
157C2×#123—2511no—
158C2×#121—2511no—
159#400—2511no—
160#399—2511no—
161#396—2512no—
162#394—2512no—
163#394—2512no—
164#394D25111yescharacteristic, complemented
165#392—2511no—
166#392—2512no—
167#380—2511no—
168C22×D4—2512no—
169C22×D4—2512no—
170C22×D4—2511no—
171C22×D4C2×D25111yescharacteristic
172C23×C4—2511no—
173#123—5024no—
174#121—5024no—
175C2×C4⋊C4—2512no—
176C2×C4⋊C4—2512no—
177C2×C4⋊C4—2511no—
178C2×C4⋊C4—2511no—
179C2×C4⋊C4C2×D25111yescharacteristic
180C2×C22⋊C4—2512no—
181C2×C22⋊C4—2512no—
182C2×C22⋊C4—2512no—
183C2×C22⋊C4—2512no—
184C2×C22⋊C4—2511no—
185C2×C22⋊C4—2511no—
186C2×C22⋊C4—2511no—
187C2×C22⋊C4C2×D25112yes—
188C2×C22⋊C4C2×D25112yes—
189#95—2511no—
190#95—2512no—
191#95—2512no—
192#95C2×D25111yescharacteristic
193C24—2511no—
194C24—2511no—
195C24C22×D25112yes—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—10042no—
199C2×D4—10042no—
200C2×D4—10042no—
201C2×D4—10042no—
202C2×D4—10042no—
203C2×D4—10042no—
204C2×D4—5022no—
205C2×D4—5024no—
206C22×C4—10042no—
207C22×C4—2514no—
208C22×C4—2512no—
209C22×C4—2512no—
210C22×C4—2511no—
211C22×C4—2511no—
212C22×C4C22×D25111yescharacteristic
213C22×C4C22×D25111yescharacteristic
214C22×C4C22×D25112yes—
215C22×C4C22×D25111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—10042no—
218C4⋊C4—10042no—
219C4⋊C4—5022no—
220C4⋊C4—5022no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—10042no—
224C22⋊C4—10042no—
225C22⋊C4—10042no—
226C22⋊C4—5022no—
227C22⋊C4—5022no—
228C22⋊C4—5024no—
229C22⋊C4—5022no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—5021no—
238C23—5022no—
239C23—5021no—
240C23—5022no—
241C23—5021no—
242C23—5021no—
243C23—5022no—
244C23—5022no—
245C23—5021no—
246C23—2512no—
247C23C23×D25111yescharacteristic
248D4—44no—
249D4—10044no—
250D4—10048no—
251D4—10048no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—10044no—
262C2×C4—5021no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5022no—
268C2×C4—5022no—
269C2×C4—5022no—
270C2×C4—5022no—
271C2×C4—2512no—
272C2×C4D4×D25112yes—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—10041no—
281C22—10041no—
282C22—10041no—
283C22—10042no—
284C22—10042no—
285C22—10042no—
286C22—10042no—
287C22—5022no—
288C22—5022no—
289C22—5022no—
290C22C2×D4×D25111yescharacteristic
291C22#2254876012yes—
292C22#2254876011yescharacteristic
293C22#2254876112yes—
294C22#2254876211yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—42no—
299C4—10042no—
300C4—10041no—
301C4—10042no—
302C4—5022no—
303C2—82no—
304C2—20081no—
305C2—10042no—
306C2#3063224711yescharacteristic
307C2#3063225511yescharacteristic
308C2#3063226611yescharacteristic
309C2#3063242512yes—
310C2#3063243412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
hypercenter#394D251
hypocenterC251#2813
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548700C2semidirect
2. C2×#22548686C2semidirect
3. #30631626C2semidirect
4. #30631624C2semidirect
5. #30631610C2semidirect
6. #30631603C2semidirect
7. #30631602C2semidirect
8. #30631596C2semidirect
9. #30631595C2semidirect
10. #30631546C2semidirect
11. #394×C251C2semidirect
52. C2×C4×D251D4semidirect
155. C251#2813semidirect
164. #394D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487002
C2×#225486862
#306316262
#306316242
#306316102
#306316032
#306316022
#306315962
#306315952
#306315462
#394×C2512
#2813251