Finite Group Databasebeta

← #44558686

Subgroups — #44558686

Order 32128.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#22548696C211yesmaximal, characteristic
2C22×C4×Dic251C211yesmaximal, characteristic, complemented
3#30631581C212yesmaximal, complemented
4#30631580C212yesmaximal, complemented
5#30631570C212yesmaximal, complemented
6#30631569C212yesmaximal, complemented
7#30631567C211yesmaximal, characteristic
8#30631562C211yesmaximal, characteristic, complemented
9#30631560C211yesmaximal, characteristic, complemented
10#30631548C211yesmaximal, characteristic, complemented
11#386×C251C211yesmaximal, characteristic
12C23×Dic251C2211yescharacteristic
13C23×Dic251C2211yescharacteristic
14#22548696—24no—
15C2×C4⋊Dic251C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×C22⋊Dic251C2211yescharacteristic
20C2×C22⋊Dic251C2211yescharacteristic
21C2×C22⋊Dic251C2212yes—
22C2×C22⋊Dic251C2211yescharacteristic
23C2×C22⋊Dic251C2211yescharacteristic
24C2×C4×Dic251—24no—
25C2×C4×Dic251C2212yes—
26C2×C4×Dic251C2212yes—
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C23×C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2212yes—
37C22×Dic251—24no—
38C22×Dic251—22no—
39C22×Dic251—22no—
40C22×Dic251—22no—
41C22×Dic251D414yes—
42C22×Dic251C2312yes—
43C22×Dic251C2311yescharacteristic
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2311yescharacteristic
46C22×Dic251C2311yescharacteristic
47C22×Dic251C2312yes—
48C4⋊Dic251—24no—
49Dic251⋊C4—22no—
50Dic251⋊C4—22no—
51Dic251⋊C4—24no—
52C22⋊Dic251—22no—
53C22⋊Dic251—22no—
54C22⋊Dic251—24no—
55C22⋊Dic251—22no—
56C22⋊Dic251—22no—
57C4×Dic251—24no—
58C4×Dic251—24no—
59C24×C251C2311yescharacteristic
60C24×C251C2311yescharacteristic
61C2×D4×C251—22no—
62C2×D4×C251—22no—
63C2×D4×C251—22no—
64C22×C4×C251—22no—
65C22×C4×C251—24no—
66C22×C4×C251C2311yescharacteristic
67C22×C4×C251C2312yes—
68C22×C4×C251C2312yes—
69C22⋊C4×C251—22no—
70C22⋊C4×C251—24no—
71C2×Dic251—28no—
72C2×Dic251—28no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—21no—
78C2×Dic251—21no—
79C2×Dic251—21no—
80C2×Dic251—21no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251C2×D412yes—
87C2×Dic251C2×D412yes—
88C2×Dic251#4114yes—
89C23×C251—22no—
90C23×C251—22no—
91C23×C251—21no—
92C23×C251—22no—
93C23×C251—21no—
94C23×C251—21no—
95C23×C251—22no—
96C23×C251C2×D412yes—
97C23×C251#4112yes—
98C23×C251#4112yes—
99C23×C251C2411yescharacteristic
100D4×C251—44no—
101C2×C4×C251—28no—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—21no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251#4114yes—
111C22×C251—22no—
112C22×C251—22no—
113C22×C251—24no—
114C22×C251—24no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251—41no—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251#12314yes—
121C22×C251C22×D411yescharacteristic
122C22×C251C2×#4112yes—
123C22×C251C2×#4111yescharacteristic
124C22×C251C2×#4112yes—
125C22×C251C2×#4111yescharacteristic
126C4×C251—24no—
127C4×C251—41no—
128C4×C251—42no—
129Dic251—22no—
130Dic251—22no—
131Dic251—24no—
132Dic251—41no—
133Dic251—41no—
134Dic251—42no—
135C2×C251—24no—
136C2×C251—81no—
137C2×C251C2×#12311yescharacteristic
138C2×C251#52511yescharacteristic
139C2×C251#52511yescharacteristic
140C2×C251#52912yes—
141C2×C251#52912yes—
142C251#285311yescharacteristic, complemented, Sylow
143#2853—2511nomaximal, Sylow
144C2×#123—2511no—
145C22×C42—2511no—
146#388—2512no—
147#388—2512no—
148#387—2512no—
149#387—2512no—
150#386—2511no—
151#386D25111yescharacteristic
152#385—2511no—
153#385—2511no—
154#381—2511no—
155C22×D4C2×D25111yescharacteristic
156C23×C4—2511no—
157C23×C4—2511no—
158C23×C4C2×D25111yescharacteristic
159#123—5024no—
160C2×C4⋊C4—2511no—
161C2×C4⋊C4—2511no—
162C2×C4⋊C4—2512no—
163C2×C4⋊C4—2512no—
164C2×C22⋊C4—2511no—
165C2×C22⋊C4—2511no—
166C2×C22⋊C4—2511no—
167C2×C22⋊C4—2511no—
168C2×C22⋊C4—2512no—
169C2×C22⋊C4C2×D25111yescharacteristic
170C2×C22⋊C4C2×D25112yes—
171C2×C42—5024no—
172C2×C42—2512no—
173C2×C42—2512no—
174#95—2512no—
175#95—2512no—
176#95—2512no—
177#95—2512no—
178#95—2512no—
179#95C2×D25112yes—
180C24C22×D25111yescharacteristic
181C24C22×D25111yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—5022no—
188C22×C4—5022no—
189C22×C4—5022no—
190C22×C4—5024no—
191C22×C4—2512no—
192C22×C4—2511no—
193C22×C4—2511no—
194C22×C4—2514no—
195C22×C4—2512no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25112yes—
199C22×C4C22×D25111yescharacteristic
200C22×C4C22×D25112yes—
201C4⋊C4—5022no—
202C4⋊C4—5022no—
203C4⋊C4—5024no—
204C4⋊C4—5024no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—5022no—
208C22⋊C4—5022no—
209C22⋊C4—5024no—
210C22⋊C4—5022no—
211C22⋊C4—5022no—
212C42—5024no—
213C42—5024no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D25112yes—
222C23#1686080912yes—
223C23#1686081012yes—
224C23C23×D25111yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—5021no—
236C2×C4—5021no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4—5022no—
241C2×C4—5022no—
242C2×C4—5028no—
243C2×C4—5022no—
244C2×C4—5022no—
245C2×C4—5021no—
246C2×C4—5021no—
247C2×C4—5021no—
248C2×C4—5021no—
249C2×C4—5028no—
250C2×C4—2512no—
251C2×C4—2512no—
252C2×C4—2514no—
253C2×C4#1686080914yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—42no—
262C22—41no—
263C22#2254869714yes—
264C22C2×D4×D25111yescharacteristic
265C22#41×D25112yes—
266C22C2×#1686080912yes—
267C22C2×#1686080911yescharacteristic
268C22C2×#1686081011yescharacteristic
269C4—24no—
270C4—42no—
271C4—41no—
272C4—10041no—
273C4—10041no—
274C4—10042no—
275C4—5024no—
276C4—5022no—
277C4—5022no—
278C2—24no—
279C2—81no—
280C2C2×#2254869711yescharacteristic
281C2#3063219311yescharacteristic
282C2#3063219511yescharacteristic
283C2#3063221712yes—
284C2#3063221812yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2853
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic251C2semidirect
3. #30631581C2semidirect
4. #30631580C2semidirect
5. #30631570C2semidirect
6. #30631569C2semidirect
8. #30631562C2semidirect
9. #30631560C2semidirect
10. #30631548C2semidirect
142. C251#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#225486962
C22×C4×Dic2512
#306315812
#306315802
#306315702
#306315692
#306315672
#306315622
#306315602
#306315482
#386×C2512
#2853251