Finite Group Databasebeta

← #44553606

Subgroups — #44553606

Order 32128.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#22548677C211yesmaximal, characteristic, complemented
2C2×#22548677C212yesmaximal, complemented
3C2×#22548673C211yesmaximal, characteristic, complemented
4#30631613C211yesmaximal, characteristic, complemented
5#30631609C212yesmaximal, complemented
6#30631605C212yesmaximal, complemented
7#30631604C211yesmaximal, characteristic, complemented
8#30631597C211yesmaximal, characteristic
9#30631538C211yesmaximal, characteristic, complemented
10#30631534C212yesmaximal, complemented
11#398×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×D1004C2212yes—
14C22×C4×D251C2212yes—
15C22×C4×D251C2211yescharacteristic
16#22548677—28no—
17#22548677—24no—
18#22548673—24no—
19C2×C4⋊Dic251C2211yescharacteristic
20C2×Dic251⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C2×C4×Dic251C2211yescharacteristic
29C2×C4×Dic251C2212yes—
30C2×C4×Dic251C2211yescharacteristic
31C251⋊#95C2212yes—
32C251⋊#95C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C4⋊C4×C251C2212yes—
35C2×C4⋊C4×C251C2211yescharacteristic
36#95×C251C2212yes—
37#95×C251C2211yescharacteristic
38C23×D251C2311yescharacteristic
39C23×D251C2312yes—
40C2×D1004—24no—
41C2×D1004—24no—
42C2×D1004—24no—
43C2×D1004—22no—
44C2×D1004—24no—
45C22×Dic251C2312yes—
46C22×Dic251C2311yescharacteristic
47C22×Dic251C2311yescharacteristic
48C22×Dic251C2311yescharacteristic
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251—24no—
52C2×C4×D251—24no—
53C2×C4×D251—24no—
54C2×C4×D251D414yescomplemented
55C4⋊Dic251—22no—
56Dic251⋊C4—24no—
57(C2×D251)⋊C4—24no—
58(C2×D251)⋊C4—24no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—24no—
61(C2×D251)⋊C4—22no—
62(C2×D251)⋊C4—24no—
63(C2×D251)⋊C4—24no—
64C4×Dic251—22no—
65C4×Dic251—24no—
66C4×Dic251D414yescomplemented
67C22×C4×C251C2312yes—
68C22×C4×C251C2311yescharacteristic
69C22×C4×C251C2311yescharacteristic
70C22×C4×C251C2311yescharacteristic
71C22×C4×C251C2312yes—
72C4⋊C4×C251—22no—
73C4⋊C4×C251—24no—
74C4⋊C4×C251—22no—
75C22×D251—21no—
76C22×D251—22no—
77C22×D251—22no—
78C22×D251—22no—
79C22×D251—21no—
80C22×D251—22no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—22no—
84C22×D251—22no—
85C22×D251C2×D412yes—
86C22×D251#4114yes—
87D1004—28no—
88D1004—216no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—21no—
92C2×Dic251—22no—
93C2×Dic251—22no—
94C2×Dic251—22no—
95C2×Dic251C2×D412yes—
96C2×Dic251C2×D412yes—
97C2×Dic251C2×D412yes—
98C2×Dic251#4112yes—
99C4×D251—28no—
100C4×D251—48no—
101C23×C251C2411yescharacteristic
102C2×C4×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251—22no—
112C2×C4×C251C2×D412yes—
113C2×C4×C251C2×D412yes—
114C2×C4×C251#4114yes—
115C22×C251C22×D411yescharacteristic
116C22×C251C22×D411yescharacteristic
117C22×C251C2×#4111yescharacteristic
118C22×C251C2×#4112yes—
119C22×C251C2×#4112yes—
120C2×D251—22no—
121C2×D251—24no—
122C2×D251—24no—
123C2×D251—24no—
124C2×D251—24no—
125C2×D251—42no—
126C2×D251—41no—
127C2×D251—42no—
128C2×D251—42no—
129C2×D251—42no—
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—22no—
133C4×C251—22no—
134C4×C251—24no—
135C4×C251—42no—
136C4×C251—41no—
137Dic251—22no—
138Dic251—42no—
139Dic251—41no—
140Dic251#12114yes—
141C2×C251C2×#12111yescharacteristic
142C2×C251#52511yescharacteristic
143C2×C251#52512yes—
144C2×C251#52912yes—
145C2×C251#53211yescharacteristic
146D251—42no—
147D251—44no—
148C251#239211yescharacteristic, complemented, Sylow
149#2392—2511nomaximal, Sylow
150C2×C4×D4—2512no—
151C2×C4×D4—2511no—
152C2×#117—2511no—
153#398D25111yescharacteristic, complemented
154#397—2511no—
155#396—2512no—
156#394—2511no—
157#394—2512no—
158#392—2511no—
159C4×C4⋊C4—2511no—
160C4×C22⋊C4—2512no—
161C22×D4—2511no—
162C22×D4—2512no—
163C23×C4—2512no—
164C23×C4—2511no—
165C4×D4—5028no—
166C4×D4—5024no—
167#117—5024no—
168C2×C4⋊C4—2511no—
169C2×C4⋊C4—2512no—
170C2×C4⋊C4C2×D25111yescharacteristic
171C2×C4⋊C4C2×D25112yes—
172C2×C4⋊C4C2×D25111yescharacteristic
173C2×C22⋊C4—2511no—
174C2×C22⋊C4—2511no—
175C2×C22⋊C4—2512no—
176C2×C22⋊C4—2512no—
177C2×C22⋊C4—2512no—
178C2×C22⋊C4—2512no—
179C2×C22⋊C4—2512no—
180C2×C42—2511no—
181C2×C42—2512no—
182C2×C42—2511no—
183#95—2512no—
184#95—2511no—
185#95C2×D25112yes—
186#95C2×D25111yescharacteristic
187C24—2511no—
188C24—2512no—
189C2×D4—5024no—
190C2×D4—5024no—
191C2×D4—5024no—
192C2×D4—5022no—
193C2×D4—5024no—
194C22×C4—5024no—
195C22×C4—5024no—
196C22×C4—5024no—
197C22×C4—5022no—
198C22×C4—5022no—
199C22×C4—2512no—
200C22×C4—2511no—
201C22×C4—2511no—
202C22×C4—2514no—
203C22×C4—2511no—
204C22×C4C22×D25112yes—
205C22×C4C22×D25111yescharacteristic
206C22×C4C22×D25111yescharacteristic
207C22×C4C22×D25111yescharacteristic
208C22×C4C22×D25112yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—5022no—
213C4⋊C4—5024no—
214C22⋊C4—5024no—
215C22⋊C4—5024no—
216C22⋊C4—5022no—
217C22⋊C4—5024no—
218C22⋊C4—5022no—
219C22⋊C4—5024no—
220C22⋊C4—5024no—
221C42—5022no—
222C42—5024no—
223C42—2514no—
224C23—5021no—
225C23—5022no—
226C23—5022no—
227C23—5021no—
228C23—5022no—
229C23—5022no—
230C23—5022no—
231C23—5022no—
232C23—5022no—
233C23—5022no—
234C23—2512no—
235C23—2514no—
236C23C23×D25111yescharacteristic
237D4—5028no—
238D4—50216no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—10048no—
250C2×C4—5028no—
251C2×C4—5022no—
252C2×C4—5021no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—2512no—
258C2×C4—2512no—
259C2×C4—2512no—
260C2×C4—2512no—
261C2×C4D4×D25112yes—
262C2×C4D4×D25112yes—
263C2×C4#1686081114yes—
264C22—10041no—
265C22—10042no—
266C22—10042no—
267C22—10042no—
268C22—10041no—
269C22—10042no—
270C22—10042no—
271C22—5024no—
272C22—5024no—
273C22—5024no—
274C22—5024no—
275C22—5022no—
276C22C2×D4×D25111yescharacteristic
277C22C2×D4×D25111yescharacteristic
278C22#41×D25112yes—
279C22#41×D25111yescharacteristic
280C22C2×#1686081112yes—
281C4—22no—
282C4—24no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—10042no—
287C4—10041no—
288C4—5022no—
289C4—2514no—
290C2—10042no—
291C2—10044no—
292C2#121×D25111yescharacteristic
293C2#3063219511yescharacteristic
294C2#3063220012yes—
295C2#3063222212yes—
296C2#3063223811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#398×C251C2
prime radical#30631597C2
hypercenter#398D251
hypocenterC251#2392
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#22548677C2semidirect
2. C2×#22548677C2semidirect
3. C2×#22548673C2semidirect
4. #30631613C2semidirect
5. #30631609C2semidirect
6. #30631605C2semidirect
7. #30631604C2semidirect
9. #30631538C2semidirect
10. #30631534C2semidirect
11. #398×C251C2semidirect
54. C2×C4×D251D4semidirect
66. C4×Dic251D4semidirect
148. C251#2392semidirect
153. #398D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486772
C2×#225486732
#306316132
#306316092
#306316052
#306316042
#306315972
#306315382
#306315342
#398×C2512
#2392251