Finite Group Databasebeta

← #44550334

Subgroups — #44550334

Order 32128.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22548761C211yesmaximal, characteristic
2#30631704C211yesmaximal, characteristic, complemented
3#30631693C211yesmaximal, characteristic, complemented
4#30631691C211yesmaximal, characteristic, complemented
5#30631669C211yesmaximal, characteristic, complemented
6#30631660C211yesmaximal, characteristic, complemented
7#421×C251C211yesmaximal, characteristic, complemented
8#22548761—22no—
9#22548761—22no—
10#22548761—22no—
11#22548761—22no—
12C2×#16860811C412yescomplemented
13C2×#16860810—21no—
14C2×#16860810C2211yescharacteristic
15C22×D1004—21no—
16C22×D1004C2211yescharacteristic
17C2×D4×D251—22no—
18C2×D4×D251C412yescomplemented
19C2×#16860794C2211yescharacteristic, complemented
20#22548674C2211yescharacteristic, complemented
21#22548642—22no—
22#22548637—22no—
23#22548635—22no—
24#22548633—21no—
25#22548631—21no—
26#22548629—22no—
27C2×#41×C251C2211yescharacteristic
28C2×M16×C251C2211yescharacteristic, complemented
29#117×C251C2211yescharacteristic, complemented
30#104×C251—22no—
31C23×D251—22no—
32C23×D251—21no—
33#16860811—24no—
34#16860811—24no—
35#16860810—24no—
36#16860810—24no—
37#16860810—21no—
38#16860810—21no—
39#16860810—21no—
40#16860810—21no—
41C2×Dic502D411yescharacteristic
42C2×D1004—24no—
43C2×D1004—24no—
44C2×D1004—22no—
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004—21no—
48C2×D1004C2×C412yes—
49C2×D1004D411yescharacteristic
50C2×D1004D412yes—
51C2×D1004D412yes—
52C2×C22⋊D251—22no—
53C2×C22⋊D251D412yes—
54D4×D251—28no—
55D4×D251—28no—
56D4×D251—24no—
57D4×D251—24no—
58C2×C4×D251—22no—
59C2×C4×D251C2×C412yes—
60#16860794—21no—
61#16860794—21no—
62C2×C251⋊C8—21no—
63C4⋊Dic251—21no—
64C22⋊Dic251—21no—
65C4×Dic251—21no—
66#41×C251—21no—
67#41×C251—21no—
68#41×C251D412yes—
69#41×C251D412yes—
70C2×Q8×C251C2×C411yescharacteristic
71C2×D4×C251—21no—
72C2×D4×C251C2×C411yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75M16×C251—21no—
76M16×C251—21no—
77C2×C8×C251—21no—
78C4⋊C4×C251—21no—
79C22⋊C4×C251—21no—
80C42×C251—21no—
81C22×D251—22no—
82C22×D251—28no—
83C22×D251—22no—
84C22×D251—22no—
85C22×D251—82no—
86C22×D251—84no—
87C22×D251C22⋊C412yes—
88Dic502—21no—
89Dic502—21no—
90D1004—28no—
91D1004—22no—
92D1004—21no—
93D1004—22no—
94D1004—21no—
95D1004—22no—
96D1004C22⋊C414yes—
97C22⋊D251—28no—
98C22⋊D251—22no—
99C22⋊D251—22no—
100C2×Dic251—41no—
101C2×Dic251C22⋊C412yes—
102C4×D251—28no—
103C4×D251—22no—
104C4×D251—22no—
105C251⋊C8—41no—
106C23×C251—21no—
107C23×C251C2×D411yescharacteristic
108Q8×C251—21no—
109Q8×C251C22⋊C412yes—
110D4×C251—22no—
111D4×C251—22no—
112D4×C251—21no—
113D4×C251C22⋊C412yes—
114C2×C4×C251—21no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251—41no—
118C2×C4×C251C22×C411yescharacteristic
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D411yescharacteristic
122C2×C4×C251C2×D411yescharacteristic
123C2×C4×C251C2×D411yescharacteristic
124C8×C251—41no—
125C22×C251—22no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251C2×C22⋊C411yescharacteristic
129C22×C251#12011yescharacteristic
130C22×C251#12011yescharacteristic
131C2×D251—22no—
132C2×D251—24no—
133C2×D251—44no—
134C2×D251—42no—
135C2×D251—82no—
136C2×D251—82no—
137C2×D251—88no—
138C4×C251—22no—
139C4×C251—81no—
140C4×C251C2×C22⋊C411yescharacteristic
141C4×C251C2×C22⋊C411yescharacteristic
142C4×C251#12011yescharacteristic
143C4×C251#12011yescharacteristic
144Dic251—22no—
145Dic251—81no—
146C2×C251—21no—
147C2×C251—21no—
148C2×C251—21no—
149C2×C251—42no—
150C2×C251#37911yescharacteristic
151D251—44no—
152D251—42no—
153C251#179511yescharacteristic, complemented, Sylow
154#1795—2511nomaximal, Sylow
155C2×H(5,𝔽2)—2511no—
156#421—2511no—
157#421D25111yescharacteristic, complemented
158#418—2511no—
159#418—2511no—
160#413—2511no—
161#410—2511no—
162H(5,𝔽2)—5022no—
163H(5,𝔽2)—5022no—
164H(5,𝔽2)—5022no—
165H(5,𝔽2)—5022no—
166C2×#41—5021no—
167C2×#41—2512no—
168C2×#41—2511no—
169C2×#41C2×D25111yescharacteristic
170C22×D4—5021no—
171C22×D4—5022no—
172C22×D4—2512no—
173C22×D4—2511no—
174C2×M16—2511no—
175C2×M16C2×D25111yescharacteristic, complemented
176#117—2511no—
177#117C2×D25111yescharacteristic, complemented
178#104—22no—
179#104—5022no—
180#102—5022no—
181#102—5022no—
182#101—5021no—
183#100—5021no—
184#99—5022no—
185C24—5022no—
186C24—5021no—
187#41—21no—
188#41—21no—
189#41—5024no—
190#41—5024no—
191#41—5024no—
192#41—5024no—
193#41—5021no—
194#41—5021no—
195#41—5021no—
196#41—5021no—
197#41C22⋊D25112yes—
198#41D100412yes—
199C2×Q8—2511no—
200C2×Q8C4×D25111yescharacteristic
201C2×D4—21no—
202C2×D4—5024no—
203C2×D4—5024no—
204C2×D4—5022no—
205C2×D4—5028no—
206C2×D4—5028no—
207C2×D4—5024no—
208C2×D4—5024no—
209C2×D4—5022no—
210C2×D4—5021no—
211C2×D4—5021no—
212C2×D4—5021no—
213C2×D4—2512no—
214C2×D4—2512no—
215C2×D4—2511no—
216C2×D4—2512no—
217C2×D4—2512no—
218C2×D4C4×D25111yescharacteristic
219C22×C4—21no—
220C22×C4—5022no—
221C22×C4—2512no—
222C22×C4C22×D25111yescharacteristic
223M16—21no—
224M16—21no—
225M16—5021no—
226M16—5021no—
227C2×C8—21no—
228C2×C8—5021no—
229C4⋊C4—21no—
230C4⋊C4—5021no—
231C22⋊C4—21no—
232C22⋊C4—5021no—
233C42—21no—
234C42—5021no—
235C23—21no—
236C23—20082no—
237C23—20084no—
238C23—5028no—
239C23—5022no—
240C23—5022no—
241C23—5022no—
242C23—2512no—
243C23D4×D25111yescharacteristic
244Q8—21no—
245Q8—5021no—
246Q8—5021no—
247Q8(C2×D251)⋊C412yes—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—5028no—
252D4—5028no—
253D4—5022no—
254D4—5022no—
255D4—5022no—
256D4—5021no—
257D4—5022no—
258D4—5021no—
259D4—5022no—
260D4—2514no—
261D4(C2×D251)⋊C412yes—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—10041no—
267C2×C4—5028no—
268C2×C4—5022no—
269C2×C4—5022no—
270C2×C4—2512no—
271C2×C4C2×C4×D25111yescharacteristic
272C2×C4D4×D25111yescharacteristic
273C2×C4D4×D25111yescharacteristic
274C2×C4D4×D25111yescharacteristic
275C2×C4C2×C22⋊D25111yescharacteristic
276C2×C4C2×D100411yescharacteristic
277C8—41no—
278C8—10041no—
279C22—22no—
280C22—41no—
281C22—42no—
282C22—20082no—
283C22—20082no—
284C22—20088no—
285C22—10042no—
286C22—10044no—
287C22—5022no—
288C22—5024no—
289C22C22⋊C4×D25111yescharacteristic
290C22#2254868311yescharacteristic
291C22#2254868511yescharacteristic
292C4—22no—
293C4—81no—
294C4—20081no—
295C4—5022no—
296C4C22⋊C4×D25111yescharacteristic
297C4C2×(C2×D251)⋊C411yescharacteristic
298C4#2254868311yescharacteristic
299C4#2254868511yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—42no—
304C2—10044no—
305C2—10042no—
306C2#3063154211yescharacteristic
subgroupG/H
centerC2#30631542
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#421×C251C2
socleC2×C251#379
prime radicalC2×#22548761C2
cosocleC22×C4×C251C23
hypercenter#421D251
hypocenterC251#1795
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #30631704C2semidirect
3. #30631693C2semidirect
4. #30631691C2semidirect
5. #30631669C2semidirect
6. #30631660C2semidirect
7. #421×C251C2semidirect
12. C2×#16860811C4semidirect
18. C2×D4×D251C4semidirect
19. C2×#16860794C22semidirect
20. #22548674C22semidirect
28. C2×M16×C251C22semidirect
29. #117×C251C22semidirect
153. C251#1795semidirect
157. #421D251semidirect
175. C2×M16C2×D251semidirect
177. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306317042
#306316932
#306316912
#306316692
#306316602
#421×C2512
#1795251