Finite Group Databasebeta

← #24988771

Subgroups — #24988771

Order 11008.

1014 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#19148239C211yesmaximal, characteristic
2#19134513C211yesmaximal, characteristic, complemented
3#19133152C214yesmaximal, complemented
4C43×#1780C211yesmaximal, characteristic, complemented
5C2×#13063370C411yescharacteristic, complemented
6#14583548—24no—
7#14583537—24no—
8#14583473C412yescomplemented
9#14583458—22no—
10#14583410C2212yes—
11C2×#13063308—22no—
12#14583374C411yescharacteristic, complemented
13C2×#13063240—22no—
14#14582762C2212yescomplemented
15#14582637—24no—
16#14582634—24no—
17C22×#41×C43C2211yescharacteristic
18C2×C22⋊C8×C43—22no—
19C2×C22⋊C8×C43C2212yescomplemented
20#13063370—44no—
21C2×#10621898C2×C412yes—
22C2×#10621900—22no—
23C22×C22⋊D43—22no—
24C2×D4×D43—22no—
25#13063323—22no—
26#13063314—22no—
27#13063308—28no—
28#13063308—24no—
29#13063308—22no—
30#13063306—24no—
31#13063304D414yes—
32#13063301—24no—
33#13063299D414yes—
34#13063297—28no—
35#13063294—22no—
36#13063293—24no—
37D4×Dic43—24no—
38C4×C22⋊D43—24no—
39#13063284C2×C412yes—
40C2×Dic43⋊C4—24no—
41C2×C22⋊Dic43—22no—
42C2×C22⋊Dic43—22no—
43C2×(C2×D43)⋊C4—24no—
44#13063240—44no—
45C43⋊#95—24no—
46C2×#41×C43—24no—
47C2×#41×C43—44no—
48C22×Q8×C43C2×C411yescharacteristic
49C22×D4×C43C2×C411yescharacteristic
50C22×D4×C43D412yes—
51C23×C4×C43D412yes—
52C23×C4×C43C2311yescharacteristic
53C22×C8×C43—22no—
54C22⋊C8×C43—24no—
55C22⋊C8×C43—44no—
56C24×C43C22⋊C412yes—
57C24×C43C2×D411yescharacteristic
58C23×D43—22no—
59#41×C43—416no—
60C2×Q8×C43—22no—
61C2×Q8×C43—42no—
62C2×D4×C43—22no—
63C2×D4×C43—24no—
64C2×D4×C43—24no—
65C2×D4×C43—24no—
66C2×D4×C43—42no—
67C22×C4×C43—24no—
68C22×C4×C43—24no—
69C22×C4×C43—42no—
70C22×C4×C43—42no—
71C22×C4×C43C22⋊C412yes—
72C22×C4×C43C22×C411yescharacteristic
73C22×C4×C43C2×D411yescharacteristic
74C22×C4×C43C2×D414yes—
75C2×C22⋊D43—24no—
76C2×C22⋊D43—22no—
77C2×C22⋊D43—24no—
78C2×C22⋊D43—44no—
79C2×C22⋊D43—44no—
80C22×Dic43—22no—
81C22×Dic43—24no—
82C22×Dic43—42no—
83#10621900—48no—
84D4×D43—48no—
85#10621898—44no—
86C2×D172C22⋊C412yes—
87C2×C4×D43—22no—
88C2×Dic86C22⋊C412yes—
89C2×C8×C43—44no—
90C2×C8×C43—42no—
91C22⋊Dic43—24no—
92C22⋊Dic43—22no—
93C22⋊Dic43—24no—
94C22⋊Dic43—44no—
95(C2×D43)⋊C4—28no—
96(C2×D43)⋊C4—24no—
97C4⋊Dic43C22⋊C414yes—
98Dic43⋊C4—28no—
99Dic43⋊C4—24no—
100C4×Dic43—22no—
101C23×C43—22no—
102C23×C43—22no—
103C23×C43—22no—
104C23×C43—42no—
105C23×C43—44no—
106C23×C43#9914yes—
107C23×C43C2×C22⋊C411yescharacteristic
108C23×C43#12012yes—
109C22×D43—22no—
110C22×D43—44no—
111C22×D43—84no—
112Q8×C43—44no—
113D4×C43—28no—
114D4×C43—48no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—22no—
118C2×C4×C43—22no—
119C2×C4×C43—22no—
120C2×C4×C43—44no—
121C2×C4×C43—48no—
122C2×C4×C43C2×C22⋊C412yes—
123C2×C4×C43#12012yes—
124C22⋊D43—48no—
125C22⋊D43—44no—
126C22⋊D43—84no—
127C2×Dic43—24no—
128C2×Dic43—22no—
129C2×Dic43—44no—
130C2×Dic43—84no—
131C2×Dic43—82no—
132C2×Dic43—84no—
133D172—42no—
134C4×D43—44no—
135Dic86—42no—
136C8×C43—82no—
137#8462—431nomaximal, Sylow
138C22×C43—22no—
139C22×C43—24no—
140C22×C43—24no—
141C22×C43—22no—
142C22×C43—44no—
143C22×C43—44no—
144C22×C43—82no—
145C22×C43—82no—
146C22×C43#37911yescharacteristic
147C22×C43C2×#9912yes—
148C2×D43—44no—
149C2×D43—42no—
150C2×D43—164no—
151C4×C43—22no—
152C4×C43—22no—
153C4×C43—42no—
154Dic43—44no—
155Dic43—42no—
156Dic43—162no—
157#3485—431no—
158#1782—431no—
159#1780D4311yescharacteristic, complemented
160#1516—434no—
161C2×C43—22no—
162C2×C43—42no—
163C2×C43—44no—
164C2×C43#178111yescharacteristic
165C2×C43#179412yes—
166D43—82no—
167C2×H(5,𝔽2)—431no—
168C22×#41C2×D4311yescharacteristic
169#547—864no—
170#546—864no—
171#536—432no—
172#534—862no—
173#525—432no—
174C2×#123—862no—
175#518—431no—
176C2×#99—862no—
177C2×C22⋊C8—22no—
178C2×C22⋊C8C2×D4312yescomplemented
179#381—432no—
180#328—864no—
181#327—864no—
182C43#846211yescharacteristic, complemented, Sylow
183H(5,𝔽2)—1724no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41—432no—
187C2×#41—862no—
188C22×Q8C4×D4311yescharacteristic
189C22×D4—862no—
190C22×D4—862no—
191C22×D4C4×D4311yescharacteristic
192C22×D4D17212yes—
193C23×C4C22⋊D4312yes—
194C23×C4C22×D4311yescharacteristic
195C22×C8—22no—
196C4⋊Q8—862no—
197#124—862no—
198#123—868no—
199#123—864no—
200#123—864no—
201#123—862no—
202C22⋊Q8—434no—
203C22⋊Q8—864no—
204#121—434no—
205#121—868no—
206#120—864no—
207#120—862no—
208C4×D4—864no—
209C4×D4—864no—
210#117—432no—
211C2×C4⋊C4—864no—
212C2×C22⋊C4—864no—
213C2×C22⋊C4—862no—
214C2×C22⋊C4—862no—
215#99—1724no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218#95—864no—
219C24—862no—
220C24(C2×D43)⋊C412yes—
221C24C2×D17211yescharacteristic
222#41—416no—
223#41—1724no—
224#41—1728no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—432no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—432no—
234C2×D4—864no—
235C2×D4—864no—
236C2×D4—862no—
237C2×D4—1728no—
238C2×D4—1724no—
239C2×D4—1724no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—864no—
245C22×C4—862no—
246C22×C4—862no—
247C22×C4—1722no—
248C22×C4(C2×D43)⋊C412yes—
249C22×C4C2×C4×D4311yescharacteristic
250C22×C4D4×D4314yes—
251C22×C4C2×C22⋊D4311yescharacteristic
252C2×C8—44no—
253C2×C8—42no—
254C4⋊C4—434no—
255C4⋊C4—868no—
256C4⋊C4—864no—
257C22⋊C4—868no—
258C22⋊C4—864no—
259C22⋊C4—862no—
260C22⋊C4—864no—
261C22⋊C4—864no—
262C22⋊C4—1724no—
263C42—862no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23—862no—
270C23—3444no—
271C23—1724no—
272C23#1306323914yes—
273C23C2×(C2×D43)⋊C411yescharacteristic
274C23#1306329512yes—
275Q8—44no—
276Q8—1722no—
277D4—28no—
278D4—48no—
279D4—3444no—
280D4—1722no—
281D4—1728no—
282D4—1724no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—864no—
291C2×C4—862no—
292C2×C4—3444no—
293C2×C4—3442no—
294C2×C4—3444no—
295C2×C4—1724no—
296C2×C4—1724no—
297C2×C4C22⋊C4×D4312yes—
298C2×C4#1306329312yes—
299C8—82no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—44no—
305C22—44no—
306C22—82no—
307C22—82no—
308C22—1722no—
309C22—1724no—
310C22—6884no—
311C22#1458275411yescharacteristic
312C22C2×#1306323912yes—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—1722no—
317C4—1724no—
318C4—6882no—
319C2—22no—
320C2—42no—
321C2—44no—
322C2—3442no—
323C2#1913450911yescharacteristic
324C2#1913455812yes—
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1780C2
socleC22×C43#379
prime radical#19148239C2
cosocleC23×C4×C43C23
hypercenter#1780D43
hypocenterC43#8462
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #19134513C2semidirect
3. #19133152C2semidirect
4. C43×#1780C2semidirect
5. C2×#13063370C4semidirect
8. #14583473C4semidirect
12. #14583374C4semidirect
14. #14582762C22semidirect
19. C2×C22⋊C8×C43C22semidirect
159. #1780D43semidirect
178. C2×C22⋊C8C2×D43semidirect
182. C43#8462semidirect

Maximal subgroups

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

subgroupindex
#191482392
#191345132
#191331522
C43×#17802
#846243