Finite Group Databasebeta

← #24997209

Subgroups — #24997209

Order 11008.

716 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 83 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#19145370C211yesmaximal, characteristic, complemented
2#19143855C212yesmaximal
3#19132958C212yesmaximal, complemented
4#19132472C211yesmaximal, characteristic, complemented
5C43×#1780C211yesmaximal, characteristic
6C2×#13063355—22no—
7C2×#13063304C2212yes—
8#14583235—22no—
9#14583140—22no—
10#14583061—24no—
11#14583057—22no—
12#14583055—22no—
13#14583050—22no—
14#14582900—22no—
15#14582863C2212yes—
16#14582624—24no—
17C22×#41×C43C2211yescharacteristic
18C2×C22⋊C8×C43—22no—
19C2×C22⋊C8×C43C2211yescharacteristic
20C2×C22⋊C8×C43C2211yescharacteristic
21C22×Dic86D412yes—
22#13063355—48no—
23C2×Dic172—22no—
24C2×#10621885—24no—
25C2×#10621883—24no—
26C2×#10621877—22no—
27#13063304—22no—
28#13063304—22no—
29#13063304—22no—
30#13063304—22no—
31C2×C4⋊Dic43D412yes—
32C2×Dic43⋊C4—22no—
33C2×C22⋊Dic43—22no—
34#13063259—22no—
35#13063250—24no—
36#13063249—24no—
37#13063246—24no—
38C22⋊(C43⋊C8)—22no—
39C22⋊(C43⋊C8)D414yes—
40C2×#41×C43—22no—
41C2×#41×C43—22no—
42C2×#41×C43—44no—
43C22×Q8×C43D411yescharacteristic
44C22×D4×C43—21no—
45C22×D4×C43D411yescharacteristic
46C23×C4×C43—21no—
47C23×C4×C43C2311yescharacteristic
48C22×C8×C43—21no—
49C22×C8×C43D412yes—
50C22⋊C8×C43—22no—
51C22⋊C8×C43—22no—
52C22⋊C8×C43—44no—
53C24×C43—21no—
54C24×C43#4111yescharacteristic
55#41×C43—28no—
56#41×C43—44no—
57#41×C43—44no—
58#41×C43—44no—
59C2×Q8×C43—22no—
60C2×Q8×C43—42no—
61C2×D4×C43—22no—
62C2×D4×C43—42no—
63C2×D4×C43—42no—
64C2×D4×C43—42no—
65C2×D4×C43—42no—
66C22×C4×C43—24no—
67C22×C4×C43—21no—
68C22×C4×C43—42no—
69C22×C4×C43—42no—
70C22×C4×C43—41no—
71C22×C4×C43—41no—
72C22×C4×C43C2×D411yescharacteristic
73C22×C4×C43C2×D412yes—
74C22×C4×C43C2×D412yes—
75C22×C4×C43C2×D411yescharacteristic
76C22×Dic43—22no—
77C22×Dic43—22no—
78C2×Dic86—22no—
79C2×Dic86—22no—
80C2×Dic86—44no—
81C2×C8×C43—22no—
82C2×C8×C43—42no—
83C2×C8×C43—42no—
84C2×C8×C43—41no—
85C22⋊Dic43—42no—
86C22⋊Dic43—42no—
87#10621885—48no—
88#10621883—48no—
89C4⋊Dic43—22no—
90C4⋊Dic43—22no—
91Dic43⋊C4—42no—
92Dic43⋊C4—42no—
93#10621877—44no—
94C2×C43⋊C8—24no—
95Dic172—82no—
96C23×C43—22no—
97C23×C43—22no—
98C23×C43—41no—
99C23×C43—41no—
100C23×C43—41no—
101C23×C43—42no—
102C23×C43—41no—
103C23×C43—42no—
104C23×C43#12112yes—
105C23×C43#12111yescharacteristic
106Q8×C43—24no—
107Q8×C43—42no—
108D4×C43—24no—
109D4×C43—42no—
110D4×C43—48no—
111C2×C4×C43—24no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—41no—
115C2×C4×C43—42no—
116C2×C4×C43—44no—
117C2×C4×C43—41no—
118C2×C4×C43—41no—
119C2×C4×C43—44no—
120C2×C4×C43#12011yescharacteristic
121C2×C4×C43#12012yes—
122C2×C4×C43#12711yescharacteristic
123C2×Dic43—44no—
124C2×Dic43—42no—
125C2×Dic43—42no—
126C2×Dic43—82no—
127Dic86—44no—
128Dic86—44no—
129C8×C43—42no—
130C8×C43—81no—
131C43⋊C8—44no—
132#9976—431nomaximal, Sylow
133C22×C43—21no—
134C22×C43—21no—
135C22×C43—22no—
136C22×C43—22no—
137C22×C43—22no—
138C22×C43—44no—
139C22×C43—42no—
140C22×C43—82no—
141C22×C43—82no—
142C22×C43—81no—
143C22×C43—81no—
144C22×C43#39211yescharacteristic
145C22×C43#47312yes—
146C4×C43—22no—
147C4×C43—21no—
148C4×C43—42no—
149C4×C43B(4,𝔽2)12yes—
150Dic43—82no—
151Dic43—82no—
152#3189—431no—
153#3007—432no—
154#1780D4311yescharacteristic
155#1472—432no—
156#1293—431no—
157C2×C43—22no—
158C2×C43—42no—
159C2×C43—82no—
160C2×C43#201711yescharacteristic
161C2×C43#202011yescharacteristic
162C2×C43#202111yescharacteristic
163C22×#41C2×D4311yescharacteristic
164C2×#137—862no—
165C2×C22⋊Q8—432no—
166#484—862no—
167#467—862no—
168#452—864no—
169#451—862no—
170#451—862no—
171#450—862no—
172#417—862no—
173#407—432no—
174C2×C22⋊C8—22no—
175C2×C22⋊C8C2×D4311yescharacteristic
176C2×C22⋊C8C2×D4311yescharacteristic
177#324—864no—
178C43#997611yescharacteristic, complemented, Sylow
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—44no—
182C22×Q8—432no—
183C22×Q8D17211yescharacteristic
184C22×D4—21no—
185C22×D4D17211yescharacteristic
186C23×C4—21no—
187C23×C4C22×D4311yescharacteristic
188#137—1728no—
189C2×Q16—862no—
190C2×Q16—864no—
191C2×SD16—864no—
192C2×M16—862no—
193C22×C8—21no—
194C22×C8C22⋊D4312yes—
195C22⋊Q8—862no—
196C22⋊Q8—862no—
197C22⋊Q8—862no—
198C22⋊Q8—862no—
199C2×C4⋊C4—432no—
200C2×C4⋊C4—862no—
201C2×C22⋊C4—862no—
202#107—862no—
203#103—864no—
204#103—864no—
205#102—864no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—44no—
209C22⋊C8—434no—
210C22⋊C8—862no—
211C24—21no—
212C24#1062189811yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—862no—
220C2×Q8—862no—
221C2×Q8—1724no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—862no—
234C22×C4—862no—
235C22×C4C2×D17211yescharacteristic
236C22×C4D4×D4312yes—
237C22×C4D4×D4312yes—
238C22×C4C2×C22⋊D4311yescharacteristic
239Q16—3442no—
240Q16—1728no—
241SD16—1728no—
242M16—1724no—
243C2×C8—22no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—864no—
248C4⋊C4—862no—
249C4⋊C4—862no—
250C4⋊C4—1722no—
251C4⋊C4—1722no—
252C22⋊C4—1722no—
253C22⋊C4—1722no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—41no—
261C23—42no—
262C23#1306329812yes—
263C23#1306329911yescharacteristic
264Q8—24no—
265Q8—42no—
266Q8—1724no—
267Q8—1724no—
268D4—24no—
269D4—42no—
270D4—48no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—3442no—
281C2×C4—1724no—
282C2×C4—1722no—
283C2×C4—1722no—
284C2×C4#1306329311yescharacteristic
285C2×C4#1306329512yes—
286C2×C4#1306332111yescharacteristic
287C8—42no—
288C8—81no—
289C8—1724no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22#1458280811yescharacteristic
302C22#1458317412yes—
303C4—21no—
304C4—22no—
305C4—42no—
306C4—3442no—
307C4—3442no—
308C4#1458308512yes—
309C2—22no—
310C2—42no—
311C2—82no—
312C2#1913569911yescharacteristic
313C2#1913571611yescharacteristic
314C2#1913572111yescharacteristic
subgroupG/H
centerC22#14582808
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#1780C2
socleC22×C43#392
prime radicalC22×#41×C43C22
hypercenter#1780D43
hypocenterC43#9976
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
1. #19145370C2semidirect
3. #19132958C2semidirect
4. #19132472C2semidirect
178. C43#9976semidirect

Maximal subgroups

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

subgroupindex
#191453702
#191438552
#191329582
#191324722
C43×#17802
#997643