Finite Group Databasebeta

← #24994421

Subgroups — #24994421

Order 11008.

726 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 103 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#19137930C211yesmaximal, characteristic
2#19134540C212yesmaximal
3#19132925C211yesmaximal, characteristic, complemented
4#19132448C212yesmaximal, complemented
5C43×#2313C211yesmaximal, characteristic, complemented
6C22×#10621898C2211yescharacteristic
7C2×#13063308—21no—
8C2×#13063253—24no—
9#14582863C2212yes—
10#14582819C412yescomplemented
11#14582816C412yescomplemented
12#14582778—21no—
13#14582770—22no—
14#14582762C2211yescharacteristic
15#14582625—22no—
16C22×C4⋊C4×C43C2211yescharacteristic
17C43×#387—24no—
18C43×#381—21no—
19C43×#380—21no—
20C4×C22⋊C4×C43C2212yes—
21C2×#10621898—22no—
22C2×#10621898—22no—
23C2×#10621898—44no—
24C22×Dic86C2×C411yescharacteristic
25C22×D172C2×C411yescharacteristic
26C22×C22⋊D43—21no—
27C22×C4×D43—21no—
28C2×#10621877—22no—
29#13063308—44no—
30C2×Dic43⋊C4—22no—
31C2×C22⋊Dic43—21no—
32C2×(C2×D43)⋊C4—22no—
33C2×(C2×D43)⋊C4—22no—
34#13063253—48no—
35C22⋊(C43⋊C8)—22no—
36C22⋊(C43⋊C8)—22no—
37C43⋊#95—22no—
38C43⋊#95—21no—
39C43⋊#95C2×C412yes—
40C23×C4×C43D411yescharacteristic
41C23×C4×C43D411yescharacteristic
42C23×C4×C43C2311yescharacteristic
43C2×C4⋊C4×C43—22no—
44C2×C4⋊C4×C43—22no—
45C2×C4⋊C4×C43—22no—
46C2×C4⋊C4×C43C2×C412yes—
47C2×C4⋊C4×C43D412yes—
48C2×C22⋊C4×C43—22no—
49C2×C42×C43D414yes—
50#95×C43—22no—
51#95×C43—24no—
52#95×C43—22no—
53C24×C43C2×D411yescharacteristic
54C23×D43—21no—
55C22×C4×C43—21no—
56C22×C4×C43—21no—
57C22×C4×C43—24no—
58C22×C4×C43—21no—
59C22×C4×C43—21no—
60C22×C4×C43—42no—
61C22×C4×C43—42no—
62C22×C4×C43—42no—
63C22×C4×C43C22⋊C412yes—
64C22×C4×C43C22⋊C412yes—
65C22×C4×C43C22×C411yescharacteristic
66C22×C4×C43C2×D411yescharacteristic
67C22×C4×C43C2×D412yes—
68C22×C4×C43#4112yes—
69C2×C22⋊D43—24no—
70C2×C22⋊D43—42no—
71C2×C22⋊D43—41no—
72C2×C22⋊D43—41no—
73C22×Dic43—21no—
74C22×Dic43—22no—
75#10621898—28no—
76#10621898—44no—
77#10621898—44no—
78#10621898—44no—
79C2×D172—22no—
80C2×D172—42no—
81C2×C4×D43—42no—
82C2×C4×D43—42no—
83C2×C4×D43—42no—
84C2×Dic86—22no—
85C2×Dic86—42no—
86C4⋊C4×C43—24no—
87C4⋊C4×C43—24no—
88C4⋊C4×C43—82no—
89C22⋊C4×C43—44no—
90C42×C43—44no—
91C22⋊Dic43—42no—
92(C2×D43)⋊C4—44no—
93(C2×D43)⋊C4—44no—
94Dic43⋊C4—44no—
95#10621877—44no—
96C2×C43⋊C8—42no—
97C23×C43—22no—
98C23×C43—22no—
99C23×C43—41no—
100C23×C43C2×C22⋊C411yescharacteristic
101C23×C43#12011yescharacteristic
102C23×C43#12311yescharacteristic
103C22×D43—22no—
104C22×D43—41no—
105C22×D43—42no—
106C22×D43—41no—
107C22×D43—42no—
108C2×C4×C43—24no—
109C2×C4×C43—22no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—22no—
113C2×C4×C43—41no—
114C2×C4×C43—41no—
115C2×C4×C43—42no—
116C2×C4×C43—42no—
117C2×C4×C43—42no—
118C2×C4×C43—42no—
119C2×C4×C43—42no—
120C2×C4×C43—44no—
121C2×C4×C43—44no—
122C2×C4×C43—44no—
123C2×C4×C43—81no—
124C2×C4×C43C4×D412yes—
125C2×C4×C43#12112yes—
126C22⋊D43—44no—
127C22⋊D43—44no—
128C2×Dic43—22no—
129C2×Dic43—41no—
130C2×Dic43—41no—
131C2×Dic43—42no—
132C2×Dic43—42no—
133C2×Dic43—42no—
134D172—24no—
135D172—42no—
136C4×D43—48no—
137Dic86—24no—
138Dic86—42no—
139C43⋊C8—82no—
140#9480—431nomaximal, Sylow
141C22×C43—21no—
142C22×C43—21no—
143C22×C43—21no—
144C22×C43—21no—
145C22×C43—21no—
146C22×C43—21no—
147C22×C43—21no—
148C22×C43—44no—
149C22×C43#38611yescharacteristic
150C22×C43#42112yes—
151C22×C43B(4,𝔽2)12yes—
152C2×D43—44no—
153C2×D43—42no—
154C2×D43—82no—
155C2×D43—81no—
156C2×D43—81no—
157C4×C43—22no—
158C4×C43—22no—
159C4×C43—42no—
160C4×C43—42no—
161C4×C43—84no—
162Dic43—42no—
163Dic43—82no—
164#2406—431no—
165#2313D4311yescharacteristic, complemented
166#1789—432no—
167#1463—431no—
168#1284—432no—
169C2×C43—22no—
170C2×C43—22no—
171C2×C43—41no—
172C2×C43#188911yescharacteristic
173C2×C43#189711yescharacteristic
174C2×C43#191211yescharacteristic
175D43—82no—
176C22×#41—431no—
177C2×#123—861no—
178C22×C4⋊C4C2×D4311yescharacteristic
179C2×#104—864no—
180#407—432no—
181#395—432no—
182#394—432no—
183#387—24no—
184#386—861no—
185#385—862no—
186#381—21no—
187#381—431no—
188#380—21no—
189C4×C22⋊C4C2×D4312yes—
190#324—862no—
191C43#948011yescharacteristic, complemented, Sylow
192C2×#41—862no—
193C2×#41—862no—
194C2×#41—1724no—
195C22×Q8—431no—
196C22×D4—431no—
197C22×D4—861no—
198C23×C4—861no—
199C23×C4C22⋊D4311yescharacteristic
200C23×C4C22⋊D4311yescharacteristic
201C23×C4C22×D4311yescharacteristic
202C2×M16—862no—
203#123—1724no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—862no—
208C2×C4⋊C4C4×D4312yes—
209C2×C4⋊C4D17212yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—861no—
212C2×C22⋊C4—862no—
213C2×C22⋊C4—862no—
214C2×C42C22⋊D4314yes—
215#104—1728no—
216C22⋊C8—862no—
217C22⋊C8—862no—
218#95—22no—
219#95—24no—
220#95—22no—
221#95—432no—
222#95—862no—
223#95—861no—
224C24—861no—
225C24C2×C22⋊D4311yescharacteristic
226#41—868no—
227#41—1724no—
228#41—1724no—
229#41—1724no—
230C2×Q8—862no—
231C2×Q8—1722no—
232C2×D4—864no—
233C2×D4—862no—
234C2×D4—1722no—
235C2×D4—1722no—
236C2×D4—1721no—
237C2×D4—1721no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—861no—
247C22×C4—862no—
248C22×C4—1722no—
249C22×C4—1722no—
250C22×C4—1722no—
251C22×C4(C2×D43)⋊C412yes—
252C22×C4(C2×D43)⋊C412yes—
253C22×C4C2×C4×D4311yescharacteristic
254C22×C4C2×D17211yescharacteristic
255C22×C4#1062189812yes—
256C22×C4C2×C22⋊D4312yes—
257M16—1724no—
258C2×C8—1722no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—82no—
262C4⋊C4—1724no—
263C22⋊C4—44no—
264C22⋊C4—1724no—
265C22⋊C4—1724no—
266C22⋊C4—1722no—
267C42—44no—
268C23—22no—
269C23—22no—
270C23—41no—
271C23—862no—
272C23—1721no—
273C23—1722no—
274C23—1722no—
275C23—1721no—
276C23C2×(C2×D43)⋊C411yescharacteristic
277C23#1306329411yescharacteristic
278C23#1306330811yescharacteristic
279Q8—864no—
280Q8—1722no—
281D4—864no—
282D4—1722no—
283D4—1724no—
284D4—1724no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—81no—
301C2×C4—862no—
302C2×C4—1728no—
303C2×C4—1722no—
304C2×C4—1722no—
305C2×C4—1722no—
306C2×C4—1721no—
307C2×C4—1721no—
308C2×C4C4×C22⋊D4312yes—
309C2×C4#1306329912yes—
310C8—3442no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—3442no—
320C22—3441no—
321C22—3441no—
322C22—1724no—
323C22—1722no—
324C22#1458277811yescharacteristic
325C22#1458291712yes—
326C22#1458308612yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—84no—
332C4—3442no—
333C4—1722no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—3442no—
338C2#1913500311yescharacteristic
339C2#1913504311yescharacteristic
340C2#1913513211yescharacteristic
subgroupG/H
centerC22#14582778
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#2313C2
socleC22×C43#386
prime radicalC22×#10621898C22
cosocleC23×C4×C43C23
hypercenter#2313D43
hypocenterC43#9480
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
3. #19132925C2semidirect
4. #19132448C2semidirect
5. C43×#2313C2semidirect
10. #14582819C4semidirect
11. #14582816C4semidirect
165. #2313D43semidirect
191. C43#9480semidirect

Maximal subgroups

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

subgroupindex
#191379302
#191345402
#191329252
#191324482
C43×#23132
#948043