Finite Group Databasebeta

← #25251833

Subgroups — #25251833

Order 11008.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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#19143506C211yesmaximal, characteristic
2#19136812C212yesmaximal, complemented
3#19136763C212yesmaximal, complemented
4#19136673C212yesmaximal, complemented
5#19136545C212yesmaximal
6#19136484C214yesmaximal, complemented
7#19136407C211yesmaximal, characteristic, complemented
8C43×#3185C211yesmaximal, characteristic, complemented
9#14583699C2211yescharacteristic
10#14583665—21no—
11#14583340C2214yes—
12#14583022—21no—
13#14583022C2212yes—
14#14583017—21no—
15#14583017C2212yes—
16#14582948C2211yescharacteristic
17C2×#13063253—21no—
18#14582721C2214yescomplemented
19#14582720C2212yescomplemented
20#14582715C2214yescomplemented
21#14582708C2214yes—
22#14582681C2212yes—
23#14582677C2212yescomplemented
24C43×#533—21no—
25C43×#498C2211yescharacteristic, complemented
26C43×#494C2211yescharacteristic, complemented
27C43×#487—21no—
28C43×#470—21no—
29C43×#470C2212yes—
30C43×#441—21no—
31C43×#441C2212yescomplemented
32C4×M16×C43C2211yescharacteristic
33C2×#10621898—21no—
34#13063358—22no—
35#13063353—22no—
36#13063347—21no—
37#13063347C2312yes—
38#13063333—42no—
39#13063331—22no—
40#13063331—22no—
41C2×#10621877—21no—
42M16×D43—42no—
43#13063271—24no—
44#13063268—24no—
45#13063265C2314yes—
46#13063261—21no—
47#13063261C2312yes—
48#13063253—22no—
49#13063253—22no—
50C2×Q16×C43—21no—
51C2×Q16×C43C2312yes—
52C2×M16×C43D411yescharacteristic
53C2×M16×C43C2311yescharacteristic
54C4⋊Q8×C43—21no—
55C4⋊Q8×C43—21no—
56#125×C43—41no—
57C22⋊Q8×C43—22no—
58C22⋊Q8×C43—41no—
59C4×Q8×C43—22no—
60C4×Q8×C43—41no—
61#117×C43—41no—
62C2×C42×C43D411yescharacteristic
63#106×C43—21no—
64#106×C43C2312yes—
65#103×C43—22no—
66#103×C43—22no—
67#97×C43D411yescharacteristic
68#97×C43C2311yescharacteristic
69C4×C8×C43D411yescharacteristic
70C4×C8×C43C2311yescharacteristic
71C2×Q8×C43—22no—
72C2×Q8×C43—41no—
73C22×C4×C43—21no—
74C22×C4×C43C2×D411yescharacteristic
75C2×C22⋊D43—41no—
76#10621898—22no—
77#10621898—22no—
78#10621898—41no—
79#10621898—41no—
80C2×D172—21no—
81C2×C4×D43—41no—
82C2×Dic86—21no—
83Q16×C43—24no—
84Q16×C43—42no—
85M16×C43—22no—
86M16×C43—22no—
87C2×C8×C43C2×D411yescharacteristic
88C2×C8×C43C2×D411yescharacteristic
89C2×C8×C43C2×D411yescharacteristic
90C2×C8×C43C2411yescharacteristic
91C4⋊C4×C43—22no—
92C4⋊C4×C43—41no—
93C4⋊C4×C43—41no—
94C4⋊C4×C43—42no—
95C4⋊C4×C43—41no—
96C4⋊C4×C43—42no—
97C22⋊C4×C43—42no—
98C42×C43—21no—
99C42×C43—42no—
100C42×C43C2×D411yescharacteristic
101C42×C43C2×D411yescharacteristic
102#10621877—22no—
103#10621877—41no—
104C2×C43⋊C8—41no—
105Dic172—21no—
106Dic172#4112yes—
107D344—21no—
108D344#4112yes—
109#10621873—22no—
110#10621873—22no—
111#10621872—42no—
112#10621872—42no—
113C8×D43—42no—
114C8×D43—42no—
115C43⋊C16—24no—
116C23×C43AGL(1,ℤ/8)11yescharacteristic
117C22×D43—41no—
118Q8×C43—42no—
119Q8×C43—42no—
120Q8×C43—82no—
121C2×C4×C43—21no—
122C2×C4×C43—21no—
123C2×C4×C43—21no—
124C2×C4×C43—21no—
125C2×C4×C43—42no—
126C2×C4×C43—42no—
127C2×C4×C43—42no—
128C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C43C22×D411yescharacteristic
130C22⋊D43—42no—
131C22⋊D43—42no—
132C2×Dic43—41no—
133D172—22no—
134D172—41no—
135C4×D43—42no—
136C4×D43—42no—
137Dic86—22no—
138Dic86—41no—
139C8×C43—21no—
140C8×C43—21no—
141C8×C43AGL(1,ℤ/8)12yes—
142C8×C43C2×#4112yes—
143C43⋊C8—42no—
144#33689—431nomaximal, Sylow
145C22×C43—21no—
146C22×C43—41no—
147C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
148C2×D43—42no—
149C2×D43—161no—
150C4×C43—22no—
151C4×C43—21no—
152C4×C43—41no—
153C4×C43—82no—
154C4×C43—82no—
155C4×C43#52511yescharacteristic
156C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
157Dic43—42no—
158#3185D4311yescharacteristic, complemented
159#2956—431no—
160#2239—432no—
161#2230—432no—
162#2214—432no—
163#2191—432no—
164#2180—434no—
165#2165—431no—
166C2×C43—21no—
167C2×C43—41no—
168C2×C43#310911yescharacteristic
169D43—82no—
170#575—431no—
171#568—861no—
172#533—21no—
173#510—434no—
174#498C2×D4311yescharacteristic, complemented
175#494C2×D4311yescharacteristic, complemented
176#487—21no—
177#470—21no—
178#470C2×D4312yes—
179#444—432no—
180#444—861no—
181#443—432no—
182#443—861no—
183#441—21no—
184#441C2×D4312yescomplemented
185#430—431no—
186C2×#104—861no—
187C4×M16C2×D4311yescharacteristic
188#365—434no—
189#364—432no—
190#362—434no—
191#360—434no—
192#349—432no—
193#347—432no—
194C43#3368911yescharacteristic, complemented, Sylow
195C2×#41—861no—
196#137—862no—
197AGL(1,ℤ/8)—862no—
198#135—432no—
199#135—861no—
200C2×Q16—21no—
201C2×Q16C22×D4312yes—
202#131—862no—
203#131—862no—
204#131—1722no—
205C2×M16—861no—
206C2×M16—1722no—
207C2×M16C22⋊D4311yescharacteristic
208C2×M16C22×D4311yescharacteristic
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211#125—41no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C4×Q8—22no—
215C4×Q8—41no—
216#117—41no—
217C2×C42C22⋊D4311yescharacteristic
218Q32—864no—
219SD32—864no—
220M32—434no—
221#108—432no—
222#108—861no—
223#106—21no—
224#106C22×D4312yes—
225#104—862no—
226#104—862no—
227#103—22no—
228#103—22no—
229#97C22⋊D4311yescharacteristic
230#97C22×D4311yescharacteristic
231C4×C8C22⋊D4311yescharacteristic
232C4×C8C22×D4311yescharacteristic
233#41—862no—
234#41—862no—
235#41—1721no—
236#41—1721no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—861no—
240C2×D4—861no—
241C2×D4—1721no—
242C22×C4—21no—
243C22×C4—1721no—
244C22×C4C2×C22⋊D4311yescharacteristic
245Q16—24no—
246Q16—42no—
247Q16—432no—
248Q16—861no—
249SD16—862no—
250SD16—862no—
251D8—432no—
252D8—861no—
253M16—22no—
254M16—22no—
255M16—862no—
256M16—1722no—
257M16—1721no—
258M16—1722no—
259C2×C8—1722no—
260C2×C8—1721no—
261C2×C8—1722no—
262C2×C8C2×C22⋊D4311yescharacteristic
263C2×C8C2×C22⋊D4311yescharacteristic
264C2×C8C2×C22⋊D4311yescharacteristic
265C2×C8C23×D4311yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—42no—
275C42C2×C22⋊D4311yescharacteristic
276C42C2×C22⋊D4311yescharacteristic
277C16—864no—
278C23—1721no—
279C23#1306335111yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—82no—
283Q8—862no—
284Q8—1721no—
285D4—862no—
286D4—1721no—
287D4—1722no—
288D4—1722no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—1722no—
297C2×C4—1721no—
298C2×C4—1722no—
299C2×C4#1306335111yescharacteristic
300C2×C4C22×C22⋊D4311yescharacteristic
301C8—21no—
302C8—21no—
303C8—1722no—
304C8#1306335012yes—
305C8#41×D4312yes—
306C22—21no—
307C22—41no—
308C22—1722no—
309C22—6881no—
310C22C2×#1306335111yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—1722no—
317C4#1458340611yescharacteristic
318C4C2×#1306335011yescharacteristic
319C2—21no—
320C2—41no—
321C2—3442no—
322C2#1914468211yescharacteristic
subgroupG/H
centerC2#19144682
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#3185C2
socleC2×C43#3109
prime radical#14583699C22
hypercenter#3185D43
hypocenterC43#33689
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
2. #19136812C2semidirect
3. #19136763C2semidirect
4. #19136673C2semidirect
6. #19136484C2semidirect
7. #19136407C2semidirect
8. C43×#3185C2semidirect
18. #14582721C22semidirect
19. #14582720C22semidirect
20. #14582715C22semidirect
23. #14582677C22semidirect
25. C43×#498C22semidirect
26. C43×#494C22semidirect
31. C43×#441C22semidirect
158. #3185D43semidirect
174. #498C2×D43semidirect
175. #494C2×D43semidirect
184. #441C2×D43semidirect
194. C43#33689semidirect

Maximal subgroups

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

subgroupindex
#191435062
#191368122
#191367632
#191366732
#191365452
#191364842
#191364072
C43×#31852
#3368943