Finite Group Databasebeta

← #25105765

Subgroups — #25105765

Order 11008.

1270 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 231 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#19143071C212yesmaximal, complemented
3#19135144C214yesmaximal, complemented
4#19135010C214yesmaximal, complemented
5#19134704C212yesmaximal, complemented
6#19134523C211yesmaximal, characteristic, complemented
7C43×#1785C211yesmaximal, characteristic, complemented
8C2×#13063370C2211yescharacteristic
9#14583665C2214yes—
10#14583548—24no—
11#14583537—24no—
12#14583473C2212yes—
13#14583458C414yes—
14#14583410C2212yes—
15C2×#13063308C414yes—
16#14583374C2211yescharacteristic
17#14582881C2214yes—
18#14582880—24no—
19C2×#13063244C2212yescomplemented
20C2×#13063242C2212yes—
21#100×D43—24no—
22#14582863C2212yescomplemented
23#14582671C2214yescomplemented
24#14582670C2214yescomplemented
25C22×#41×C43C2211yescharacteristic
26C2×#101×C43C2212yescomplemented
27C2×#100×C43C2212yes—
28C43×#407C2212yescomplemented
29#13063370—42no—
30#13063370—42no—
31C2×#10621898C2312yes—
32C2×#10621900D414yes—
33C22×C22⋊D43C2×C414yes—
34C2×D4×D43D414yes—
35#13063333—44no—
36#13063331—44no—
37C2×#10621877C2314yes—
38M16×D43—28no—
39#13063323—22no—
40#13063314—22no—
41#13063308—28no—
42#13063308C2×C414yes—
43#13063308C2×C418yes—
44#13063306—24no—
45#13063304—22no—
46#13063301—24no—
47#13063299—22no—
48#13063297—28no—
49#13063294C2×C414yes—
50#13063293—24no—
51D4×Dic43—24no—
52C4×C22⋊D43—24no—
53#13063284C2312yes—
54C2×Dic43⋊C4—24no—
55C2×C22⋊Dic43C2×C414yes—
56C2×(C2×D43)⋊C4—24no—
57#13063244—42no—
58#13063243—24no—
59#13063242—24no—
60#13063241—24no—
61C22⋊(C43⋊C8)—42no—
62C2×#41×C43—24no—
63C2×#41×C43—42no—
64C2×#41×C43—42no—
65C22×Q8×C43C2311yescharacteristic
66C22×D4×C43C2×C412yes—
67C22×D4×C43C2311yescharacteristic
68C23×C4×C43C2×C412yes—
69C23×C4×C43C2311yescharacteristic
70C2×M16×C43C2314yes—
71#101×C43—42no—
72#100×C43—24no—
73C22⋊C8×C43—42no—
74C24×C43C22×C411yescharacteristic
75C24×C43C22×C412yes—
76C23×D43C22⋊C414yes—
77#41×C43—416no—
78C2×Q8×C43—22no—
79C2×Q8×C43—22no—
80C2×Q8×C43—22no—
81C2×D4×C43—22no—
82C2×D4×C43—22no—
83C2×D4×C43—24no—
84C2×D4×C43—42no—
85C2×D4×C43—42no—
86C2×D4×C43C2×D414yes—
87C22×C4×C43—22no—
88C22×C4×C43—22no—
89C22×C4×C43—22no—
90C22×C4×C43—24no—
91C22×C4×C43—42no—
92C22×C4×C43—42no—
93C22×C4×C43C22×C411yescharacteristic
94C22×C4×C43C22×C412yes—
95C22×C4×C43C2411yescharacteristic
96C2×C22⋊D43—24no—
97C2×C22⋊D43—44no—
98C2×C22⋊D43—44no—
99C2×C22⋊D43C22⋊C418yes—
100C2×C22⋊D43C22×C414yes—
101C22×Dic43—24no—
102C22×Dic43C22⋊C414yes—
103#10621900—48no—
104D4×D43—48no—
105#10621898—24no—
106#10621898—24no—
107C2×D172C2×D412yes—
108C2×C4×D43C2×D414yes—
109C2×Dic86C2×D412yes—
110M16×C43—24no—
111M16×C43—24no—
112C2×C8×C43—24no—
113C22⋊Dic43—24no—
114C22⋊Dic43—24no—
115C22⋊Dic43C22×C414yes—
116(C2×D43)⋊C4—24no—
117(C2×D43)⋊C4—28no—
118C4⋊Dic43—22no—
119Dic43⋊C4—24no—
120Dic43⋊C4—28no—
121C4×Dic43—22no—
122#10621877—24no—
123#10621877—24no—
124C2×C43⋊C8—24no—
125#10621872—48no—
126C8×D43—48no—
127C23×C43—22no—
128C23×C43—22no—
129C23×C43—21no—
130C23×C43—21no—
131C23×C43—21no—
132C23×C43—81no—
133C23×C43—82no—
134C23×C43C2×C22⋊C414yes—
135C23×C43C23×C411yescharacteristic
136C22×D43—44no—
137C22×D43—84no—
138C22×D43C2×C22⋊C414yes—
139Q8×C43—44no—
140D4×C43—28no—
141D4×C43—48no—
142C2×C4×C43—21no—
143C2×C4×C43—22no—
144C2×C4×C43—21no—
145C2×C4×C43—21no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—48no—
149C2×C4×C43—44no—
150C2×C4×C43C22×D412yes—
151C2×C4×C43H(5,𝔽2)12yes—
152C2×C4×C43H(5,𝔽2)12yes—
153C22⋊D43—24no—
154C22⋊D43—24no—
155C22⋊D43—48no—
156C22⋊D43—84no—
157C2×Dic43—24no—
158C2×Dic43—44no—
159C2×Dic43—84no—
160C2×Dic43C2×C22⋊C414yes—
161D172—24no—
162C4×D43—28no—
163Dic86—24no—
164C8×C43—44no—
165C43⋊C8—44no—
166#21779—431nomaximal, Sylow
167C22×C43—24no—
168C22×C43—41no—
169C22×C43—44no—
170C22×C43—42no—
171C22×C43—42no—
172C22×C43—42no—
173C22×C43—41no—
174C22×C43—82no—
175C22×C43—82no—
176C22×C43C22×C22⋊C411yescharacteristic
177C22×C43#51811yescharacteristic
178C22×C43#51811yescharacteristic
179C2×D43—24no—
180C2×D43—44no—
181C2×D43—164no—
182C4×C43—41no—
183C4×C43—42no—
184C4×C43#53414yes—
185Dic43—24no—
186Dic43—44no—
187#3485—431no—
188#2888—432no—
189#1915—434no—
190#1891—434no—
191#1829—432no—
192#1785—431no—
193#1785D4311yescharacteristic, complemented
194C2×C43—21no—
195C2×C43—21no—
196C2×C43—21no—
197C2×C43—44no—
198C2×C43—81no—
199C2×C43#230811yescharacteristic
200D43—44no—
201C2×H(5,𝔽2)—431no—
202C22×#41C2×D4311yescharacteristic
203#568—434no—
204#547—864no—
205#546—864no—
206#536—432no—
207#534—434no—
208#525—432no—
209C2×#123—434no—
210#518—431no—
211#413—434no—
212#413—864no—
213C2×#101—432no—
214C2×#101C2×D4312yescomplemented
215C2×#100—432no—
216C2×#100—864no—
217C2×#100C2×D4312yes—
218#407—432no—
219#407C2×D4312yescomplemented
220#344—434no—
221#344—434no—
222C43#2177911yescharacteristic, complemented, Sylow
223H(5,𝔽2)—1722no—
224H(5,𝔽2)—1722no—
225C2×#41—24no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—434no—
229C2×#41—432no—
230C22×Q8C22×D4311yescharacteristic
231C22×D4—434no—
232C22×D4—434no—
233C22×D4C4×D4312yes—
234C22×D4C22×D4311yescharacteristic
235C23×C4C4×D4312yes—
236C23×C4C22×D4311yescharacteristic
237#131—1724no—
238#131—1724no—
239C2×M16—434no—
240C2×M16—868no—
241C2×M16C22×D4314yes—
242C4⋊Q8—862no—
243#124—862no—
244#123—434no—
245#123—438no—
246#123—868no—
247#123—864no—
248C22⋊Q8—862no—
249C22⋊Q8—864no—
250#121—862no—
251#121—868no—
252#120—434no—
253#120—864no—
254C4×D4—864no—
255C4×D4—864no—
256#117—432no—
257C2×C4⋊C4—864no—
258C2×C22⋊C4—434no—
259C2×C22⋊C4—864no—
260#101—42no—
261#101—864no—
262#101—1722no—
263#100—24no—
264#100—864no—
265#100—864no—
266C22⋊C8—42no—
267C22⋊C8—1722no—
268C24—434no—
269C24C2×C4×D4311yescharacteristic
270C24C2×C4×D4312yes—
271#41—416no—
272#41—864no—
273#41—864no—
274#41—1728no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—432no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—434no—
285C2×D4—438no—
286C2×D4—432no—
287C2×D4—864no—
288C2×D4—1728no—
289C2×D4—1724no—
290C2×D4—1724no—
291C2×D4D4×D4314yes—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—434no—
299C22×C4—434no—
300C22×C4—864no—
301C22×C4C2×C4×D4311yescharacteristic
302C22×C4C2×C4×D4312yes—
303C22×C4C23×D4311yescharacteristic
304M16—24no—
305M16—24no—
306M16—864no—
307M16—864no—
308M16—1728no—
309C2×C8—24no—
310C2×C8—864no—
311C2×C8—1728no—
312C4⋊C4—864no—
313C4⋊C4—868no—
314C4⋊C4—862no—
315C22⋊C4—434no—
316C22⋊C4—864no—
317C22⋊C4—868no—
318C22⋊C4—864no—
319C22⋊C4—864no—
320C42—862no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—81no—
327C23—82no—
328C23—434no—
329C23—3444no—
330C23—1724no—
331C23C22⋊C4×D4314yes—
332C23C22×C4×D4311yescharacteristic
333Q8—44no—
334Q8—864no—
335D4—28no—
336D4—48no—
337D4—864no—
338D4—864no—
339D4—864no—
340D4—3444no—
341D4—1728no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—434no—
351C2×C4—868no—
352C2×C4—864no—
353C2×C4—3444no—
354C2×C4—1724no—
355C2×C4C2×D4×D4312yes—
356C2×C4#1306337012yes—
357C2×C4#1306337012yes—
358C8—44no—
359C8—1724no—
360C22—24no—
361C22—41no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—82no—
368C22—82no—
369C22—864no—
370C22—1724no—
371C22—6884no—
372C22C2×C22⋊C4×D4311yescharacteristic
373C22#1458337211yescharacteristic
374C22#1458337211yescharacteristic
375C4—41no—
376C4—42no—
377C4—864no—
378C4—1724no—
379C4#1458345914yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—44no—
384C2—81no—
385C2—1724no—
386C2#1913717111yescharacteristic
subgroupG/H
centerC2#19137171
commutatorC23×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1785C2
socleC2×C43#2308
prime radical#19148239C2
cosocleC22×C4×C43C24
hypercenter#1785D43
hypocenterC43#21779
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #19143071C2semidirect
3. #19135144C2semidirect
4. #19135010C2semidirect
5. #19134704C2semidirect
6. #19134523C2semidirect
7. C43×#1785C2semidirect
19. C2×#13063244C22semidirect
22. #14582863C22semidirect
23. #14582671C22semidirect
24. #14582670C22semidirect
26. C2×#101×C43C22semidirect
28. C43×#407C22semidirect
193. #1785D43semidirect
214. C2×#101C2×D43semidirect
219. #407C2×D43semidirect
222. C43#21779semidirect

Maximal subgroups

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

subgroupindex
#191482392
#191430712
#191351442
#191350102
#191347042
#191345232
C43×#17852
#2177943