Finite Group Databasebeta

← #29571479

Subgroups — #29571479

Order 15616.

1170 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×#14519765C211yesmaximal, characteristic
2#21445581C211yesmaximal, characteristic, complemented
3C2×#16839501C214yesmaximal, complemented
4C61×#1749C211yesmaximal, characteristic, complemented
5C23×Dic122C411yescharacteristic
6C23×D244C411yescharacteristic
7C2×#14519765—28no—
8C2×#14519765—21no—
9C2×#14519765—21no—
10C2×#14519765C412yes—
11C2×#14519765C2212yes—
12C22×#11758198—21no—
13C22×#11758198—21no—
14C2×C22⋊#92418—21no—
15C2×#14519688C2212yescomplemented
16#16839517—21no—
17#16839501—28no—
18C22×C42×C61C2211yescharacteristic
19C2×C22⋊C8×C61—21no—
20C2×#97×C61C2212yescomplemented
21C61×#336—21no—
22C24×D61—21no—
23C22×Dic122—21no—
24C22×Dic122—24no—
25C22×Dic122—42no—
26C22×Dic122—42no—
27C22×Dic122C2×C412yes—
28C22×D244—21no—
29C22×D244—24no—
30C22×D244—42no—
31C22×D244—42no—
32C22×D244C2×C412yes—
33C23×Dic61—21no—
34C22×#92418—22no—
35#14519765—28no—
36#14519765—28no—
37#14519765—24no—
38#14519765—24no—
39#14519765—24no—
40#14519765D418yes—
41C2×#11758198—28no—
42C2×#11758198—28no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C22⋊#92418—24no—
48#14519688—24no—
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C23×C4×C61D411yescharacteristic
52C22×C8×C61—22no—
53C2×C42×C61—24no—
54C2×C42×C61C2×C411yescharacteristic
55C2×C42×C61D411yescharacteristic
56C2×C42×C61D411yescharacteristic
57C2×C42×C61C2311yescharacteristic
58C22⋊C8×C61—24no—
59#97×C61—24no—
60C24×C61C22⋊C411yescharacteristic
61C23×D61—24no—
62C23×D61—22no—
63C23×D61—42no—
64C23×D61—42no—
65C23×D61—42no—
66C23×D61—42no—
67C23×D61—42no—
68C23×D61—42no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—41no—
77C22×C4×C61—41no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—24no—
85C22×Dic61—22no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C2×D244—24no—
89C2×D244—22no—
90C2×D244—416no—
91C2×D244—42no—
92C2×D244—42no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244C22⋊C414yes—
96C2×Dic122—24no—
97C2×Dic122—22no—
98C2×Dic122—42no—
99C2×Dic122—416no—
100C2×Dic122—42no—
101C2×Dic122—42no—
102C2×Dic122—42no—
103C2×Dic122C22⋊C414yes—
104C2×C8×C61—24no—
105C2×C8×C61—42no—
106C2×C8×C61—42no—
107C42×C61—22no—
108C42×C61—22no—
109C42×C61—22no—
110C42×C61C2×D414yes—
111#11758198—28no—
112#11758198—28no—
113#11758198—28no—
114#11758198—28no—
115C2×#92418—24no—
116C2×#92418—42no—
117C2×#92418—42no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61#10012yes—
123C23×C61#10412yes—
124C23×C61#10412yes—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—24no—
127C22×D61—48no—
128C22×D61—48no—
129C22×D61—48no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—84no—
137C22×D61—84no—
138C22×D61—84no—
139C22×D61—84no—
140C2×C4×C61—28no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—44no—
156C2×C4×C61—44no—
157C2×C4×C61#10012yes—
158C2×C4×C61C2×C22⋊C412yes—
159C2×C4×C61#12012yes—
160C2×C4×C61#12012yes—
161C2×Dic61—24no—
162C2×Dic61—48no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165D244—44no—
166D244—48no—
167D244—44no—
168Dic122—48no—
169Dic122—44no—
170Dic122—44no—
171C8×C61—44no—
172#92418—44no—
173#8334—611nomaximal, Sylow
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—22no—
178C22×C61—22no—
179C22×C61—22no—
180C22×C61#32914yescomplemented
181C22×C61#37911yescharacteristic
182C22×C61C2×#10011yescharacteristic
183C22×C61C2×#10011yescharacteristic
184C22×C61C2×#10411yescharacteristic
185C22×C61C2×#10411yescharacteristic
186C22×C61#42111yescharacteristic
187C22×C61#42111yescharacteristic
188C2×D61—44no—
189C2×D61—44no—
190C2×D61—44no—
191C2×D61—816no—
192C2×D61—82no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C4×C61—22no—
197C4×C61—22no—
198C4×C61—22no—
199C4×C61—22no—
200C4×C61—41no—
201C4×C61—41no—
202Dic61—44no—
203C22×#124—611no—
204#1749—611no—
205#1749D6111yescharacteristic, complemented
206C2×#329—614no—
207C2×C61—24no—
208C2×C61C2×#32911yescharacteristic
209C2×C61#152711yescharacteristic
210C2×C61#152711yescharacteristic
211C2×C61#153311yescharacteristic
212C2×C61#178311yescharacteristic
213C2×C61#178911yescharacteristic
214C2×C61#178911yescharacteristic
215D61—84no—
216C23×Q8—611no—
217C23×D4—611no—
218C2×#124—612no—
219C2×#124—612no—
220C2×#124—1228no—
221C2×#124—1221no—
222C2×#124—1221no—
223C22×C22⋊C4—1221no—
224C22×C22⋊C4—1221no—
225C22×C42C2×D6111yescharacteristic
226C2×C22⋊C8—21no—
227C2×C22⋊C8—1221no—
228C2×#97—612no—
229C2×#97C2×D6112yescomplemented
230#336—21no—
231#336—1221no—
232#329—1228no—
233C61#833411yescharacteristic, complemented, Sylow
234C25—1221no—
235C22×Q8—612no—
236C22×Q8—1224no—
237C22×Q8—1221no—
238C22×Q8—2442no—
239C22×Q8—2442no—
240C22×D4—612no—
241C22×D4—1224no—
242C22×D4—1221no—
243C22×D4—2442no—
244C22×D4—2442no—
245C23×C4—1221no—
246C23×C4C4×D6111yescharacteristic
247C23×C4D24411yescharacteristic
248C23×C4C22⋊D6111yescharacteristic
249C22×C8—22no—
250C22×C8—1222no—
251#124—618no—
252#124—1224no—
253#124—1224no—
254#124—1228no—
255#124—1228no—
256#124—1224no—
257C2×C22⋊C4—1228no—
258C2×C22⋊C4—1228no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1222no—
262C2×C22⋊C4—1222no—
263C2×C42—24no—
264C2×C42C4×D6111yescharacteristic
265C2×C42D24411yescharacteristic
266C2×C42C22⋊D6111yescharacteristic
267C2×C42C22×D6111yescharacteristic
268C22⋊C8—24no—
269C22⋊C8—1224no—
270#97—24no—
271#97—1224no—
272C24—1224no—
273C24—1222no—
274C24—2442no—
275C24—2442no—
276C24—2442no—
277C24—2442no—
278C24—2442no—
279C24—2442no—
280C24#1175819811yescharacteristic
281C2×Q8—614no—
282C2×Q8—1224no—
283C2×Q8—1222no—
284C2×Q8—24416no—
285C2×Q8—2442no—
286C2×Q8—2442no—
287C2×Q8—2442no—
288C2×Q8—2442no—
289C2×D4—614no—
290C2×D4—1224no—
291C2×D4—1222no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—24416no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—1224no—
307C22×C4—1222no—
308C22×C4—2442no—
309C22×C4—2442no—
310C22×C4#1175819811yescharacteristic
311C22×C4#1175819811yescharacteristic
312C22×C4#1175819811yescharacteristic
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4C2×D24411yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—1224no—
320C2×C8—2442no—
321C2×C8—2442no—
322C22⋊C4—1228no—
323C22⋊C4—1228no—
324C22⋊C4—1228no—
325C22⋊C4—1228no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42D4×D6114yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—1224no—
335C23—4884no—
336C23—4884no—
337C23—4884no—
338C23—4884no—
339C23—2448no—
340C23—2448no—
341C23—2448no—
342C23—2442no—
343C23—2442no—
344C23—2442no—
345C23—2442no—
346C23—2442no—
347C23—2442no—
348C23#1451969312yes—
349C23#1451970312yes—
350C23#1451970512yes—
351C23C2×#1175819811yescharacteristic
352Q8—2444no—
353Q8—2448no—
354Q8—2444no—
355D4—2444no—
356D4—2448no—
357D4—2444no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—1224no—
376C2×C4—2448no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4#1451969312yes—
380C2×C4C22⋊C4×D6112yes—
381C2×C4#1451974512yes—
382C2×C4#1451974712yes—
383C8—44no—
384C8—2444no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—48816no—
392C22—4882no—
393C22—4882no—
394C22—4882no—
395C22—4882no—
396C22—2444no—
397C22—2444no—
398C22—2444no—
399C22#1683950114yescomplemented
400C22#1683961711yescharacteristic
401C22C2×#1451969311yescharacteristic
402C22C2×#1451969311yescharacteristic
403C22C2×#1451970311yescharacteristic
404C22C2×#1451970511yescharacteristic
405C22#1683977811yescharacteristic
406C22#1683978011yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—41no—
413C4—2444no—
414C2—24no—
415C2—4884no—
416C2C2×#1683950111yescharacteristic
417C2#2144441611yescharacteristic
418C2#2144441811yescharacteristic
419C2#2144444211yescharacteristic
420C2#2144573911yescharacteristic
421C2#2144576211yescharacteristic
422C2#2144576411yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D244C4
cosocleC2×C42×C61C23
hypercenter#1749D61
hypocenterC61#8334
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #21445581C2semidirect
3. C2×#16839501C2semidirect
4. C61×#1749C2semidirect
15. C2×#14519688C22semidirect
20. C2×#97×C61C22semidirect
180. C22×C61#329semidirect
205. #1749D61semidirect
229. C2×#97C2×D61semidirect
233. C61#8334semidirect
399. C22#16839501semidirect

Maximal subgroups

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

subgroupindex
C22×#145197652
#214455812
C2×#168395012
C61×#17492
#833461