Finite Group Databasebeta

← #29577775

Subgroups — #29577775

Order 15616.

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

normal subgroups only

This group has 934 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#21454843C211yesmaximal, characteristic, complemented
2#21449175C211yesmaximal, characteristic
3#21445749C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6C23×D244C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519745—21no—
9D4×#92418—24no—
10#16839815—21no—
11C2×#14519697—24no—
12C2×C22⋊#92418—21no—
13C2×C22⋊#92418C2212yes—
14#16839679C412yescomplemented
15#16839671C412yescomplemented
16#16839664—21no—
17#16839622C2211yescharacteristic
18#16839617—21no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C61×#380C2212yes—
22C61×#342—21no—
23C24×D61—21no—
24C22×D244—22no—
25C22×D244—21no—
26C22×D244—22no—
27C22×D244—44no—
28C22×D244C2×C412yes—
29C22×C22⋊D61—22no—
30C22×#92418—22no—
31C22×#92418D414yescomplemented
32#14519751—44no—
33#14519745—42no—
34#14519745—42no—
35C2×C4⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C4⋊#92418—22no—
41#14519697—48no—
42C22⋊#92418—22no—
43C22⋊#92418—22no—
44C22⋊#92418—22no—
45C4×#92418—22no—
46#14519684C2×C412yes—
47C22×D4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C23×C4×C61C2311yescharacteristic
50C4×D4×C61—22no—
51C4×D4×C61—22no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C42×C61D411yescharacteristic
58#99×C61—42no—
59#95×C61—22no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C2×D411yescharacteristic
62C23×D61—22no—
63C23×D61—44no—
64C23×D61—42no—
65C23×D61—41no—
66C23×D61—41no—
67C23×D61—42no—
68C23×D61—82no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61D812yes—
72C2×D4×C61SD1612yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61#4112yes—
85C2×C22⋊D61—44no—
86C2×C22⋊D61—44no—
87C2×C22⋊D61—82no—
88C22×Dic61—22no—
89C2×D244—28no—
90C2×D244—24no—
91C2×D244—24no—
92C2×D244—44no—
93C2×D244—48no—
94C2×D244—44no—
95C2×D244—42no—
96C2×D244—41no—
97C2×D244—41no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—44no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—22no—
102C22⋊C4×C61—42no—
103C22⋊C4×C61—42no—
104C42×C61—21no—
105C42×C61—21no—
106C22⋊Dic61—41no—
107C22⋊Dic61—41no—
108#11758198—44no—
109#11758198—44no—
110#11758198—82no—
111C4⋊Dic61—42no—
112C2×#92418—24no—
113C2×#92418—44no—
114C2×#92418—82no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61#10212yescomplemented
122C23×C61C2×C22⋊C411yescharacteristic
123C23×C61#12112yes—
124C23×C61C2×D811yescharacteristic
125C23×C61C2×SD1611yescharacteristic
126C23×C61#13512yes—
127C22×D61—24no—
128C22×D61—42no—
129C22×D61—48no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—84no—
136C22×D61—84no—
137C22×D61—88no—
138C22×D61—84no—
139C22×D61—82no—
140C22×D61—82no—
141D4×C61—24no—
142D4×C61—24no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—24no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—81no—
155C2×C4×C61—82no—
156C2×C4×C61—82no—
157C2×C4×C61—82no—
158C2×C4×C61#10212yescomplemented
159C2×C4×C61C4×D412yes—
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12311yescharacteristic
162C22⋊D61—88no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166D244—28no—
167D244—44no—
168D244—44no—
169D244—44no—
170#92418—44no—
171#9468—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—21no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—81no—
184C22×C61#38611yescharacteristic
185C22×C61C2×#10211yescharacteristic
186C22×C61#41611yescharacteristic
187C22×C61#46512yes—
188C22×C61#46612yes—
189C2×D61—44no—
190C2×D61—44no—
191C2×D61—44no—
192C2×D61—82no—
193C2×D61—82no—
194C2×D61—88no—
195C2×D61—88no—
196C2×D61—82no—
197C2×D61—82no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—22no—
201C4×C61—41no—
202C4×C61—41no—
203C4×C61—162no—
204C4×C61B(4,𝔽2)12yes—
205Dic61—82no—
206#2972—611no—
207#2408—611no—
208#1799D6111yescharacteristic, complemented
209#1786—612no—
210#1289—612no—
211C2×C61—21no—
212C2×C61—22no—
213C2×C61—21no—
214C2×C61—22no—
215C2×C61—42no—
216C2×C61#188911yescharacteristic
217C2×C61#189311yescharacteristic
218C2×C61#191011yescharacteristic
219D61—84no—
220C23×D4—611no—
221C2×#121—1221no—
222C2×#120—1221no—
223C2×C4×D4C2×D6111yescharacteristic
224C8×D4—1224no—
225#431—1221no—
226C2×#102—1224no—
227C2×#99—21no—
228C2×C22⋊C8—612no—
229C2×C22⋊C8—1221no—
230#394—612no—
231#392—612no—
232#390—1221no—
233#380—611no—
234#380C2×D6112yes—
235#379—1221no—
236#342—21no—
237C61#946811yescharacteristic, complemented, Sylow
238C25—1221no—
239C22×D4—612no—
240C22×D4—1222no—
241C22×D4—1221no—
242C22×D4—1222no—
243C22×D4—1222no—
244C22×D4—2444no—
245C22×D4C22⋊D6111yescharacteristic
246C23×C4D24411yescharacteristic
247C23×C4C22×D6111yescharacteristic
248C22×C8—614no—
249C22×C8—1222no—
250#121—2444no—
251#120—2442no—
252#120—2442no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4C4×D6111yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1222no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4C4×D6111yescharacteristic
264C2×C22⋊C4C22⋊D6111yescharacteristic
265C2×C42D24411yescharacteristic
266C4⋊C8—1222no—
267#102—2448no—
268#99—42no—
269C22⋊C8—1222no—
270C22⋊C8—1222no—
271C22⋊C8—1222no—
272C4×C8—1222no—
273#95—22no—
274#95—612no—
275C24—1222no—
276C24—4882no—
277C24—2444no—
278C24—2441no—
279C24—2441no—
280C24—2442no—
281C24—2442no—
282C24#1175819811yescharacteristic
283C24C2×C22⋊D6111yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—1228no—
287C2×D4—1224no—
288C2×D4—1224no—
289C2×D4—4882no—
290C2×D4—2444no—
291C2×D4—2442no—
292C2×D4—2444no—
293C2×D4—2448no—
294C2×D4—2444no—
295C2×D4—2444no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4#1175819912yes—
299C2×D4#1175820112yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—1222no—
306C22×C4#1175819811yescharacteristic
307C22×C4#1175819811yescharacteristic
308C22×C4#1175819811yescharacteristic
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4C2×D24411yescharacteristic
311C22×C4D4×D6112yes—
312C22×C4#1175822412yes—
313C2×C8—1224no—
314C2×C8—4882no—
315C2×C8—2444no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—2442no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—4882no—
324C22⋊C4—2444no—
325C22⋊C4—2441no—
326C22⋊C4—2441no—
327C22⋊C4—2444no—
328C42—21no—
329C42—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—1224no—
337C23—4884no—
338C23—4884no—
339C23—4884no—
340C23—4888no—
341C23—4882no—
342C23—4882no—
343C23—2442no—
344C23—2448no—
345C23—2442no—
346C23—2442no—
347C23—2442no—
348C23—2442no—
349C23—2442no—
350C23#1451969712yescomplemented
351C23C2×#1175819811yescharacteristic
352C23#1451974912yes—
353C23C2×#1175819911yescharacteristic
354C23C2×#1175820111yescharacteristic
355C23#1451979712yes—
356D4—24no—
357D4—24no—
358D4—1228no—
359D4—4888no—
360D4—2444no—
361D4—2444no—
362D4—2444no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4#1451969712yescomplemented
382C2×C4#1451973712yes—
383C2×C4#1451974711yescharacteristic
384C2×C4#1451976111yescharacteristic
385C8—2444no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22—4888no—
399C22—4888no—
400C22—4882no—
401C22—4882no—
402C22—4882no—
403C22—4882no—
404C22—2444no—
405C22—2444no—
406C22—2444no—
407C22#1683963911yescharacteristic
408C22C2×#1451969711yescharacteristic
409C22#1683975611yescharacteristic
410C22#1683999312yes—
411C22#1683999612yes—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—162no—
418C4—4882no—
419C4#1683995012yes—
420C2—22no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—42no—
425C2—4884no—
426C2#2144622511yescharacteristic
427C2#2144624511yescharacteristic
428C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radical#21449175C2
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9468
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
1. #21454843C2semidirect
3. #21445749C2semidirect
4. #21443686C2semidirect
5. C61×#1799C2semidirect
14. #16839679C4semidirect
15. #16839671C4semidirect
31. C22×#92418D4semidirect
121. C23×C61#102semidirect
158. C2×C4×C61#102semidirect
208. #1799D61semidirect
237. C61#9468semidirect
350. C23#14519697semidirect
381. C2×C4#14519697semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214491752
#214457492
#214436862
C61×#17992
#946861