Finite Group Databasebeta

← #29590620

Subgroups — #29590620

Order 15616.

590 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 73 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#21449177C211yesmaximal, characteristic
2#21445606C211yesmaximal, characteristic
3#21444158C211yesmaximal, characteristic
4#21443841C211yesmaximal, characteristic, complemented
5#21443840C211yesmaximal, characteristic, complemented
6#21443839C211yesmaximal, characteristic, complemented
7C61×#1351C211yesmaximal, characteristic, complemented
8C22×Q8×D61C411yescharacteristic, complemented
9C2×#14519754—21no—
10C2×#14519737C2211yescharacteristic
11#16839816—21no—
12C2×#14519691—21no—
13#16839683C411yescharacteristic, complemented
14#16839681C411yescharacteristic, complemented
15#16839674C2211yescharacteristic
16#16839666—21no—
17#16839638—21no—
18#16839636—21no—
19#16839621C411yescharacteristic, complemented
20C22⋊C4×Dic61—21no—
21#16839529C2211yescharacteristic
22#16839486C2211yescharacteristic
23#16839485—21no—
24C61×#393C2211yescharacteristic
25C61×#342C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×Dic122—21no—
28C22×Dic122C2×C411yescharacteristic
29C2×Q8×D61—44no—
30C2×Q8×D61—41no—
31C2×Q8×D61—41no—
32C22×C22⋊D61D411yescharacteristic
33C23×Dic61D411yescharacteristic
34C22×C4×D61—21no—
35C22×C4×D61C2×C411yescharacteristic
36#14519754—44no—
37#14519737—41no—
38#14519737—41no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196C2×C411yescharacteristic
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46C2×#11758198D411yescharacteristic
47C2×C4×Dic61—41no—
48C2×C4×Dic61D411yescharacteristic
49#14519706—41no—
50#14519691—24no—
51C22⋊#92418—21no—
52#14519687—41no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—41no—
57#14519684C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61C2311yescharacteristic
62C22⋊C8×C61—21no—
63#95×C61—21no—
64#95×C61C2×C411yescharacteristic
65C24×C61C2×D411yescharacteristic
66C23×D61C22⋊C411yescharacteristic
67C2×Q8×C61—42no—
68C2×Q8×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—41no—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—21no—
75C2×C22⋊D61—21no—
76C2×C22⋊D61—22no—
77C2×C22⋊D61—41no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—42no—
83C22×Dic61—41no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87Q8×D61—44no—
88Q8×D61—44no—
89Q8×D61—44no—
90Q8×D61—44no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C4×D61—42no—
96C2×C4×D61—42no—
97C2×Dic122—24no—
98C2×Dic122—42no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102C2×Dic122—42no—
103C2×C8×C61—41no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—41no—
108C22⋊Dic61—22no—
109C22⋊Dic61—42no—
110C22⋊Dic61—41no—
111#11758198—41no—
112#11758198—42no—
113C4⋊Dic61—42no—
114#11758196—41no—
115#11758196—42no—
116#11758196—42no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—82no—
120C2×#92418—41no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61#13711yescharacteristic
127C23×C61#13711yescharacteristic
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61#10412yes—
133Q8×C61—42no—
134Q8×C61—42no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—81no—
147C2×C4×C61—81no—
148C22⋊D61—42no—
149C22⋊D61—42no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—42no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—81no—
165C2×Dic61—81no—
166C2×Dic61—81no—
167C2×Dic61#9911yescharacteristic
168C2×Dic61#9911yescharacteristic
169C4×D61—42no—
170C4×D61—48no—
171C4×D61—42no—
172Dic122—44no—
173Dic122—42no—
174Dic122—44no—
175Dic122—42no—
176C8×C61—81no—
177#92418—81no—
178#11785—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61C2×#9911yescharacteristic
189C22×C61#41911yescharacteristic
190C22×C61C2×#10411yescharacteristic
191C2×D61—22no—
192C2×D61—42no—
193C2×D61—41no—
194C2×D61—41no—
195C2×D61—42no—
196C4×C61—42no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—81no—
200C4×C61—82no—
201Dic61—21no—
202Dic61—42no—
203Dic61—41no—
204Dic61—41no—
205Dic61—81no—
206Dic61—82no—
207Dic61AΣL(1,𝔽16)12yes—
208#2409—611no—
209C4×#99—611no—
210#1466—611no—
211#1351—611no—
212#1351—611no—
213#1351—611no—
214#1351D6111yescharacteristic, complemented
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—41no—
218C2×C61—41no—
219C2×C61#150811yescharacteristic
220C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
221C2×C61#212911yescharacteristic
222D61—42no—
223C23×Q8—611no—
224C2×C22⋊Q8—1221no—
225C2×C4×D4—611no—
226#432—1221no—
227C2×#99—1221no—
228#395—611no—
229#395—611no—
230#393—611no—
231#393C2×D6111yescharacteristic
232#391—1221no—
233#386—1221no—
234#385—1221no—
235#380—611no—
236C4×C22⋊C4—1221no—
237#342—611no—
238#342C2×D6111yescharacteristic
239#323—611no—
240#323—1221no—
241#323C2×D6111yescharacteristic
242C61#1178511yescharacteristic, complemented, Sylow
243C22×Q8—611no—
244C22×Q8—1221no—
245C22×Q8—2441no—
246C22×Q8—2441no—
247C22×Q8—2444no—
248C22×Q8C4×D6111yescharacteristic
249C22×D4—611no—
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—1221no—
253C22⋊Q8—2444no—
254C4×D4—2441no—
255C4×D4—2441no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—611no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4C22×D6111yescharacteristic
267C2×C42—611no—
268C2×C42—2441no—
269C4⋊C8—2441no—
270#99—1224no—
271C22⋊C8—21no—
272C22⋊C8—1221no—
273#97—2441no—
274#95—21no—
275#95—611no—
276#95—1221no—
277#95—1221no—
278#95—1221no—
279#95—2441no—
280#95C4×D6111yescharacteristic
281C24—611no—
282C24D4×D6111yescharacteristic
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—1224no—
286C2×Q8—2441no—
287C2×Q8—2441no—
288C2×Q8—2441no—
289C2×Q8—2442no—
290C2×Q8—2442no—
291C2×Q8—2444no—
292C2×Q8—2444no—
293C2×Q8—2444no—
294C2×Q8—2444no—
295C2×D4—1222no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—2441no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—2442no—
312C22×C4—2442no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2441no—
317C22×C4C2×C4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C2×C8—41no—
320C2×C8—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4—2442no—
324C4⋊C4—2442no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—1222no—
330C22⋊C4—2441no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2441no—
334C42—1221no—
335C42—1221no—
336C42—4882no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—612no—
342C23—1221no—
343C23—1221no—
344C23—2441no—
345C23—2441no—
346C23C22⋊C4×D6111yescharacteristic
347C23#1451980611yescharacteristic
348C23#1451980911yescharacteristic
349Q8—42no—
350Q8—42no—
351Q8—2444no—
352Q8—2444no—
353Q8—2442no—
354Q8—2442no—
355D4—2442no—
356D4—2442no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—4881no—
379C2×C4—4881no—
380C2×C4—4881no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2448no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C8—81no—
393C8—4881no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—1222no—
404C22—2442no—
405C22—2442no—
406C22—2441no—
407C22—2441no—
408C22#99×D6111yescharacteristic
409C22#1683976911yescharacteristic
410C22#104×D6111yescharacteristic
411C4—42no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—82no—
416C4—612no—
417C4—1221no—
418C4—4881no—
419C4—4882no—
420C4—2441no—
421C4—2442no—
422C4—2441no—
423C2—21no—
424C2—21no—
425C2—41no—
426C2—41no—
427C2—2442no—
428C2#2144433711yescharacteristic
429C2AΣL(1,𝔽16)×D6111yescharacteristic
430C2#2144749211yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1351C2
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1351D61
hypocenterC61#11785
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
4. #21443841C2semidirect
5. #21443840C2semidirect
6. #21443839C2semidirect
7. C61×#1351C2semidirect
8. C22×Q8×D61C4semidirect
13. #16839683C4semidirect
14. #16839681C4semidirect
19. #16839621C4semidirect
214. #1351D61semidirect
242. C61#11785semidirect

Maximal subgroups

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

subgroupindex
#214491772
#214456062
#214441582
#214438412
#214438402
#214438392
C61×#13512
#1178561