Finite Group Databasebeta

← #29597331

Subgroups — #29597331

Order 15616.

732 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448993C211yesmaximal, characteristic, complemented
2C2×#16839557C211yesmaximal, characteristic, complemented
3#21445767C212yesmaximal
4#21443938C212yesmaximal, complemented
5C2×C61×#455C211yesmaximal, characteristic
6C2×#14519692—22no—
7C2×#14519692C2212yes—
8#16839669—21no—
9#16839655—22no—
10#16839642—22no—
11#16839642C2212yes—
12#16839637—22no—
13#16839628—21no—
14#16839557—22no—
15C2×H(5,𝔽2)×C61C2211yescharacteristic
16C2×#137×C61—21no—
17C2×#124×C61C2211yescharacteristic
18C61×#455—22no—
19C61×#455C414yescomplemented
20C2×#104×C61—21no—
21C2×#100×C61C2211yescharacteristic
22C23×Dic61D412yes—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—42no—
29C2×C22⋊Dic61D412yes—
30C2×C4×Dic61—21no—
31#14519692—22no—
32#14519692—22no—
33#14519692—44no—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—42no—
38H(5,𝔽2)×C61—22no—
39H(5,𝔽2)×C61—21no—
40H(5,𝔽2)×C61—42no—
41H(5,𝔽2)×C61C2×C412yes—
42C2×#41×C61—21no—
43C2×#41×C61—41no—
44C22×Q8×C61D411yescharacteristic
45C22×D4×C61—22no—
46C22×D4×C61—41no—
47C22×D4×C61C2311yescharacteristic
48#137×C61—24no—
49#137×C61—42no—
50C2×Q16×C61—41no—
51C2×SD16×C61—41no—
52C2×M16×C61—21no—
53#124×C61—22no—
54#124×C61C2×C412yes—
55#124×C61D412yes—
56C2×C22⋊C4×C61—22no—
57C2×C42×C61D411yescharacteristic
58#104×C61—24no—
59#100×C61—21no—
60#100×C61C2×C412yes—
61C24×C61—22no—
62C24×C61C2×D412yes—
63#41×C61—22no—
64#41×C61—22no—
65#41×C61—42no—
66#41×C61—41no—
67#41×C61—44no—
68#41×C61—42no—
69#41×C61—41no—
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C2×Q8×C61C22⋊C412yescomplemented
75C2×D4×C61—21no—
76C2×D4×C61—24no—
77C2×D4×C61—24no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C2×D4×C61—42no—
81C2×D4×C61—42no—
82C2×D4×C61—42no—
83C2×D4×C61—42no—
84C2×D4×C61—44no—
85C2×D4×C61—41no—
86C2×D4×C61—42no—
87C2×D4×C61C22×C411yescharacteristic
88C2×D4×C61C2×D411yescharacteristic
89C2×D4×C61#4112yes—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—82no—
100Q16×C61—44no—
101SD16×C61—44no—
102M16×C61—22no—
103M16×C61—41no—
104C2×C8×C61—41no—
105C22⋊C4×C61—24no—
106C22⋊C4×C61—24no—
107C42×C61—21no—
108C42×C61C22⋊C412yescomplemented
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111C22⋊Dic61—42no—
112C22⋊Dic61—82no—
113C22⋊Dic61—84no—
114C4⋊Dic61—81no—
115C4⋊Dic61—81no—
116C4⋊Dic61—81no—
117C4×Dic61—81no—
118C23×C61—22no—
119C23×C61—22no—
120C23×C61—24no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—41no—
125C23×C61—82no—
126C23×C61—82no—
127C23×C61C4×D412yes—
128C23×C61#12011yescharacteristic
129C23×C61#12112yes—
130Q8×C61—22no—
131Q8×C61—41no—
132Q8×C61—42no—
133Q8×C61—81no—
134D4×C61—22no—
135D4×C61—41no—
136D4×C61—41no—
137D4×C61—42no—
138D4×C61—41no—
139D4×C61—44no—
140D4×C61—44no—
141D4×C61—42no—
142D4×C61—42no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—24no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—44no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—82no—
166C2×Dic61—82no—
167C2×Dic61—82no—
168C2×Dic61—82no—
169C8×C61—42no—
170#13050—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—82no—
181C22×C61—84no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—162no—
185C22×C61#38611yescharacteristic
186C22×C61B(4,𝔽2)11yescharacteristic
187C22×C61B(4,𝔽2)11yescharacteristic
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—42no—
191C4×C61—42no—
192C4×C61—42no—
193Dic61—82no—
194Dic61—82no—
195Dic61—162no—
196C2×#455D6111yescharacteristic
197#2379—611no—
198C2×#354—611no—
199#1790—612no—
200#1390—612no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—42no—
204C2×C61—42no—
205C2×C61—82no—
206C2×C61#188911yescharacteristic
207C2×C61#219911yescharacteristic
208C2×C61#220111yescharacteristic
209C2×H(5,𝔽2)C2×D6111yescharacteristic
210C2×#137—21no—
211C2×#124C2×D6111yescharacteristic
212#455—22no—
213#455Dic6114yescomplemented
214C2×#104—21no—
215C2×#100C2×D6111yescharacteristic
216C2×#99—612no—
217C2×#99—1222no—
218#391—1221no—
219#388—1222no—
220#386—612no—
221#386—1222no—
222#385—1222no—
223#382—1221no—
224#354—1222no—
225C61#1305011yescharacteristic, complemented, Sylow
226H(5,𝔽2)—22no—
227H(5,𝔽2)—21no—
228H(5,𝔽2)—42no—
229H(5,𝔽2)C2×Dic6112yes—
230C2×#41—21no—
231C2×#41—41no—
232C22×Q8C22⋊D6111yescharacteristic
233C22×D4—22no—
234C22×D4—41no—
235C22×D4C22×D6111yescharacteristic
236C23×C4—612no—
237#137—24no—
238#137—42no—
239C2×Q16—41no—
240C2×SD16—41no—
241C2×M16—21no—
242#124—22no—
243#124C2×Dic6112yes—
244#124C22⋊D6112yes—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—612no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—2442no—
253C2×C42—1221no—
254C2×C42C22⋊D6111yescharacteristic
255#104—24no—
256#100—21no—
257#100C2×Dic6112yes—
258#99—1222no—
259#99—1222no—
260#99—2444no—
261#95—1222no—
262#95—1222no—
263#95—1222no—
264#95—2442no—
265C24—22no—
266C24D4×D6112yes—
267#41—22no—
268#41—22no—
269#41—42no—
270#41—41no—
271#41—44no—
272#41—42no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8C22⋊Dic6112yescomplemented
279C2×D4—21no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4#1175822412yes—
292C2×D4C22×Dic6111yescharacteristic
293C2×D4C2×C22⋊D6111yescharacteristic
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—1222no—
299C22×C4—1222no—
300C22×C4—4882no—
301C22×C4—2442no—
302C22×C4—2442no—
303C22×C4C2×C22⋊D6111yescharacteristic
304Q16—44no—
305SD16—44no—
306M16—22no—
307M16—41no—
308C2×C8—41no—
309C4⋊C4—4881no—
310C4⋊C4—4881no—
311C4⋊C4—4881no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—4884no—
315C22⋊C4—4882no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C42—21no—
320C42—4881no—
321C42C22⋊Dic6112yescomplemented
322C23—22no—
323C23—22no—
324C23—24no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—82no—
330C23—82no—
331C23D4×Dic6112yes—
332C23#1451974511yescharacteristic
333C23#1451974912yes—
334Q8—22no—
335Q8—42no—
336Q8—41no—
337Q8—81no—
338D4—22no—
339D4—42no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—44no—
344D4—44no—
345D4—42no—
346D4—42no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—4882no—
361C2×C4—4882no—
362C2×C4—4882no—
363C2×C4—4882no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2444no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4C2×C22⋊Dic6111yescharacteristic
372C2×C4#1451975811yescharacteristic
373C8—42no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22—162no—
388C22#1683964211yescharacteristic
389C22#1683994811yescharacteristic
390C22#1683994811yescharacteristic
391C4—21no—
392C4—21no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—9762no—
397C4—4882no—
398C4—4882no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—82no—
404C2#2144622811yescharacteristic
405C2#2144781911yescharacteristic
406C2#2144783111yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×#455C2
socleC22×C61B(4,𝔽2)
prime radicalC2×H(5,𝔽2)×C61C22
cosocleC22×D4×C61C23
hypercenterC2×#455D61
hypocenterC61#13050
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#455

Complements

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

NH ≅ G/Ntype
1. #21448993C2semidirect
2. C2×#16839557C2semidirect
4. #21443938C2semidirect
19. C61×#455C4semidirect
74. C2×Q8×C61C22⋊C4semidirect
108. C42×C61C22⋊C4semidirect
213. #455Dic61semidirect
225. C61#13050semidirect
278. C2×Q8C22⋊Dic61semidirect
321. C42C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489932
C2×#168395572
#214457672
#214439382
C2×C61×#4552
#1305061