Finite Group Databasebeta

← #29590187

Subgroups — #29590187

Order 15616.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449055C211yesmaximal, characteristic
2#21444133C211yesmaximal, characteristic, complemented
3#21443818C212yesmaximal, complemented
4#21443815C212yesmaximal
5C61×#1740C211yesmaximal, characteristic, complemented
6C2×C4×D244C411yescharacteristic, complemented
7C2×C4×C22⋊D61—21no—
8#16839815—21no—
9#16839692C411yescharacteristic, complemented
10#16839687C2211yescharacteristic
11#16839684—21no—
12#16839679C411yescharacteristic, complemented
13#16839678—21no—
14#16839671C2211yescharacteristic
15C4×C4⋊Dic61C411yescharacteristic, complemented
16C4×#11758198—21no—
17#16839488—21no—
18#16839486C2212yes—
19C2×#117×C61—21no—
20C2×#117×C61C2211yescharacteristic
21C61×#342—21no—
22C61×#342C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C4×D61—21no—
26C4×D244—41no—
27C4×D244—41no—
28C4×C22⋊D61—42no—
29C4×C22⋊D61—42no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×C4×Dic61—21no—
39C4⋊#92418—41no—
40C22⋊#92418—22no—
41C4×#92418—41no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C23×C4×C61D411yescharacteristic
47#117×C61—24no—
48#117×C61—41no—
49#117×C61—41no—
50#117×C61—42no—
51C2×C4⋊C4×C61—41no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2311yescharacteristic
57C2×C42×C61—41no—
58C2×C42×C61C2×C411yescharacteristic
59C2×C42×C61D411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C23×D61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61D812yes—
71C22×C4×C61SD1612yes—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—42no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—41no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C2×D244—41no—
82C2×D244—42no—
83C2×C4×D61—41no—
84C2×C4×D61—41no—
85C2×C4×D61—42no—
86C2×C4×D61—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—44no—
90C22⋊C4×C61—24no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C42×C61—21no—
95C42×C61—21no—
96C42×C61—21no—
97C42×C61—21no—
98C42×C61—44no—
99C22⋊Dic61—41no—
100C22⋊Dic61—41no—
101#11758198—42no—
102#11758198—42no—
103#11758198—42no—
104#11758198—42no—
105C4⋊Dic61—42no—
106C4⋊Dic61—41no—
107#11758196—42no—
108C4×Dic61—42no—
109C2×#92418—42no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61C2×C22⋊C411yescharacteristic
115C23×C61C2×D811yescharacteristic
116C23×C61C2×SD1611yescharacteristic
117C22×D61—22no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—44no—
139C2×C4×C61—82no—
140C2×C4×C61—82no—
141C2×C4×C61#9911yescharacteristic
142C2×C4×C61#9911yescharacteristic
143C2×C4×C61#10212yes—
144C2×C4×C61#10212yes—
145C2×C4×C61#10412yes—
146C22⋊D61—44no—
147C22⋊D61—44no—
148C2×Dic61—22no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157D244—42no—
158D244—42no—
159C4×D61—82no—
160C4×D61—82no—
161#92418—82no—
162#11691—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—81no—
172C22×C61C2×#9911yescharacteristic
173C22×C61C2×#10211yescharacteristic
174C22×C61C2×#10411yescharacteristic
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61—81no—
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—22no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—41no—
187C4×C61—84no—
188C4×C61#32712yes—
189C4×C61#32712yes—
190Dic61—42no—
191Dic61—81no—
192Dic61—81no—
193#2392—611no—
194#1740D6111yescharacteristic, complemented
195#1458—611no—
196#1344—612no—
197#1343—612no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—41no—
201C2×C61—41no—
202C2×C61C2×#32711yescharacteristic
203C2×C61#212011yescharacteristic
204C2×C61#212611yescharacteristic
205D61—82no—
206C2×C4×D4—611no—
207C2×C4×D4—1221no—
208C2×#117—21no—
209C2×#117C2×D6111yescharacteristic
210#431—1221no—
211#398—611no—
212#397—611no—
213#396—1221no—
214#394—611no—
215#394—1221no—
216#392—611no—
217C4×C4⋊C4—611no—
218C4×C22⋊C4—1221no—
219#342—21no—
220#342C2×D6112yes—
221#324—1221no—
222#323—612no—
223C61#1169111yescharacteristic, complemented, Sylow
224C22×D4—611no—
225C22×D4—1221no—
226C23×C4—1221no—
227C23×C4C22⋊D6111yescharacteristic
228C4×D4—2441no—
229C4×D4—2441no—
230C4×D4—2442no—
231C4×D4—2442no—
232#117—24no—
233#117—41no—
234#117—41no—
235#117—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4C4×D6111yescharacteristic
241C2×C4⋊C4D24411yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—1221no—
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4C22⋊D6111yescharacteristic
249C2×C22⋊C4C22×D6111yescharacteristic
250C2×C42—41no—
251C2×C42—1221no—
252C2×C42C4×D6111yescharacteristic
253C2×C42D24411yescharacteristic
254C4⋊C8—2441no—
255C22⋊C8—1222no—
256C4×C8—2441no—
257#95—611no—
258#95—611no—
259#95—1221no—
260#95—1221no—
261C24—1221no—
262C24C2×C22⋊D6111yescharacteristic
263C2×D4—2441no—
264C2×D4—2442no—
265C2×D4—2442no—
266C2×D4—2442no—
267C2×D4—2441no—
268C2×D4—2441no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—2441no—
277C22×C4—2441no—
278C22×C4—2442no—
279C22×C4—2442no—
280C22×C4#1175819811yescharacteristic
281C22×C4#1175819811yescharacteristic
282C22×C4#1175819811yescharacteristic
283C22×C4#1175819811yescharacteristic
284C22×C4#1175819912yes—
285C22×C4#1175820112yes—
286C22×C4C2×C4×D6111yescharacteristic
287C22×C4C2×D24411yescharacteristic
288C2×C8—2442no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—44no—
292C4⋊C4—2441no—
293C4⋊C4—2442no—
294C4⋊C4—2442no—
295C22⋊C4—24no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C22⋊C4—2441no—
303C22⋊C4—2441no—
304C22⋊C4—2442no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—44no—
310C42—2442no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—1222no—
316C23—2441no—
317C23—2441no—
318C23—2441no—
319C23—2441no—
320C23—2441no—
321C23—2441no—
322C23C2×#1175819811yescharacteristic
323C23C2×#1175819911yescharacteristic
324C23C2×#1175820111yescharacteristic
325D4—2442no—
326D4—2442no—
327D4—2444no—
328D4—2444no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—1222no—
347C2×C4—4882no—
348C2×C4—4882no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4#1451969111yescharacteristic
358C2×C4#1451969111yescharacteristic
359C2×C4#1451969712yes—
360C2×C4#1451969712yes—
361C2×C4#1451970512yes—
362C8—4882no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—4881no—
373C22—4881no—
374C22—4881no—
375C22—4881no—
376C22—2442no—
377C22—2442no—
378C22—2442no—
379C22C2×#1451969111yescharacteristic
380C22C2×#1451969711yescharacteristic
381C22C2×#1451970511yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—84no—
388C4—4881no—
389C4—4881no—
390C4—2442no—
391C4#1683949512yes—
392C4#1683949512yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—4882no—
398C2C2×#1683949511yescharacteristic
399C2#2144745811yescharacteristic
400C2#2144748211yescharacteristic
subgroupG/H
centerC2×C4#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1740C2
socleC22×C61C2×#99
prime radical#16839671C22
cosocleC2×C22⋊C4×C61C23
hypercenter#1740D61
hypocenterC61#11691
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. #21444133C2semidirect
3. #21443818C2semidirect
5. C61×#1740C2semidirect
6. C2×C4×D244C4semidirect
9. #16839692C4semidirect
12. #16839679C4semidirect
15. C4×C4⋊Dic61C4semidirect
194. #1740D61semidirect
223. C61#11691semidirect

Maximal subgroups

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

subgroupindex
#214490552
#214441332
#214438182
#214438152
C61×#17402
#1169161