Finite Group Databasebeta

← #29598299

Subgroups — #29598299

Order 15616.

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

normal subgroups only

This group has 840 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#21449175C211yesmaximal, characteristic
2C2×#16839548C214yesmaximal, complemented
3C4×#14519694C211yesmaximal, characteristic, complemented
4C61×#1799C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic, complemented
6C2×#14519751—21no—
7C2×#14519745C2212yes—
8C2×#14519694C2212yes—
9C4×#11758194—21no—
10#16839679—21no—
11#16839671—21no—
12#16839664—21no—
13#16839622C411yescharacteristic, complemented
14#16839617C412yescomplemented
15#16839548—28no—
16#16839531—21no—
17C2×C4×D4×C61C2211yescharacteristic
18C2×#99×C61C2212yes—
19C61×#380—21no—
20C61×#342—21no—
21C24×D61C2×C412yes—
22C22×D244—21no—
23C22×D244—21no—
24C22×D244—44no—
25C22×D244—41no—
26C22×D244—41no—
27C22×C22⋊D61—22no—
28C2×#11758194—22no—
29#14519751—44no—
30#14519745—24no—
31#14519745—24no—
32C2×C4⋊Dic61—21no—
33C2×C22⋊Dic61C2×C412yes—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37#14519694—22no—
38#14519694D414yescomplemented
39#14519688—41no—
40C4×#92418—41no—
41#14519684—21no—
42C22×D4×C61C2311yescharacteristic
43C23×C4×C61C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C4×D4×C61—41no—
46C4×D4×C61—41no—
47C2×C4⋊C4×C61—41no—
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C42×C61D411yescharacteristic
53#99×C61—24no—
54#95×C61—41no—
55C24×C61C22×C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C23×D61—22no—
58C23×D61—24no—
59C23×D61—44no—
60C23×D61—42no—
61C23×D61—42no—
62C23×D61—82no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61C2×D412yes—
66C2×D4×C61#4112yes—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—44no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—82no—
83C22×Dic61—22no—
84C2×D244—24no—
85C2×D244—42no—
86C2×D244—42no—
87C2×D244—44no—
88C2×D244—48no—
89C2×D244—44no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—44no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—82no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C42×C61—21no—
100C42×C61—21no—
101C22⋊Dic61—24no—
102#11758198—44no—
103#11758198—44no—
104#11758198—82no—
105C4⋊Dic61—42no—
106#11758194—24no—
107#11758194—42no—
108C2×#92418—42no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61C2×C22⋊C411yescharacteristic
118C23×C61C4×D412yes—
119C23×C61#12411yescharacteristic
120C23×C61#12711yescharacteristic
121C22×D61—24no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—44no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—44no—
130C22×D61—88no—
131C22×D61—88no—
132C22×D61—84no—
133C22×D61—84no—
134D4×C61—42no—
135D4×C61—42no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—81no—
149C2×C4×C61—82no—
150C2×C4×C61—82no—
151C2×C4×C61—82no—
152C2×C4×C61#9911yescharacteristic
153C2×C4×C61#9911yescharacteristic
154C2×C4×C61#12112yes—
155C22⋊D61—84no—
156C22⋊D61—84no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160D244—44no—
161D244—48no—
162D244—44no—
163#92418—44no—
164#13218—611nomaximal, Sylow
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—22no—
169C22×C61—21no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—42no—
173C22×C61—42no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—81no—
179C22×C61#39011yescharacteristic
180C22×C61C2×#9911yescharacteristic
181C22×C61#41011yescharacteristic
182C2×D61—44no—
183C2×D61—44no—
184C2×D61—44no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—88no—
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—88no—
191C4×C61—21no—
192C4×C61—22no—
193C4×C61—41no—
194C4×C61—41no—
195C4×C61—41no—
196C4×C61—162no—
197C4×C61AΣL(1,𝔽16)12yescomplemented
198Dic61—82no—
199#2408—611no—
200C2×AΣL(1,𝔽16)—614no—
201#1799D6111yescharacteristic, complemented
202C4×#100—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—41no—
206C2×C61—42no—
207C2×C61—41no—
208C2×C61#196411yescharacteristic
209C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
210C2×C61#212211yescharacteristic
211D61—84no—
212C23×D4—611no—
213C2×#121—1221no—
214C2×#120—612no—
215C2×C4×D4C2×D6111yescharacteristic
216C2×#100—612no—
217C2×#99C2×D6112yes—
218C4×M16—1221no—
219#394—1221no—
220#392—1221no—
221#390—1221no—
222#380—21no—
223#380—611no—
224#379—612no—
225AΣL(1,𝔽16)—1228no—
226#343—1221no—
227#342—21no—
228C61#1321811yescharacteristic, complemented, Sylow
229C25—612no—
230C22×D4—1222no—
231C22×D4—1221no—
232C22×D4—1221no—
233C22×D4—2441no—
234C22×D4—2441no—
235C22×D4—2444no—
236C22×D4C22×D6111yescharacteristic
237C23×C4C4×D6111yescharacteristic
238C23×C4C22⋊D6111yescharacteristic
239C2×M16—1222no—
240#121—2444no—
241#120—1224no—
242#120—1224no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4D24411yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—612no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C22⋊C4C4×D6111yescharacteristic
254C2×C22⋊C4C22⋊D6111yescharacteristic
255C2×C42D24411yescharacteristic
256#100—614no—
257#100—1222no—
258#99—24no—
259#97—2441no—
260C4×C8—2441no—
261#95—41no—
262#95—1221no—
263C24—1222no—
264C24—1224no—
265C24—4882no—
266C24—2442no—
267C24—2442no—
268C24—2444no—
269C24C2×C4×D6111yescharacteristic
270C24C2×C22⋊D6111yescharacteristic
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—1224no—
274C2×D4—4882no—
275C2×D4—2442no—
276C2×D4—2442no—
277C2×D4—2444no—
278C2×D4—2448no—
279C2×D4—2444no—
280C2×D4—2444no—
281C2×D4—2444no—
282C2×D4—2442no—
283C2×D4—2442no—
284C2×D4—2441no—
285C2×D4—2441no—
286C2×D4D4×D6112yes—
287C2×D4#1175822612yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—1222no—
296C22×C4#1175819811yescharacteristic
297C22×C4#1175819811yescharacteristic
298C22×C4#1175819811yescharacteristic
299C22×C4#1175819811yescharacteristic
300C22×C4C2×D24411yescharacteristic
301M16—1224no—
302M16—2442no—
303C2×C8—2442no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C4⋊C4—2442no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—1224no—
312C22⋊C4—4882no—
313C22⋊C4—2444no—
314C22⋊C4—2444no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—1224no—
326C23—4884no—
327C23—4884no—
328C23—4888no—
329C23—4888no—
330C23—2442no—
331C23—2442no—
332C23—2442no—
333C23—2444no—
334C23—2444no—
335C23—2442no—
336C23—2442no—
337C23—2442no—
338C23C2×#1175819811yescharacteristic
339C23#1451973912yes—
340C23#1451976611yescharacteristic
341C23#1451977311yescharacteristic
342D4—42no—
343D4—42no—
344D4—4884no—
345D4—4884no—
346D4—2444no—
347D4—2448no—
348D4—2444no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4#1451969111yescharacteristic
369C2×C4#1451969111yescharacteristic
370C2×C4#1451975212yes—
371C8—2444no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—4882no—
387C22—4882no—
388C22—4888no—
389C22—4888no—
390C22—4882no—
391C22—4882no—
392C22—2444no—
393C22—2444no—
394C22—2444no—
395C22#1683966311yescharacteristic
396C22C2×#1451969111yescharacteristic
397C22#1683973511yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—162no—
404C4—4882no—
405C4#1683954812yescomplemented
406C2—21no—
407C2—21no—
408C2—41no—
409C2—42no—
410C2—41no—
411C2—4884no—
412C2#2144665411yescharacteristic
413C2C2×#1683954811yescharacteristic
414C2#2144746811yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radical#21449175C2
cosocleC22×D4×C61C23
hypercenter#1799D61
hypocenterC61#13218
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. C2×#16839548C2semidirect
3. C4×#14519694C2semidirect
4. C61×#1799C2semidirect
5. C23×D244C4semidirect
13. #16839622C4semidirect
14. #16839617C4semidirect
38. #14519694D4semidirect
197. C4×C61AΣL(1,𝔽16)semidirect
201. #1799D61semidirect
228. C61#13218semidirect
405. C4#16839548semidirect

Maximal subgroups

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

subgroupindex
#214491752
C2×#168395482
C4×#145196942
C61×#17992
#1321861