Finite Group Databasebeta

← #29689176

Subgroups — #29689176

Order 15616.

1440 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 231 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#21459455C211yesmaximal, characteristic
2#21454295C212yesmaximal, complemented
3#21446365C214yesmaximal, complemented
4#21446236C214yesmaximal, complemented
5#21445930C211yesmaximal, characteristic, complemented
6#21445738C212yesmaximal, complemented
7C61×#1829C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9C22×D4×D61C2211yescharacteristic
10#16840528C2214yescomplemented
11#16840407—24no—
12D4×D244—24no—
13#16840331C2212yes—
14#16840324C414yes—
15#16840274C2212yes—
16C2×#14519747C414yes—
17#16839744C2214yescomplemented
18#16839743—24no—
19C2×#14519694C2212yes—
20C2×#14519693C2214yescomplemented
21#100×D61—24no—
22#16839725C2214yescomplemented
23#16839532C2214yescomplemented
24C61×#518C2211yescharacteristic
25C2×#100×C61C2212yes—
26C61×#344C2214yescomplemented
27C24×D61C2×C412yes—
28#14519822—44no—
29C2×#11758222C2312yes—
30C2×#11758224D414yes—
31C22×D244—24no—
32C22×D244C2311yescharacteristic
33C22×C22⋊D61C2×C414yes—
34C22×C22⋊D61C2×C412yes—
35C2×D4×D61—24no—
36C2×D4×D61—44no—
37C2×D4×D61D414yes—
38C22×C4×D61C2311yescharacteristic
39#14519785—44no—
40#14519783—44no—
41C2×#11758194C2314yes—
42M16×D61—28no—
43#14519774—22no—
44#14519765—22no—
45#14519761—24no—
46#14519761C2×C414yes—
47#14519757—22no—
48#14519752—22no—
49#14519752—24no—
50#14519751—28no—
51#14519747—24no—
52#14519747—28no—
53#14519747C2×C414yes—
54#14519747C2×C418yes—
55C4×D244—24no—
56C2×#11758198—24no—
57#14519695—24no—
58#14519694—24no—
59#14519693—24no—
60#14519693—44no—
61#14519689—44no—
62C22×D4×C61C2311yescharacteristic
63C2×M16×C61C2314yes—
64C4×D4×C61—24no—
65#117×C61C2312yes—
66C2×C22⋊C4×C61C2×C414yes—
67#100×C61—24no—
68C24×C61C22×C412yes—
69C23×D61—24no—
70C23×D61—44no—
71C23×D61—84no—
72C23×D61C22⋊C414yes—
73C23×D61C22×C411yescharacteristic
74C23×D61C22×C412yes—
75C2×D4×C61—24no—
76C2×D4×C61C2×D414yes—
77C22×C4×C61—24no—
78C22×C4×C61C2411yescharacteristic
79C2×C22⋊D61—24no—
80C2×C22⋊D61—44no—
81C2×C22⋊D61—48no—
82C2×C22⋊D61C22⋊C418yes—
83C2×C22⋊D61C22×C414yes—
84C22×Dic61C22⋊C414yes—
85C22×Dic61C22×C411yescharacteristic
86#11758224—48no—
87D4×D61—48no—
88D4×D61—416no—
89#11758222—24no—
90#11758222—24no—
91C2×D244—24no—
92C2×D244—22no—
93C2×D244—24no—
94C2×D244—28no—
95C2×D244—88no—
96C2×D244C2×D412yes—
97C2×C4×D61—22no—
98C2×C4×D61—24no—
99C2×C4×D61C2×D414yes—
100C2×Dic122C2×D412yes—
101M16×C61—24no—
102M16×C61—24no—
103C2×C8×C61—24no—
104C4⋊C4×C61—22no—
105C22⋊C4×C61—24no—
106C22⋊C4×C61C22×C414yes—
107C42×C61—22no—
108#11758198—24no—
109#11758198—24no—
110#11758198—28no—
111C4⋊Dic61—24no—
112#11758194—24no—
113#11758194—24no—
114C2×#92418—24no—
115#11758189—48no—
116C8×D61—48no—
117C23×C61—24no—
118C23×C61—82no—
119C23×C61C2×C22⋊C414yes—
120C23×C61C23×C411yescharacteristic
121C22×D61—21no—
122C22×D61—22no—
123C22×D61—22no—
124C22×D61—24no—
125C22×D61—42no—
126C22×D61—44no—
127C22×D61—48no—
128C22×D61—48no—
129C22×D61—88no—
130C22×D61—84no—
131C22×D61—82no—
132C22×D61—84no—
133C22×D61—88no—
134C22×D61C2×C22⋊C414yes—
135D4×C61—28no—
136C2×C4×C61—24no—
137C2×C4×C61—84no—
138C2×C4×C61C22×D412yes—
139C2×C4×C61H(5,𝔽2)14yes—
140C22⋊D61—24no—
141C22⋊D61—24no—
142C22⋊D61—48no—
143C22⋊D61—48no—
144C2×Dic61—21no—
145C2×Dic61—22no—
146C2×Dic61—44no—
147C2×Dic61C2×C22⋊C414yes—
148D244—24no—
149D244—48no—
150D244—44no—
151D244—88no—
152C4×D61—28no—
153C4×D61—44no—
154Dic122—24no—
155C8×C61—44no—
156#92418—44no—
157#21778—611nomaximal, Sylow
158C22×C61—24no—
159C22×C61—41no—
160C22×C61—44no—
161C22×C61—84no—
162C22×C61C22×C22⋊C411yescharacteristic
163C22×C61#51812yes—
164C2×D61—22no—
165C2×D61—24no—
166C2×D61—24no—
167C2×D61—41no—
168C2×D61—42no—
169C2×D61—41no—
170C2×D61—44no—
171C2×D61—44no—
172C2×D61—42no—
173C2×D61—816no—
174C2×D61—84no—
175C2×D61—164no—
176C4×C61—44no—
177C4×C61#53414yes—
178Dic61—24no—
179Dic61—41no—
180#3484—611no—
181#2888—612no—
182#1914—614no—
183#1891—614no—
184#1829—611no—
185#1829D6111yescharacteristic, complemented
186#1783—612no—
187C2×C61—21no—
188C2×C61—22no—
189C2×C61—44no—
190C2×C61#230811yescharacteristic
191D61—42no—
192D61—44no—
193D61—82no—
194C2×H(5,𝔽2)—611no—
195C23×D4—611no—
196#568—614no—
197#546—1224no—
198D42—1224no—
199#535—612no—
200#534—614no—
201#525—612no—
202C2×#120—614no—
203#518C2×D6111yescharacteristic
204#413—614no—
205#413—1224no—
206C2×#100—612no—
207C2×#100—614no—
208C2×#100—1224no—
209C2×#100C2×D6112yes—
210#407—614no—
211#344—614no—
212#344C2×D6114yescomplemented
213C61#2177811yescharacteristic, complemented, Sylow
214C25—612no—
215H(5,𝔽2)—2444no—
216C2×#41—614no—
217C2×#41—612no—
218C22×D4—614no—
219C22×D4—614no—
220C22×D4—612no—
221C22×D4—611no—
222C22×D4—1224no—
223C22×D4—1224no—
224C22×D4—2444no—
225C22×D4C22×D6111yescharacteristic
226C23×C4—611no—
227#131—2444no—
228#131—2444no—
229C2×M16—614no—
230C2×M16—1228no—
231C2×M16C22×D6114yes—
232#127—1222no—
233#124—1222no—
234#123—614no—
235#123—1224no—
236C22⋊Q8—1222no—
237#121—1224no—
238#121—1222no—
239#121—1228no—
240#120—614no—
241#120—618no—
242#120—1228no—
243#120—1224no—
244C4×D4—24no—
245C4×D4—1224no—
246#117C22×D6112yes—
247C2×C22⋊C4—1224no—
248C2×C22⋊C4C4×D6114yes—
249#101—1224no—
250#100—24no—
251#100—1224no—
252#100—1224no—
253#100—2444no—
254C22⋊C8—2444no—
255C24—614no—
256C24—612no—
257C24—611no—
258C24—1224no—
259C24—4884no—
260C24—2444no—
261C24C2×C4×D6112yes—
262#41—1224no—
263#41—1224no—
264#41—2448no—
265C2×Q8—612no—
266C2×D4—24no—
267C2×D4—614no—
268C2×D4—618no—
269C2×D4—612no—
270C2×D4—1228no—
271C2×D4—1224no—
272C2×D4—1224no—
273C2×D4—1222no—
274C2×D4—1224no—
275C2×D4—4888no—
276C2×D4—2448no—
277C2×D4—2444no—
278C2×D4—24416no—
279C2×D4—2448no—
280C2×D4D4×D6114yes—
281C22×C4—24no—
282C22×C4—614no—
283C22×C4—614no—
284C22×C4—611no—
285C22×C4—1222no—
286C22×C4—1224no—
287C22×C4C23×D6111yescharacteristic
288M16—24no—
289M16—24no—
290M16—1224no—
291M16—1224no—
292M16—2448no—
293C2×C8—24no—
294C2×C8—1224no—
295C2×C8—2448no—
296C4⋊C4—22no—
297C4⋊C4—1224no—
298C22⋊C4—24no—
299C22⋊C4—1224no—
300C22⋊C4—1228no—
301C22⋊C4—1224no—
302C22⋊C4C2×C4×D6114yes—
303C42—22no—
304C23—24no—
305C23—82no—
306C23—614no—
307C23—1221no—
308C23—1222no—
309C23—1224no—
310C23—1222no—
311C23—4884no—
312C23—4884no—
313C23—4882no—
314C23—4888no—
315C23—4888no—
316C23—2444no—
317C23—2448no—
318C23—2442no—
319C23—2448no—
320C23C22⋊C4×D6114yes—
321C23C22×C4×D6111yescharacteristic
322Q8—1224no—
323D4—28no—
324D4—1224no—
325D4—1224no—
326D4—1224no—
327D4—4888no—
328D4—2448no—
329D4—2448no—
330D4—2448no—
331D4—2444no—
332C2×C4—24no—
333C2×C4—84no—
334C2×C4—614no—
335C2×C4—1228no—
336C2×C4—1221no—
337C2×C4—1222no—
338C2×C4—2444no—
339C2×C4—2444no—
340C2×C4C2×D4×D6112yes—
341C2×C4#1451982314yes—
342C8—44no—
343C8—2444no—
344C22—24no—
345C22—41no—
346C22—44no—
347C22—84no—
348C22—1222no—
349C22—1224no—
350C22—1224no—
351C22—9764no—
352C22—4884no—
353C22—48816no—
354C22—2444no—
355C22—2444no—
356C22—2441no—
357C22—2442no—
358C22—2442no—
359C22—2441no—
360C22C2×C22⋊C4×D6111yescharacteristic
361C22#1684023612yes—
362C4—44no—
363C4—1224no—
364C4—2441no—
365C4#1684032314yes—
366C2—21no—
367C2—22no—
368C2—44no—
369C2—4882no—
370C2—2442no—
371C2—2444no—
372C2#2144839611yescharacteristic
subgroupG/H
centerC2#21448396
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1829C2
socleC2×C61#2308
prime radical#21459455C2
cosocleC22×C4×C61C24
hypercenter#1829D61
hypocenterC61#21778
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1829

Complements

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

NH ≅ G/Ntype
2. #21454295C2semidirect
3. #21446365C2semidirect
4. #21446236C2semidirect
5. #21445930C2semidirect
6. #21445738C2semidirect
7. C61×#1829C2semidirect
10. #16840528C22semidirect
17. #16839744C22semidirect
20. C2×#14519693C22semidirect
22. #16839725C22semidirect
23. #16839532C22semidirect
26. C61×#344C22semidirect
185. #1829D61semidirect
212. #344C2×D61semidirect
213. C61#21778semidirect

Maximal subgroups

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

subgroupindex
#214594552
#214542952
#214463652
#214462362
#214459302
#214457382
C61×#18292
#2177861