Finite Group Databasebeta

← #29779549

Subgroups — #29779549

Order 15616.

1078 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 191 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#21454129C211yesmaximal, characteristic, complemented
2C2×#16839664C211yesmaximal, characteristic
3#21447234C212yesmaximal, complemented
4#21446966C212yesmaximal, complemented
5#21446516C214yesmaximal, complemented
6#21445906C212yesmaximal, complemented
7#21445751C212yesmaximal
8C61×#1749C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C2×#14519769C2211yescharacteristic, complemented
11C22×C4⋊Dic61C2211yescharacteristic
12C22×#11758198—22no—
13C2×#14519699C2214yes—
14C2×#14519699C2214yes—
15#16839664—22no—
16#16839664C2212yes—
17#16839664C2212yes—
18#16839660C2212yescomplemented
19#16839646—22no—
20#16839635—22no—
21#16839628C2211yescharacteristic, complemented
22#16839622C2212yes—
23#16839525C2214yescomplemented
24#16839525C2214yescomplemented
25C22×C42×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2212yes—
27C2×#97×C61C2212yescomplemented
28C61×#336C2212yescomplemented
29C24×D61—21no—
30C22×D244—21no—
31C22×D244—24no—
32C22×D244—24no—
33C22×D244—42no—
34C22×D244C2312yes—
35C23×Dic61—21no—
36#14519769—42no—
37C2×C4⋊Dic61—24no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2312yes—
40C2×C4⋊Dic61C2312yes—
41C2×C4⋊Dic61C2312yes—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—22no—
45C2×#11758198—24no—
46C2×#11758198—24no—
47C2×#11758198—48no—
48#14519699—28no—
49#14519699—44no—
50#14519684—22no—
51#14519684—22no—
52#14519684—22no—
53C23×C4×C61D412yes—
54C23×C4×C61C2311yescharacteristic
55C22×C8×C61C2314yes—
56C2×C42×C61—42no—
57C2×C42×C61D412yes—
58C2×C42×C61C2311yescharacteristic
59C2×C42×C61C2311yescharacteristic
60C22⋊C8×C61—24no—
61#97×C61—42no—
62C24×C61C2×D411yescharacteristic
63C23×D61—24no—
64C23×D61—22no—
65C23×D61—42no—
66C23×D61—44no—
67C23×D61—42no—
68C23×D61—44no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—24no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C2×D412yes—
75C22×C4×C61C2×D412yes—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61#4114yes—
78C22×C4×C61C2411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61—22no—
81C22×Dic61—22no—
82C22×Dic61—42no—
83C22×Dic61—44no—
84C2×D244—24no—
85C2×D244—24no—
86C2×D244—24no—
87C2×D244—22no—
88C2×D244—416no—
89C2×D244—42no—
90C2×D244—42no—
91C2×D244#4114yes—
92C2×C8×C61—24no—
93C2×C8×C61—24no—
94C2×C8×C61—24no—
95C42×C61—22no—
96C42×C61—22no—
97C42×C61—42no—
98C22⋊Dic61—44no—
99#11758198—48no—
100#11758198—48no—
101C4⋊Dic61—24no—
102C4⋊Dic61—24no—
103C4⋊Dic61—24no—
104C4⋊Dic61—42no—
105#11758196—44no—
106#11758196—44no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—22no—
110C23×C61#12412yes—
111C23×C61AGL(1,ℤ/8)14yes—
112C23×C61C22×D411yescharacteristic
113C22×D61—24no—
114C22×D61—48no—
115C22×D61—416no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—44no—
119C22×D61—44no—
120C22×D61—84no—
121C22×D61—88no—
122C22×D61—84no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—28no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—48no—
133C2×C4×C61#12412yes—
134C2×C4×C61C2×#4112yes—
135C2×C4×C61C2×#4112yes—
136C2×C4×C61H(5,𝔽2)12yes—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—44no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—84no—
146D244—28no—
147D244—48no—
148D244—44no—
149C8×C61—44no—
150#29110—611nomaximal, Sylow
151C22×C61—22no—
152C22×C61—24no—
153C22×C61—22no—
154C22×C61—24no—
155C22×C61#49014yes—
156C22×C61#52511yescharacteristic
157C22×C61C2×#12411yescharacteristic
158C22×C61#53411yescharacteristic
159C22×C61C2×AGL(1,ℤ/8)12yes—
160C22×C61#57512yes—
161C2×D61—44no—
162C2×D61—48no—
163C2×D61—816no—
164C2×D61—82no—
165C2×D61—82no—
166C2×D61—84no—
167C4×C61—22no—
168C4×C61—42no—
169C4×C61—42no—
170C4×C61#55114yes—
171Dic61—82no—
172Dic61—82no—
173Dic61—82no—
174#2851—611no—
175C2×#390—611no—
176#2073—612no—
177#2025—612no—
178#1936—614no—
179#1823—612no—
180#1786—612no—
181#1749D6111yescharacteristic, complemented
182C2×C61—24no—
183C2×C61#256111yescharacteristic
184C2×C61#311412yes—
185C2×C61C2×#49011yescharacteristic
186C2×C61#314011yescharacteristic
187C2×C61#319412yes—
188D61—84no—
189C23×D4—611no—
190C2×#125—611no—
191C22×C4⋊C4—611no—
192C22×C22⋊C4—1222no—
193C22×C42C2×D6111yescharacteristic
194C2×#102—614no—
195C2×#102—614no—
196C2×C22⋊C8C2×D6112yes—
197C2×#97C2×D6112yescomplemented
198#390—612no—
199#390—612no—
200#390—1222no—
201#389—612no—
202#387—1222no—
203#385—1222no—
204#382—611no—
205#380—612no—
206#340—614no—
207#340—614no—
208#336C2×D6112yescomplemented
209C61#2911011yescharacteristic, complemented, Sylow
210C25—1221no—
211C22×D4—612no—
212C22×D4—1224no—
213C22×D4—1221no—
214C22×D4—1224no—
215C22×D4—2442no—
216C23×C4—1221no—
217C23×C4D24412yes—
218C23×C4C22×D6111yescharacteristic
219C22×C8C22×D6114yes—
220#125—2442no—
221C2×C4⋊C4—612no—
222C2×C4⋊C4—612no—
223C2×C4⋊C4—612no—
224C2×C4⋊C4—1224no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1222no—
228C2×C22⋊C4—1222no—
229C2×C22⋊C4—1224no—
230C2×C22⋊C4—1224no—
231C2×C22⋊C4—2448no—
232C2×C42—42no—
233C2×C42D24412yes—
234C2×C42C22×D6111yescharacteristic
235C2×C42C22×D6111yescharacteristic
236#102—1228no—
237#102—2444no—
238C22⋊C8—24no—
239#97—42no—
240#95—1222no—
241#95—1222no—
242#95—1222no—
243C24—1224no—
244C24—1222no—
245C24—2442no—
246C24—2444no—
247C24—2442no—
248C24—2444no—
249C24C2×D24411yescharacteristic
250C2×D4—614no—
251C2×D4—1224no—
252C2×D4—1222no—
253C2×D4—1224no—
254C2×D4—1224no—
255C2×D4—24416no—
256C2×D4—2442no—
257C2×D4—2442no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—1222no—
266C22×C4—2442no—
267C22×C4—2444no—
268C22×C4C2×D24412yes—
269C22×C4C2×D24412yes—
270C22×C4C2×D24411yescharacteristic
271C22×C4#1175822214yes—
272C22×C4C23×D6111yescharacteristic
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C4⋊C4—1224no—
277C4⋊C4—1224no—
278C4⋊C4—1224no—
279C4⋊C4—2444no—
280C4⋊C4—2444no—
281C4⋊C4—2442no—
282C22⋊C4—2448no—
283C22⋊C4—2448no—
284C22⋊C4—2444no—
285C42—22no—
286C42—22no—
287C42—42no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—1224no—
292C23—4884no—
293C23—4888no—
294C23—4884no—
295C23—2448no—
296C23—24416no—
297C23—2442no—
298C23—2444no—
299C23—2444no—
300C23—2442no—
301C23#1451976512yes—
302C23#1451980514yes—
303C23C22×D24411yescharacteristic
304D4—1228no—
305D4—2448no—
306D4—2444no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—48no—
317C2×C4—4884no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4—2444no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4#1451976512yes—
327C2×C4#41×D6112yes—
328C2×C4C2×#1175822212yes—
329C2×C4#1451982212yes—
330C8—44no—
331C22—22no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—48816no—
336C22—4882no—
337C22—4882no—
338C22—4884no—
339C22—2448no—
340C22—2444no—
341C22#1684012714yes—
342C22#1684027411yescharacteristic
343C22C2×#1451976511yescharacteristic
344C22#1684032411yescharacteristic
345C22C2×#1451980512yes—
346C22#1684056212yes—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—4882no—
351C4—4882no—
352C4—4882no—
353C4#1684044114yes—
354C2—24no—
355C2—4884no—
356C2#2145097211yescharacteristic
357C2#2145595912yes—
358C2C2×#1684012711yescharacteristic
359C2#2145620611yescharacteristic
360C2#2145663512yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1749D61
hypocenterC61#29110
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. #21454129C2semidirect
3. #21447234C2semidirect
4. #21446966C2semidirect
5. #21446516C2semidirect
6. #21445906C2semidirect
8. C61×#1749C2semidirect
10. C2×#14519769C22semidirect
18. #16839660C22semidirect
21. #16839628C22semidirect
23. #16839525C22semidirect
24. #16839525C22semidirect
27. C2×#97×C61C22semidirect
28. C61×#336C22semidirect
181. #1749D61semidirect
197. C2×#97C2×D61semidirect
208. #336C2×D61semidirect
209. C61#29110semidirect

Maximal subgroups

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

subgroupindex
#214541292
C2×#168396642
#214472342
#214469662
#214465162
#214459062
#214457512
C61×#17492
#2911061