Finite Group Databasebeta

← #29571481

Subgroups — #29571481

Order 15616.

826 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#125×D61C211yesmaximal, characteristic
2#21445582C212yesmaximal
3C2×#16839504C211yesmaximal, characteristic, complemented
4C2×#16839503C212yesmaximal, complemented
5C2×C61×#330C211yesmaximal, characteristic, complemented
6C2×#14519769C2211yescharacteristic
7C2×#14519768C412yes—
8C2×#14519767—22no—
9#125×D61—24no—
10#125×D61—22no—
11#125×D61—22no—
12C2×C4⋊C4×D61—22no—
13C2×C4⋊C4×D61C412yes—
14C2×C42×D61C2211yescharacteristic
15C2×#14519689—22no—
16C2×#14519688C2212yes—
17#16839516—22no—
18#16839504—22no—
19#16839503—24no—
20C2×#125×C61C2211yescharacteristic
21C2×#97×C61C2212yes—
22C61×#330—22no—
23C22×C4×D61—22no—
24C22×C4×D61C2×C411yescharacteristic
25C22×C4×D61D412yes—
26C22×#92418—22no—
27#14519769—21no—
28#14519769—21no—
29#14519769D414yes—
30#14519768—24no—
31#14519768—24no—
32#14519767—28no—
33#14519767—28no—
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61C2×C412yes—
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758196—22no—
39C4⋊C4×D61—28no—
40C4⋊C4×D61—28no—
41C4⋊C4×D61—24no—
42C4⋊C4×D61—24no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61D412yes—
45C42×D61—21no—
46C42×D61—22no—
47C42×D61—21no—
48#14519689—28no—
49#14519688—22no—
50#14519688—22no—
51C22×C8×C61—22no—
52#125×C61—22no—
53#125×C61D412yes—
54#125×C61D412yes—
55C2×C4⋊C4×C61—22no—
56C2×C4⋊C4×C61C2×C412yes—
57C2×C42×C61C2311yescharacteristic
58#97×C61—22no—
59#97×C61—22no—
60C23×D61C22⋊C411yescharacteristic
61C22×C4×C61—22no—
62C22×C4×C61C22×C411yescharacteristic
63C22×C4×C61C2×D412yes—
64C22×Dic61—22no—
65C22×Dic61C22⋊C412yes—
66C22×Dic61C22⋊C411yescharacteristic
67C2×C4×D61—22no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—22no—
71C2×C4×D61—28no—
72C2×C4×D61—24no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—42no—
76C2×C4×D61—44no—
77C2×C4×D61—44no—
78C2×C8×C61—24no—
79C2×C8×C61—42no—
80C2×C8×C61—42no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—24no—
84C4⋊C4×C61C22⋊C414yes—
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D412yes—
87C42×C61C2×D411yescharacteristic
88C4⋊Dic61—28no—
89C4⋊Dic61—22no—
90C4⋊Dic61C22⋊C414yes—
91#11758196—28no—
92#11758196—28no—
93#11758196—24no—
94#11758196—24no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—22no—
98C4×Dic61—22no—
99C2×#92418—24no—
100C2×#92418—42no—
101C2×#92418—42no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—22no—
106C22×D61#10112yes—
107C22×D61#10414yes—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—24no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C2×C4×C61#12012yes—
118C2×C4×C61#12012yes—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—22no—
122C2×Dic61—24no—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61#10112yes—
128C4×D61—24no—
129C4×D61—24no—
130C4×D61—44no—
131C4×D61—44no—
132C4×D61—416no—
133C8×C61—44no—
134#92418—44no—
135#8335—611nomaximal, Sylow
136C22×C61#37911yescharacteristic
137C22×C61C2×#10111yescharacteristic
138C22×C61C2×#10111yescharacteristic
139C22×C61C2×#10412yes—
140C22×C61#42112yes—
141C2×D61—24no—
142C2×D61—22no—
143C2×D61—24no—
144C2×D61—22no—
145C2×D61#33014yescomplemented
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—44no—
151Dic61—22no—
152Dic61—42no—
153Dic61—44no—
154C22×#125—611no—
155#1749—612no—
156C2×#330—612no—
157C2×#330—611no—
158C2×#330D6111yescharacteristic, complemented
159C2×C61C2×#33011yescharacteristic
160C2×C61#152812yes—
161C2×C61#153411yescharacteristic
162C2×C61#178411yescharacteristic
163C2×C61#178912yes—
164D61—24no—
165C2×#125—612no—
166C2×#125—611no—
167C2×#125—1222no—
168C2×#125—1224no—
169C2×#125—1222no—
170C2×#125—1222no—
171C2×#125C2×D6111yescharacteristic
172C22×C4⋊C4—612no—
173C22×C4⋊C4—1222no—
174C22×C42—611no—
175C2×C22⋊C8—1222no—
176C2×#97—612no—
177C2×#97C2×D6112yes—
178#336—1222no—
179#330—22no—
180#330—1224no—
181#330—1222no—
182C61#833511yescharacteristic, complemented, Sylow
183C23×C4—611no—
184C23×C4—612no—
185C23×C4—1222no—
186C22×C8—22no—
187C22×C8—1222no—
188#125—22no—
189#125—614no—
190#125—1224no—
191#125—1228no—
192#125—1228no—
193#125—1224no—
194#125—1221no—
195#125—1221no—
196#125D24412yes—
197#125C22⋊D6112yes—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—612no—
200C2×C4⋊C4—1224no—
201C2×C4⋊C4—1228no—
202C2×C4⋊C4—1228no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4—1222no—
207C2×C4⋊C4—1224no—
208C2×C4⋊C4C4×D6112yes—
209C2×C42—611no—
210C2×C42—612no—
211C2×C42—1221no—
212C2×C42—1222no—
213C2×C42—1221no—
214C2×C42C22×D6111yescharacteristic
215C22⋊C8—1228no—
216#97—22no—
217#97—22no—
218#97—1222no—
219#97—1222no—
220C24—611no—
221C22×C4—22no—
222C22×C4—611no—
223C22×C4—612no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—1224no—
227C22×C4—1228no—
228C22×C4—1221no—
229C22×C4—1221no—
230C22×C4—1222no—
231C22×C4—1222no—
232C22×C4—1222no—
233C22×C4—2442no—
234C22×C4—2444no—
235C22×C4—2444no—
236C22×C4C2×C4×D6111yescharacteristic
237C22×C4D4×D6112yes—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—1224no—
242C2×C8—2442no—
243C2×C8—2442no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—614no—
248C4⋊C4—1228no—
249C4⋊C4—1228no—
250C4⋊C4—1228no—
251C4⋊C4—1222no—
252C4⋊C4—1224no—
253C4⋊C4—1224no—
254C4⋊C4#1175819814yes—
255C42—1221no—
256C42—1221no—
257C42—1222no—
258C42—1222no—
259C42C2×D24411yescharacteristic
260C42D4×D6112yes—
261C42C2×C22⋊D6111yescharacteristic
262C23—612no—
263C23—614no—
264C23—1221no—
265C23—1222no—
266C23—1221no—
267C23C22⋊C4×D6111yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—612no—
276C2×C4—1224no—
277C2×C4—1224no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—1222no—
283C2×C4—1224no—
284C2×C4—2444no—
285C2×C4—2444no—
286C2×C4—24416no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4C22⋊C4×D6111yescharacteristic
290C2×C4C2×#1175819811yescharacteristic
291C2×C4#1451974512yes—
292C2×C4#1451974712yes—
293C8—44no—
294C8—2444no—
295C22—614no—
296C22—1224no—
297C22—1222no—
298C22—1222no—
299C22—1224no—
300C22#1683961711yescharacteristic
301C22#101×D6111yescharacteristic
302C22#101×D6111yescharacteristic
303C22#104×D6112yes—
304C22#1683977712yes—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—1222no—
311C4—2442no—
312C4—2444no—
313C2—1224no—
314C2#330×D6111yescharacteristic
315C2#2144441912yes—
316C2#2144444411yescharacteristic
317C2#2144574111yescharacteristic
318C2#2144576112yes—
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#330C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenterC2×#330D61
hypocenterC61#8335
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
3. C2×#16839504C2semidirect
4. C2×#16839503C2semidirect
5. C2×C61×#330C2semidirect
145. C2×D61#330semidirect
158. C2×#330D61semidirect
182. C61#8335semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D612
#214455822
C2×#168395042
C2×#168395032
C2×C61×#3302
#833561