Finite Group Databasebeta

← #29591140

Subgroups — #29591140

Order 15616.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450136C211yesmaximal, characteristic
2#21444140C211yesmaximal, characteristic
3#21443842C214yesmaximal, complemented
4C61×#1799C211yesmaximal, characteristic
5C22×C4⋊Dic61C411yescharacteristic, complemented
6#16839815—21no—
7#16839701C412yescomplemented
8#16839698C2212yes—
9#16839660—21no—
10#16839642—21no—
11#16839634—21no—
12#16839622—21no—
13#16839622C411yescharacteristic, complemented
14#16839486—21no—
15#16839486C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C2×#99×C61—21no—
18C61×#380—21no—
19C61×#342C2212yes—
20C23×Dic61—21no—
21C2×C4⋊Dic61—21no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—41no—
24C2×C4⋊Dic61—41no—
25C2×C4⋊Dic61C2×C412yes—
26C2×#11758196—22no—
27C2×C22⋊Dic61—21no—
28C2×C22⋊Dic61—22no—
29C4⋊#92418—41no—
30C22⋊#92418—22no—
31C4×#92418—41no—
32#14519684—22no—
33#14519684—22no—
34#14519684C2×C412yes—
35C22×D4×C61D411yescharacteristic
36C23×C4×C61C2×C411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C4×D4×C61—41no—
39C4×D4×C61—41no—
40C2×C4⋊C4×C61—41no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61C2311yescharacteristic
45C2×C42×C61D411yescharacteristic
46#99×C61—42no—
47#95×C61—41no—
48C24×C61C22⋊C411yescharacteristic
49C24×C61C2×D411yescharacteristic
50C2×D4×C61—41no—
51C2×D4×C61—41no—
52C2×D4×C61SD1612yes—
53C2×D4×C61Q1612yes—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—42no—
69C22×Dic61—82no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—82no—
72C22⋊C4×C61—41no—
73C22⋊C4×C61—41no—
74C22⋊C4×C61—42no—
75C22⋊C4×C61—42no—
76C42×C61—21no—
77C42×C61—21no—
78C22⋊Dic61—42no—
79C22⋊Dic61—82no—
80C4⋊Dic61—42no—
81C4⋊Dic61—41no—
82C4⋊Dic61—41no—
83C4⋊Dic61—42no—
84#11758196—82no—
85C2×#92418—42no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61#10312yes—
95C23×C61C2×C22⋊C411yescharacteristic
96C23×C61C2×SD1611yescharacteristic
97C23×C61C2×Q1611yescharacteristic
98D4×C61—42no—
99D4×C61—42no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—81no—
113C2×C4×C61—82no—
114C2×C4×C61—82no—
115C2×C4×C61—82no—
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61#10312yes—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—84no—
126#92418—82no—
127#11899—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—21no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—81no—
142C22×C61C2×#9911yescharacteristic
143C22×C61C2×#10311yescharacteristic
144C22×C61#42111yescharacteristic
145C4×C61—21no—
146C4×C61—22no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—162no—
151C4×C61#35212yes—
152Dic61—82no—
153Dic61—82no—
154#2487—611no—
155#1799D6111yescharacteristic
156#1460—611no—
157#1352—614no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61—41no—
161C2×C61—42no—
162C2×C61—41no—
163C2×C61#152011yescharacteristic
164C2×C61C2×#35211yescharacteristic
165C2×C61#212811yescharacteristic
166C2×C4×D4C2×D6111yescharacteristic
167C22×C4⋊C4—611no—
168#431—1221no—
169C2×#99—21no—
170#400—612no—
171#399—612no—
172#389—1221no—
173#386—1221no—
174#385—1221no—
175#380—21no—
176#380—611no—
177#380—1221no—
178#342C2×D6112yes—
179#323—612no—
180#323—1221no—
181C61#1189911yescharacteristic, complemented, Sylow
182C22×D4C22⋊D6111yescharacteristic
183C23×C4—1221no—
184C23×C4C4×D6111yescharacteristic
185C23×C4C22⋊D6111yescharacteristic
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—612no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—2441no—
194C2×C4⋊C4—2441no—
195C2×C4⋊C4C4×D6111yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1221no—
199C2×C22⋊C4D24411yescharacteristic
200C2×C22⋊C4C22×D6111yescharacteristic
201C2×C42D24411yescharacteristic
202C4⋊C8—2441no—
203#99—42no—
204C22⋊C8—1222no—
205C4×C8—2441no—
206#95—41no—
207#95—612no—
208#95—1222no—
209#95—1222no—
210C24#1175819811yescharacteristic
211C24C2×C22⋊D6111yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#1175820012yes—
215C2×D4#1175820212yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—4882no—
226C22×C4—2442no—
227C22×C4#1175819811yescharacteristic
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4C2×C4×D6111yescharacteristic
231C22×C4C2×D24411yescharacteristic
232C2×C8—2442no—
233C4⋊C4—41no—
234C4⋊C4—82no—
235C4⋊C4—4882no—
236C4⋊C4—2442no—
237C4⋊C4—2442no—
238C4⋊C4—2441no—
239C4⋊C4—2441no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—4882no—
245C22⋊C4—2442no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1451970012yes—
257C23C2×#1175819811yescharacteristic
258C23C2×#1175820011yescharacteristic
259C23C2×#1175820211yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—4884no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2442no—
285C2×C4#1451969111yescharacteristic
286C2×C4#1451969111yescharacteristic
287C2×C4#1451970012yes—
288C8—4882no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1451969111yescharacteristic
304C22C2×#1451970011yescharacteristic
305C22#1683978011yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—162no—
312C4—4882no—
313C4—4882no—
314C4#1683955112yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#2144439011yescharacteristic
321C2C2×#1683955111yescharacteristic
322C2#2144749011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1799D61
hypocenterC61#11899
LCSG ▷ C22×C4×C61 ▷ C23×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
3. #21443842C2semidirect
5. C22×C4⋊Dic61C4semidirect
7. #16839701C4semidirect
13. #16839622C4semidirect
181. C61#11899semidirect

Maximal subgroups

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

subgroupindex
#214501362
#214441402
#214438422
C61×#17992
#1189961