Finite Group Databasebeta

← #29590537

Subgroups — #29590537

Order 15616.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#21451517C211yesmaximal, characteristic
2#21444153C211yesmaximal, characteristic
3#21443845C212yesmaximal, complemented
4#21443824C212yesmaximal, complemented
5C61×#1742C211yesmaximal, characteristic
6C2×#14519769C411yescharacteristic
7#16839819—21no—
8#16839701C2211yescharacteristic
9#16839698C411yescharacteristic
10#16839692C2211yescharacteristic
11#16839687C411yescharacteristic
12#16839657—21no—
13#16839656—21no—
14#16839646—21no—
15#16839635—21no—
16#16839608C411yescharacteristic
17#16839488C2212yes—
18#16839486—21no—
19C22×M16×C61—21no—
20C2×#117×C61C2211yescharacteristic
21C61×#343—21no—
22C61×#343C2212yes—
23#14519769—41no—
24#14519769—41no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×#11758196—21no—
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×C4×Dic61—21no—
34C4⋊#92418—41no—
35C22⋊#92418—22no—
36#14519688—41no—
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684C2×C411yescharacteristic
43#14519684C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C2×M16×C61—24no—
46C2×M16×C61—22no—
47C2×M16×C61—41no—
48C2×M16×C61—42no—
49C22×C8×C61—41no—
50#117×C61—41no—
51#117×C61—41no—
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C4⋊C4×C61C2311yescharacteristic
54C2×C22⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C24×C61C22⋊C411yescharacteristic
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74M16×C61—24no—
75M16×C61—44no—
76M16×C61—42no—
77C2×C8×C61—44no—
78C2×C8×C61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C22⋊Dic61—42no—
88C22⋊Dic61—42no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91#11758196—42no—
92#11758196—42no—
93#11758196—41no—
94#11758196—41no—
95#11758196—41no—
96#11758196—41no—
97C4×Dic61—42no—
98C2×#92418—42no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61#10412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C61#13711yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C8×C61—44no—
138#92418—82no—
139#11767—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—42no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—81no—
147C22×C61C2×#9911yescharacteristic
148C22×C61#41711yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—41no—
158Dic61—42no—
159Dic61—81no—
160Dic61—81no—
161Dic61—81no—
162#2611—611no—
163#1742D6111yescharacteristic
164#1464—611no—
165#1353—612no—
166#1346—612no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—42no—
170C2×C61#150411yescharacteristic
171C2×C61#212311yescharacteristic
172C2×C61#213211yescharacteristic
173C22×M16—21no—
174C2×#125—611no—
175C2×#117C2×D6111yescharacteristic
176#432—1221no—
177#400—611no—
178#399—611no—
179#398—611no—
180#397—611no—
181#389—1221no—
182#388—1221no—
183#387—1221no—
184#385—1221no—
185#376—611no—
186#343—21no—
187#343C2×D6112yes—
188#324—612no—
189#323—1221no—
190C61#1176711yescharacteristic, complemented, Sylow
191C23×C4C22⋊D6111yescharacteristic
192C2×M16—24no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—42no—
196C22×C8—41no—
197#125—2441no—
198#125—2441no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4C22⋊D6111yescharacteristic
208C2×C4⋊C4C22×D6111yescharacteristic
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4C4×D6111yescharacteristic
212C2×C22⋊C4D24411yescharacteristic
213C2×C42—1221no—
214C2×C42C4×D6111yescharacteristic
215C2×C42D24411yescharacteristic
216C4⋊C8—2441no—
217C22⋊C8—1222no—
218#97—2441no—
219#95—611no—
220#95—611no—
221#95—1221no—
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226C24#1175819811yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4—1221no—
236C22×C4#1175819811yescharacteristic
237C22×C4#1175819811yescharacteristic
238C22×C4#1175819811yescharacteristic
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×D24411yescharacteristic
241C22×C4C2×C22⋊D6111yescharacteristic
242M16—24no—
243M16—44no—
244M16—42no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—2442no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C4⋊C4—2442no—
253C4⋊C4—2442no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—2442no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1451970512yes—
272C23C2×#1175819811yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—1222no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4#1451969111yescharacteristic
302C2×C4#1451969111yescharacteristic
303C2×C4#1451980311yescharacteristic
304C2×C4#1451980811yescharacteristic
305C8—44no—
306C8—4882no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1451969111yescharacteristic
315C22#1683976111yescharacteristic
316C22C2×#1451970511yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—4881no—
326C4—4881no—
327C4—4881no—
328C4—2442no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2144432111yescharacteristic
333C2#2144747411yescharacteristic
334C2#2144750211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1742C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1742D61
hypocenterC61#11767
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #21443845C2semidirect
4. #21443824C2semidirect
190. C61#11767semidirect

Maximal subgroups

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

subgroupindex
#214515172
#214441532
#214438452
#214438242
C61×#17422
#1176761