Finite Group Databasebeta

← #29578007

Subgroups — #29578007

Order 15616.

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

normal subgroups only

This group has 478 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#21450102C211yesmaximal, characteristic
2#21445902C211yesmaximal, characteristic
3#21444148C212yesmaximal
4#21443728C212yesmaximal
5C61×#2921C211yesmaximal, characteristic
6C22×#11758196C2211yescharacteristic
7#16839726C2212yes—
8#16839700—21no—
9#16839693—21no—
10#16839646—21no—
11#16839646—21no—
12#16839634C411yescharacteristic
13#16839634C411yescharacteristic
14#16839625C2211yescharacteristic
15#16839623C411yescharacteristic
16#16839623C411yescharacteristic
17#16839531—21no—
18#16839531—21no—
19#16839487—24no—
20C2×#117×C61C2211yescharacteristic
21C61×#433—24no—
22C61×#423—21no—
23C61×#407C2212yes—
24C4×M16×C61—21no—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C2×#11758194—22no—
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—22no—
32C2×#11758196—21no—
33C2×#11758196—22no—
34C2×C22⋊Dic61C2×C411yescharacteristic
35C2×C22⋊Dic61C2×C411yescharacteristic
36C22⋊#92418—22no—
37C22⋊#92418—22no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C23×C4×C61C2311yescharacteristic
47C2×M16×C61—22no—
48#117×C61—22no—
49#117×C61—22no—
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56C4⋊C8×C61—24no—
57C22⋊C8×C61—22no—
58C22⋊C8×C61D414yes—
59#97×C61—22no—
60C4×C8×C61—22no—
61C24×C61C22×C411yescharacteristic
62C22×C4×C61—41no—
63C22×C4×C61—41no—
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×C4×C61C2×D412yes—
71C22×C4×C61#4112yes—
72C22×Dic61—21no—
73C22×Dic61—22no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—22no—
78C22×Dic61—42no—
79C22×Dic61—41no—
80C22×Dic61—41no—
81M16×C61—42no—
82M16×C61—42no—
83C2×C8×C61—24no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—22no—
86C22⋊C4×C61—22no—
87C22⋊C4×C61—22no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—21no—
92C22⋊Dic61—41no—
93C22⋊Dic61—41no—
94#11758196—42no—
95#11758196—42no—
96#11758196—41no—
97#11758196—42no—
98#11758196—41no—
99#11758194—42no—
100#11758194—42no—
101C2×#92418—42no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61C4×D412yes—
108C2×C4×C61—21no—
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—42no—
117C2×C4×C61—42no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61#12011yescharacteristic
123C2×C4×C61#12112yes—
124C2×C4×C61#12311yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—24no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—82no—
140C8×C61—44no—
141#92418—82no—
142#9514—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—22no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61#38611yescharacteristic
151C22×C61#41011yescharacteristic
152C22×C61#41011yescharacteristic
153C4×C61—22no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—41no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—41no—
160Dic61—42no—
161Dic61—81no—
162Dic61—81no—
163Dic61—81no—
164#2921D6111yescharacteristic
165#2484—611no—
166#1822—611no—
167#1463—612no—
168#1306—612no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—42no—
172C2×C61#189111yescharacteristic
173C2×C61#189911yescharacteristic
174C2×C61#191511yescharacteristic
175C2×#117C2×D6111yescharacteristic
176C22×C4⋊C4—611no—
177#433—24no—
178#423—21no—
179#407—612no—
180#407C2×D6112yes—
181C4×M16—21no—
182#400—1221no—
183#398—1221no—
184#387—1221no—
185#387—1221no—
186#385—611no—
187#385—611no—
188#381—611no—
189#381—611no—
190#381—611no—
191#343—1221no—
192#343—1221no—
193#324—1224no—
194C61#951411yescharacteristic, complemented, Sylow
195C23×C4—611no—
196C23×C4—611no—
197C23×C4C22×D6111yescharacteristic
198C2×M16—22no—
199C2×M16—1222no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4—1222no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4D24411yescharacteristic
209C2×C4⋊C4D24411yescharacteristic
210C2×C22⋊C4—611no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4C4×D6111yescharacteristic
213C2×C22⋊C4C4×D6111yescharacteristic
214C2×C42C22⋊D6111yescharacteristic
215C2×C42C22⋊D6111yescharacteristic
216C4⋊C8—24no—
217C22⋊C8—22no—
218C22⋊C8—1222no—
219C22⋊C8—1222no—
220C22⋊C8C22⋊D6114yes—
221#97—22no—
222C4×C8—22no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24C2×C4×D6111yescharacteristic
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—1221no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—2442no—
241C22×C4—2441no—
242C22×C4—2441no—
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4#1175819811yescharacteristic
246C22×C4#1175819811yescharacteristic
247C22×C4C2×D24411yescharacteristic
248C22×C4#1175822212yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C22×C4C2×C22⋊D6112yes—
251M16—42no—
252M16—42no—
253M16—2442no—
254M16—2442no—
255C2×C8—24no—
256C2×C8—2442no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2441no—
262C4⋊C4—2442no—
263C4⋊C4—2441no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—2441no—
267C22⋊C4—2441no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×#1175819811yescharacteristic
277C23C4×C22⋊D6112yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1222no—
293C2×C4—1224no—
294C2×C4—4882no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2442no—
306C2×C4—2441no—
307C2×C4#1451974611yescharacteristic
308C2×C4#1451975112yes—
309C2×C4#1451976011yescharacteristic
310C8—44no—
311C8—4882no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#1683964111yescharacteristic
320C22#1683973511yescharacteristic
321C22#1683973511yescharacteristic
322C4—21no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4881no—
330C4—4881no—
331C4—4881no—
332C4—2442no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2144623711yescharacteristic
337C2#2144627711yescharacteristic
338C2#2144637111yescharacteristic
subgroupG/H
centerC22#16839641
commutator, prime radicalC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2921C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2921D61
hypocenterC61#9514
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
194. C61#9514semidirect

Maximal subgroups

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

subgroupindex
#214501022
#214459022
#214441482
#214437282
C61×#29212
#951461