Finite Group Databasebeta

← #29590647

Subgroups — #29590647

Order 15616.

572 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14519694C211yesmaximal, characteristic
2#21444157C211yesmaximal, characteristic
3#21443849C214yesmaximal, complemented
4C61×#2409C211yesmaximal, characteristic
5C2×#14519694—21no—
6#16839726—21no—
7C4×#11758194—21no—
8#16839531C2212yes—
9#16839488C2212yes—
10#16839486—21no—
11C23×Q8×C61C411yescharacteristic
12C2×C22⋊Q8×C61—21no—
13C2×C4×D4×C61C2211yescharacteristic
14C61×#395C2212yes—
15C61×#393C412yes—
16C61×#391—21no—
17C61×#386—21no—
18C61×#385—21no—
19C61×#380C411yescharacteristic
20C2×#11758194—22no—
21C2×#11758194—41no—
22#14519694—24no—
23C22⋊#92418—22no—
24C22⋊#92418—41no—
25#14519688—41no—
26C4×#92418—41no—
27C22×Q8×C61—21no—
28C22×Q8×C61—41no—
29C22×Q8×C61—41no—
30C22×Q8×C61—44no—
31C22×Q8×C61C2×C412yes—
32C22×D4×C61D411yescharacteristic
33C23×C4×C61—21no—
34C23×C4×C61C2×C411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C22⋊Q8×C61—44no—
37C4×D4×C61—41no—
38C4×D4×C61—41no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61D411yescharacteristic
46C2×C42×C61D411yescharacteristic
47#95×C61—22no—
48#95×C61—22no—
49#95×C61C2×C412yes—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C22⋊C411yescharacteristic
52C2×Q8×C61—24no—
53C2×Q8×C61—42no—
54C2×Q8×C61—44no—
55C2×Q8×C61—42no—
56C2×Q8×C61—48no—
57C2×Q8×C61—44no—
58C2×Q8×C61—44no—
59C2×Q8×C61—41no—
60C2×Q8×C61—41no—
61C2×D4×C61—22no—
62C2×D4×C61—22no—
63C2×D4×C61—41no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—44no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C4⋊C4×C61—41no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—44no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—44no—
85C22⋊C4×C61—42no—
86C42×C61—21no—
87C42×C61—21no—
88#11758194—24no—
89#11758194—42no—
90#11758194—82no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61#10412yes—
103C23×C61C2×C22⋊C411yescharacteristic
104Q8×C61—44no—
105Q8×C61—44no—
106Q8×C61—44no—
107Q8×C61—44no—
108D4×C61—44no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—24no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—48no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—81no—
129C2×C4×C61#9911yescharacteristic
130C2×C4×C61#9911yescharacteristic
131C2×C4×C61#13712yes—
132#92418—44no—
133#92418—82no—
134#11791—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—22no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—42no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—81no—
147C22×C61C2×#9911yescharacteristic
148C22×C61#41911yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C4×C61—21no—
151C4×C61—22no—
152C4×C61—41no—
153C4×C61—44no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—82no—
157C4×C61AΣL(1,𝔽16)12yes—
158#2409D6111yescharacteristic
159C4×#100—611no—
160#1465—611no—
161#1354—614no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—42no—
165C2×C61—81no—
166C2×C61#150811yescharacteristic
167C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
168C2×C61#213511yescharacteristic
169C23×Q8Dic6111yescharacteristic
170C2×C22⋊Q8—21no—
171C2×C4×D4C2×D6111yescharacteristic
172C2×#100—1221no—
173#407—1221no—
174C4×M16—1221no—
175#395C2×D6112yes—
176#393Dic6112yes—
177#391—21no—
178#386—21no—
179#385—21no—
180#380Dic6111yescharacteristic
181#343—612no—
182#324—612no—
183#323—1221no—
184C61#1179111yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8C2×Dic6112yes—
190C22×D4C22⋊D6111yescharacteristic
191C23×C4—21no—
192C23×C4C2×Dic6111yescharacteristic
193C23×C4C22⋊D6111yescharacteristic
194C2×M16—1222no—
195C2×M16—2441no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22×D6111yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×Dic6111yescharacteristic
205C2×C22⋊C4C22⋊D6111yescharacteristic
206C2×C42C22⋊D6111yescharacteristic
207#100—1224no—
208C22⋊C8—1222no—
209C22⋊C8—2441no—
210#97—2441no—
211C4×C8—2441no—
212#95—22no—
213#95—22no—
214#95C2×Dic6112yes—
215C24C22⋊Dic6111yescharacteristic
216C24C22⋊Dic6111yescharacteristic
217C2×Q8—24no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic6111yescharacteristic
240C22×C4C22⋊Dic6111yescharacteristic
241C22×C4C22×Dic6111yescharacteristic
242C22×C4C2×C22⋊D6111yescharacteristic
243C22×C4C2×C22⋊D6111yescharacteristic
244M16—1224no—
245M16—4882no—
246M16—2442no—
247C2×C8—2442no—
248C2×C8—2442no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1451970412yes—
268C23C2×C22⋊Dic6111yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—44no—
272Q8—44no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#1451969211yescharacteristic
295C2×C4#1451969211yescharacteristic
296C2×C4#1451980812yes—
297C8—4882no—
298C8—2444no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1451969211yescharacteristic
312C22#1683977111yescharacteristic
313C22C2×#1451970411yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4#1683954912yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#2144433911yescharacteristic
327C2C2×#1683954911yescharacteristic
328C2#2144751411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2409C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C4⋊C4×C61C23
hypercenter#2409D61
hypocenterC61#11791
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #21443849C2semidirect
184. C61#11791semidirect

Maximal subgroups

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

subgroupindex
C4×#145196942
#214441572
#214438492
C61×#24092
#1179161