Finite Group Databasebeta

← #29591148

Subgroups — #29591148

Order 15616.

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

normal subgroups only

This group has 508 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#21450772C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic
3#21443842C214yesmaximal, complemented
4C61×#1799C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic, complemented
6C2×#14519756—21no—
7C2×#14519756—21no—
8C2×C22⋊#92418—21no—
9#16839698C412yescomplemented
10#16839698C2212yes—
11#16839665—21no—
12#16839631—21no—
13C4×C4⋊Dic61C411yescharacteristic, complemented
14#16839488—21no—
15#16839486C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C2×#99×C61—21no—
18C61×#380—21no—
19C61×#342C2212yes—
20C22×Dic122—21no—
21C22×#92418—41no—
22#14519777—41no—
23#14519777—41no—
24#14519756—44no—
25#14519756—44no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C4×Dic61—21no—
34C22⋊#92418—22no—
35C22⋊#92418—41no—
36#14519684C2×C412yes—
37C22×D4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C4×D4×C61—41no—
41C4×D4×C61—41no—
42C2×C4⋊C4×C61—41no—
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61C2311yescharacteristic
47C2×C42×C61C2×C411yescharacteristic
48#99×C61—42no—
49#95×C61—41no—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C2×D411yescharacteristic
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61SD1612yes—
55C2×D4×C61Q1612yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—22no—
69C22×Dic61—22no—
70C2×Dic122—42no—
71C2×Dic122—42no—
72C2×Dic122—41no—
73C2×Dic122—41no—
74C4⋊C4×C61—41no—
75C4⋊C4×C61—82no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C22⋊C4×C61—42no—
79C22⋊C4×C61—42no—
80C42×C61—21no—
81C42×C61—21no—
82C22⋊Dic61—44no—
83C22⋊Dic61—44no—
84C4⋊Dic61—24no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87#11758196—44no—
88#11758196—44no—
89C4×Dic61—42no—
90C2×#92418—42no—
91C2×#92418—82no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61#10312yes—
101C23×C61C2×C22⋊C411yescharacteristic
102C23×C61C2×SD1611yescharacteristic
103C23×C61C2×Q1611yescharacteristic
104D4×C61—42no—
105D4×C61—42no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61—82no—
120C2×C4×C61—82no—
121C2×C4×C61—82no—
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61#9911yescharacteristic
124C2×C4×C61#10312yes—
125C2×Dic61—24no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131Dic122—82no—
132Dic122—82no—
133#92418—82no—
134#11901—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—22no—
139C22×C61—21no—
140C22×C61—42no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61—81no—
148C22×C61—81no—
149C22×C61C2×#9911yescharacteristic
150C22×C61C2×#10311yescharacteristic
151C22×C61#42111yescharacteristic
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—162no—
158C4×C61#35312yes—
159Dic61—44no—
160Dic61—82no—
161#2542—611no—
162#1799D6111yescharacteristic
163#1459—611no—
164#1352—614no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—41no—
168C2×C61—42no—
169C2×C61—41no—
170C2×C61#152011yescharacteristic
171C2×C61#212311yescharacteristic
172C2×C61C2×#35311yescharacteristic
173C2×C4⋊Q8—611no—
174C2×C22⋊Q8—1221no—
175C2×C22⋊Q8—1221no—
176C2×C4×D4C2×D6111yescharacteristic
177C2×#99—21no—
178C2×C22⋊C8—1221no—
179#399—612no—
180#399—612no—
181#390—1221no—
182#384—1221no—
183#380—21no—
184C4×C4⋊C4—611no—
185#342C2×D6112yes—
186#324—1221no—
187#323—612no—
188C61#1190111yescharacteristic, complemented, Sylow
189C22×Q8—1221no—
190C22×D4C22⋊D6111yescharacteristic
191C23×C4D24411yescharacteristic
192C23×C4C22⋊D6111yescharacteristic
193C22×C8—2441no—
194C4⋊Q8—2441no—
195C4⋊Q8—2441no—
196C22⋊Q8—2444no—
197C22⋊Q8—2444no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4D24411yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4C4×D6111yescharacteristic
211C2×C22⋊C4C22×D6111yescharacteristic
212C2×C42—1221no—
213C2×C42C4×D6111yescharacteristic
214#99—42no—
215C22⋊C8—1222no—
216C22⋊C8—2441no—
217#95—41no—
218#95—612no—
219C24#1175819811yescharacteristic
220C24C2×C22⋊D6111yescharacteristic
221C2×Q8—2442no—
222C2×Q8—2442no—
223C2×Q8—2441no—
224C2×Q8—2441no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1175820012yes—
228C2×D4#1175820212yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4#1175819811yescharacteristic
239C22×C4#1175819811yescharacteristic
240C22×C4#1175819811yescharacteristic
241C22×C4C2×C4×D6111yescharacteristic
242C22×C4C2×D24411yescharacteristic
243C2×C8—4882no—
244C2×C8—2442no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C4⋊C4—1224no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C4⋊C4—2444no—
251C4⋊C4—2444no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—2444no—
257C22⋊C4—2444no—
258C42—21no—
259C42—21no—
260C42—2442no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1451970012yes—
270C23C2×#1175819811yescharacteristic
271C23C2×#1175820011yescharacteristic
272C23C2×#1175820211yescharacteristic
273Q8—4882no—
274Q8—4882no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—1224no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4#1451969111yescharacteristic
300C2×C4#1451969111yescharacteristic
301C2×C4#1451970012yes—
302C8—4882no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1451969111yescharacteristic
318C22C2×#1451970011yescharacteristic
319C22#1683978011yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—162no—
326C4—4882no—
327C4—2444no—
328C4#1683955412yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2144439011yescharacteristic
335C2#2144747211yescharacteristic
336C2C2×#1683955411yescharacteristic
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#11901
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. C2×#14519777C4semidirect
9. #16839698C4semidirect
13. C4×C4⋊Dic61C4semidirect
188. C61#11901semidirect

Maximal subgroups

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

subgroupindex
#214507722
#214441372
#214438422
C61×#17992
#1190161