Finite Group Databasebeta

← #29572156

Subgroups — #29572156

Order 15616.

1026 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 119 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#21459583C211yesmaximal, characteristic
2#21446618C211yesmaximal, characteristic, complemented
3#21444515C212yesmaximal, complemented
4#21444431C212yesmaximal, complemented
5C61×#1466C211yesmaximal, characteristic, complemented
6C2×#14519822C412yes—
7#16840416—22no—
8#16840407—24no—
9D4×D244—22no—
10#16840343C412yes—
11#16840324—22no—
12#16840312C2212yes—
13C2×#14519751—22no—
14C2×C4⋊#92418C2211yescharacteristic, complemented
15C2×C22⋊#92418—22no—
16C2×#14519688C2211yescharacteristic, complemented
17#16839510—22no—
18#16839505—22no—
19#16839501—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#432C2212yescomplemented
22C61×#323—22no—
23#14519822—48no—
24C2×#11758222C2×C414yes—
25C2×#11758224—24no—
26C22×D244—22no—
27C22×C22⋊D61—24no—
28C2×D4×D61—24no—
29C22×#92418—21no—
30C22×#92418—21no—
31#14519777D412yes—
32#14519774D412yes—
33#14519765D414yes—
34#14519761—28no—
35#14519757—24no—
36#14519752—24no—
37#14519751—24no—
38#14519751—24no—
39#14519751—24no—
40#14519751—22no—
41#14519751—22no—
42#14519747—28no—
43C4×D244—22no—
44C4×D244—22no—
45C2×C4⋊Dic61—22no—
46C2×#11758198—24no—
47C4⋊#92418—22no—
48C22⋊#92418—44no—
49#14519688—22no—
50C22×D4×C61C2×C411yescharacteristic
51C23×C4×C61D412yes—
52C4×D4×C61—22no—
53C4×D4×C61—22no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D412yes—
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—22no—
58C22⋊C8×C61—22no—
59#97×C61—22no—
60C24×C61C22⋊C412yes—
61C23×D61—24no—
62C2×D4×C61—24no—
63C2×D4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—28no—
73C2×C22⋊D61—24no—
74C2×C22⋊D61—48no—
75C2×C22⋊D61—48no—
76C22×Dic61—24no—
77#11758224—416no—
78D4×D61—416no—
79#11758222—48no—
80C2×D244—22no—
81C2×D244—24no—
82C2×D244—22no—
83C2×D244—82no—
84C2×D244—82no—
85C2×D244C22⋊C414yes—
86C2×C4×D61—24no—
87C2×Dic122C22⋊C414yes—
88C2×C8×C61—42no—
89C4⋊C4×C61—22no—
90C22⋊C4×C61—24no—
91C42×C61C2×D412yes—
92C42×C61C2×D412yes—
93#11758198—28no—
94#11758198—24no—
95#11758198—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61—22no—
98C4⋊Dic61—22no—
99C2×#92418—22no—
100C2×#92418—41no—
101C2×#92418—41no—
102C2×#92418—42no—
103C2×#92418—41no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61#10014yes—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—24no—
111C22×D61—48no—
112C22×D61—84no—
113C22×D61—84no—
114D4×C61—28no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—22no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—82no—
124C2×C4×C61C2×C22⋊C412yes—
125C2×C4×C61#12012yes—
126C2×C4×C61#12011yescharacteristic
127C2×C4×C61#12011yescharacteristic
128C22⋊D61—416no—
129C22⋊D61—48no—
130C2×Dic61—24no—
131C2×Dic61—48no—
132D244—44no—
133D244—42no—
134D244—42no—
135D244—82no—
136D244—84no—
137C4×D61—48no—
138Dic122—44no—
139C8×C61—82no—
140#92418—42no—
141#92418—81no—
142#8456—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—24no—
146C22×C61—44no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—82no—
150C22×C61#37911yescharacteristic
151C22×C61C2×#10012yes—
152C2×D61—44no—
153C2×D61—44no—
154C2×D61—44no—
155C2×D61—168no—
156C4×C61—21no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—42no—
160C4×C61#45311yescharacteristic
161C4×C61#45511yescharacteristic
162Dic61—44no—
163#3496—611no—
164#1955—611no—
165#1551—612no—
166#1530—612no—
167#1466D6111yescharacteristic, complemented
168C2×C61—22no—
169C2×C61—44no—
170C2×C61#178311yescharacteristic
171C2×C61#179211yescharacteristic
172C2×C61#179411yescharacteristic
173D61—84no—
174C2×H(5,𝔽2)—612no—
175#547—1222no—
176#546—1224no—
177D42—1222no—
178#537—612no—
179#534—1222no—
180#532—612no—
181C2×#121—1222no—
182C2×C4×D4C2×D6111yescharacteristic
183#432C2×D6112yescomplemented
184C2×C4⋊C8—611no—
185C2×C22⋊C8—1222no—
186C2×#97—611no—
187#333—1222no—
188#331—1222no—
189#329—1224no—
190#323—22no—
191C61#845611yescharacteristic, complemented, Sylow
192H(5,𝔽2)—2448no—
193C2×#41—614no—
194C2×#41—1224no—
195C22×D4—1222no—
196C22×D4—1224no—
197C22×D4—1224no—
198C22×D4C4×D6111yescharacteristic
199C23×C4D24412yes—
200C22×C8—1221no—
201C22×C8—1221no—
202C4⋊Q8—612no—
203#127—612no—
204#124—614no—
205#123—1228no—
206C22⋊Q8—1224no—
207#121—1224no—
208#121—1224no—
209#121—1224no—
210#121—1224no—
211#121—1222no—
212#121—1222no—
213#120—1228no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—1222no—
217C4×D4—1222no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4C4×D6111yescharacteristic
220C2×C22⋊C4—1224no—
221C2×C22⋊C4C22⋊D6112yes—
222C2×C42C22×D6111yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—1222no—
225C22⋊C8—22no—
226C22⋊C8—2444no—
227#97—22no—
228#97—1222no—
229C24—1224no—
230C24#1175819812yes—
231#41—2448no—
232#41—24416no—
233C2×Q8—614no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—614no—
237C2×D4—1224no—
238C2×D4—1228no—
239C2×D4—1224no—
240C2×D4—1222no—
241C2×D4—1222no—
242C2×D4—4882no—
243C2×D4—4882no—
244C2×D4—24416no—
245C2×D4—2448no—
246C2×D4—2448no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—1224no—
252C22×C4—1224no—
253C22×C4#1175819812yes—
254C22×C4C2×C4×D6111yescharacteristic
255C22×C4C2×D24411yescharacteristic
256C22×C4C2×C22⋊D6111yescharacteristic
257C2×C8—42no—
258C2×C8—1222no—
259C2×C8—2442no—
260C2×C8—2441no—
261C2×C8—2441no—
262C2×C8—2441no—
263C4⋊C4—22no—
264C4⋊C4—1224no—
265C4⋊C4—1222no—
266C4⋊C4—1222no—
267C22⋊C4—24no—
268C22⋊C4—1228no—
269C22⋊C4—1224no—
270C22⋊C4—1224no—
271C42D4×D6112yes—
272C42D4×D6112yes—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—1224no—
278C23—4884no—
279C23—4884no—
280C23—2448no—
281C23#1451969314yes—
282C23C2×#1175819811yescharacteristic
283Q8—2444no—
284D4—28no—
285D4—4882no—
286D4—4884no—
287D4—2444no—
288D4—24416no—
289D4—2448no—
290D4—2442no—
291D4—2442no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—1224no—
302C2×C4—2448no—
303C2×C4—2448no—
304C2×C4C22⋊C4×D6112yes—
305C2×C4#1451974512yes—
306C2×C4#1451974711yescharacteristic
307C2×C4#1451974711yescharacteristic
308C8—82no—
309C8—4881no—
310C8—2442no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—9768no—
319C22—2444no—
320C22—2444no—
321C22—2444no—
322C22#1683961711yescharacteristic
323C22C2×#1451969312yes—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—42no—
328C4—2444no—
329C4#1683993211yescharacteristic
330C4#1683994211yescharacteristic
331C2—22no—
332C2—44no—
333C2—4884no—
334C2#2144573911yescharacteristic
335C2#2144577511yescharacteristic
336C2#2144578511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1466C2
socleC22×C61#379
prime radical#21459583C2
cosocleC2×C42×C61C23
hypercenter#1466D61
hypocenterC61#8456
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #21446618C2semidirect
3. #21444515C2semidirect
4. #21444431C2semidirect
5. C61×#1466C2semidirect
14. C2×C4⋊#92418C22semidirect
16. C2×#14519688C22semidirect
21. C61×#432C22semidirect
167. #1466D61semidirect
183. #432C2×D61semidirect
191. C61#8456semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214466182
#214445152
#214444312
C61×#14662
#845661