Finite Group Databasebeta

← #29584355

Subgroups — #29584355

Order 15616.

500 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 77 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#21456570C211yesmaximal, characteristic
2#21446280C212yesmaximal, complemented
3#21446001C211yesmaximal, characteristic, complemented
4#21444336C212yesmaximal, complemented
5C61×#1293C211yesmaximal, characteristic, complemented
6C22×#11758194C2211yescharacteristic
7C2×#14519751C2212yes—
8#16840076—22no—
9#16839998—21no—
10#16839998—21no—
11#16839898—22no—
12C2×#14519706—21no—
13C2×#14519693—22no—
14C4×C22⋊Dic61C2211yescharacteristic
15#16839530—22no—
16#16839516—21no—
17#16839497—22no—
18#16839497—22no—
19C61×#407C2212yes—
20C2×C22⋊C8×C61C2211yescharacteristic
21C22×D244D412yes—
22C22×C22⋊D61—22no—
23C2×D488—21no—
24C2×D488—21no—
25C2×#11758194—22no—
26C2×#11758194—42no—
27C22×#92418—21no—
28#14519751—22no—
29#14519751—22no—
30#14519751—22no—
31#14519751—22no—
32C2×C4⋊Dic61D412yes—
33C2×C22⋊Dic61—21no—
34C2×#11758198—22no—
35C2×C4×Dic61—21no—
36#14519711—21no—
37#14519711—21no—
38#14519706—42no—
39#14519699—22no—
40#14519699—22no—
41#14519693—44no—
42#14519684D411yescharacteristic
43#14519684D411yescharacteristic
44C23×C4×C61C2311yescharacteristic
45C2×M16×C61—22no—
46C22×C8×C61D411yescharacteristic
47C22×C8×C61D411yescharacteristic
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61#4111yescharacteristic
52C23×D61—22no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C2×D412yes—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61#4112yes—
59C2×C22⋊D61—42no—
60C2×C22⋊D61—42no—
61C2×C22⋊D61—82no—
62C2×C22⋊D61—82no—
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61—22no—
66C2×D244—22no—
67C2×D244—22no—
68C2×D244—42no—
69C2×D244—42no—
70M16×C61—42no—
71M16×C61—42no—
72C2×C8×C61—21no—
73C2×C8×C61—21no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C2×C8×C61—42no—
77C22⋊Dic61—42no—
78#11758198—42no—
79#11758198—42no—
80C4⋊Dic61—22no—
81C4⋊Dic61—22no—
82C4×Dic61—41no—
83C4×Dic61—41no—
84#11758194—24no—
85#11758194—42no—
86C2×#92418—22no—
87C2×#92418—41no—
88C2×#92418—41no—
89D488—81no—
90D488—81no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61#12111yescharacteristic
96C23×C61#12312yes—
97C22×D61—42no—
98C22×D61—44no—
99C22×D61—82no—
100C22×D61—82no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61—22no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61#12011yescharacteristic
108C2×C4×C61#12112yes—
109C2×C4×C61#12411yescharacteristic
110C22⋊D61—84no—
111C22⋊D61—82no—
112C22⋊D61—82no—
113C2×Dic61—22no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121D244—42no—
122D244—42no—
123D244—82no—
124C8×C61—41no—
125C8×C61—41no—
126C8×C61—82no—
127#92418—42no—
128#10557—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61#39411yescharacteristic
137C22×C61#47211yescharacteristic
138C22×C61#47211yescharacteristic
139C2×D61—82no—
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—164no—
143C4×C61—22no—
144C4×C61—21no—
145C4×C61B(4,𝔽2)11yescharacteristic
146C4×C61#45811yescharacteristic
147Dic61—42no—
148Dic61—81no—
149Dic61—82no—
150#3186—611no—
151#1900—612no—
152#1851—611no—
153#1507—612no—
154#1293D6111yescharacteristic, complemented
155C2×C61—21no—
156C2×C61—21no—
157C2×C61—42no—
158C2×C61#204411yescharacteristic
159C2×C61#204511yescharacteristic
160C2×C61#205011yescharacteristic
161D61—162no—
162C22×M16—611no—
163C2×#121—612no—
164#480—1222no—
165#466—1221no—
166#466—1221no—
167#447—1222no—
168C2×C4⋊C8—1221no—
169C2×#100—1222no—
170#407C2×D6112yes—
171C2×C22⋊C8C2×D6111yescharacteristic
172C4×C22⋊C4—611no—
173#343—1222no—
174#336—1221no—
175#327—1222no—
176#327—1222no—
177C61#1055711yescharacteristic, complemented, Sylow
178C22×D4—612no—
179C22×D4—1222no—
180C23×C4C22×D6111yescharacteristic
181C2×D8—1221no—
182C2×D8—1221no—
183C2×M16—22no—
184C2×M16—1222no—
185C2×M16—2442no—
186C22×C8—1221no—
187C22×C8D24411yescharacteristic
188C22×C8D24411yescharacteristic
189#121—1222no—
190#121—1222no—
191#121—1222no—
192#121—1222no—
193C2×C4⋊C4—612no—
194C2×C22⋊C4—1221no—
195C2×C22⋊C4—1222no—
196C2×C42—1221no—
197#107—1221no—
198#107—1221no—
199C4⋊C8—2442no—
200#102—1222no—
201#102—1222no—
202#100—2444no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#95—611no—
207#95—611no—
208C24—1222no—
209C24#1175822611yescharacteristic
210C2×D4—1222no—
211C2×D4—1222no—
212C2×D4—4882no—
213C2×D4—4882no—
214C2×D4—2442no—
215C2×D4—2442no—
216C2×D4—2442no—
217C2×D4—2442no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—1221no—
221C22×C4—1222no—
222C22×C4—1221no—
223C22×C4C2×D24411yescharacteristic
224C22×C4#1175822212yes—
225C22×C4D4×D6112yes—
226C22×C4D4×D6111yescharacteristic
227D8—4881no—
228D8—4881no—
229M16—42no—
230M16—42no—
231M16—1224no—
232M16—2442no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—1222no—
239C2×C8—2441no—
240C2×C8—2441no—
241C4⋊C4—1222no—
242C4⋊C4—1222no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C22⋊C4—2442no—
246C42—2441no—
247C42—2441no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—4882no—
253C23—4882no—
254C23—2444no—
255C23—2442no—
256C23#1451975211yescharacteristic
257C23#1451976212yes—
258D4—4882no—
259D4—4884no—
260D4—4882no—
261D4—4882no—
262D4—2442no—
263D4—2442no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—1222no—
271C2×C4—4882no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4—2442no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4#1451974711yescharacteristic
279C2×C4#1451975012yes—
280C2×C4#1451976511yescharacteristic
281C8—41no—
282C8—41no—
283C8—82no—
284C8—2442no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—9764no—
293C22—4882no—
294C22—4882no—
295C22—4882no—
296C22#1683968011yescharacteristic
297C22#1684003311yescharacteristic
298C22#1684003311yescharacteristic
299C4—22no—
300C4—21no—
301C4—4881no—
302C4—4882no—
303C4—2442no—
304C4#1683995011yescharacteristic
305C4#1683995311yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—9762no—
310C2#2144707111yescharacteristic
311C2#2144707611yescharacteristic
312C2#2144710711yescharacteristic
subgroupG/H
centerC22#16839680
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1293C2
socleC22×C61#394
prime radical#21456570C2
hypercenter#1293D61
hypocenterC61#10557
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #21446280C2semidirect
3. #21446001C2semidirect
4. #21444336C2semidirect
5. C61×#1293C2semidirect
154. #1293D61semidirect
177. C61#10557semidirect

Maximal subgroups

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

subgroupindex
#214565702
#214462802
#214460012
#214443362
C61×#12932
#1055761