Finite Group Databasebeta

← #29593076

Subgroups — #29593076

Order 15616.

352 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 57 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#21451521C211yesmaximal, characteristic
2#21447166C211yesmaximal, characteristic
3#21444861C211yesmaximal, characteristic
4#21443849C211yesmaximal, characteristic
5#21443847C211yesmaximal, characteristic
6#21443762C211yesmaximal, characteristic
7C61×#1605C211yesmaximal, characteristic
8#16840095—21no—
9#16839805—21no—
10#16839701C2211yescharacteristic
11#16839694—21no—
12#16839669—21no—
13#16839661—21no—
14#16839647—21no—
15#16839643—21no—
16#16839627C2211yescharacteristic
17#16839607—21no—
18#16839531C2211yescharacteristic
19#16839521—21no—
20#16839499—21no—
21#16839488C2211yescharacteristic
22C61×#544C2211yescharacteristic
23C61×#451—21no—
24C61×#395C2211yescharacteristic
25C61×#328—21no—
26C61×#324C2211yescharacteristic
27C2×#11758194—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61D411yescharacteristic
31C2×C4⋊Dic61D411yescharacteristic
32C2×#11758196—21no—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61D411yescharacteristic
36#14519710—41no—
37#14519710—41no—
38#14519708—41no—
39#14519701—41no—
40C22⋊#92418—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684D411yescharacteristic
49C22×Q8×C61D411yescharacteristic
50C2×Q16×C61—41no—
51C4⋊Q8×C61—41no—
52#125×C61—41no—
53C22⋊Q8×C61—21no—
54C22⋊Q8×C61—21no—
55#117×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2311yescharacteristic
58#103×C61—41no—
59C22⋊C8×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C2×Q8×C61—42no—
65C2×Q8×C61—41no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75Q16×C61—82no—
76C2×C8×C61—41no—
77C4⋊C4×C61—21no—
78C4⋊C4×C61—21no—
79C4⋊C4×C61—21no—
80C4⋊C4×C61—21no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C42×C61—41no—
87C4⋊Dic61—42no—
88C4⋊Dic61—41no—
89C4⋊Dic61—41no—
90C4⋊Dic61—81no—
91#11758196—42no—
92C4×Dic61—21no—
93C4×Dic61—42no—
94C4×Dic61—42no—
95C4×Dic61Q1612yescomplemented
96#11758194—22no—
97#11758194—41no—
98C2×#92418—41no—
99C2×#92418—41no—
100C23×C61#12011yescharacteristic
101Q8×C61—42no—
102Q8×C61—81no—
103Q8×C61—81no—
104Q8×C61—81no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61C2×Q1611yescharacteristic
120C2×C4×C61#13711yescharacteristic
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C8×C61—81no—
134#92418—42no—
135#92418—81no—
136#12258—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61#45111yescharacteristic
140C22×C61#45611yescharacteristic
141C22×C61B(4,𝔽2)11yescharacteristic
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—42no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61—42no—
148C4×C61—81no—
149C4×C61—81no—
150Dic61—42no—
151Dic61—81no—
152Dic61—81no—
153Dic61—81no—
154#2612—611no—
155#2061—611no—
156#1617—611no—
157#1605D6111yescharacteristic
158#1354—611no—
159#1353—611no—
160#1324—611no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61#160511yescharacteristic
164C2×C61#219711yescharacteristic
165C2×C61#220411yescharacteristic
166#544C2×D6111yescharacteristic
167#484—1221no—
168#451—21no—
169#428—1221no—
170#400—611no—
171#398—1221no—
172#395C2×D6111yescharacteristic
173#391—1221no—
174#389—1221no—
175#387—1221no—
176#387—1221no—
177#382—611no—
178#376—1221no—
179#343—611no—
180#339—1221no—
181#328—21no—
182#328—1221no—
183#324—611no—
184#324C2×D6111yescharacteristic
185C61#1225811yescharacteristic, complemented, Sylow
186C22×Q8C22⋊D6111yescharacteristic
187C2×Q16—41no—
188C2×M16—1221no—
189C4⋊Q8—41no—
190#125—41no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—611no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4C22×D6111yescharacteristic
201C2×C42—611no—
202C2×C42—1221no—
203C2×C42—1221no—
204#107—2441no—
205#107—2441no—
206#106—2441no—
207#103—41no—
208#103—2441no—
209C22⋊C8—21no—
210C22⋊C8—1221no—
211#95—21no—
212#95—611no—
213#95—1221no—
214#95—1221no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95C22⋊D6111yescharacteristic
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—1221no—
230C22×C4—1221no—
231C22×C4D4×D6111yescharacteristic
232C22×C4D4×D6111yescharacteristic
233C22×C4C2×C22⋊D6111yescharacteristic
234Q16—82no—
235M16—1222no—
236M16—2441no—
237C2×C8—41no—
238C2×C8—2441no—
239C2×C8—2441no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—4881no—
249C4⋊C4—2441no—
250C4⋊C4—2441no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C22⋊C4—41no—
254C42—41no—
255C42—612no—
256C42—1221no—
257C42—2442no—
258C42—2442no—
259C23#1451974511yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—81no—
263Q8—81no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—1222no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4Q16×D6111yescharacteristic
291C2×C4#1451980911yescharacteristic
292C8—81no—
293C8—4881no—
294C8—2442no—
295C22—21no—
296C22—21no—
297C22#1683991611yescharacteristic
298C22#1683994311yescharacteristic
299C22#1683994811yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—4881no—
309C4—4881no—
310C4—4881no—
311C4—2442no—
312C2—21no—
313C2—21no—
314C2#2144479111yescharacteristic
315C2#2144780711yescharacteristic
316C2#2144784411yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1605C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#1605D61
hypocenterC61#12258
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1605

Complements

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

NH ≅ G/Ntype
95. C4×Dic61Q16semidirect
185. C61#12258semidirect

Maximal subgroups

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

subgroupindex
#214515212
#214471662
#214448612
#214438492
#214438472
#214437622
C61×#16052
#1225861