Finite Group Databasebeta

← #29592182

Subgroups — #29592182

Order 15616.

422 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 55 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#21456643C211yesmaximal, characteristic, complemented
2#21447770C212yesmaximal, complemented
3#21443806C212yesmaximal, complemented
4#21443755C211yesmaximal, characteristic, complemented
5C61×#2010C211yesmaximal, characteristic
6C2×#14519756—21no—
7#16840202—22no—
8#16840093—22no—
9#16840020—21no—
10#16840020C2212yes—
11#16839920—22no—
12#16839572—22no—
13#16839544C2212yes—
14C61×#575C2211yescharacteristic
15C61×#525—21no—
16C61×#421—21no—
17C61×#407C2211yescharacteristic
18C61×H(3,ℤ/4)C2211yescharacteristic
19C22×Dic122—21no—
20C2×Dic244—21no—
21C2×Dic244D412yescomplemented
22#14519756—22no—
23#14519756—22no—
24#14519756—41no—
25#14519756—41no—
26C2×C4⋊Dic61—21no—
27C2×#11758196—41no—
28C2×C22⋊Dic61—41no—
29#14519723—42no—
30#14519721—42no—
31#14519717—22no—
32#14519709—21no—
33#14519709D412yescomplemented
34#14519702—22no—
35#14519702—22no—
36C2×#41×C61—21no—
37C23×C4×C61D411yescharacteristic
38#137×C61—41no—
39AGL(1,ℤ/8)×C61—41no—
40#135×C61—22no—
41C2×M16×C61D411yescharacteristic
42C2×M16×C61C2311yescharacteristic
43#123×C61—41no—
44C22⋊Q8×C61—41no—
45#121×C61—41no—
46#120×C61—41no—
47C4×D4×C61—42no—
48#117×C61—21no—
49#104×C61—42no—
50C22⋊C8×C61—22no—
51C24×C61D811yescharacteristic
52#41×C61—41no—
53#41×C61—41no—
54#41×C61—41no—
55#41×C61—41no—
56C2×Q8×C61—21no—
57C2×D4×C61—21no—
58C2×D4×C61—42no—
59C2×D4×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61D811yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—41no—
68C22×Dic61—41no—
69C2×Dic122—22no—
70C2×Dic122—41no—
71C2×Dic122—81no—
72C2×Dic122—81no—
73Q16×C61—22no—
74SD16×C61—42no—
75D8×C61—22no—
76M16×C61—41no—
77M16×C61—41no—
78C2×C8×C61—21no—
79C2×C8×C61C2×D412yes—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—41no—
85C42×C61—22no—
86C22⋊Dic61—42no—
87C22⋊Dic61—42no—
88C4⋊Dic61—22no—
89C4⋊Dic61—41no—
90#11758196—42no—
91#11758196—42no—
92Dic244—42no—
93Dic244—81no—
94#11758185—42no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—42no—
98C23×C61—41no—
99C23×C61C2×D811yescharacteristic
100Q8×C61—41no—
101Q8×C61—41no—
102D4×C61—41no—
103D4×C61—41no—
104D4×C61—81no—
105D4×C61—84no—
106D4×C61—81no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—81no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—82no—
123C2×Dic61—82no—
124Dic122—42no—
125Dic122—42no—
126Dic122—82no—
127C8×C61—22no—
128C8×C61—41no—
129#12115—611nomaximal, Sylow
130C22×C61—22no—
131C22×C61—21no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61—42no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61—82no—
138C22×C61—81no—
139C22×C61#44711yescharacteristic
140C4×C61—21no—
141C4×C61—41no—
142C4×C61—82no—
143C4×C61—81no—
144C4×C61#45311yescharacteristic
145C4×C61B(4,𝔽2)11yescharacteristic
146Dic61—82no—
147Dic61—82no—
148#3195—611no—
149#2191—612no—
150#2010D6111yescharacteristic
151#1340—612no—
152#1320—611no—
153C2×C61—21no—
154C2×C61—41no—
155C2×C61—42no—
156C2×C61—161no—
157C2×C61#159511yescharacteristic
158#575C2×D6111yescharacteristic
159#525—21no—
160C2×C22⋊Q8—1221no—
161#510—1222no—
162#483—1222no—
163#470—612no—
164#470—1221no—
165#451—1222no—
166#421—21no—
167#407C2×D6111yescharacteristic
168#360—1222no—
169#349—612no—
170H(3,ℤ/4)C2×D6111yescharacteristic
171C61#1211511yescharacteristic, complemented, Sylow
172C2×#41—21no—
173C22×Q8—1221no—
174C23×C4C22⋊D6111yescharacteristic
175#137—41no—
176AGL(1,ℤ/8)—41no—
177#135—22no—
178C2×Q16—612no—
179C2×Q16—1221no—
180C2×M16C22⋊D6111yescharacteristic
181C2×M16C22×D6111yescharacteristic
182#123—41no—
183C22⋊Q8—41no—
184C22⋊Q8—1222no—
185C22⋊Q8—1222no—
186C22⋊Q8—2441no—
187C22⋊Q8—2441no—
188#121—41no—
189#120—41no—
190C4×D4—42no—
191#117—21no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—2441no—
194C2×C22⋊C4—2441no—
195Q32—2442no—
196SD32—2442no—
197M32—1222no—
198#106—612no—
199#106—1221no—
200#104—42no—
201#103—1222no—
202#103—1222no—
203C22⋊C8—22no—
204C24#1175819911yescharacteristic
205#41—41no—
206#41—41no—
207#41—41no—
208#41—41no—
209C2×Q8—21no—
210C2×Q8—1222no—
211C2×Q8—4881no—
212C2×Q8—4881no—
213C2×Q8—2441no—
214C2×D4—21no—
215C2×D4—42no—
216C2×D4—41no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—2441no—
223C22×C4—2441no—
224C22×C4#1175819911yescharacteristic
225C22×C4C2×C22⋊D6111yescharacteristic
226Q16—22no—
227Q16—4881no—
228Q16—2442no—
229SD16—42no—
230D8—22no—
231M16—41no—
232M16—41no—
233C2×C8—21no—
234C2×C8D4×D6112yes—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—1222no—
238C4⋊C4—2441no—
239C4⋊C4—2442no—
240C4⋊C4—2442no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—41no—
244C22⋊C4—2442no—
245C22⋊C4—2442no—
246C42—22no—
247C16—2442no—
248C23—21no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23C2×#1175819911yescharacteristic
253Q8—41no—
254Q8—41no—
255Q8—4882no—
256Q8—2442no—
257Q8—2442no—
258D4—41no—
259D4—41no—
260D4—81no—
261D4—84no—
262D4—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4#1451974511yescharacteristic
279C2×C4#1451980211yescharacteristic
280C8—22no—
281C8—41no—
282C22—22no—
283C22—21no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22#1683989511yescharacteristic
292C4—21no—
293C4—41no—
294C4—82no—
295C4—81no—
296C4—4882no—
297C4—4882no—
298C4#1683992911yescharacteristic
299C4#1683994811yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—161no—
304C2#2144473611yescharacteristic
subgroupG/H
centerC2#21444736
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
Fitting, prime radicalC61×#2010C2
socleC2×C61#1595
hypercenter#2010D61
hypocenterC61#12115
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #21456643C2semidirect
2. #21447770C2semidirect
3. #21443806C2semidirect
4. #21443755C2semidirect
21. C2×Dic244D4semidirect
33. #14519709D4semidirect
171. C61#12115semidirect

Maximal subgroups

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

subgroupindex
#214566432
#214477702
#214438062
#214437552
C61×#20102
#1211561