Finite Group Databasebeta

← #29835253

Subgroups — #29835253

Order 15616.

520 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 135 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#21456564C211yesmaximal, characteristic, complemented
2#21448038C212yesmaximal, complemented
3#21447990C212yesmaximal, complemented
4#21447900C212yesmaximal, complemented
5#21447770C212yesmaximal
6#21447710C214yesmaximal, complemented
7#21447631C211yesmaximal, characteristic, complemented
8C61×#2956C211yesmaximal, characteristic
9#16840319—21no—
10#16840202C2214yes—
11#16840160C2211yescharacteristic, complemented
12#16840142C2211yescharacteristic, complemented
13#16840113—21no—
14#16840020—21no—
15#16840020C2212yes—
16#16839870—21no—
17#16839870C2212yescomplemented
18#16839586C2214yescomplemented
19#16839583C2212yescomplemented
20#16839577C2214yescomplemented
21#16839572C2214yes—
22#16839544C2212yes—
23#16839540C2212yescomplemented
24C61×#575C2211yescharacteristic
25C61×#568—21no—
26C61×#444—21no—
27C61×#444C2212yes—
28C61×#443—21no—
29C61×#443C2212yes—
30C61×#430C2211yescharacteristic
31C2×#104×C61—21no—
32C4×M16×C61C2211yescharacteristic
33C2×Dic244—21no—
34C2×Dic244C2312yes—
35#14519777—21no—
36#14519777—21no—
37#14519769—41no—
38#14519756—22no—
39#14519756—41no—
40C4×Dic122—22no—
41C4×Dic122—41no—
42#14519736—41no—
43#14519723—24no—
44#14519721—24no—
45#14519717C2314yes—
46#14519709—21no—
47#14519709C2312yes—
48#14519702—22no—
49#14519702—22no—
50C2×#41×C61—21no—
51#137×C61—22no—
52AGL(1,ℤ/8)×C61—22no—
53#135×C61—21no—
54#135×C61C2312yes—
55#131×C61—22no—
56#131×C61—22no—
57#131×C61—42no—
58C2×M16×C61—21no—
59C2×M16×C61—42no—
60C2×M16×C61D411yescharacteristic
61C2×M16×C61C2311yescharacteristic
62C2×C42×C61D411yescharacteristic
63#108×C61—21no—
64#108×C61C2312yes—
65#104×C61—22no—
66#104×C61—22no—
67#97×C61D411yescharacteristic
68#97×C61C2311yescharacteristic
69C4×C8×C61D411yescharacteristic
70C4×C8×C61C2311yescharacteristic
71#41×C61—22no—
72#41×C61—22no—
73#41×C61—41no—
74#41×C61—41no—
75C2×Q8×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C2×Dic122—22no—
82C2×Dic122—41no—
83Q16×C61—21no—
84Q16×C61#4112yes—
85SD16×C61—22no—
86SD16×C61—22no—
87D8×C61—21no—
88D8×C61#4112yes—
89M16×C61—22no—
90M16×C61—22no—
91M16×C61—22no—
92M16×C61—42no—
93M16×C61—41no—
94M16×C61—42no—
95C2×C8×C61—42no—
96C2×C8×C61—41no—
97C2×C8×C61—42no—
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61C2411yescharacteristic
102C42×C61—21no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C22⋊Dic61—42no—
106C4⋊Dic61—22no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110#11758196—42no—
111#11758196—42no—
112C4×Dic61—42no—
113Dic244—24no—
114Dic244—42no—
115#11758185—24no—
116C23×C61—41no—
117C23×C61AGL(1,ℤ/8)11yescharacteristic
118Q8×C61—22no—
119Q8×C61—41no—
120D4×C61—22no—
121D4×C61—41no—
122D4×C61—42no—
123D4×C61—42no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C61C22×D411yescharacteristic
134C2×Dic61—42no—
135C2×Dic61—42no—
136Dic122—42no—
137Dic122—42no—
138Dic122—82no—
139C8×C61—21no—
140C8×C61—21no—
141C8×C61—42no—
142C8×C61AGL(1,ℤ/8)12yes—
143C8×C61C2×#4112yes—
144#33689—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—41no—
147C22×C61—42no—
148C22×C61—161no—
149C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61—42no—
154C4×C61#52511yescharacteristic
155C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
156Dic61—82no—
157Dic61—82no—
158#3185—611no—
159#2956D6111yescharacteristic
160#2239—612no—
161#2230—612no—
162#2214—612no—
163#2191—612no—
164#2180—614no—
165#2165—611no—
166C2×C61—21no—
167C2×C61—41no—
168C2×C61—82no—
169C2×C61#310911yescharacteristic
170#575C2×D6111yescharacteristic
171#568—21no—
172#533—1221no—
173#510—614no—
174#498—611no—
175#494—611no—
176#487—1221no—
177#470—612no—
178#470—1221no—
179#444—21no—
180#444C2×D6112yes—
181#443—21no—
182#443C2×D6112yes—
183#441—612no—
184#441—1221no—
185#430C2×D6111yescharacteristic
186C2×#104—21no—
187C4×M16C2×D6111yescharacteristic
188#365—614no—
189#364—612no—
190#362—614no—
191#360—614no—
192#349—612no—
193#347—612no—
194C61#3368911yescharacteristic, complemented, Sylow
195C2×#41—21no—
196#137—22no—
197AGL(1,ℤ/8)—22no—
198#135—21no—
199#135C22×D6112yes—
200C2×Q16—612no—
201C2×Q16—1221no—
202#131—22no—
203#131—22no—
204#131—42no—
205C2×M16—21no—
206C2×M16—42no—
207C2×M16C22⋊D6111yescharacteristic
208C2×M16C22×D6111yescharacteristic
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211#125—2441no—
212C22⋊Q8—1222no—
213C22⋊Q8—2441no—
214C4×Q8—1222no—
215C4×Q8—2441no—
216#117—2441no—
217C2×C42C22⋊D6111yescharacteristic
218Q32—1224no—
219SD32—1224no—
220M32—614no—
221#108—21no—
222#108C22×D6112yes—
223#106—612no—
224#106—1221no—
225#104—22no—
226#104—22no—
227#103—1222no—
228#103—1222no—
229#97C22⋊D6111yescharacteristic
230#97C22×D6111yescharacteristic
231C4×C8C22⋊D6111yescharacteristic
232C4×C8C22×D6111yescharacteristic
233#41—22no—
234#41—22no—
235#41—41no—
236#41—41no—
237C2×Q8—21no—
238C2×Q8—1222no—
239C2×Q8—2441no—
240C2×D4—21no—
241C2×D4—41no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4C2×C22⋊D6111yescharacteristic
245Q16—21no—
246Q16—1224no—
247Q16—2442no—
248Q16#1175822212yes—
249SD16—22no—
250SD16—22no—
251D8—21no—
252D8#1175822212yes—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—42no—
257M16—41no—
258M16—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8C2×C22⋊D6111yescharacteristic
263C2×C8C2×C22⋊D6111yescharacteristic
264C2×C8C2×C22⋊D6111yescharacteristic
265C2×C8C23×D6111yescharacteristic
266C4⋊C4—1222no—
267C4⋊C4—2442no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4—2441no—
271C4⋊C4—2442no—
272C22⋊C4—2442no—
273C42—21no—
274C42—2442no—
275C42C2×C22⋊D6111yescharacteristic
276C42C2×C22⋊D6111yescharacteristic
277C16—1224no—
278C23—41no—
279C23#1451980211yescharacteristic
280Q8—22no—
281Q8—41no—
282Q8—4882no—
283Q8—2442no—
284Q8—2442no—
285D4—22no—
286D4—41no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4#1451980211yescharacteristic
300C2×C4C22×C22⋊D6111yescharacteristic
301C8—21no—
302C8—21no—
303C8—42no—
304C8#1451980312yes—
305C8C2×#1175822212yes—
306C22—21no—
307C22—41no—
308C22—42no—
309C22—161no—
310C22C2×#1451980211yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—42no—
315C4—4882no—
316C4—4882no—
317C4#1684027311yescharacteristic
318C4C2×#1451980311yescharacteristic
319C2—21no—
320C2—41no—
321C2—82no—
322C2#2145591011yescharacteristic
subgroupG/H
centerC2#21455910
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2956C2
socleC2×C61#3109
prime radicalC61×#575C22
hypercenter#2956D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. #21456564C2semidirect
2. #21448038C2semidirect
3. #21447990C2semidirect
4. #21447900C2semidirect
6. #21447710C2semidirect
7. #21447631C2semidirect
11. #16840160C22semidirect
12. #16840142C22semidirect
17. #16839870C22semidirect
18. #16839586C22semidirect
19. #16839583C22semidirect
20. #16839577C22semidirect
23. #16839540C22semidirect
194. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214565642
#214480382
#214479902
#214479002
#214477702
#214477102
#214476312
C61×#29562
#3368961