Finite Group Databasebeta

← #29593861

Subgroups — #29593861

Order 15616.

372 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 65 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#21457431C211yesmaximal, characteristic
2#21448058C211yesmaximal, characteristic, complemented
3#21447963C211yesmaximal, characteristic, complemented
4#21443861C211yesmaximal, characteristic, complemented
5#21443785C211yesmaximal, characteristic, complemented
6#21443777C211yesmaximal, characteristic, complemented
7C61×#1693C211yesmaximal, characteristic, complemented
8C2×#14519805—21no—
9#16840408—21no—
10#16840146C2211yescharacteristic
11#16840008—21no—
12#16839978—21no—
13#16839926—21no—
14#16839898—21no—
15C4×D488C2211yescharacteristic
16#16839589—21no—
17C4⋊#11758185C2211yescharacteristic, complemented
18#16839565—21no—
19#16839562—21no—
20#16839562—21no—
21#16839538C2211yescharacteristic, complemented
22C61×#500C2211yescharacteristic, complemented
23C61×#461—21no—
24C61×#436C2211yescharacteristic
25C61×#334C2211yescharacteristic, complemented
26C61×#332—21no—
27C2×#11758222—21no—
28C22×D244—21no—
29#14519805—42no—
30#14519805—42no—
31C2×Dic244D411yescharacteristic
32C2×#11758190—21no—
33C2×#11758190—21no—
34C2×D488—21no—
35C2×D488D411yescharacteristic
36C2×D488D411yescharacteristic
37#14519765—21no—
38#14519762—41no—
39#14519756—21no—
40#14519751—21no—
41#14519750—41no—
42#14519745—41no—
43C4×D244—21no—
44C22⋊C4×D61—41no—
45C2×#11758185—21no—
46C2×#11758185—21no—
47#14519711D411yescharacteristic
48#14519702—21no—
49#14519699—21no—
50#14519699—21no—
51C2×SD16×C61—41no—
52C2×M16×C61—21no—
53C4⋊Q8×C61—21no—
54C4×D4×C61D411yescharacteristic
55#107×C61—21no—
56#107×C61—21no—
57C4⋊C8×C61—21no—
58C4⋊C8×C61D411yescharacteristic
59#103×C61—41no—
60C22⋊C8×C61—21no—
61C4×C8×C61C2311yescharacteristic
62C23×D61—41no—
63C2×Q8×C61—41no—
64C2×D4×C61D811yescharacteristic
65C22×C4×C61—21no—
66C2×C22⋊D61—41no—
67C2×C22⋊D61—41no—
68#11758222—42no—
69#11758222—81no—
70C2×D244—21no—
71C2×D244—21no—
72C2×D244—42no—
73C2×D244—41no—
74C2×D244—81no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×Dic122—21no—
78SD16×C61—82no—
79M16×C61—41no—
80M16×C61—41no—
81C2×C8×C61—21no—
82C2×C8×C61—41no—
83C2×C8×C61C2×D411yescharacteristic
84C2×C8×C61C2×D411yescharacteristic
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61D811yescharacteristic
88C22⋊C4×C61—21no—
89C42×C61C2×D411yescharacteristic
90C22⋊Dic61—41no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94#11758198—41no—
95C4⋊Dic61—21no—
96#11758196—41no—
97Dic244—41no—
98D488—21no—
99D488—41no—
100D488—42no—
101D488SD1612yescomplemented
102#11758190—42no—
103#11758190—41no—
104#11758190—41no—
105#11758185—41no—
106#11758185—41no—
107C23×C61—21no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61—82no—
111C22×D61—82no—
112C22×D61—161no—
113Q8×C61—81no—
114Q8×C61—81no—
115D4×C61—41no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—81no—
120C2×C4×C61#12011yescharacteristic
121C2×C4×C61C2×D811yescharacteristic
122C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
123C22⋊D61—82no—
124C22⋊D61—161no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127D244—22no—
128D244—41no—
129D244—41no—
130D244—41no—
131D244—82no—
132C4×D61—82no—
133C4×D61—161no—
134Dic122—41no—
135Dic122—41no—
136C8×C61—21no—
137C8×C61—41no—
138C8×C61—81no—
139C8×C61C2×SD1611yescharacteristic
140C8×C61AGL(1,ℤ/8)11yescharacteristic
141#12378—611nomaximal, Sylow
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61#44711yescharacteristic
146C2×D61—42no—
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—164no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—41no—
156C4×C61—161no—
157C4×C61#45011yescharacteristic
158C4×C61#45511yescharacteristic
159Dic61—81no—
160Dic61—81no—
161#3273—611no—
162#2243—611no—
163#2226—611no—
164#1693D6111yescharacteristic, complemented
165#1360—611no—
166#1333—611no—
167#1330—611no—
168C2×C61—81no—
169C2×C61#163911yescharacteristic
170C2×C61#219011yescharacteristic
171C2×C61#219411yescharacteristic
172D61—82no—
173D61—161no—
174C2×AGL(1,ℤ/8)—1221no—
175#546—1221no—
176#500C2×D6111yescharacteristic, complemented
177#495—611no—
178#468—1221no—
179#463—1221no—
180#461—21no—
181#452—1221no—
182#447—1221no—
183C4×D8—611no—
184#436C2×D6111yescharacteristic
185#367—1221no—
186C4⋊C16—611no—
187#358—1221no—
188#357—1221no—
189#357—1221no—
190#346—611no—
191#334C2×D6111yescharacteristic, complemented
192#332—21no—
193C61#1237811yescharacteristic, complemented, Sylow
194C2×#41—1221no—
195C22×D4—1221no—
196AGL(1,ℤ/8)—2442no—
197AGL(1,ℤ/8)—2442no—
198C2×Q16—611no—
199C2×SD16—41no—
200C2×SD16—1221no—
201C2×SD16—1221no—
202C2×D8—611no—
203C2×D8—611no—
204C2×D8—1221no—
205C2×M16—21no—
206C4⋊Q8—21no—
207#124—1221no—
208#123—2441no—
209C22⋊Q8—1221no—
210#121—1221no—
211#121—2441no—
212#120—2441no—
213C4×D4—1221no—
214C4×D4C22⋊D6111yescharacteristic
215C2×C22⋊C4—2441no—
216C2×C16—1221no—
217C2×C16—1221no—
218#107—21no—
219#107—21no—
220#107—611no—
221C4⋊C8—21no—
222C4⋊C8C22⋊D6111yescharacteristic
223#103—41no—
224#103—1221no—
225#102—1221no—
226#102—1221no—
227C22⋊C8—21no—
228C4×C8C22×D6111yescharacteristic
229C24—2441no—
230#41—4881no—
231#41—2442no—
232C2×Q8—41no—
233C2×Q8—1221no—
234C2×D4—1221no—
235C2×D4—1221no—
236C2×D4—4881no—
237C2×D4—2441no—
238C2×D4—2442no—
239C2×D4—2441no—
240C2×D4—2441no—
241C2×D4#1175819911yescharacteristic
242C22×C4—21no—
243C22×C4—2441no—
244C22×C4—2441no—
245Q16—2441no—
246SD16—82no—
247SD16—2442no—
248SD16—2441no—
249SD16—2441no—
250D8—612no—
251D8—1221no—
252D8—2441no—
253D8—2442no—
254M16—41no—
255M16—41no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8D4×D6111yescharacteristic
259C2×C8D4×D6111yescharacteristic
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—1221no—
263C4⋊C4—2441no—
264C4⋊C4#1175819911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2441no—
267C22⋊C4—2441no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C42C2×C22⋊D6111yescharacteristic
272C16—2441no—
273C16—2441no—
274C23—21no—
275C23—9761no—
276C23—4882no—
277C23—4882no—
278C23—2441no—
279C23—2441no—
280Q8—81no—
281Q8—81no—
282Q8—2441no—
283Q8—2441no—
284D4—41no—
285D4—1222no—
286D4—9761no—
287D4—4882no—
288D4—4882no—
289D4—2441no—
290D4—2441no—
291D4—2441no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—9761no—
297C2×C4—4882no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4#1451974511yescharacteristic
301C2×C4C2×#1175819911yescharacteristic
302C2×C4#1451980211yescharacteristic
303C8—21no—
304C8—41no—
305C8—81no—
306C8SD16×D6111yescharacteristic
307C8#1451980411yescharacteristic
308C22—41no—
309C22—41no—
310C22—41no—
311C22—9764no—
312C22—4881no—
313C22—4881no—
314C22—4881no—
315C22—4881no—
316C22—4881no—
317C22—2442no—
318C22#1683989511yescharacteristic
319C4—21no—
320C4—21no—
321C4—41no—
322C4—161no—
323C4—4881no—
324C4—4881no—
325C4#1683991111yescharacteristic
326C4#1683993811yescharacteristic
327C2—81no—
328C2—9761no—
329C2—4882no—
330C2#2144499611yescharacteristic
331C2#2144776411yescharacteristic
332C2#2144778811yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1693C2
socleC22×C61#447
prime radical#21457431C2
hypercenter#1693D61
hypocenterC61#12378
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ #1693

Complements

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

NH ≅ G/Ntype
2. #21448058C2semidirect
3. #21447963C2semidirect
4. #21443861C2semidirect
5. #21443785C2semidirect
6. #21443777C2semidirect
7. C61×#1693C2semidirect
17. C4⋊#11758185C22semidirect
21. #16839538C22semidirect
22. C61×#500C22semidirect
25. C61×#334C22semidirect
101. D488SD16semidirect
164. #1693D61semidirect
176. #500C2×D61semidirect
191. #334C2×D61semidirect
193. C61#12378semidirect

Maximal subgroups

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

subgroupindex
#214574312
#214480582
#214479632
#214438612
#214437852
#214437772
C61×#16932
#1237861