Finite Group Databasebeta

← #29593248

Subgroups — #29593248

Order 15616.

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

normal subgroups only

This group has 338 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#21456679C211yesmaximal, characteristic
2#21448004C211yesmaximal, characteristic, complemented
3#21447260C211yesmaximal, characteristic, complemented
4#21443835C211yesmaximal, characteristic, complemented
5#21443834C211yesmaximal, characteristic, complemented
6#21443744C211yesmaximal, characteristic, complemented
7C61×#2232C211yesmaximal, characteristic, complemented
8#16840328—21no—
9#16840089—21no—
10#16840084—21no—
11#16840013—21no—
12#16839998C2211yescharacteristic
13#16839897—21no—
14#16839894—21no—
15#16839761—21no—
16#16839728C2211yescharacteristic
17#16839589—21no—
18#16839562—21no—
19C22⋊#11758185C2211yescharacteristic, complemented
20#16839531—21no—
21#16839527C2211yescharacteristic, complemented
22C61×#466C2211yescharacteristic
23C2×#107×C61C2211yescharacteristic, complemented
24C61×#366—21no—
25C61×#357—21no—
26C61×C22⋊C16C2211yescharacteristic, complemented
27C2×#11758222D411yescharacteristic
28C2×D4×D61—21no—
29C2×D488D411yescharacteristic
30C2×#11758199—21no—
31C2×#11758194—21no—
32C2×#11758194D411yescharacteristic
33#14519773—41no—
34#14519763—41no—
35#14519752—21no—
36#14519751—21no—
37#14519748—21no—
38#14519747—41no—
39#14519745—41no—
40#14519735—21no—
41C2×#11758185—21no—
42#14519711D411yescharacteristic
43#14519708—21no—
44#14519700—41no—
45#14519699—21no—
46#14519698—21no—
47#14519697—21no—
48#14519697—41no—
49#14519689—21no—
50#14519689—21no—
51C2×D8×C61D411yescharacteristic
52C22×C8×C61C2311yescharacteristic
53#121×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C16×C61—21no—
56#107×C61—21no—
57#107×C61—21no—
58#107×C61D411yescharacteristic
59#102×C61—21no—
60C23×D61—41no—
61C2×D4×C61—21no—
62C2×D4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61C2×D411yescharacteristic
65C2×C22⋊D61—21no—
66C2×C22⋊D61—41no—
67C2×C22⋊D61—41no—
68C2×C22⋊D61—41no—
69D4×D61—81no—
70D4×D61—81no—
71#11758222—41no—
72#11758222—41no—
73C2×D244—21no—
74C2×D244—41no—
75C2×D244D811yescharacteristic
76C2×C4×D61—21no—
77C2×Dic122D811yescharacteristic
78D8×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61C2×D411yescharacteristic
81C2×C8×C61C2×D411yescharacteristic
82C4⋊C4×C61—21no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊Dic61—41no—
87#11758199—81no—
88#11758198—41no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4×Dic61—41no—
93#11758194—41no—
94#11758194—41no—
95#11758194—41no—
96C2×#92418—21no—
97C2×#92418—41no—
98D488—41no—
99C16×C61—41no—
100#11758185—41no—
101C23×C61—41no—
102C23×C61AGL(1,ℤ/8)11yescharacteristic
103C22×D61—21no—
104C22×D61—41no—
105C22×D61—82no—
106C22×D61—161no—
107C22×D61—161no—
108D4×C61—41no—
109D4×C61—41no—
110D4×C61—161no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—81no—
115C2×C4×C61—161no—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61C2×D811yescharacteristic
118C22⋊D61—81no—
119C22⋊D61—82no—
120C22⋊D61—161no—
121C22⋊D61—161no—
122C2×Dic61—21no—
123C2×Dic61—41no—
124D244—41no—
125D244—41no—
126D244—41no—
127D244—161no—
128C4×D61—41no—
129C4×D61—41no—
130Dic122—41no—
131C8×C61—21no—
132C8×C61—21no—
133#92418—41no—
134#92418—81no—
135#12284—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—81no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61#44711yescharacteristic
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—81no—
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—164no—
150C4×C61—21no—
151C4×C61—81no—
152C4×C61—161no—
153C4×C61#45511yescharacteristic
154C4×C61#45811yescharacteristic
155Dic61—41no—
156Dic61—81no—
157#3199—611no—
158#2233—611no—
159#2232D6111yescharacteristic, complemented
160#2078—611no—
161#1349—611no—
162#1349—611no—
163#1315—611no—
164C2×C61—41no—
165C2×C61—161no—
166C2×C61#160711yescharacteristic
167C2×C61#219211yescharacteristic
168C2×C61#219411yescharacteristic
169D61—81no—
170D61—161no—
171#535—1221no—
172#482—1221no—
173#482—1221no—
174#469—1221no—
175#466—611no—
176#466C2×D6111yescharacteristic
177#447—1221no—
178#447—1221no—
179C2×#107C2×D6111yescharacteristic, complemented
180#417—1221no—
181#408—611no—
182#367—1221no—
183#366—21no—
184#357—21no—
185#357—1221no—
186C22⋊C16—611no—
187C22⋊C16C2×D6111yescharacteristic, complemented
188#343—1221no—
189#341—611no—
190C61#1228411yescharacteristic, complemented, Sylow
191C2×#41—611no—
192C22×D4—1221no—
193C2×D8—611no—
194C2×D8—1221no—
195C2×D8C22⋊D6111yescharacteristic
196C2×M16—611no—
197C2×M16—1221no—
198C22×C8C22×D6111yescharacteristic
199#127—2441no—
200#124—2441no—
201#121—21no—
202#121—1221no—
203#121—1221no—
204#121—1221no—
205#120—2441no—
206#120—2441no—
207#117—1221no—
208C2×C4⋊C4—21no—
209C2×C16—21no—
210C2×C16—1221no—
211#107—21no—
212#107—21no—
213#107—611no—
214#107C22⋊D6111yescharacteristic
215#106—1221no—
216#103—2441no—
217#102—21no—
218#102—1221no—
219#102—1221no—
220#102—1221no—
221#102—2441no—
222C22⋊C8—1221no—
223C22⋊C8—1221no—
224C24—2441no—
225#41—2441no—
226#41—2441no—
227C2×Q8—611no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—611no—
231C2×D4—1221no—
232C2×D4—1221no—
233C2×D4—4881no—
234C2×D4—4881no—
235C2×D4—2441no—
236C2×D4—2441no—
237C2×D4—2441no—
238C2×D4—2441no—
239C22×C4—41no—
240C22×C4—1221no—
241C22×C4D4×D6111yescharacteristic
242D8—41no—
243D8—4881no—
244D8—2441no—
245M16—2441no—
246M16—2441no—
247M16—2441no—
248C2×C8—41no—
249C2×C8—1221no—
250C2×C8—2441no—
251C2×C8D4×D6111yescharacteristic
252C2×C8C2×C22⋊D6111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—1221no—
257C22⋊C4—41no—
258C22⋊C4—2441no—
259C22⋊C4—2441no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C42—2441no—
263C16—41no—
264C16—2441no—
265C23—41no—
266C23—1221no—
267C23—9761no—
268C23—9761no—
269C23—4882no—
270C23—2441no—
271C23#1451980111yescharacteristic
272Q8—2441no—
273D4—41no—
274D4—41no—
275D4—161no—
276D4—9761no—
277D4—9761no—
278D4—9761no—
279D4—4881no—
280D4—4882no—
281D4—2441no—
282D4—2441no—
283D4—2441no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—161no—
289C2×C4—1221no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4#1451974511yescharacteristic
294C2×C4D8×D6111yescharacteristic
295C8—21no—
296C8—21no—
297C8—4881no—
298C8—2441no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—9764no—
306C22—4881no—
307C22—4881no—
308C22—4881no—
309C22—2441no—
310C22—2441no—
311C22—2441no—
312C22#1683989411yescharacteristic
313C4—21no—
314C4—81no—
315C4—161no—
316C4—4881no—
317C4—2441no—
318C4#1683993811yescharacteristic
319C4#1683995111yescharacteristic
320C2—41no—
321C2—161no—
322C2—9761no—
323C2—4881no—
324C2#2144480311yescharacteristic
325C2#2144777511yescharacteristic
326C2#2144778711yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#2232C2
socleC22×C61#447
prime radical#21456679C2
hypercenter#2232D61
hypocenterC61#12284
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2232

Complements

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

NH ≅ G/Ntype
2. #21448004C2semidirect
3. #21447260C2semidirect
4. #21443835C2semidirect
5. #21443834C2semidirect
6. #21443744C2semidirect
7. C61×#2232C2semidirect
19. C22⋊#11758185C22semidirect
21. #16839527C22semidirect
23. C2×#107×C61C22semidirect
26. C61×C22⋊C16C22semidirect
159. #2232D61semidirect
179. C2×#107C2×D61semidirect
187. C22⋊C16C2×D61semidirect
190. C61#12284semidirect

Maximal subgroups

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

subgroupindex
#214566792
#214480042
#214472602
#214438352
#214438342
#214437442
C61×#22322
#1228461