Finite Group Databasebeta

← #29593826

Subgroups — #29593826

Order 15616.

360 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 69 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#21457579C211yesmaximal, characteristic
2#21448044C211yesmaximal, characteristic, complemented
3#21447968C211yesmaximal, characteristic, complemented
4#21443864C211yesmaximal, characteristic, complemented
5#21443782C211yesmaximal, characteristic, complemented
6D4⋊#11758185C211yesmaximal, characteristic, complemented
7C61×#1675C211yesmaximal, characteristic, complemented
8C2×#14519799—21no—
9#16840433—21no—
10#16840139C2211yescharacteristic
11#16839998—21no—
12#16839963—21no—
13#16839909—22no—
14C4×Dic244C2211yescharacteristic
15#16839587—21no—
16C4⋊#11758185C2211yescharacteristic
17#16839565—21no—
18#16839562—22no—
19C4×#11758185C2211yescharacteristic
20C61×#500C2211yescharacteristic
21C61×#476—21no—
22C8×D4×C61C2211yescharacteristic
23C61×#335C2211yescharacteristic
24C61×#333—21no—
25C2×#11758222—22no—
26#14519799—42no—
27#14519799—42no—
28C2×Dic244D411yescharacteristic
29C2×#11758190—22no—
30C2×D488—21no—
31C2×D488D412yes—
32#14519774—21no—
33#14519773—41no—
34#14519751—22no—
35#14519750—42no—
36C4×Dic122—21no—
37#14519739—41no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40#14519711D411yescharacteristic
41#14519702—21no—
42#14519699—22no—
43C22×C8×C61—21no—
44C4⋊Q8×C61—21no—
45C4×D4×C61D411yescharacteristic
46#107×C61—21no—
47#107×C61—21no—
48#106×C61—41no—
49C4⋊C8×C61—21no—
50C4⋊C8×C61D411yescharacteristic
51#102×C61—41no—
52C22⋊C8×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—41no—
55C2×D4×C61D811yescharacteristic
56C22×C4×C61—21no—
57C2×C22⋊D61—42no—
58C2×C22⋊D61—42no—
59#11758222—44no—
60#11758222—82no—
61C2×D244—22no—
62C2×D244—41no—
63C2×D244—41no—
64C2×C4×D61—42no—
65C2×Dic122—21no—
66C2×C8×C61—21no—
67C2×C8×C61—41no—
68C2×C8×C61—41no—
69C2×C8×C61—41no—
70C2×C8×C61C2×D411yescharacteristic
71C2×C8×C61C2×D411yescharacteristic
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C4⋊C4×C61D811yescharacteristic
75C22⋊C4×C61—21no—
76C42×C61C2×D411yescharacteristic
77#11758198—42no—
78C4⋊Dic61—21no—
79#11758196—41no—
80C4×Dic61—41no—
81Dic244—21no—
82Dic244SD1612yes—
83D488—42no—
84D488—41no—
85D488—41no—
86#11758190—44no—
87#11758185—41no—
88#11758185—41no—
89C23×C61—21no—
90C22×D61—42no—
91Q8×C61—161no—
92D4×C61—21no—
93D4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—81no—
98C2×C4×C61#12011yescharacteristic
99C2×C4×C61C2×D811yescharacteristic
100C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
101C22⋊D61—84no—
102C22⋊D61—162no—
103C2×Dic61—41no—
104C2×Dic61—41no—
105D244—42no—
106D244—42no—
107D244—81no—
108D244—81no—
109D244—161no—
110C4×D61—84no—
111Dic122—22no—
112Dic122—41no—
113C8×C61—21no—
114C8×C61—41no—
115C8×C61—81no—
116C8×C61C2×SD1611yescharacteristic
117C8×C61AGL(1,ℤ/8)11yescharacteristic
118#12373—611nomaximal, Sylow
119C22×C61—41no—
120C22×C61—41no—
121C22×C61—41no—
122C22×C61#44711yescharacteristic
123C2×D61—82no—
124C2×D61—82no—
125C2×D61—82no—
126C4×C61—21no—
127C4×C61—41no—
128C4×C61—161no—
129C4×C61#45011yescharacteristic
130C4×C61#45511yescharacteristic
131C4×C61#50811yescharacteristic
132C4×C61#51011yescharacteristic
133Dic61—42no—
134Dic61—81no—
135#3287—611no—
136#2240—611no—
137#2227—611no—
138#1675D6111yescharacteristic, complemented
139#1361—611no—
140#1332—611no—
141D4⋊C16—611no—
142C2×C61—81no—
143C2×C61#163911yescharacteristic
144C2×C61#218911yescharacteristic
145C2×C61#219211yescharacteristic
146D61—162no—
147C2×#135—1221no—
148#550—1221no—
149#500C2×D6111yescharacteristic
150#493—611no—
151#476—21no—
152#466—1221no—
153#460—1221no—
154#449—1222no—
155C4×Q16—611no—
156C8×D4C2×D6111yescharacteristic
157#366—1221no—
158C4⋊C16—611no—
159#358—1221no—
160#357—1222no—
161C4×C16—611no—
162#335C2×D6111yescharacteristic
163#333—21no—
164C61#1237311yescharacteristic, complemented, Sylow
165C2×#41—1222no—
166#135—2442no—
167#135—2442no—
168C2×Q16—611no—
169C2×SD16—1222no—
170C2×D8—612no—
171C2×D8—1221no—
172C22×C8—21no—
173C4⋊Q8—21no—
174#127—1221no—
175#127—2441no—
176#121—1222no—
177#121—2442no—
178C4×Q8—1221no—
179C4×D4—2441no—
180C4×D4C22⋊D6111yescharacteristic
181C2×C16—1221no—
182C2×C16—1221no—
183#107—21no—
184#107—21no—
185#107—611no—
186#106—41no—
187C4⋊C8—21no—
188C4⋊C8C22⋊D6111yescharacteristic
189#103—1221no—
190#102—41no—
191#102—1222no—
192C22⋊C8—21no—
193C4×C8C22×D6111yescharacteristic
194#41—4882no—
195#41—2444no—
196C2×Q8—41no—
197C2×Q8—1221no—
198C2×D4—1222no—
199C2×D4—2442no—
200C2×D4—2442no—
201C2×D4—2441no—
202C2×D4—2441no—
203C2×D4#1175819911yescharacteristic
204C22×C4—21no—
205C22×C4—2442no—
206Q16—612no—
207Q16—1221no—
208SD16—2444no—
209D8—2442no—
210D8—2441no—
211D8—2441no—
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8D4×D6111yescharacteristic
217C2×C8D4×D6111yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—1221no—
221C4⋊C4—2441no—
222C4⋊C4#1175819911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2442no—
225C42—2441no—
226C42C2×C22⋊D6111yescharacteristic
227C16—2441no—
228C16—2441no—
229C23—21no—
230C23—2442no—
231Q8—161no—
232Q8—1222no—
233Q8—2441no—
234D4—21no—
235D4—21no—
236D4—9761no—
237D4—9762no—
238D4—4884no—
239D4—4881no—
240D4—4881no—
241D4—2442no—
242D4—2442no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—4884no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4#1451974511yescharacteristic
251C2×C4C2×#1175819911yescharacteristic
252C2×C4#1451980211yescharacteristic
253C8—21no—
254C8—41no—
255C8—81no—
256C8SD16×D6111yescharacteristic
257C8#1451980411yescharacteristic
258C22—41no—
259C22—41no—
260C22—41no—
261C22—4882no—
262C22—4882no—
263C22—4882no—
264C22#1683989511yescharacteristic
265C4—21no—
266C4—41no—
267C4—161no—
268C4—4881no—
269C4—2442no—
270C4#1683991111yescharacteristic
271C4#1683993811yescharacteristic
272C4#1684019211yescharacteristic
273C4#1684020311yescharacteristic
274C2—81no—
275C2—9762no—
276C2#2144499611yescharacteristic
277C2#2144775711yescharacteristic
278C2#2144777411yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1675C2
socleC22×C61#447
prime radical#21457579C2
hypercenter#1675D61
hypocenterC61#12373
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. #21448044C2semidirect
3. #21447968C2semidirect
4. #21443864C2semidirect
5. #21443782C2semidirect
6. D4⋊#11758185C2semidirect
7. C61×#1675C2semidirect
138. #1675D61semidirect
164. C61#12373semidirect

Maximal subgroups

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

subgroupindex
#214575792
#214480442
#214479682
#214438642
#214437822
D4⋊#117581852
C61×#16752
#1237361