Finite Group Databasebeta

← #29593827

Subgroups — #29593827

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#21457578C211yesmaximal, characteristic
2#21448044C211yesmaximal, characteristic, complemented
3#21445215C211yesmaximal, characteristic, complemented
4#21443866C211yesmaximal, characteristic, complemented
5#21443783C211yesmaximal, characteristic, complemented
6#21443773C211yesmaximal, characteristic, complemented
7C61×#2227C211yesmaximal, characteristic, complemented
8C2×#14519800—21no—
9#16840434—21no—
10#16840139C2211yescharacteristic
11#16840049—21no—
12#16839999—21no—
13#16839962—21no—
14#16839910—22no—
15C8×D244C2211yescharacteristic
16#16839562—22no—
17C4×#11758185C2211yescharacteristic
18#16839515C2211yescharacteristic
19#16839511—21no—
20C61×#500C2211yescharacteristic
21C4×Q16×C61C2211yescharacteristic
22C61×#366—21no—
23C61×C4⋊C16C2211yescharacteristic
24C61×#358—21no—
25C2×#11758226—22no—
26#14519800—42no—
27#14519800—42no—
28C2×#11758201—22no—
29C2×D488—21no—
30C2×D488D412yes—
31C2×C8×D61—21no—
32#14519774—21no—
33#14519774—41no—
34#14519752—22no—
35#14519752—42no—
36C4×D244—41no—
37C4×D244D411yescharacteristic
38C2×#11758185—21no—
39#14519708—41no—
40C4⋊#92418—21no—
41C4⋊#92418D411yescharacteristic
42#14519697—22no—
43#14519697—41no—
44#14519689—21no—
45C2×Q16×C61D411yescharacteristic
46C4⋊Q8×C61—21no—
47C4×Q8×C61—21no—
48C2×C16×C61—21no—
49#107×C61—21no—
50#107×C61—21no—
51#107×C61D411yescharacteristic
52#103×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—21no—
55C2×Q8×C61—41no—
56#11758226—44no—
57#11758226—82no—
58C2×D244—22no—
59C2×D244—42no—
60C2×D244—41no—
61C2×D244—42no—
62C2×D244—41no—
63C2×D244D811yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—42no—
66Q16×C61—21no—
67Q16×C61SD1612yes—
68C2×C8×C61C2×D411yescharacteristic
69C2×C8×C61C2×D411yescharacteristic
70C4⋊C4×C61—21no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75C42×C61C2×D411yescharacteristic
76#11758201—44no—
77#11758198—21no—
78#11758198—42no—
79C4⋊Dic61D811yescharacteristic
80C2×#92418—21no—
81C2×#92418—41no—
82D488—42no—
83D488—41no—
84D488—41no—
85C8×D61—41no—
86C8×D61—41no—
87C16×C61—41no—
88#11758185—41no—
89C22×D61—21no—
90C22×D61—42no—
91Q8×C61—22no—
92Q8×C61—41no—
93Q8×C61—161no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—81no—
97C2×C4×C61#12011yescharacteristic
98C2×C4×C61C2×D811yescharacteristic
99C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
100C2×Dic61—21no—
101D244—21no—
102D244—21no—
103D244—42no—
104D244—42no—
105D244—84no—
106D244—81no—
107D244—81no—
108D244—161no—
109D244—162no—
110C4×D61—41no—
111C4×D61—41no—
112C4×D61—84no—
113C8×C61—21no—
114C8×C61C2×SD1611yescharacteristic
115C8×C61AGL(1,ℤ/8)11yescharacteristic
116#92418—41no—
117#92418—81no—
118#12373—611nomaximal, Sylow
119C22×C61#44711yescharacteristic
120C2×D61—41no—
121C2×D61—41no—
122C2×D61—41no—
123C2×D61—82no—
124C2×D61—82no—
125C2×D61—82no—
126C4×C61—21no—
127C4×C61—42no—
128C4×C61—81no—
129C4×C61—161no—
130C4×C61#45011yescharacteristic
131C4×C61#45511yescharacteristic
132C4×C61#50811yescharacteristic
133C4×C61#51011yescharacteristic
134Dic61—41no—
135#3287—611no—
136#2240—611no—
137#2227D6111yescharacteristic, complemented
138#1675—611no—
139#1361—611no—
140#1332—611no—
141D4⋊C16—611no—
142C2×C61#163911yescharacteristic
143C2×C61#218911yescharacteristic
144C2×C61#219211yescharacteristic
145D61—81no—
146D61—162no—
147C2×#135—1221no—
148#550—1221no—
149#500C2×D6111yescharacteristic
150#493—611no—
151#476—1221no—
152#466—1221no—
153#460—1221no—
154#449—1222no—
155C4×Q16C2×D6111yescharacteristic
156C8×D4—611no—
157#366—21no—
158C4⋊C16C2×D6111yescharacteristic
159#358—21no—
160#357—1222no—
161C4×C16—611no—
162#335—611no—
163#333—1221no—
164C61#1237311yescharacteristic, complemented, Sylow
165C2×#41—1222no—
166#135—2442no—
167#135—2442no—
168C2×Q16C22⋊D6111yescharacteristic
169C2×SD16—1222no—
170C2×D8—612no—
171C2×D8—1221no—
172C22×C8—1221no—
173C4⋊Q8—21no—
174#127—1221no—
175#127—2441no—
176#121—1222no—
177#121—2442no—
178C4×Q8—21no—
179C4×D4—611no—
180C4×D4—2441no—
181C2×C16—21no—
182C2×C16—1221no—
183#107—21no—
184#107—21no—
185#107C22⋊D6111yescharacteristic
186#106—2441no—
187C4⋊C8—611no—
188C4⋊C8—1221no—
189#103—21no—
190#102—1222no—
191#102—2441no—
192C22⋊C8—1221no—
193C4×C8C22×D6111yescharacteristic
194#41—4882no—
195#41—2444no—
196C2×Q8—21no—
197C2×Q8—41no—
198C2×D4—611no—
199C2×D4—1222no—
200C2×D4—2442no—
201C2×D4—2442no—
202C2×D4—2441no—
203C2×D4—2441no—
204C22×C4—1221no—
205C22×C4—2442no—
206Q16—21no—
207Q16#1175820112yes—
208SD16—2444no—
209D8—2442no—
210D8—2441no—
211D8—2441no—
212C2×C8—1221no—
213C2×C8—2441no—
214C2×C8—2441no—
215C2×C8—2441no—
216C2×C8D4×D6111yescharacteristic
217C2×C8C2×C22⋊D6111yescharacteristic
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—611no—
223C22⋊C4—1221no—
224C22⋊C4—2442no—
225C42—41no—
226C42D4×D6111yescharacteristic
227C16—41no—
228C16—2441no—
229C23—1221no—
230C23—2442no—
231Q8—22no—
232Q8—41no—
233Q8—161no—
234D4—1221no—
235D4—1221no—
236D4—9761no—
237D4—9762no—
238D4—4884no—
239D4—4881no—
240D4—4881no—
241D4—2442no—
242D4—2442no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4—1221no—
247C2×C4—4884no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4#1451974511yescharacteristic
251C2×C4D8×D6111yescharacteristic
252C2×C4#1451980111yescharacteristic
253C8—21no—
254C8—4881no—
255C8—2441no—
256C8C2×#1175820111yescharacteristic
257C8#1451980211yescharacteristic
258C22—4882no—
259C22—4882no—
260C22—4882no—
261C22—2441no—
262C22—2441no—
263C22—2441no—
264C22#1683989411yescharacteristic
265C4—21no—
266C4—42no—
267C4—81no—
268C4—161no—
269C4—2441no—
270C4#1683991211yescharacteristic
271C4#1683993811yescharacteristic
272C4#1684019511yescharacteristic
273C4#1684020411yescharacteristic
274C2—9762no—
275C2—4881no—
276C2#2144499711yescharacteristic
277C2#2144775811yescharacteristic
278C2#2144777311yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2227C2
socleC22×C61#447
prime radical#21457578C2
hypercenter#2227D61
hypocenterC61#12373
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
2. #21448044C2semidirect
3. #21445215C2semidirect
4. #21443866C2semidirect
5. #21443783C2semidirect
6. #21443773C2semidirect
7. C61×#2227C2semidirect
137. #2227D61semidirect
164. C61#12373semidirect

Maximal subgroups

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

subgroupindex
#214575782
#214480442
#214452152
#214438662
#214437832
#214437732
C61×#22272
#1237361