Finite Group Databasebeta

← #29591362

Subgroups — #29591362

Order 15616.

350 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456399C211yesmaximal, characteristic
2#21447621C211yesmaximal, characteristic, complemented
3#335×D61C211yesmaximal, characteristic, complemented
4#21443877C211yesmaximal, characteristic, complemented
5#21443866C211yesmaximal, characteristic, complemented
6#21443864C211yesmaximal, characteristic, complemented
7C61×#1365C211yesmaximal, characteristic, complemented
8#16840313—21no—
9#16840139C2211yescharacteristic
10#16840110—21no—
11#16839999—21no—
12#16839999C412yes—
13#16839844—21no—
14#16839844C412yes—
15C4⋊C8×D61—21no—
16C4×C8×D61C2211yescharacteristic
17C4⋊#11758185C2211yescharacteristic
18#16839541—21no—
19#16839537—21no—
20#16839515C2211yescharacteristic
21#16839514—21no—
22C61×#500C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C61×#335C2211yescharacteristic
25C2×#11758226—41no—
26C2×D488—21no—
27C2×D488C2×C412yes—
28C2×C8×D61—41no—
29C2×C8×D61C2×C411yescharacteristic
30C2×C8×D61D411yescharacteristic
31#14519774—21no—
32#14519766—41no—
33#14519752—22no—
34#14519752—41no—
35#14519739—22no—
36#14519739—41no—
37C42×D61D411yescharacteristic
38C2×#11758185—21no—
39C4⋊#92418—21no—
40#14519706—41no—
41#14519697—22no—
42#14519697—22no—
43C4×#92418D411yescharacteristic
44C8×Dic61C2×C411yescharacteristic
45C8×Dic61D411yescharacteristic
46C4⋊Q8×C61—21no—
47C2×C16×C61—21no—
48#107×C61—21no—
49#107×C61C2×C412yes—
50C4⋊C8×C61—21no—
51C4×C8×C61C2311yescharacteristic
52C2×Q8×C61—41no—
53#11758226—84no—
54C2×D244—22no—
55C2×D244—42no—
56C2×D244—41no—
57C2×D244—41no—
58C2×C4×D61—42no—
59C2×C4×D61C22⋊C411yescharacteristic
60C2×C4×D61D811yescharacteristic
61C2×C4×D61SD1611yescharacteristic
62C2×C8×C61—41no—
63C2×C8×C61C22×C411yescharacteristic
64C2×C8×C61C2×D411yescharacteristic
65C4⋊C4×C61—22no—
66C4⋊C4×C61—41no—
67C42×C61C2×D411yescharacteristic
68#11758198—42no—
69#11758198—42no—
70C4×Dic61C22⋊C411yescharacteristic
71C4×Dic61D811yescharacteristic
72C4×Dic61SD1611yescharacteristic
73C2×#92418—41no—
74C2×#92418C22⋊C411yescharacteristic
75C2×#92418C22⋊C411yescharacteristic
76D488—42no—
77D488—41no—
78D488—41no—
79C8×D61—21no—
80C8×D61—21no—
81C8×D61—41no—
82C8×D61—82no—
83C16×C61—41no—
84#11758185—41no—
85C22×D61—42no—
86C22×D61#10211yescharacteristic
87Q8×C61—82no—
88C2×C4×C61—42no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C2×C4×C61C2×D811yescharacteristic
91C2×C4×C61C2×SD1611yescharacteristic
92C2×Dic61#10211yescharacteristic
93C2×Dic61#10211yescharacteristic
94C2×Dic61#10211yescharacteristic
95D244—42no—
96D244—42no—
97D244—82no—
98D244—84no—
99D244—81no—
100D244—81no—
101C4×D61—22no—
102C4×D61—21no—
103C4×D61—21no—
104C4×D61—21no—
105C4×D61—21no—
106C4×D61—84no—
107C8×C61—21no—
108C8×C61—81no—
109C8×C61#13511yescharacteristic
110C8×C61#13511yescharacteristic
111#92418—21no—
112#92418—21no—
113#92418—81no—
114#11952—611nomaximal, Sylow
115C22×C61C2×#10211yescharacteristic
116C2×D61—21no—
117C2×D61—21no—
118C2×D61—82no—
119C2×D61—82no—
120C2×D61—82no—
121C2×D61#36112yes—
122C4×C61—21no—
123C4×C61—82no—
124C4×C61#41311yescharacteristic
125C4×C61#41611yescharacteristic
126C4×C61#50911yescharacteristic
127C4×C61#51011yescharacteristic
128Dic61—21no—
129Dic61—21no—
130Dic61—21no—
131#3169—611no—
132#2163—611no—
133C2×#335—611no—
134#1365—611no—
135#1365D6111yescharacteristic, complemented
136#1361—611no—
137#1361—611no—
138C2×C61#154011yescharacteristic
139C2×C61#213911yescharacteristic
140C2×C61C2×#36111yescharacteristic
141D61—22no—
142D61—162no—
143#532—1221no—
144#500C2×D6111yescharacteristic
145#493—611no—
146#486—1221no—
147#466—612no—
148#466—1221no—
149C4×D8—612no—
150C4×D8—1221no—
151C2×C4⋊C8—1221no—
152C2×C4×C8—611no—
153C4⋊C16—611no—
154C4⋊C16C2×D6111yescharacteristic
155C22⋊C16—1221no—
156#346—1221no—
157#335—611no—
158#335—1221no—
159#335C2×D6111yescharacteristic
160C61#1195211yescharacteristic, complemented, Sylow
161C2×#41—2441no—
162C2×D8—612no—
163C2×D8—1221no—
164C22×C8—611no—
165C22×C8—611no—
166C22×C8—2441no—
167C4⋊Q8—21no—
168#127—1221no—
169#124—2441no—
170#121—1222no—
171#121—2441no—
172C4×D4—1222no—
173C4×D4—2441no—
174C2×C42—611no—
175C2×C16—21no—
176C2×C16—1221no—
177#107—21no—
178#107C4×D6112yes—
179C4⋊C8—21no—
180C4⋊C8—1221no—
181C4⋊C8—2441no—
182#102—1222no—
183#102—1222no—
184C4×C8—611no—
185C4×C8—611no—
186C4×C8—611no—
187C4×C8C22×D6111yescharacteristic
188#41—4884no—
189C2×Q8—41no—
190C2×D4—1222no—
191C2×D4—2442no—
192C2×D4—2441no—
193C2×D4—2441no—
194C22×C4—611no—
195C22×C4—611no—
196C22×C4—611no—
197C22×C4—2442no—
198D8—2442no—
199D8—2441no—
200D8—2441no—
201C2×C8—41no—
202C2×C8—611no—
203C2×C8—611no—
204C2×C8—1221no—
205C2×C8—1221no—
206C2×C8—4882no—
207C2×C8—2441no—
208C2×C8—2441no—
209C2×C8C2×C4×D6111yescharacteristic
210C2×C8D4×D6111yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—41no—
213C22⋊C4—2442no—
214C22⋊C4—2442no—
215C42—611no—
216C42—611no—
217C42—611no—
218C42D4×D6111yescharacteristic
219C16—41no—
220C16—2441no—
221C23—611no—
222C23—2442no—
223Q8—82no—
224D4—4882no—
225D4—4884no—
226D4—4881no—
227D4—4881no—
228D4—2442no—
229D4—2442no—
230C2×C4—42no—
231C2×C4—611no—
232C2×C4—611no—
233C2×C4—611no—
234C2×C4—1222no—
235C2×C4—1221no—
236C2×C4—1221no—
237C2×C4—1221no—
238C2×C4—1221no—
239C2×C4—4884no—
240C2×C4C22⋊C4×D6111yescharacteristic
241C2×C4D8×D6111yescharacteristic
242C2×C4SD16×D6111yescharacteristic
243C8—21no—
244C8—81no—
245C8—1221no—
246C8—1221no—
247C8—4881no—
248C8#1451979811yescharacteristic
249C8#1451980011yescharacteristic
250C22—612no—
251C22—1221no—
252C22—1221no—
253C22—4882no—
254C22—4882no—
255C22—4882no—
256C22#102×D6111yescharacteristic
257C4—21no—
258C4—82no—
259C4—1221no—
260C4—1221no—
261C4—1221no—
262C4#1683974511yescharacteristic
263C4#1683975711yescharacteristic
264C4#1684019911yescharacteristic
265C4#1684020411yescharacteristic
266C2—1222no—
267C2—9762no—
268C2#2144446711yescharacteristic
269C2#2144753011yescharacteristic
270C2#361×D6111yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#1365C2
socleC22×C61C2×#102
prime radical#21456399C2
cosocleC4×C8×C61C23
hypercenter#1365D61
hypocenterC61#11952
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
2. #21447621C2semidirect
3. #335×D61C2semidirect
4. #21443877C2semidirect
5. #21443866C2semidirect
6. #21443864C2semidirect
7. C61×#1365C2semidirect
135. #1365D61semidirect
160. C61#11952semidirect

Maximal subgroups

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

subgroupindex
#214563992
#214476212
#335×D612
#214438772
#214438662
#214438642
C61×#13652
#1195261