Finite Group Databasebeta

← #29591367

Subgroups — #29591367

Order 15616.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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#21456406C211yesmaximal, characteristic
2#21447621C211yesmaximal, characteristic
3#21444566C211yesmaximal, characteristic
4#21443877C211yesmaximal, characteristic, complemented
5#21443869C211yesmaximal, characteristic, complemented
6#21443867C211yesmaximal, characteristic, complemented
7C61×#1365C211yesmaximal, characteristic, complemented
8#16840317—21no—
9#16840142C2211yescharacteristic
10#16840114—21no—
11#16840004—21no—
12#16840004C412yes—
13#16839856—21no—
14#16839856C412yes—
15#16839814—21no—
16C4×C8×D61C2211yescharacteristic
17C4⋊#11758185C2211yescharacteristic
18#16839541—21no—
19#16839537—21no—
20#16839515C2211yescharacteristic
21#16839512—21no—
22C61×#500C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C61×#335C2211yescharacteristic
25C2×Dic244—21no—
26C2×Dic244C2×C412yes—
27C2×C8×D61C2×C411yescharacteristic
28C2×C8×D61D411yescharacteristic
29#14519777—21no—
30#14519768—41no—
31#14519757—22no—
32#14519753—41no—
33Q8×Dic61—41no—
34#14519742—22no—
35#14519735—41no—
36C42×D61D411yescharacteristic
37C2×#11758185—21no—
38C4⋊#92418—21no—
39#14519700—22no—
40#14519700—22no—
41#14519689—41no—
42C4×#92418D411yescharacteristic
43C8×Dic61—41no—
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—
53C2×C4×D61C22⋊C411yescharacteristic
54C2×C4×D61D811yescharacteristic
55C2×C4×D61SD1611yescharacteristic
56C2×Dic122—22no—
57C2×C8×C61—41no—
58C2×C8×C61C22×C411yescharacteristic
59C2×C8×C61C2×D411yescharacteristic
60C4⋊C4×C61—22no—
61C4⋊C4×C61—41no—
62C42×C61C2×D411yescharacteristic
63#11758198—42no—
64C4⋊Dic61—41no—
65C4⋊Dic61—41no—
66C4⋊Dic61—42no—
67#11758196—42no—
68C4×Dic61—42no—
69C4×Dic61C22⋊C411yescharacteristic
70C4×Dic61D811yescharacteristic
71C4×Dic61SD1611yescharacteristic
72C2×#92418—41no—
73C2×#92418C22⋊C411yescharacteristic
74C2×#92418C22⋊C411yescharacteristic
75Dic244—42no—
76Dic244—41no—
77Dic244—41no—
78C8×D61—21no—
79C8×D61—21no—
80C8×D61—41no—
81C16×C61—41no—
82#11758185—41no—
83C22×D61#10211yescharacteristic
84Q8×C61—82no—
85C2×C4×C61—42no—
86C2×C4×C61C2×C22⋊C411yescharacteristic
87C2×C4×C61C2×D811yescharacteristic
88C2×C4×C61C2×SD1611yescharacteristic
89C2×Dic61—42no—
90C2×Dic61#10211yescharacteristic
91C2×Dic61#10211yescharacteristic
92C2×Dic61#10211yescharacteristic
93C4×D61—21no—
94C4×D61—21no—
95C4×D61—21no—
96C4×D61—21no—
97C4×D61—41no—
98Dic122—42no—
99Dic122—42no—
100C8×C61—21no—
101C8×C61—81no—
102C8×C61#13511yescharacteristic
103C8×C61#13511yescharacteristic
104#92418—21no—
105#92418—21no—
106#92418—81no—
107#11953—611nomaximal, Sylow
108C22×C61C2×#10211yescharacteristic
109C2×D61—21no—
110C2×D61—21no—
111C2×D61—21no—
112C4×C61—21no—
113C4×C61—82no—
114C4×C61#41311yescharacteristic
115C4×C61#41611yescharacteristic
116C4×C61#51011yescharacteristic
117C4×C61#51011yescharacteristic
118Dic61—21no—
119Dic61—21no—
120Dic61—21no—
121Dic61—82no—
122#3170—611no—
123#2163—611no—
124#1564—611no—
125#1365—611no—
126#1365D6111yescharacteristic, complemented
127#1362—611no—
128#1362—611no—
129C2×C61#154011yescharacteristic
130C2×C61#214111yescharacteristic
131C2×C61#214811yescharacteristic
132D61—41no—
133#533—1221no—
134#500C2×D6111yescharacteristic
135#494—611no—
136#487—1221no—
137#467—612no—
138#467—1221no—
139C4×Q16—612no—
140C4×Q16—1221no—
141#431—1221no—
142C2×C4×C8—611no—
143C4⋊C16—611no—
144C4⋊C16C2×D6111yescharacteristic
145C22⋊C16—1221no—
146#346—1221no—
147#335—611no—
148#335C2×D6111yescharacteristic
149#334—1221no—
150C61#1195311yescharacteristic, complemented, Sylow
151C2×Q16—612no—
152C2×Q16—1221no—
153C22×C8—611no—
154C22×C8—611no—
155C4⋊Q8—21no—
156C4⋊Q8—1221no—
157#125—2441no—
158C22⋊Q8—1222no—
159C22⋊Q8—2441no—
160C4×Q8—1222no—
161C4×Q8—2441no—
162#117—2441no—
163C2×C42—611no—
164C2×C16—21no—
165C2×C16—1221no—
166#107—21no—
167#107C4×D6112yes—
168C4⋊C8—21no—
169C4⋊C8—1221no—
170#103—1222no—
171#103—1222no—
172C22⋊C8—2441no—
173C4×C8—611no—
174C4×C8—611no—
175C4×C8—611no—
176C4×C8—2441no—
177C4×C8C22×D6111yescharacteristic
178C2×Q8—41no—
179C2×Q8—1222no—
180C22×C4—611no—
181C22×C4—611no—
182C22×C4—611no—
183Q16—2442no—
184Q16—2441no—
185Q16—2441no—
186C2×C8—41no—
187C2×C8—611no—
188C2×C8—611no—
189C2×C8—1221no—
190C2×C8—1221no—
191C2×C8—2441no—
192C2×C8—2441no—
193C2×C8C2×C4×D6111yescharacteristic
194C2×C8D4×D6111yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—2442no—
198C4⋊C4—2442no—
199C4⋊C4—2441no—
200C4⋊C4—2441no—
201C22⋊C4—2442no—
202C42—611no—
203C42—611no—
204C42—611no—
205C42—2442no—
206C42D4×D6111yescharacteristic
207C16—41no—
208C16—2441no—
209C23—611no—
210Q8—82no—
211Q8—2442no—
212Q8—2442no—
213C2×C4—42no—
214C2×C4—611no—
215C2×C4—611no—
216C2×C4—611no—
217C2×C4—1221no—
218C2×C4—1221no—
219C2×C4—1221no—
220C2×C4—1221no—
221C2×C4—2441no—
222C2×C4—2442no—
223C2×C4C22⋊C4×D6111yescharacteristic
224C2×C4D8×D6111yescharacteristic
225C2×C4SD16×D6111yescharacteristic
226C8—21no—
227C8—81no—
228C8—1221no—
229C8—1221no—
230C8—4881no—
231C8#1451979811yescharacteristic
232C8#1451980011yescharacteristic
233C22—1221no—
234C22—1221no—
235C22—1221no—
236C22#102×D6111yescharacteristic
237C4—21no—
238C4—82no—
239C4—1221no—
240C4—1221no—
241C4—1221no—
242C4—4882no—
243C4#1683974511yescharacteristic
244C4#1683975711yescharacteristic
245C4#1684020111yescharacteristic
246C4#1684020411yescharacteristic
247C2—2441no—
248C2#2144446711yescharacteristic
249C2#2144753711yescharacteristic
250C2#2144756511yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#1365C2
socleC22×C61C2×#102
prime radicalC22×D61#102
cosocleC4×C8×C61C23
hypercenter#1365D61
hypocenterC61#11953
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
4. #21443877C2semidirect
5. #21443869C2semidirect
6. #21443867C2semidirect
7. C61×#1365C2semidirect
126. #1365D61semidirect
150. C61#11953semidirect

Maximal subgroups

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

subgroupindex
#214564062
#214476212
#214445662
#214438772
#214438692
#214438672
C61×#13652
#1195361