Finite Group Databasebeta

← #29677063

Subgroups — #29677063

Order 15616.

690 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 131 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#21458060C214yesmaximal, complemented
2#21445349C214yesmaximal, complemented
3#21444962C212yesmaximal, complemented
4#21444871C212yesmaximal, complemented
5#21444705C212yesmaximal, complemented
6C61×#1709C211yesmaximal, characteristic, complemented
7#16840478C2212yes—
8#16840397C2212yescomplemented
9#16840084—24no—
10#16840049C2214yescomplemented
11#16840014—24no—
12#16839954C2214yes—
13#16839897—24no—
14C4×D488C2212yes—
15#16839841C2214yescomplemented
16#16839765—24no—
17#16839515C2212yescomplemented
18#16839511C2212yescomplemented
19#16839506C2212yescomplemented
20#16839491C2214yescomplemented
21C61×#494C2211yescharacteristic, complemented
22C61×#493C2211yescharacteristic
23C61×#486C2214yescomplemented
24C82×C61C2211yescharacteristic, complemented
25C2×D4×D61—24no—
26C2×D488D412yes—
27C2×#11758199—24no—
28C2×#11758189—24no—
29#14519766—42no—
30#14519752—24no—
31#14519748—24no—
32#14519745—42no—
33C4×D244C2314yes—
34#14519735—24no—
35#14519711D412yes—
36#14519708—24no—
37C4⋊#92418C2314yes—
38#14519699D414yes—
39#14519698—24no—
40#14519697—24no—
41#14519689—24no—
42C2×Q16×C61—41no—
43C2×D8×C61—41no—
44C4⋊Q8×C61C2312yes—
45#127×C61C2312yes—
46#102×C61—24no—
47#102×C61—24no—
48C4×C8×C61C2311yescharacteristic
49C4×C8×C61C2312yes—
50C23×D61—42no—
51C2×Q8×C61—22no—
52C2×D4×C61—24no—
53C2×D4×C61—22no—
54C2×C22⋊D61—44no—
55D4×D61—48no—
56D4×D61—84no—
57C2×D244—42no—
58C2×D244C2×D414yes—
59C2×C4×D61—24no—
60Q16×C61—82no—
61D8×C61—82no—
62C2×C8×C61—21no—
63C2×C8×C61C2×D412yes—
64C2×C8×C61C2×D412yes—
65C4⋊C4×C61—24no—
66C42×C61C2411yescharacteristic
67C22⋊Dic61—42no—
68#11758199—84no—
69#11758198—24no—
70#11758198—44no—
71C4⋊Dic61C2×D414yes—
72C4×Dic61—42no—
73C2×#92418—24no—
74D488—42no—
75#11758189—84no—
76C23×C61—42no—
77C22×D61—24no—
78C22×D61—88no—
79C22×D61—162no—
80Q8×C61—82no—
81D4×C61—44no—
82D4×C61—44no—
83D4×C61—82no—
84C2×C4×C61—42no—
85C2×C4×C61C22×D412yes—
86C2×C4×C61H(5,𝔽2)11yescharacteristic
87C22⋊D61—88no—
88C2×Dic61—24no—
89D244—24no—
90D244—24no—
91D244—162no—
92C4×D61—44no—
93C4×D61—44no—
94C8×C61—22no—
95C8×C61—22no—
96C8×C61—22no—
97#92418—44no—
98#20302—611nomaximal, Sylow
99C22×C61—84no—
100C22×C61—82no—
101C22×C61D4211yescharacteristic
102C2×D61—44no—
103C2×D61—44no—
104C2×D61—44no—
105C2×D61—168no—
106C4×C61—21no—
107C4×C61—82no—
108C4×C61#57512yes—
109C4×C61#57612yes—
110Dic61—44no—
111#3325—614no—
112#1709D6111yescharacteristic, complemented
113#1697—614no—
114#1634—612no—
115#1619—612no—
116#1589—612no—
117C2×C61—162no—
118C2×C61#327312yes—
119C2×C61#329211yescharacteristic
120D61—84no—
121#555—612no—
122D42—612no—
123#494C2×D6111yescharacteristic, complemented
124#493C2×D6111yescharacteristic
125#486C2×D6114yescomplemented
126#482—1224no—
127#476—614no—
128#469—1224no—
129#459—614no—
130#447—1224no—
131C4×D8—612no—
132#436—614no—
133#418—1224no—
134#335—612no—
135#333—612no—
136#331—612no—
137D4⋊C8—614no—
138C82C2×D6111yescharacteristic, complemented
139C61#2030211yescharacteristic, complemented, Sylow
140C22×D4—1224no—
141C2×Q16—41no—
142C2×D8—41no—
143C2×D8—612no—
144C2×D8—1224no—
145C2×M16—1224no—
146C4⋊Q8C22×D6112yes—
147#127C22×D6112yes—
148#124—2442no—
149#121—1224no—
150#121—1224no—
151#120—2442no—
152C4×D4—614no—
153#117—1224no—
154#107—612no—
155#106—1224no—
156C4⋊C8—614no—
157#102—24no—
158#102—24no—
159#102—614no—
160#102—1224no—
161#102—1224no—
162C22⋊C8—1224no—
163C4×C8C22×D6111yescharacteristic
164C4×C8C22×D6112yes—
165C24—2442no—
166C2×Q8—22no—
167C2×D4—24no—
168C2×D4—22no—
169C2×D4—614no—
170C2×D4—4884no—
171C2×D4—2442no—
172C2×D4—2444no—
173C2×D4—2448no—
174C22×C4—1224no—
175Q16—82no—
176D8—82no—
177D8—4884no—
178D8—2442no—
179M16—4884no—
180C2×C8—21no—
181C2×C8—1224no—
182C2×C8D4×D6112yes—
183C2×C8D4×D6112yes—
184C4⋊C4—24no—
185C4⋊C4—614no—
186C22⋊C4—1224no—
187C22⋊C4—2444no—
188C22⋊C4—2442no—
189C42—2442no—
190C42C23×D6111yescharacteristic
191C23—42no—
192C23—1224no—
193C23—9762no—
194C23—4888no—
195Q8—82no—
196D4—44no—
197D4—44no—
198D4—82no—
199D4—1224no—
200D4—1224no—
201D4—9762no—
202D4—4888no—
203C2×C4—42no—
204C2×C4—1224no—
205C2×C4—2444no—
206C2×C4—2444no—
207C2×C4C2×D4×D6112yes—
208C2×C4#1451982211yescharacteristic
209C8—22no—
210C8—22no—
211C8—22no—
212C8—2444no—
213C22—84no—
214C22—82no—
215C22—9768no—
216C22—2444no—
217C22—2444no—
218C22—2444no—
219C22#1684039711yescharacteristic
220C4—21no—
221C4—82no—
222C4—2444no—
223C4#1684055912yes—
224C4#1684056412yes—
225C2—162no—
226C2—4884no—
227C2#2145742812yes—
228C2#2145761811yescharacteristic
subgroupG/H
centerC22#16840397
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1709C2
socleC22×C61D42
hypercenter#1709D61
hypocenterC61#20302
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1709

Complements

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

NH ≅ G/Ntype
1. #21458060C2semidirect
2. #21445349C2semidirect
3. #21444962C2semidirect
4. #21444871C2semidirect
5. #21444705C2semidirect
6. C61×#1709C2semidirect
8. #16840397C22semidirect
10. #16840049C22semidirect
15. #16839841C22semidirect
17. #16839515C22semidirect
18. #16839511C22semidirect
19. #16839506C22semidirect
20. #16839491C22semidirect
21. C61×#494C22semidirect
23. C61×#486C22semidirect
24. C82×C61C22semidirect
112. #1709D61semidirect
123. #494C2×D61semidirect
125. #486C2×D61semidirect
138. C82C2×D61semidirect
139. C61#20302semidirect

Maximal subgroups

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

subgroupindex
#214580602
#214453492
#214449622
#214448712
#214447052
C61×#17092
#2030261