Finite Group Databasebeta

← #29579396

Subgroups — #29579396

Order 15616.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#21456142C211yesmaximal, characteristic
2#21445950C211yesmaximal, characteristic, complemented
3#21444384C212yesmaximal, complemented
4#21444380C212yesmaximal
5C61×#1450C211yesmaximal, characteristic, complemented
6#16840319C2211yescharacteristic
7#16840312C2211yescharacteristic
8#16840117—21no—
9#16840117—21no—
10#16840113C412yescomplemented
11#16840109C412yescomplemented
12#16839759—22no—
13#16839628C2212yes—
14#16839525—22no—
15#16839500—24no—
16#16839497—24no—
17C61×#431C2212yes—
18C2×C4×C8×C61C2211yescharacteristic
19C61×#322—21no—
20C82×C61—21no—
21C2×#11758222—21no—
22#14519777C2×C411yescharacteristic
23#14519777C2×C412yes—
24#14519774C2×C411yescharacteristic
25#14519769—21no—
26#14519765—21no—
27#14519756—22no—
28#14519751—22no—
29C4×Dic122—22no—
30C4×D244—22no—
31#14519736—21no—
32C2×C4⋊Dic61—22no—
33#14519702—22no—
34#14519702—22no—
35#14519699—22no—
36#14519699—22no—
37#14519684—22no—
38#14519684—22no—
39C22×C8×C61D412yes—
40C2×C42×C61C2311yescharacteristic
41C4⋊C8×C61—22no—
42C22⋊C8×C61D414yes—
43C4×C8×C61—22no—
44C4×C8×C61C2×C412yes—
45C4×C8×C61D411yescharacteristic
46C4×C8×C61D411yescharacteristic
47C22×C4×C61C2×D412yes—
48C22×C4×C61C2×D411yescharacteristic
49C2×C22⋊D61—41no—
50C22×Dic61—42no—
51#11758222—44no—
52C2×D244—21no—
53C2×D244—22no—
54C2×C4×D61—41no—
55C2×Dic122—21no—
56C2×Dic122—22no—
57C2×C8×C61—24no—
58C2×C8×C61—24no—
59C2×C8×C61C22⋊C412yes—
60C2×C8×C61C22⋊C412yes—
61C42×C61C22×C411yescharacteristic
62C42×C61C2×D411yescharacteristic
63C42×C61#4112yes—
64C22⋊Dic61—41no—
65#11758198—42no—
66C4⋊Dic61—22no—
67C4⋊Dic61—22no—
68C4⋊Dic61—22no—
69C4⋊Dic61—44no—
70#11758196—42no—
71C4×Dic61—41no—
72C23×C61#12711yescharacteristic
73C22×D61—41no—
74C2×C4×C61—42no—
75C2×C4×C61C2×C22⋊C411yescharacteristic
76C2×C4×C61C4×D412yes—
77C2×C4×C61#12112yes—
78C2×C4×C61#12411yescharacteristic
79C2×C4×C61#13514yes—
80C22⋊D61—82no—
81C2×Dic61—42no—
82C2×Dic61—41no—
83C2×Dic61—84no—
84C2×Dic61—82no—
85D244—42no—
86D244—82no—
87C4×D61—82no—
88Dic122—42no—
89Dic122—82no—
90C8×C61—22no—
91C8×C61—22no—
92C8×C61—44no—
93#9757—611nomaximal, Sylow
94C22×C61—22no—
95C22×C61#39011yescharacteristic
96C22×C61#49512yes—
97C2×D61—81no—
98C2×D61—82no—
99C4×C61—22no—
100C4×C61—22no—
101C4×C61#41612yes—
102C4×C61#48812yes—
103Dic61—81no—
104Dic61—82no—
105Dic61—162no—
106#3131—611no—
107#1835—611no—
108#1518—612no—
109#1517—612no—
110#1450D6111yescharacteristic, complemented
111C2×C61—22no—
112C2×C61#195811yescharacteristic
113C2×C61#197612yes—
114D61—161no—
115#533—611no—
116#532—611no—
117#488—1221no—
118#488—1221no—
119#487—612no—
120#486—612no—
121#431C2×D6112yes—
122#416—1222no—
123C2×C4×C8C2×D6111yescharacteristic
124#382—612no—
125#340—1222no—
126#328—1224no—
127#327—1224no—
128#322—21no—
129C82—21no—
130C61#975711yescharacteristic, complemented, Sylow
131C2×#41—1221no—
132C22×C8D24412yes—
133C4⋊Q8—611no—
134C4⋊Q8—612no—
135#127—611no—
136#125—1221no—
137#124—1221no—
138C22⋊Q8—1222no—
139#121—1222no—
140C4×Q8—1222no—
141C4×D4—1222no—
142#117—1221no—
143C2×C4⋊C4—1222no—
144C2×C42C22×D6111yescharacteristic
145C4⋊C8—22no—
146#103—1222no—
147#103—1222no—
148#102—1222no—
149#102—1222no—
150C22⋊C8D24414yes—
151C4×C8—22no—
152C4×C8C4×D6112yes—
153C4×C8C22⋊D6111yescharacteristic
154C4×C8C22⋊D6111yescharacteristic
155#95—1222no—
156#95—1222no—
157#41—2444no—
158C2×Q8—1221no—
159C2×Q8—1222no—
160C2×D4—1221no—
161C2×D4—1222no—
162C2×D4—2441no—
163C22×C4—2442no—
164C22×C4—2441no—
165C22×C4C2×D24412yes—
166C22×C4C2×D24411yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8#1175819812yes—
170C2×C8#1175819812yes—
171C4⋊C4—1222no—
172C4⋊C4—1222no—
173C4⋊C4—1222no—
174C4⋊C4—2444no—
175C4⋊C4—2442no—
176C22⋊C4—2441no—
177C22⋊C4—2442no—
178C42—2441no—
179C42C2×C4×D6111yescharacteristic
180C42#1175822212yes—
181C42C2×C22⋊D6111yescharacteristic
182C23—2441no—
183C23#1451977411yescharacteristic
184Q8—4882no—
185Q8—2442no—
186D4—4882no—
187D4—4882no—
188D4—2442no—
189C2×C4—42no—
190C2×C4—4884no—
191C2×C4—4882no—
192C2×C4—4882no—
193C2×C4—2442no—
194C2×C4—2441no—
195C2×C4C2×#1175819811yescharacteristic
196C2×C4C4×D24412yes—
197C2×C4#1451975112yes—
198C2×C4#1451976511yescharacteristic
199C2×C4#1451979914yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—4882no—
205C22—4881no—
206C22#1683966411yescharacteristic
207C22#1684014612yes—
208C4—22no—
209C4—22no—
210C4—9762no—
211C4—4882no—
212C4—4881no—
213C4#1683975912yes—
214C4#1684011712yes—
215C2—22no—
216C2—9761no—
217C2#2144663011yescharacteristic
218C2#2144671712yes—
subgroupG/H
centerC22#16839664
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1450C2
socleC22×C61#390
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1450D61
hypocenterC61#9757
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #21445950C2semidirect
3. #21444384C2semidirect
5. C61×#1450C2semidirect
10. #16840113C4semidirect
11. #16840109C4semidirect
110. #1450D61semidirect
130. C61#9757semidirect

Maximal subgroups

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

subgroupindex
#214561422
#214459502
#214443842
#214443802
C61×#14502
#975761