Finite Group Databasebeta

← #29592591

Subgroups — #29592591

Order 15616.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#21451061C211yesmaximal, characteristic, complemented
2#21447043C211yesmaximal, characteristic
3#21444858C212yesmaximal
4#21443847C212yesmaximal, complemented
5C61×#1325C211yesmaximal, characteristic, complemented
6#16840343C2211yescharacteristic
7#16840081—22no—
8C2×#14519705—21no—
9#16839761—21no—
10#16839701—21no—
11#16839701C2212yes—
12#16839661—22no—
13#16839627—22no—
14C4×C4⋊Dic61—21no—
15#16839531C2211yescharacteristic
16#16839497—22no—
17C61×#384C2211yescharacteristic
18C61×#324C2212yes—
19C2×#11758222—21no—
20C2×#11758194—21no—
21#14519757—41no—
22#14519752—41no—
23#14519751—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61D412yes—
28C2×C4×Dic61—22no—
29#14519709—42no—
30#14519705—42no—
31#14519700—41no—
32#14519699—42no—
33#14519697—41no—
34#14519684—21no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684D412yes—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C42×C61D411yescharacteristic
43C22⋊C8×C61—22no—
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61C2×D412yes—
48C2×C22⋊D61—41no—
49C22×Dic61—22no—
50C22×Dic61—22no—
51#11758222—41no—
52#11758222—41no—
53#11758222—41no—
54#11758222—41no—
55C2×D244—21no—
56C2×D244—22no—
57C2×C4×D61—41no—
58C2×Dic122—21no—
59C2×C8×C61—42no—
60C4⋊C4×C61—41no—
61C4⋊C4×C61—44no—
62C4⋊C4×C61—41no—
63C42×C61—21no—
64C42×C61SD1612yes—
65#11758198—42no—
66C4⋊Dic61—22no—
67C4⋊Dic61—42no—
68C4⋊Dic61—42no—
69C4⋊Dic61—42no—
70C4⋊Dic61—81no—
71C4×Dic61—44no—
72#11758194—41no—
73#11758194—41no—
74C2×#92418—41no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—41no—
86C2×C4×C61C2×SD1611yescharacteristic
87C2×C4×C61#13711yescharacteristic
88C22⋊D61—81no—
89C22⋊D61—81no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—41no—
96C2×Dic61—42no—
97D244—41no—
98D244—41no—
99D244—82no—
100C4×D61—81no—
101C4×D61—81no—
102Dic122—41no—
103Dic122—41no—
104C8×C61—82no—
105#92418—81no—
106#12181—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—21no—
109C22×C61#44811yescharacteristic
110C22×C61#45411yescharacteristic
111C22×C61B(4,𝔽2)11yescharacteristic
112C2×D61—81no—
113C2×D61—81no—
114C2×D61—81no—
115C4×C61—24no—
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—42no—
119C4×C61—81no—
120Dic61—82no—
121Dic61—81no—
122Dic61—82no—
123#2571—611no—
124#2039—611no—
125#1616—612no—
126#1353—612no—
127#1325D6111yescharacteristic, complemented
128C2×C61—21no—
129C2×C61—21no—
130C2×C61#159911yescharacteristic
131C2×C61#219811yescharacteristic
132C2×C61#220411yescharacteristic
133D61—161no—
134#537—611no—
135#481—1222no—
136C2×#104—1221no—
137#417—1221no—
138#400—612no—
139#400—1221no—
140#389—1222no—
141#384C2×D6111yescharacteristic
142#382—1222no—
143C4×C4⋊C4—1221no—
144#343—611no—
145#327—1222no—
146#324C2×D6112yes—
147C61#1218111yescharacteristic, complemented, Sylow
148C2×#41—1221no—
149C2×M16—1221no—
150C22⋊Q8—2441no—
151#121—1222no—
152#121—2441no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—612no—
155C2×C4⋊C4—1221no—
156C2×C4⋊C4—1221no—
157C2×C4⋊C4—1221no—
158C2×C4⋊C4D24411yescharacteristic
159C2×C4⋊C4C22×D6111yescharacteristic
160C2×C42—1222no—
161C2×C42D24411yescharacteristic
162#106—2442no—
163#104—2442no—
164#103—2441no—
165#102—2441no—
166#102—2442no—
167C22⋊C8—22no—
168#95—612no—
169#95—1221no—
170#95—1222no—
171#95—1222no—
172#95—1222no—
173#41—2441no—
174#41—2441no—
175#41—2441no—
176#41—2441no—
177C2×Q8—1221no—
178C2×D4—1222no—
179C2×D4—1221no—
180C2×D4—2441no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—2441no—
186C22×C4C2×D24411yescharacteristic
187C22×C4D4×D6112yes—
188M16—2441no—
189M16—2441no—
190C2×C8—42no—
191C2×C8—2441no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—44no—
195C4⋊C4—1222no—
196C4⋊C4—4881no—
197C4⋊C4—2442no—
198C4⋊C4—2442no—
199C4⋊C4—2442no—
200C22⋊C4—2442no—
201C42—21no—
202C42—2444no—
203C42#1175819012yes—
204C23—2441no—
205C23#1451974711yescharacteristic
206Q8—2441no—
207Q8—2441no—
208D4—4882no—
209D4—4881no—
210D4—4881no—
211D4—2441no—
212D4—2441no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—4881no—
223C2×C4—4881no—
224C2×C4—2442no—
225C2×C4—2442no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2441no—
230C2×C4—2442no—
231C2×C4C2×#1175819011yescharacteristic
232C2×C4#1451981011yescharacteristic
233C8—82no—
234C8—4881no—
235C22—21no—
236C22—21no—
237C22—4881no—
238C22—4881no—
239C22—4881no—
240C22#1683990311yescharacteristic
241C22#1683993711yescharacteristic
242C22#1683995011yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—81no—
248C4—4882no—
249C4—4881no—
250C4—4882no—
251C2—21no—
252C2—21no—
253C2—9761no—
254C2#2144476111yescharacteristic
255C2#2144781711yescharacteristic
256C2#2144784711yescharacteristic
subgroupG/H
centerC22#16839950
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1325C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#1325D61
hypocenterC61#12181
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
1. #21451061C2semidirect
4. #21443847C2semidirect
5. C61×#1325C2semidirect
127. #1325D61semidirect
147. C61#12181semidirect

Maximal subgroups

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

subgroupindex
#214510612
#214470432
#214448582
#214438472
C61×#13252
#1218161