Finite Group Databasebeta

← #29592761

Subgroups — #29592761

Order 15616.

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

normal subgroups only

This group has 388 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#21453856C211yesmaximal, characteristic
2#21446986C211yesmaximal, characteristic
3#21444812C212yesmaximal
4#21443820C212yesmaximal
5C61×#1317C211yesmaximal, characteristic
6#16840388C2211yescharacteristic
7C2×#14519756—21no—
8#16840093—22no—
9#16839702—22no—
10#16839687—21no—
11#16839687C2212yes—
12#16839683—22no—
13#16839625—21no—
14#16839529C2211yescharacteristic
15#16839528—21no—
16#16839500—22no—
17C61×#381C2211yescharacteristic
18C61×#323C2212yes—
19C22×Dic122—21no—
20#14519756—22no—
21#14519756—42no—
22#14519756—42no—
23#14519754—42no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61D412yes—
27C2×#11758196—22no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31#14519709—42no—
32#14519702—42no—
33#14519684—22no—
34#14519684—21no—
35#14519684—22no—
36#14519684—42no—
37#14519684D412yes—
38C23×C4×C61D411yescharacteristic
39C2×C22⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61C2311yescharacteristic
41C22⋊C8×C61—22no—
42#95×C61—22no—
43C24×C61C2×D411yescharacteristic
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61—41no—
48C22×C4×C61—41no—
49C22×C4×C61SD1612yes—
50C22×C4×C61C2×D412yes—
51C22×Dic61—22no—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C22×Dic61—41no—
55C2×Dic122—22no—
56C2×Dic122—42no—
57C2×Dic122—81no—
58C2×Dic122—81no—
59C2×C8×C61—42no—
60C22⋊C4×C61—41no—
61C22⋊C4×C61—41no—
62C22⋊Dic61—41no—
63C22⋊Dic61—41no—
64C22⋊Dic61—42no—
65C22⋊Dic61—42no—
66C22⋊Dic61—41no—
67C22⋊Dic61—41no—
68C4⋊Dic61—22no—
69C4⋊Dic61—42no—
70C4⋊Dic61—42no—
71C4⋊Dic61—81no—
72#11758196—44no—
73#11758196—42no—
74C23×C61—21no—
75C23×C61—21no—
76C23×C61—42no—
77C23×C61#12011yescharacteristic
78C23×C61C2×SD1611yescharacteristic
79C23×C61AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—41no—
83C2×C4×C61—41no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—41no—
88C2×C4×C61—81no—
89C2×Dic61—44no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—82no—
96C2×Dic61—81no—
97C2×Dic61—82no—
98Dic122—84no—
99Dic122—82no—
100C8×C61—82no—
101#12207—611nomaximal, Sylow
102C22×C61—22no—
103C22×C61—21no—
104C22×C61—21no—
105C22×C61—41no—
106C22×C61—41no—
107C22×C61—42no—
108C22×C61—81no—
109C22×C61#45011yescharacteristic
110C22×C61#45511yescharacteristic
111C22×C61B(4,𝔽2)11yescharacteristic
112C4×C61—42no—
113C4×C61—42no—
114C4×C61—81no—
115Dic61—82no—
116Dic61—82no—
117Dic61—82no—
118Dic61—161no—
119#2817—611no—
120#2030—611no—
121#1608—612no—
122#1344—612no—
123#1317D6111yescharacteristic
124C2×C61—21no—
125C2×C61—21no—
126C2×C61—41no—
127C2×C61—41no—
128C2×C61#160111yescharacteristic
129C2×C61#220112yes—
130#543—611no—
131C2×C22⋊Q8—1221no—
132#483—1222no—
133#400—1222no—
134#397—612no—
135#397—1221no—
136#395—1222no—
137#381—1221no—
138#381C2×D6111yescharacteristic
139#342—611no—
140#342—1221no—
141#328—1222no—
142#323C2×D6112yes—
143C61#1220711yescharacteristic, complemented, Sylow
144C22×Q8—1221no—
145C23×C4D24411yescharacteristic
146C22⋊Q8—1222no—
147C22⋊Q8—2442no—
148C22⋊Q8—2442no—
149C22⋊Q8—2442no—
150C2×C4⋊C4—612no—
151C2×C4⋊C4—1221no—
152C2×C4⋊C4—1222no—
153C2×C4⋊C4—1221no—
154C2×C22⋊C4—1222no—
155C2×C22⋊C4—1221no—
156C2×C22⋊C4—1221no—
157C2×C22⋊C4D24411yescharacteristic
158C2×C22⋊C4C22×D6111yescharacteristic
159#106—2442no—
160#103—2442no—
161C22⋊C8—22no—
162#95—22no—
163#95—612no—
164#95—1222no—
165#95—1221no—
166#95—1222no—
167#95—2442no—
168C24C2×D24411yescharacteristic
169C2×Q8—1222no—
170C2×Q8—4881no—
171C2×Q8—4881no—
172C2×Q8—2442no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—1222no—
179C22×C4—1222no—
180C22×C4—1222no—
181C22×C4—2441no—
182C22×C4#1175819012yes—
183C22×C4D4×D6112yes—
184C2×C8—42no—
185C4⋊C4—1222no—
186C4⋊C4—4881no—
187C4⋊C4—2442no—
188C4⋊C4—2444no—
189C4⋊C4—2442no—
190C4⋊C4—2442no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—2441no—
194C22⋊C4—2441no—
195C22⋊C4—2441no—
196C22⋊C4—2441no—
197C22⋊C4—2442no—
198C22⋊C4—2442no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1451974711yescharacteristic
203C23C2×#1175819011yescharacteristic
204C23#1451980511yescharacteristic
205Q8—4882no—
206Q8—4884no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—81no—
216C2×C4—4881no—
217C2×C4—4882no—
218C2×C4—4882no—
219C2×C4—2444no—
220C2×C4—2442no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2442no—
224C2×C4—2442no—
225C8—82no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1683991511yescharacteristic
234C22#1683994211yescharacteristic
235C22#1683995011yescharacteristic
236C4—42no—
237C4—42no—
238C4—81no—
239C4—9761no—
240C4—4882no—
241C4—4882no—
242C4—4882no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2144477311yescharacteristic
248C2#2144783312yes—
subgroupG/H
centerC22#16839950
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1317C2
socleC22×C61B(4,𝔽2)
prime radicalC24×C61C2×D4
hypercenter#1317D61
hypocenterC61#12207
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1317

Complements

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

NH ≅ G/Ntype
143. C61#12207semidirect

Maximal subgroups

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

subgroupindex
#214538562
#214469862
#214448122
#214438202
C61×#13172
#1220761