Finite Group Databasebeta

← #29647764

Subgroups — #29647764

Order 15616.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#21459629C211yesmaximal, characteristic
2#21454900C211yesmaximal, characteristic
3#21444382C216yesmaximal
4#21444317C216yesmaximal
5C61×#2351C211yesmaximal, characteristic
6Q8×Dic122C414yescomplemented
7#16840491C414yescomplemented
8#16840488—24no—
9#16840393—23no—
10#16840319C2216yes—
11#16840240—21no—
12Q8×#92418—24no—
13#16839815C2216yes—
14#16839498C22112yes—
15#16839498C2214yes—
16C2×C4×Q8×C61—21no—
17C2×C4×Q8×C61C2211yescharacteristic
18C61×#389—23no—
19C61×#382C2213yes—
20C4×C4⋊C4×C61C2213yes—
21#14519777C2×C416yes—
22#14519777C2×C416yes—
23#14519776—212no—
24#14519769—26no—
25#14519768—212no—
26#14519756—26no—
27#14519756C2314yes—
28Q8×Dic61—24no—
29Q8×Dic61—24no—
30C4×Dic122—26no—
31C4×Dic122C2×C4112yes—
32#14519736—26no—
33C4⋊#92418—26no—
34C22⋊#92418C2314yes—
35C4×#92418—26no—
36C22×Q8×C61D411yescharacteristic
37C4×Q8×C61—44no—
38C4×Q8×C61C2×C414yes—
39C4×Q8×C61D414yes—
40C2×C4⋊C4×C61—23no—
41C2×C4⋊C4×C61—23no—
42C2×C4⋊C4×C61D413yes—
43C2×C42×C61—23no—
44C2×C42×C61C2313yes—
45#95×C61—23no—
46#95×C61C2311yescharacteristic
47#95×C61C2313yes—
48C2×Q8×C61C22⋊C414yes—
49C2×Q8×C61SD1614yes—
50C2×Q8×C61Q1614yes—
51C22×C4×C61—21no—
52C22×C4×C61—23no—
53C22×C4×C61C2×D413yes—
54C22×C4×C61C2×D413yes—
55C22×C4×C61C2411yescharacteristic
56C2×Dic122—26no—
57C2×Dic122C22×C414yes—
58C4⋊C4×C61—46no—
59C4⋊C4×C61—46no—
60C4⋊C4×C61C22⋊C4112yes—
61C42×C61—46no—
62C42×C61C22×C416yes—
63C42×C61C2×D416yes—
64C22⋊Dic61—24no—
65C4⋊Dic61—26no—
66C4⋊Dic61—26no—
67C4⋊Dic61—26no—
68C4⋊Dic61C22×C414yes—
69#11758196—212no—
70#11758196—24no—
71C4×Dic61—212no—
72C2×#92418—24no—
73C23×C61C22×D411yescharacteristic
74Q8×C61#103116yes—
75C2×C4×C61—22no—
76C2×C4×C61—43no—
77C2×C4×C61—41no—
78C2×C4×C61—41no—
79C2×C4×C61—41no—
80C2×C4×C61—43no—
81C2×C4×C61—43no—
82C2×C4×C61—43no—
83C2×C4×C61C2×C22⋊C416yes—
84C2×C4×C61C2×C22⋊C416yes—
85C2×C4×C61C2×SD1616yes—
86C2×C4×C61C2×Q1616yes—
87C2×C4×C61C23×C411yescharacteristic
88C2×C4×C61C22×D411yescharacteristic
89C2×Dic61—24no—
90C2×Dic61—24no—
91Dic122—44no—
92Dic122—86no—
93#92418—44no—
94#17619—611nomaximal, Sylow
95C22×C61—21no—
96C22×C61—21no—
97C22×C61C22×C22⋊C411yescharacteristic
98C22×C61C22×SD1611yescharacteristic
99C22×C61C22×Q1611yescharacteristic
100C4×C61—21no—
101C4×C61—21no—
102C4×C61—26no—
103C4×C61—42no—
104C4×C61—83no—
105C4×C61C2×#103112yes—
106Dic61—44no—
107Dic61—44no—
108#3502—611no—
109#2982—611no—
110#2351D6111yescharacteristic
111#1518—616no—
112#1503—616no—
113C2×C61—21no—
114C2×C61—21no—
115C2×C61#288411yescharacteristic
116C2×C61C22×#10311yescharacteristic
117C2×C61#290111yescharacteristic
118Q82—614no—
119#557—614no—
120#557—1224no—
121#544—1223no—
122#533—616no—
123#519—1221no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2×D6111yescharacteristic
126C8×Q8—1224no—
127#431—616no—
128#389—23no—
129#382C2×D6113yes—
130C4×C4⋊C4C2×D6113yes—
131#328—614no—
132#328—6112no—
133C61#1761911yescharacteristic, complemented, Sylow
134C22×Q8C22⋊D6111yescharacteristic
135C4⋊Q8—616no—
136C4⋊Q8—616no—
137C4⋊Q8—12212no—
138#125—12212no—
139#125—1226no—
140C22⋊Q8—614no—
141C22⋊Q8—1226no—
142C4×Q8—44no—
143C4×Q8—6112no—
144C4×Q8—1224no—
145C4×Q8—1224no—
146C4×Q8—1226no—
147C4×Q8C4×D6114yes—
148C4×Q8D24414yes—
149#117—1226no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4—23no—
152C2×C4⋊C4C22⋊D6113yes—
153C2×C42—23no—
154C2×C42C22×D6113yes—
155C4⋊C8—1226no—
156C22⋊C8—614no—
157C4×C8—1226no—
158#95—23no—
159#95C22×D6111yescharacteristic
160#95C22×D6113yes—
161C2×Q8—614no—
162C2×Q8—1226no—
163C2×Q8#1175819814yes—
164C2×Q8#1175820014yes—
165C2×Q8#1175820214yes—
166C22×C4—21no—
167C22×C4—23no—
168C22×C4C2×C22⋊D6113yes—
169C22×C4C2×C22⋊D6113yes—
170C22×C4C23×D6111yescharacteristic
171C2×C8—1224no—
172C4⋊C4—46no—
173C4⋊C4—46no—
174C4⋊C4—614no—
175C4⋊C4—1224no—
176C4⋊C4—12212no—
177C4⋊C4—1226no—
178C4⋊C4—1226no—
179C4⋊C4—1226no—
180C4⋊C4#11758198112yes—
181C22⋊C4—1224no—
182C42—46no—
183C42—12212no—
184C42C2×C4×D6116yes—
185C42C2×D24416yes—
186C23C22×C22⋊D6111yescharacteristic
187Q8—4886no—
188Q8—2444no—
189Q8#14519700116yes—
190C2×C4—22no—
191C2×C4—43no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—43no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—1224no—
199C2×C4—1224no—
200C2×C4C2×#1175819816yes—
201C2×C4C2×#1175819816yes—
202C2×C4C2×#1175820016yes—
203C2×C4C2×#1175820216yes—
204C2×C4C22×C4×D6111yescharacteristic
205C2×C4C22×D24411yescharacteristic
206C8—2444no—
207C22—21no—
208C22—21no—
209C22C22×#1175819811yescharacteristic
210C22C22×#1175820011yescharacteristic
211C22C22×#1175820211yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—83no—
217C4—2444no—
218C4—2444no—
219C4C2×#14519700112yes—
220C2—21no—
221C2—21no—
222C2#2145428211yescharacteristic
223C2C22×#1451970011yescharacteristic
224C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2351C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenter#2351D61
hypocenterC61#17619
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
6. Q8×Dic122C4semidirect
7. #16840491C4semidirect
133. C61#17619semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214549002
#214443822
#214443172
C61×#23512
#1761961