Finite Group Databasebeta

← #29647767

Subgroups — #29647767

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#21448740C211yesmaximal, characteristic
3#21444384C216yesmaximal
4#21444319C216yesmaximal
5C61×#2982C211yesmaximal, characteristic
6Q8×Dic122C414yescomplemented
7#16840491C414yescomplemented
8#16840488—24no—
9#16840393—23no—
10#16840319C2216yes—
11#16840240—21no—
12C2×Q8×Dic61—21no—
13#16839661—23no—
14#16839628C2213yes—
15C4×C4⋊Dic61C2213yes—
16#16839500C22112yes—
17#16839500C2214yes—
18C2×C4×Q8×C61C2211yescharacteristic
19C8×Q8×C61—24no—
20C61×#431C2216yes—
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—
30Q8×Dic61—44no—
31C4×Dic122—26no—
32C4×Dic122C2×C4112yes—
33#14519736—26no—
34C2×C4⋊Dic61—23no—
35C2×C4⋊Dic61—23no—
36C2×C4×Dic61—23no—
37#14519684—23no—
38#14519684C2311yescharacteristic
39#14519684C2313yes—
40C22×Q8×C61D411yescharacteristic
41C4×Q8×C61C2×C414yes—
42C4×Q8×C61D414yes—
43C2×C4⋊C4×C61D413yes—
44C2×C42×C61C2313yes—
45C4⋊C8×C61—26no—
46C22⋊C8×C61C2314yes—
47C4×C8×C61—26no—
48C2×Q8×C61C22⋊C414yes—
49C2×Q8×C61SD1614yes—
50C2×Q8×C61Q1614yes—
51C22×C4×C61C2×D413yes—
52C22×C4×C61C2×D413yes—
53C22×C4×C61C2411yescharacteristic
54C22×Dic61—21no—
55C22×Dic61—23no—
56C2×Dic122—26no—
57C2×Dic122C22×C414yes—
58C2×C8×C61—24no—
59C4⋊C4×C61C22⋊C4112yes—
60C42×C61C22×C416yes—
61C42×C61C2×D416yes—
62C22⋊Dic61—24no—
63C4⋊Dic61—26no—
64C4⋊Dic61—26no—
65C4⋊Dic61—26no—
66C4⋊Dic61—46no—
67C4⋊Dic61—46no—
68C4⋊Dic61C22×C414yes—
69#11758196—212no—
70#11758196—24no—
71C4×Dic61—212no—
72C4×Dic61—46no—
73C23×C61C22×D411yescharacteristic
74Q8×C61#103116yes—
75C2×C4×C61—43no—
76C2×C4×C61—41no—
77C2×C4×C61C2×C22⋊C416yes—
78C2×C4×C61C2×C22⋊C416yes—
79C2×C4×C61C2×SD1616yes—
80C2×C4×C61C2×Q1616yes—
81C2×C4×C61C23×C411yescharacteristic
82C2×C4×C61C22×D411yescharacteristic
83C2×Dic61—22no—
84C2×Dic61—24no—
85C2×Dic61—24no—
86C2×Dic61—41no—
87C2×Dic61—41no—
88C2×Dic61—43no—
89C2×Dic61—43no—
90C2×Dic61—43no—
91Dic122—44no—
92Dic122—86no—
93C8×C61—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×C61C2×#103112yes—
104Dic61—42no—
105Dic61—44no—
106Dic61—44no—
107Dic61—83no—
108#3502—611no—
109#2982D6111yescharacteristic
110#2351—611no—
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—1221no—
125C2×C4×Q8C2×D6111yescharacteristic
126C8×Q8—24no—
127#431C2×D6116yes—
128#389—1223no—
129#382—613no—
130C4×C4⋊C4—613no—
131#328—614no—
132#328—6112no—
133C61#1761911yescharacteristic, complemented, Sylow
134C22×Q8D24411yescharacteristic
135C4⋊Q8—616no—
136C4⋊Q8—616no—
137C4⋊Q8—12212no—
138#125—12212no—
139#125—1226no—
140C22⋊Q8—614no—
141C22⋊Q8—1226no—
142C4×Q8—6112no—
143C4×Q8—1224no—
144C4×Q8—1224no—
145C4×Q8—1226no—
146C4×Q8—2444no—
147C4×Q8C4×D6114yes—
148C4×Q8C22⋊D6114yes—
149#117—1226no—
150C2×C4⋊C4—1223no—
151C2×C4⋊C4—1223no—
152C2×C4⋊C4D24413yes—
153C2×C42—1223no—
154C2×C42C22×D6113yes—
155C4⋊C8—26no—
156C22⋊C8C22×D6114yes—
157C4×C8—26no—
158#95—611no—
159#95—613no—
160#95—1223no—
161C2×Q8—614no—
162C2×Q8—1226no—
163C2×Q8#1175819014yes—
164C2×Q8Dic24414yes—
165C2×Q8#1175819814yes—
166C22×C4—1221no—
167C22×C4—1223no—
168C22×C4C2×D24413yes—
169C22×C4C2×D24413yes—
170C22×C4C23×D6111yescharacteristic
171C2×C8—24no—
172C4⋊C4—614no—
173C4⋊C4—1224no—
174C4⋊C4—12212no—
175C4⋊C4—1226no—
176C4⋊C4—1226no—
177C4⋊C4—1226no—
178C4⋊C4—2446no—
179C4⋊C4—2446no—
180C4⋊C4#11758198112yes—
181C22⋊C4—1224no—
182C42—12212no—
183C42—2446no—
184C42C2×C4×D6116yes—
185C42C2×C22⋊D6116yes—
186C23C22×D24411yescharacteristic
187Q8—4886no—
188Q8—2444no—
189Q8#14519702116yes—
190C2×C4—43no—
191C2×C4—41no—
192C2×C4—1222no—
193C2×C4—1224no—
194C2×C4—1224no—
195C2×C4—2443no—
196C2×C4—2443no—
197C2×C4—2443no—
198C2×C4—2441no—
199C2×C4—2441no—
200C2×C4C2×#1175819816yes—
201C2×C4C2×#1175819816yes—
202C2×C4C2×#1175819016yes—
203C2×C4C2×Dic24416yes—
204C2×C4C22×C4×D6111yescharacteristic
205C2×C4C22×C22⋊D6111yescharacteristic
206C8—44no—
207C22—21no—
208C22—21no—
209C22C22×#1175819811yescharacteristic
210C22C22×#1175819011yescharacteristic
211C22C22×Dic24411yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—4883no—
216C4—2442no—
217C4—2444no—
218C4—2444no—
219C4C2×#14519702112yes—
220C2—21no—
221C2—21no—
222C2#2145428211yescharacteristic
223C2C22×#1451970211yescharacteristic
224C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2982C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenter#2982D61
hypocenterC61#17619
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

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
#214487402
#214443842
#214443192
C61×#29822
#1761961