Finite Group Databasebeta

← #27205916

Subgroups — #27205916

Order 13184.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#15693000C211yesmaximal, characteristic
2#20268126C212yesmaximal, complemented
3#20268124C212yesmaximal, complemented
4#20268030C214yesmaximal, complemented
5#20267947C212yesmaximal, complemented
6#20267946C212yesmaximal, complemented
7C2×C22⋊(C103⋊C8)C211yesmaximal, characteristic
8C2×C22⋊Q8×C103C211yesmaximal, characteristic
9C22×Dic206C2211yescharacteristic
10C2×#13110095C2214yescomplemented
11C22×C103⋊C8C2212yes—
12#15693000—21no—
13#15693000—21no—
14#15693000C2212yes—
15#15693000C2212yes—
16C2×C4⋊Dic103C2211yescharacteristic
17C2×Dic103⋊C4—21no—
18C2×C22⋊Dic103—21no—
19#15692954C2214yescomplemented
20#15692945C2214yescomplemented
21#15692944C2214yescomplemented
22C22⋊(C103⋊C8)C2212yes—
23C22⋊(C103⋊C8)C2212yes—
24C22×Q8×C103C2211yescharacteristic
25C23×C4×C103C2211yescharacteristic
26C22⋊Q8×C103—21no—
27C22⋊Q8×C103—21no—
28C22⋊Q8×C103C2212yes—
29C22⋊Q8×C103C2212yes—
30C2×C4⋊C4×C103—21no—
31C2×C4⋊C4×C103C2211yescharacteristic
32C2×C22⋊C4×C103—21no—
33C2×Dic206—21no—
34C2×Dic206—42no—
35C2×Dic206C2312yes—
36C22×Dic103—21no—
37C22×Dic103—21no—
38#13110095—44no—
39C2×C103⋊C8—42no—
40C2×C103⋊C8C2314yes—
41C4⋊Dic103—21no—
42C4⋊Dic103C2312yes—
43Dic103⋊C4—22no—
44Dic103⋊C4—22no—
45C22⋊Dic103—22no—
46C22⋊Dic103—22no—
47C24×C103D411yescharacteristic
48C2×Q8×C103—21no—
49C2×Q8×C103—42no—
50C2×Q8×C103C2312yes—
51C22×C4×C103—21no—
52C22×C4×C103—21no—
53C22×C4×C103—42no—
54C22×C4×C103D411yescharacteristic
55C22×C4×C103D411yescharacteristic
56C22×C4×C103D411yescharacteristic
57C22×C4×C103C2311yescharacteristic
58C22×C4×C103C2311yescharacteristic
59C22×C4×C103C2311yescharacteristic
60C4⋊C4×C103—22no—
61C4⋊C4×C103—22no—
62C4⋊C4×C103—21no—
63C4⋊C4×C103C2312yes—
64C22⋊C4×C103—22no—
65C22⋊C4×C103—22no—
66C23×C103—41no—
67C23×C103—41no—
68C23×C103Q1614yescomplemented
69C23×C103C2×D411yescharacteristic
70C23×C103C2×D411yescharacteristic
71C23×C103C2×D411yescharacteristic
72Q8×C103—24no—
73Q8×C103—42no—
74C2×C4×C103—22no—
75C2×C4×C103—22no—
76C2×C4×C103—22no—
77C2×C4×C103—22no—
78C2×C4×C103—22no—
79C2×C4×C103—41no—
80C2×C4×C103—41no—
81C2×C4×C103—41no—
82C2×C4×C103—42no—
83C2×C4×C103—42no—
84C2×C4×C103C2×D411yescharacteristic
85C2×C4×C103C2×D411yescharacteristic
86C2×C4×C103C2×D411yescharacteristic
87C2×C4×C103C2411yescharacteristic
88C2×Dic103—22no—
89C2×Dic103—22no—
90C2×Dic103—42no—
91C2×Dic103—42no—
92Dic206—24no—
93Dic206—42no—
94C103⋊C8—24no—
95C22×C103—21no—
96C22×C103—21no—
97C22×C103—21no—
98C22×C103—21no—
99C22×C103—21no—
100C22×C103—21no—
101C22×C103—44no—
102C22×C103C2×Q1612yes—
103C22×C103C2×Q1612yes—
104C22×C103C2×Q1612yes—
105C22×C103C22×D411yescharacteristic
106C4×C103—21no—
107C4×C103—21no—
108C4×C103—21no—
109C4×C103—42no—
110C4×C103—42no—
111C4×C103H(5,𝔽2)12yes—
112Dic103—42no—
113Dic103—42no—
114C2×C103—22no—
115C2×C103—22no—
116C2×C103—22no—
117C2×C103#53411yescharacteristic
118C2×C103C22×Q1611yescharacteristic
119C2×C103#57511yescharacteristic
120#3189—1031nomaximal, Sylow
121C103#318911yescharacteristic, complemented, Sylow
122C2×C22⋊Q8—1031no—
123C2×C22⋊Q8D10311yescharacteristic
124#484—1032no—
125#484—1032no—
126#467—1034no—
127#451—1032no—
128#451—1032no—
129C2×C22⋊C8—1031no—
130C22×Q8—1031no—
131C22×Q8C2×D10311yescharacteristic
132C23×C4C2×D10311yescharacteristic
133C2×Q16—1034no—
134C22×C8—1032no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—1032no—
138C22⋊Q8—1032no—
139C22⋊Q8—2061no—
140C22⋊Q8—2061no—
141C22⋊Q8C2×D10312yes—
142C22⋊Q8C2×D10312yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—1031no—
145C2×C4⋊C4—2061no—
146C2×C4⋊C4C2×D10311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—2061no—
149#107—1034no—
150#103—1034no—
151#103—1034no—
152C22⋊C8—1032no—
153C22⋊C8—1032no—
154C24C22⋊D10311yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1032no—
158C2×Q8—4122no—
159C2×Q8—2061no—
160C2×Q8C22×D10312yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—2061no—
165C22×C4—2061no—
166C22×C4C22⋊D10311yescharacteristic
167C22×C4C22⋊D10311yescharacteristic
168C22×C4C22⋊D10311yescharacteristic
169C22×C4C22×D10311yescharacteristic
170C22×C4C22×D10311yescharacteristic
171C22×C4C22×D10311yescharacteristic
172Q16—4124no—
173C2×C8—1034no—
174C2×C8—4122no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—1032no—
179C4⋊C4—2062no—
180C4⋊C4—2062no—
181C4⋊C4—2061no—
182C4⋊C4C22×D10312yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—2062no—
186C22⋊C4—2062no—
187C23—41no—
188C23—41no—
189C23#1311009514yescomplemented
190C23C2×C22⋊D10311yescharacteristic
191C23C2×C22⋊D10311yescharacteristic
192C23C2×C22⋊D10311yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—4122no—
196Q8—2064no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—4122no—
208C2×C4—4122no—
209C2×C4—2062no—
210C2×C4—2062no—
211C2×C4C2×C22⋊D10311yescharacteristic
212C2×C4C2×C22⋊D10311yescharacteristic
213C2×C4C2×C22⋊D10311yescharacteristic
214C2×C4C23×D10311yescharacteristic
215C8—2064no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22C2×#1311009512yes—
224C22C2×#1311009512yes—
225C22C2×#1311009512yes—
226C22C22×C22⋊D10311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—4122no—
233C4—4122no—
234C4#1569306612yes—
235C2—22no—
236C2—22no—
237C2—22no—
238C2#2026835011yescharacteristic
239C2C22×#1311009511yescharacteristic
240C2#2026858911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC2×C22⋊Q8×C103C2
socleC22×C103C22×D4
prime radicalC24×C103D4
hypercenterC2×C22⋊Q8D103
hypocenterC103#3189
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20268126C2semidirect
3. #20268124C2semidirect
4. #20268030C2semidirect
5. #20267947C2semidirect
6. #20267946C2semidirect
10. C2×#13110095C22semidirect
19. #15692954C22semidirect
20. #15692945C22semidirect
21. #15692944C22semidirect
68. C23×C103Q16semidirect
121. C103#3189semidirect
189. C23#13110095semidirect

Maximal subgroups

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

subgroupindex
C2×#156930002
#202681262
#202681242
#202680302
#202679472
#202679462
C2×C22⋊(C103⋊C8)2
C2×C22⋊Q8×C1032
#3189103