Finite Group Databasebeta

← #27195645

Subgroups — #27195645

Order 13184.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#20268589C211yesmaximal, characteristic
2#20267808C211yesmaximal, characteristic, complemented
3C2×#15692949C211yesmaximal, characteristic, complemented
4#20267764C211yesmaximal, characteristic, complemented
5#20267561C211yesmaximal, characteristic, complemented
6#20267551C211yesmaximal, characteristic, complemented
7C4×#41×C103C211yesmaximal, characteristic, complemented
8C2×#13110105C2211yescharacteristic
9#15693052—21no—
10#15693051C412yescomplemented
11#15693047—21no—
12#15693046C412yescomplemented
13#15693041—22no—
14C2×#13110089C2211yescharacteristic
15C2×#13110089C2211yescharacteristic
16#15692980C2211yescharacteristic
17#15692949—21no—
18#15692949—21no—
19#15692948—22no—
20#15692935—22no—
21C2×#41×C103C2211yescharacteristic
22C4×Q8×C103—21no—
23C4×D4×C103—21no—
24C4×D4×C103—22no—
25#117×C103—21no—
26#117×C103C2211yescharacteristic
27C2×C42×C103—21no—
28C2×C42×C103C2211yescharacteristic
29#13110105—21no—
30#13110105—21no—
31#13110105—21no—
32#13110105—21no—
33C2×Dic206C2×C411yescharacteristic, complemented
34C2×D412C2×C411yescharacteristic, complemented
35C2×C22⋊D103—21no—
36C2×C4×D103—21no—
37#13110095—22no—
38#13110093—22no—
39#13110092—22no—
40#13110090—22no—
41#13110089—21no—
42#13110089—21no—
43#13110089—21no—
44#13110089C2×C412yescomplemented
45C2×C103⋊C8—21no—
46C2×C103⋊C8—21no—
47C4⋊Dic103—21no—
48C22⋊Dic103—21no—
49C4×Dic103—21no—
50#41×C103—21no—
51#41×C103—21no—
52#41×C103C2×C412yes—
53#41×C103D412yes—
54C2×Q8×C103D411yescharacteristic
55C2×D4×C103—21no—
56C2×D4×C103D411yescharacteristic
57C22×C4×C103—21no—
58C22×C4×C103—22no—
59C22×C4×C103—21no—
60C22×C4×C103C2311yescharacteristic
61C4⋊C4×C103—22no—
62C4⋊C4×C103—21no—
63C22⋊C4×C103—22no—
64C22⋊C4×C103D412yes—
65C42×C103—22no—
66C42×C103—21no—
67C42×C103—21no—
68C42×C103D411yescharacteristic
69C42×C103D411yescharacteristic
70C23×C103—21no—
71C23×C103C2×D411yescharacteristic
72C22×D103—21no—
73Q8×C103—21no—
74Q8×C103C22⋊C412yes—
75D4×C103—22no—
76D4×C103—22no—
77D4×C103—21no—
78D4×C103C22⋊C412yes—
79C2×C4×C103—21no—
80C2×C4×C103—21no—
81C2×C4×C103—22no—
82C2×C4×C103—22no—
83C2×C4×C103—22no—
84C2×C4×C103—21no—
85C2×C4×C103—42no—
86C2×C4×C103—44no—
87C2×C4×C103C22×C411yescharacteristic
88C2×C4×C103C2×D411yescharacteristic
89C2×C4×C103C2×D411yescharacteristic
90C2×C4×C103C2×D411yescharacteristic
91C2×C4×C103#4111yescharacteristic
92C2×C4×C103#4111yescharacteristic
93C22⋊D103—41no—
94C22⋊D103—41no—
95C2×Dic103—21no—
96C2×Dic103—41no—
97D412—21no—
98D412—21no—
99C4×D103—41no—
100C4×D103—41no—
101Dic206—21no—
102Dic206—21no—
103C103⋊C8—22no—
104C103⋊C8—41no—
105C22×C103—22no—
106C22×C103—41no—
107C22×C103—42no—
108C22×C103C4×D411yescharacteristic
109C22×C103#12011yescharacteristic
110C22×C103#12111yescharacteristic
111C2×D103—41no—
112C2×D103—81no—
113C4×C103—22no—
114C4×C103—22no—
115C4×C103—42no—
116C4×C103—41no—
117C4×C103C2×C22⋊C411yescharacteristic
118C4×C103C4×D411yescharacteristic
119C4×C103#12111yescharacteristic
120C4×C103#12311yescharacteristic
121Dic103—41no—
122Dic103—81no—
123C2×C103—21no—
124C2×C103—21no—
125C2×C103—21no—
126C2×C103—42no—
127C2×C103#38611yescharacteristic
128D103—81no—
129#1906—1031nomaximal, Sylow
130C103#190611yescharacteristic, complemented, Sylow
131#575—1031no—
132C4×#41D10311yescharacteristic, complemented
133#421—1031no—
134C2×#104—1031no—
135#410—1031no—
136#344—1031no—
137#339—1031no—
138C2×#41—1031no—
139C2×#41C2×D10311yescharacteristic
140#137—1032no—
141#137—2061no—
142AGL(1,ℤ/8)—1032no—
143AGL(1,ℤ/8)—2061no—
144#135—2062no—
145C2×M16—1031no—
146C2×M16—1031no—
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117—1031no—
152#117C2×D10311yescharacteristic
153C2×C42—21no—
154C2×C42C2×D10311yescharacteristic
155#104—2062no—
156#104—2061no—
157#104—2061no—
158#99—2062no—
159#41—21no—
160#41—21no—
161#41—2061no—
162#41—2061no—
163#41—2061no—
164#41—2061no—
165#41C4×D10312yes—
166#41D41212yes—
167C2×Q8—1031no—
168C2×Q8C22⋊D10311yescharacteristic
169C2×D4—21no—
170C2×D4—1031no—
171C2×D4—2061no—
172C2×D4C22⋊D10311yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—2061no—
177C22×C4C22×D10311yescharacteristic
178Q16—2062no—
179SD16—2062no—
180SD16—2062no—
181D8—2062no—
182M16—1032no—
183M16—2061no—
184M16—2061no—
185M16—2061no—
186C2×C8—2061no—
187C2×C8—2061no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—2061no—
191C22⋊C4—22no—
192C22⋊C4—2061no—
193C22⋊C4C22⋊D10312yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—2061no—
198C42C22⋊D10311yescharacteristic
199C42C22⋊D10311yescharacteristic
200C23—21no—
201C23—2061no—
202C23C2×C22⋊D10311yescharacteristic
203Q8—21no—
204Q8—2061no—
205Q8—2061no—
206Q8(C2×D103)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—4121no—
211D4—4121no—
212D4—2061no—
213D4—2061no—
214D4(C2×D103)⋊C412yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—4121no—
224C2×C4—4121no—
225C2×C4—4121no—
226C2×C4—2061no—
227C2×C4C2×C4×D10311yescharacteristic
228C2×C4C2×C22⋊D10311yescharacteristic
229C2×C4C2×C22⋊D10311yescharacteristic
230C2×C4C2×D41211yescharacteristic
231C2×C4#1311010511yescharacteristic
232C2×C4#1311010511yescharacteristic
233C8—4121no—
234C8—2062no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—8241no—
239C22—4121no—
240C22C4×C22⋊D10311yescharacteristic
241C22#1569299011yescharacteristic
242C22#1569299511yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—8241no—
248C4—4121no—
249C4C2×(C2×D103)⋊C411yescharacteristic
250C4C4×C22⋊D10311yescharacteristic
251C4#1569299511yescharacteristic
252C4#1569300411yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—8241no—
258C2#2026767011yescharacteristic
subgroupG/H
centerC4#15692995
commutatorC2×C4×C103C22×C4
FrattiniC22×C4C22×D103
FittingC4×#41×C103C2
socleC2×C103#386
prime radical#20268589C2
cosocleC22×C4×C103C23
hypercenterC4×#41D103
hypocenterC103#1906
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #20267808C2semidirect
3. C2×#15692949C2semidirect
4. #20267764C2semidirect
5. #20267561C2semidirect
6. #20267551C2semidirect
7. C4×#41×C103C2semidirect
10. #15693051C4semidirect
12. #15693046C4semidirect
33. C2×Dic206C2×C4semidirect
34. C2×D412C2×C4semidirect
44. #13110089C2×C4semidirect
130. C103#1906semidirect
132. C4×#41D103semidirect

Maximal subgroups

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

subgroupindex
#202685892
#202678082
C2×#156929492
#202677642
#202675612
#202675512
C4×#41×C1032
#1906103