Finite Group Databasebeta

← #27195576

Subgroups — #27195576

Order 13184.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#13110088C211yesmaximal, characteristic
2C2×#15693001C211yesmaximal, characteristic
3C2×#15692946C211yesmaximal, characteristic, complemented
4C2×#15692944C211yesmaximal, characteristic, complemented
5#20267648C211yesmaximal, characteristic
6#20267553C211yesmaximal, characteristic, complemented
7#340×C103C211yesmaximal, characteristic, complemented
8C22×Dic206C2211yescharacteristic
9C22×C4×D103C2211yescharacteristic
10C2×#13110088—22no—
11C2×#13110088—22no—
12C2×#13110088—22no—
13C22×C103⋊C8C2211yescharacteristic
14#15693001—22no—
15#15693001C414yescomplemented
16C2×C4⋊Dic103—21no—
17C2×C4⋊Dic103C2211yescharacteristic
18C2×(C2×D103)⋊C4—21no—
19C2×(C2×D103)⋊C4—21no—
20#15692946—22no—
21#15692944—22no—
22C103⋊#95—21no—
23C22×C8×C103C2211yescharacteristic
24C2×C4⋊C4×C103C2211yescharacteristic
25C2×C4⋊C4×C103C2211yescharacteristic
26C23×D103D411yescharacteristic
27C2×Dic206—21no—
28C2×Dic206—21no—
29C2×Dic206—21no—
30C2×Dic206—21no—
31C2×Dic206—21no—
32C2×Dic206C2×C412yes—
33C22×Dic103—21no—
34C22×Dic103—21no—
35C22×Dic103D411yescharacteristic
36C2×C4×D103—21no—
37C2×C4×D103—21no—
38C2×C4×D103—21no—
39C2×C4×D103—21no—
40C2×C4×D103C2×C412yes—
41C2×C4×D103D412yes—
42#13110088—28no—
43C2×C103⋊C8—21no—
44C2×C103⋊C8—21no—
45C2×C103⋊C8D412yes—
46C4⋊Dic103—24no—
47C4⋊Dic103—21no—
48C4⋊Dic103—21no—
49(C2×D103)⋊C4—24no—
50(C2×D103)⋊C4—42no—
51C22×C4×C103—21no—
52C22×C4×C103—21no—
53C22×C4×C103C2311yescharacteristic
54C2×C8×C103—21no—
55C2×C8×C103—21no—
56C2×C8×C103D412yes—
57C4⋊C4×C103—21no—
58C4⋊C4×C103—21no—
59C4⋊C4×C103—21no—
60C4⋊C4×C103C2×C412yes—
61C23×C103C2×D411yescharacteristic
62C22×D103—21no—
63C22×D103—21no—
64C22×D103—21no—
65C22×D103—21no—
66C22×D103—21no—
67C22×D103—21no—
68C22×D103C22⋊C412yes—
69C2×C4×C103—22no—
70C2×C4×C103—41no—
71C2×C4×C103—41no—
72C2×C4×C103—41no—
73C2×C4×C103—41no—
74C2×C4×C103—41no—
75C2×C4×C103C22×C411yescharacteristic
76C2×C4×C103C2×D411yescharacteristic
77C2×C4×C103C2×D411yescharacteristic
78C2×C4×C103C2×D411yescharacteristic
79C2×C4×C103#4111yescharacteristic
80C2×C4×C103#4111yescharacteristic
81C2×Dic103—21no—
82C2×Dic103—21no—
83C2×Dic103—22no—
84C2×Dic103—41no—
85C2×Dic103—41no—
86C2×Dic103—41no—
87C2×Dic103—41no—
88C2×Dic103—41no—
89C2×Dic103C22⋊C412yes—
90C4×D103—22no—
91C4×D103—22no—
92C4×D103—22no—
93C4×D103—22no—
94Dic206—22no—
95Dic206—22no—
96Dic206—41no—
97Dic206—41no—
98C8×C103—22no—
99C103⋊C8—22no—
100C22×C103C2×C22⋊C411yescharacteristic
101C22×C103#12111yescharacteristic
102C22×C103#12111yescharacteristic
103C22×C103AGL(1,ℤ/8)11yescharacteristic
104C22×C103#13711yescharacteristic
105C22×C103#13711yescharacteristic
106C22×C103#13711yescharacteristic
107C2×D103—22no—
108C2×D103—22no—
109C2×D103—22no—
110C2×D103—41no—
111C2×D103—41no—
112C2×D103—41no—
113C2×D103—41no—
114C4×C103—42no—
115C4×C103—81no—
116C4×C103C4×D411yescharacteristic
117C4×C103C4×D411yescharacteristic
118C4×C103#12011yescharacteristic
119C4×C103#12311yescharacteristic
120Dic103—22no—
121Dic103—42no—
122Dic103—81no—
123C2×C103#38611yescharacteristic
124C2×C103#41711yescharacteristic
125C2×C103#41911yescharacteristic
126C2×C103#46811yescharacteristic
127C2×C103#46811yescharacteristic
128C2×C103#47011yescharacteristic
129C2×C103#47011yescharacteristic
130D103—42no—
131#1894—1031nomaximal, Sylow
132C103#189411yescharacteristic, complemented, Sylow
133C22×M16—1031no—
134C2×C22⋊Q8—1031no—
135C2×#103—1031no—
136C2×#103—1031no—
137#380—1031no—
138#340—1031no—
139#340D10311yescharacteristic, complemented
140C22×Q8—1031no—
141C23×C4—1031no—
142C2×M16—2062no—
143C2×M16—2062no—
144C2×M16—2062no—
145C22×C8—1031no—
146C22×C8C2×D10311yescharacteristic
147C22⋊Q8—1034no—
148C22⋊Q8—2062no—
149C2×C4⋊C4—1031no—
150C2×C4⋊C4—2061no—
151C2×C4⋊C4C2×D10311yescharacteristic
152C2×C4⋊C4C2×D10311yescharacteristic
153C2×C22⋊C4—2061no—
154C2×C22⋊C4—2061no—
155#103—2062no—
156#103—2062no—
157#95—2061no—
158C24—1031no—
159C2×Q8—1032no—
160C2×Q8—2061no—
161C2×Q8—2061no—
162C2×Q8—2061no—
163C2×Q8—2061no—
164C2×Q8—2061no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—1031no—
168C22×C4—1032no—
169C22×C4—1032no—
170C22×C4—2061no—
171C22×C4—2061no—
172C22×C4—2061no—
173C22×C4—2061no—
174C22×C4—2061no—
175C22×C4—2061no—
176C22×C4C22×D10311yescharacteristic
177M16—2068no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—1032no—
181C2×C8—2061no—
182C2×C8—2061no—
183C2×C8D41212yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—2064no—
188C4⋊C4—2061no—
189C4⋊C4—2061no—
190C4⋊C4C4×D10312yes—
191C22⋊C4—4122no—
192C22⋊C4—2064no—
193C23—1032no—
194C23—2061no—
195C23—2061no—
196C23—2061no—
197C23—2061no—
198C23—2061no—
199C23—2061no—
200C23D4×D10311yescharacteristic
201Q8—4121no—
202Q8—4121no—
203Q8—2062no—
204Q8—2062no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1032no—
212C2×C4—4121no—
213C2×C4—4121no—
214C2×C4—4121no—
215C2×C4—4121no—
216C2×C4—4121no—
217C2×C4—2062no—
218C2×C4—2061no—
219C2×C4—2061no—
220C2×C4—2062no—
221C2×C4—2062no—
222C2×C4—2062no—
223C2×C4—2062no—
224C2×C4C2×C4×D10311yescharacteristic
225C2×C4D4×D10311yescharacteristic
226C2×C4D4×D10311yescharacteristic
227C2×C4C2×D41211yescharacteristic
228C2×C4#1311010511yescharacteristic
229C2×C4#1311010611yescharacteristic
230C8—22no—
231C8—2062no—
232C22—4121no—
233C22—4121no—
234C22—4121no—
235C22—4121no—
236C22—2062no—
237C22—2062no—
238C22—2062no—
239C22C22⋊C4×D10311yescharacteristic
240C22#1569299411yescharacteristic
241C22#1569299611yescharacteristic
242C22#1569304511yescharacteristic
243C22#1569305011yescharacteristic
244C22#1569305011yescharacteristic
245C22#1569305311yescharacteristic
246C4—42no—
247C4—81no—
248C4—8241no—
249C4—4122no—
250C4—2062no—
251C4#1569298311yescharacteristic
252C4C4×D41211yescharacteristic
253C4#1569299111yescharacteristic
254C4#1569300611yescharacteristic
255C2—4122no—
256C2#2026766911yescharacteristic
257C2#2026778911yescharacteristic
258C2#2026779811yescharacteristic
259C2#2026803611yescharacteristic
260C2#2026803811yescharacteristic
261C2#2026804811yescharacteristic
262C2#2026805011yescharacteristic
subgroupG/H
centerC23D4×D103
commutatorC2×C4×C103C22×C4
FrattiniC22×C4C22×D103
Fitting#340×C103C2
socleC23×C103C2×D4
prime radicalC23×D103D4
cosocleC22×C4×C103C23
hypercenter#340D103
hypocenterC103#1894
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
3. C2×#15692946C2semidirect
4. C2×#15692944C2semidirect
6. #20267553C2semidirect
7. #340×C103C2semidirect
15. #15693001C4semidirect
132. C103#1894semidirect
139. #340D103semidirect

Maximal subgroups

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

subgroupindex
C22×#131100882
C2×#156930012
C2×#156929462
C2×#156929442
#202676482
#202675532
#340×C1032
#1894103