Finite Group Databasebeta

← #28372410

Subgroups — #28372410

Order 13952.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#20295862C211yesmaximal, characteristic
2C2×#16812730C212yesmaximal, complemented
3#20295208C214yesmaximal, complemented
4#20295203C214yesmaximal, complemented
5#20295044C212yesmaximal, complemented
6#20295010C211yesmaximal, characteristic, complemented
7#536×C109C211yesmaximal, characteristic, complemented
8#16812747C2212yes—
9#16812746—22no—
10#16812745—22no—
11#16812744—22no—
12#16812744C2212yes—
13C2×#13182029—22no—
14C2×#13182029C2212yes—
15#41×D109—24no—
16#16812730—24no—
17#16812730C2218yescomplemented
18C2×#13182019—22no—
19C2×#13182016—22no—
20C2×#13182013C2212yescomplemented
21#16812627C2218yescomplemented
22#16812618C2212yescomplemented
23#16812616C2212yescomplemented
24C2×#41×C109C2211yescharacteristic
25C22×Q8×C109C2211yescharacteristic, complemented
26C4⋊Q8×C109C2212yescomplemented
27#124×C109C2212yescomplemented
28#123×C109—22no—
29C22⋊Q8×C109—22no—
30#117×C109C2211yescharacteristic, complemented
31#13182030—24no—
32#13182029—24no—
33#13182029—24no—
34#13182029—22no—
35#13182029—24no—
36#13182029—22no—
37#13182029D414yes—
38#13182029C2314yes—
39#13182028—28no—
40#13182028—24no—
41C2×Dic218—22no—
42C2×Dic218D412yes—
43Q8×D109—24no—
44C2×D436—22no—
45C2×D436D412yes—
46C2×C22⋊D109—24no—
47C2×C22⋊D109—22no—
48D4×D109—28no—
49D4×D109—24no—
50C2×C4×D109—24no—
51C2×C4×D109—22no—
52#13182019—28no—
53#13182016—28no—
54#13182013—22no—
55#13182013C2314yes—
56C2×#11724784—22no—
57#41×C109—22no—
58#41×C109—22no—
59C2×Q8×C109—21no—
60C2×Q8×C109—22no—
61C2×Q8×C109—22no—
62C2×Q8×C109C2312yes—
63C2×Q8×C109C2311yescharacteristic
64C2×D4×C109D412yes—
65C2×D4×C109C2311yescharacteristic
66C22×C4×C109—21no—
67C22×C4×C109D412yes—
68C22×C4×C109C2311yescharacteristic
69C4⋊C4×C109—21no—
70C4⋊C4×C109—24no—
71C22⋊C4×C109—21no—
72C22⋊C4×C109—24no—
73C42×C109C2312yes—
74C23×C109C2×D412yes—
75C23×C109C2×D411yescharacteristic
76C22×D109—24no—
77C22×D109—22no—
78Q8×C109—22no—
79Q8×C109—24no—
80Q8×C109—41no—
81Q8×C109—41no—
82D4×C109—24no—
83D4×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—22no—
87C2×C4×C109—22no—
88C2×C4×C109—42no—
89C2×C4×C109C2×D412yes—
90C2×C4×C109C2×D412yes—
91C2×C4×C109C2×D412yes—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—28no—
95C22⋊D109—24no—
96C22⋊D109—24no—
97C2×Dic109—24no—
98C2×Dic109—22no—
99D436—24no—
100D436—22no—
101D436C2×D414yes—
102C4×D109—28no—
103C4×D109—24no—
104C4×D109—24no—
105Dic218—24no—
106Dic218—22no—
107Dic218C2×D414yes—
108#11724784—24no—
109C22×C109—22no—
110C22×C109—41no—
111C22×C109—42no—
112C22×C109#12012yes—
113C22×C109C22×D411yescharacteristic
114C2×D109—24no—
115C2×D109—82no—
116C2×D109—84no—
117C4×C109—22no—
118C4×C109—42no—
119C4×C109—42no—
120C4×C109#12011yescharacteristic
121C4×C109#12011yescharacteristic
122C4×C109C22×D412yes—
123Dic109—24no—
124C2×C109—21no—
125C2×C109—22no—
126C2×C109—42no—
127C2×C109C2×#12011yescharacteristic
128D109—44no—
129#3022—1091nomaximal, Sylow
130C109#302211yescharacteristic, complemented, Sylow
131#585—1091no—
132C2×#137—1092no—
133#536D10911yescharacteristic, complemented
134#456—1094no—
135#455—1094no—
136#421—1092no—
137#413—1091no—
1382−1+4—1092no—
1392−1+4—2182no—
140H(5,𝔽2)—1092no—
141H(5,𝔽2)—2182no—
142H(5,𝔽2)—2182no—
143C2×#41—1092no—
144C2×#41—2184no—
145C2×#41—2182no—
146C2×#41C2×D10911yescharacteristic
147C22×Q8C2×D10911yescharacteristic, complemented
148#137—1098no—
149#137—2184no—
150C2×Q16—2182no—
151C2×SD16—2182no—
152C2×M16—1092no—
153C4⋊Q8C2×D10912yescomplemented
154#124C2×D10912yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D10911yescharacteristic, complemented
158#104—1098no—
159#101—1092no—
160#100—1092no—
161#41—22no—
162#41—22no—
163#41—1094no—
164#41—1094no—
165#41—2184no—
166#41—2184no—
167#41—2184no—
168#41—2188no—
169#41—2184no—
170#41—2182no—
171#41—2182no—
172#41—2184no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—1092no—
177C2×Q8—2182no—
178C2×Q8—2184no—
179C2×Q8C22×D10912yes—
180C2×Q8C22×D10911yescharacteristic
181C2×D4—1092no—
182C2×D4—2182no—
183C2×D4—2184no—
184C2×D4—2188no—
185C2×D4—2182no—
186C2×D4—2184no—
187C2×D4C22⋊D10912yes—
188C2×D4C22×D10911yescharacteristic
189C22×C4—21no—
190C22×C4—2184no—
191C22×C4—2182no—
192C22×C4C22⋊D10912yes—
193C22×C4C22×D10911yescharacteristic
194Q16—2188no—
195SD16—2188no—
196M16—1094no—
197M16—2182no—
198C2×C8—2182no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D10912yes—
204C23—2184no—
205C23—2182no—
206C23C2×C22⋊D10912yes—
207C23C2×C22⋊D10911yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—1094no—
213Q8—2184no—
214Q8—2182no—
215D4—24no—
216D4—22no—
217D4—1094no—
218D4—2184no—
219D4—2188no—
220D4—2184no—
221D4—2184no—
222D4—2182no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—2188no—
229C2×C4—2184no—
230C2×C4—2182no—
231C2×C4—2184no—
232C2×C4—2184no—
233C2×C4D4×D10912yes—
234C2×C4D4×D10912yes—
235C2×C4C2×C22⋊D10912yes—
236C2×C4C2×C22⋊D10911yescharacteristic
237C2×C4C23×D10911yescharacteristic
238C8—2184no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8722no—
243C22—8724no—
244C22—2184no—
245C22#1681266712yes—
246C22C22×C22⋊D10911yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—2184no—
251C4#1681266711yescharacteristic
252C4#1681266711yescharacteristic
253C4C2×D4×D10912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—4364no—
258C2C2×#1681266711yescharacteristic
subgroupG/H
centerC2C2×#16812667
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#536×C109C2
socleC2×C109C2×#120
prime radical#20295862C2
hypercenter#536D109
hypocenterC109#3022
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16812730C2semidirect
3. #20295208C2semidirect
4. #20295203C2semidirect
5. #20295044C2semidirect
6. #20295010C2semidirect
7. #536×C109C2semidirect
17. #16812730C22semidirect
20. C2×#13182013C22semidirect
21. #16812627C22semidirect
22. #16812618C22semidirect
23. #16812616C22semidirect
25. C22×Q8×C109C22semidirect
26. C4⋊Q8×C109C22semidirect
27. #124×C109C22semidirect
30. #117×C109C22semidirect
130. C109#3022semidirect
133. #536D109semidirect
147. C22×Q8C2×D109semidirect
153. C4⋊Q8C2×D109semidirect
154. #124C2×D109semidirect
157. #117C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
C2×#168127302
#202952082
#202952032
#202950442
#202950102
#536×C1092
#3022109