Finite Group Databasebeta

← #28372403

Subgroups — #28372403

Order 13952.

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

normal subgroups only

This group has 736 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, complemented
2C2×#16812725C212yesmaximal, complemented
3#20295198C214yesmaximal, complemented
4#20295193C214yesmaximal, complemented
5#20295044C212yesmaximal, complemented
6#20295010C211yesmaximal, characteristic, complemented
7#535×C109C211yesmaximal, characteristic, complemented
8#16812747C2212yescomplemented
9#16812746—22no—
10#16812745—22no—
11#16812744—22no—
12#16812744C2212yescomplemented
13C2×#13182029—22no—
14C2×#13182029C2212yes—
15#41×D109—24no—
16#16812725—24no—
17#16812725C2218yescomplemented
18C2×#13182017—22no—
19C2×#13182014—22no—
20C2×#13182013C2212yescomplemented
21#16812627C2218yescomplemented
22#16812618C2212yescomplemented
23#16812616C2212yescomplemented
24C2×#41×C109C2211yescharacteristic
25C22×D4×C109C2211yescharacteristic, complemented
26#127×C109C2212yescomplemented
27#124×C109C2212yescomplemented
28#121×C109—22no—
29#120×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#13182029D414yescomplemented
38#13182029C2314yes—
39#13182028—28no—
40#13182028—24no—
41C2×Dic218—22no—
42C2×Dic218D412yescomplemented
43Q8×D109—24no—
44C2×D436—22no—
45C2×D436D412yescomplemented
46C2×C22⋊D109—24no—
47C2×C22⋊D109—22no—
48D4×D109—28no—
49D4×D109—24no—
50C2×C4×D109—24no—
51C2×C4×D109—22no—
52#13182017—28no—
53#13182014—28no—
54#13182013—22no—
55#13182013C2314yes—
56C2×#11724784—22no—
57C24×C109—21no—
58#41×C109—22no—
59#41×C109—22no—
60C2×Q8×C109C2311yescharacteristic
61C2×D4×C109—24no—
62C2×D4×C109—22no—
63C2×D4×C109—22no—
64C2×D4×C109—21no—
65C2×D4×C109D412yes—
66C2×D4×C109C2312yes—
67C2×D4×C109C2311yescharacteristic
68C22×C4×C109D412yes—
69C22×C4×C109C2311yescharacteristic
70C4⋊C4×C109—21no—
71C22⋊C4×C109—21no—
72C22⋊C4×C109—24no—
73C42×C109C2312yes—
74C23×C109—22no—
75C23×C109—42no—
76C23×C109—82no—
77C23×C109C2×D412yes—
78C23×C109C2×D411yescharacteristic
79C22×D109—24no—
80C22×D109—22no—
81Q8×C109—22no—
82D4×C109—24no—
83D4×C109—22no—
84D4×C109—24no—
85D4×C109—41no—
86D4×C109—41no—
87D4×C109—84no—
88C2×C4×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—22no—
91C2×C4×C109C2×D412yes—
92C2×C4×C109C2×D412yes—
93C2×C4×C109C2×D412yes—
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—28no—
97C22⋊D109—24no—
98C22⋊D109—24no—
99C2×Dic109—24no—
100C2×Dic109—22no—
101D436—24no—
102D436—22no—
103D436C2×D414yes—
104C4×D109—28no—
105C4×D109—24no—
106C4×D109—24no—
107Dic218—24no—
108Dic218—22no—
109Dic218C2×D414yes—
110#11724784—24no—
111C22×C109—22no—
112C22×C109—41no—
113C22×C109—42no—
114C22×C109—44no—
115C22×C109—42no—
116C22×C109—84no—
117C22×C109#12012yes—
118C22×C109C22×D411yescharacteristic
119C2×D109—24no—
120C2×D109—82no—
121C2×D109—84no—
122C4×C109—22no—
123C4×C109—42no—
124C4×C109#12011yescharacteristic
125C4×C109#12011yescharacteristic
126C4×C109C22×D412yes—
127Dic109—24no—
128C2×C109—21no—
129C2×C109—22no—
130C2×C109—42no—
131C2×C109—82no—
132C2×C109C2×#12011yescharacteristic
133D109—44no—
134#3021—1091nomaximal, Sylow
135C109#302111yescharacteristic, complemented, Sylow
136#585—1091no—
137C2×AGL(1,ℤ/8)—1092no—
138#535D10911yescharacteristic, complemented
139#454—1094no—
140#453—1094no—
141#421—1092no—
142#413—1091no—
1432−1+4—1092no—
1442−1+4—2182no—
145H(5,𝔽2)—1092no—
146H(5,𝔽2)—2182no—
147H(5,𝔽2)—2182no—
148C2×#41—1092no—
149C2×#41—2184no—
150C2×#41—2182no—
151C2×#41C2×D10911yescharacteristic
152C22×D4C2×D10911yescharacteristic, complemented
153AGL(1,ℤ/8)—1098no—
154AGL(1,ℤ/8)—2184no—
155C2×SD16—2182no—
156C2×D8—2182no—
157C2×M16—1092no—
158#127C2×D10912yescomplemented
159#124C2×D10912yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D10911yescharacteristic, complemented
163#104—1098no—
164#101—1092no—
165#100—1092no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—1094no—
170#41—1094no—
171#41—2184no—
172#41—2184no—
173#41—2184no—
174#41—2188no—
175#41—2184no—
176#41—2182no—
177#41—2182no—
178#41—2184no—
179C2×Q8—1092no—
180C2×Q8—2182no—
181C2×Q8—2184no—
182C2×Q8C22×D10911yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—1092no—
188C2×D4—2182no—
189C2×D4—2184no—
190C2×D4—2188no—
191C2×D4—2182no—
192C2×D4—2184no—
193C2×D4C22⋊D10912yes—
194C2×D4C22×D10912yes—
195C2×D4C22×D10911yescharacteristic
196C22×C4—2184no—
197C22×C4—2182no—
198C22×C4C22⋊D10912yes—
199C22×C4C22×D10911yescharacteristic
200SD16—2188no—
201D8—2188no—
202M16—1094no—
203M16—2182no—
204C2×C8—2182no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D10912yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—2184no—
213C23—2182no—
214C23C2×C22⋊D10912yes—
215C23C2×C22⋊D10911yescharacteristic
216Q8—22no—
217Q8—1094no—
218Q8—2184no—
219Q8—2182no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—1094no—
227D4—2184no—
228D4—2188no—
229D4—2184no—
230D4—2184no—
231D4—2182no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—2188no—
236C2×C4—2184no—
237C2×C4—2182no—
238C2×C4—2184no—
239C2×C4—2184no—
240C2×C4D4×D10912yes—
241C2×C4D4×D10912yes—
242C2×C4C2×C22⋊D10912yes—
243C2×C4C2×C22⋊D10911yescharacteristic
244C2×C4C23×D10911yescharacteristic
245C8—2184no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—8722no—
253C22—8724no—
254C22—2184no—
255C22#1681266712yes—
256C22C22×C22⋊D10911yescharacteristic
257C4—22no—
258C4—42no—
259C4—2184no—
260C4#1681266711yescharacteristic
261C4#1681266711yescharacteristic
262C4C2×D4×D10912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—4364no—
268C2C2×#1681266711yescharacteristic
subgroupG/H
centerC2C2×#16812667
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#535×C109C2
socleC2×C109C2×#120
hypercenter#535D109
hypocenterC109#3021
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. C2×#16812725C2semidirect
3. #20295198C2semidirect
4. #20295193C2semidirect
5. #20295044C2semidirect
6. #20295010C2semidirect
7. #535×C109C2semidirect
8. #16812747C22semidirect
12. #16812744C22semidirect
17. #16812725C22semidirect
20. C2×#13182013C22semidirect
21. #16812627C22semidirect
22. #16812618C22semidirect
23. #16812616C22semidirect
25. C22×D4×C109C22semidirect
26. #127×C109C22semidirect
27. #124×C109C22semidirect
30. #117×C109C22semidirect
37. #13182029D4semidirect
42. C2×Dic218D4semidirect
45. C2×D436D4semidirect
135. C109#3021semidirect
138. #535D109semidirect
152. C22×D4C2×D109semidirect
158. #127C2×D109semidirect
159. #124C2×D109semidirect
162. #117C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
C2×#168127252
#202951982
#202951932
#202950442
#202950102
#535×C1092
#3021109