Finite Group Databasebeta

← #28373382

Subgroups — #28373382

Order 13952.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#20295654C211yesmaximal, characteristic, complemented
2#20295593C211yesmaximal, characteristic
3#20295389C212yesmaximal, complemented
4#20295384C212yesmaximal, complemented
5#20295379C212yesmaximal, complemented
6#20295371C212yesmaximal, complemented
7#20295028C211yesmaximal, characteristic, complemented
8#20295020C211yesmaximal, characteristic, complemented
9C2×#16812620C211yesmaximal, characteristic, complemented
10#20294984C211yesmaximal, characteristic, complemented
11#418×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×#11724784C2211yescharacteristic
14#16812699—21no—
15#16812698C2212yescomplemented
16#16812695C2212yes—
17#16812691—21no—
18#16812690C2212yescomplemented
19#16812686C2212yes—
20#16812676—22no—
21#16812670—21no—
22#16812670—21no—
23#16812668—22no—
24#16812658C2211yescharacteristic
25C2×C4⋊Dic109C2211yescharacteristic
26#16812624C2212yescomplemented
27#16812622C2211yescharacteristic, complemented
28#16812622C2211yescharacteristic, complemented
29#16812621C2212yescomplemented
30#16812620C2212yes—
31#16812620C2212yes—
32#16812619C2211yescharacteristic, complemented
33#16812619C2211yescharacteristic, complemented
34#16812609C2212yescomplemented
35C8×Dic109C2212yescomplemented
36C22×D4×C109C2211yescharacteristic
37C2×M16×C109C2211yescharacteristic
38#117×C109C2211yescharacteristic
39#102×C109C2212yes—
40#102×C109C2212yes—
41#13182029—22no—
42#13182029—22no—
43C2×Dic218—22no—
44C2×Dic218C2311yescharacteristic
45C2×D436C2311yescharacteristic
46C2×C22⋊D109—21no—
47C2×C22⋊D109—22no—
48C2×C22⋊D109—22no—
49C22×Dic109—21no—
50C2×C4×D109—21no—
51C2×#11724784—22no—
52C2×#11724784C2311yescharacteristic
53C2×#11724784C2311yescharacteristic
54C4⋊Dic109—21no—
55C4⋊Dic109—22no—
56C4⋊Dic109D411yescharacteristic
57C4⋊Dic109D411yescharacteristic
58C4⋊Dic109C2312yes—
59#13182008—22no—
60#13182008—22no—
61C22⋊Dic109—22no—
62C22⋊Dic109—22no—
63C22⋊Dic109D412yes—
64C4×Dic109C2312yes—
65C24×C109—21no—
66C2×D4×C109—21no—
67C2×D4×C109—22no—
68C2×D4×C109—22no—
69C2×D4×C109C2312yes—
70C22×C4×C109C2311yescharacteristic
71M16×C109—22no—
72C2×C8×C109C2312yes—
73C4⋊C4×C109C2311yescharacteristic
74C4⋊C4×C109C2311yescharacteristic
75C22⋊C4×C109—21no—
76C42×C109—21no—
77C23×C109—22no—
78C23×C109—42no—
79C23×C109—81no—
80C23×C109—81no—
81C23×C109C2×D411yescharacteristic
82C22×D109—21no—
83D4×C109—22no—
84D4×C109—22no—
85D4×C109—42no—
86C2×C4×C109—21no—
87C2×C4×C109—21no—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109#4112yes—
90C2×C4×C109#4112yes—
91C2×C4×C109C2411yescharacteristic
92C22⋊D109—42no—
93C22⋊D109—82no—
94C22⋊D109—82no—
95C2×Dic109—21no—
96C2×Dic109—22no—
97C2×Dic109—81no—
98C2×Dic109C2×D412yes—
99C2×Dic109C2×D412yes—
100D436—21no—
101D436—21no—
102C4×D109—42no—
103Dic218—21no—
104Dic218—21no—
105Dic218—82no—
106C8×C109—22no—
107#11724784—21no—
108#11724784—21no—
109C22×C109—21no—
110C22×C109—21no—
111C22×C109—42no—
112C22×C109—42no—
113C22×C109—42no—
114C22×C109—84no—
115C22×C109#12412yes—
116C22×C109C22×D411yescharacteristic
117C2×D109—41no—
118C2×D109—41no—
119C2×D109—41no—
120C4×C109—41no—
121C4×C109—41no—
122C4×C109#12412yes—
123C4×C109C2×#4111yescharacteristic
124C4×C109C2×#4111yescharacteristic
125Dic109—22no—
126Dic109—22no—
127Dic109—41no—
128Dic109—42no—
129C2×C109—22no—
130C2×C109—82no—
131C2×C109C2×#12411yescharacteristic
132C2×C109#57611yescharacteristic
133C2×C109#57711yescharacteristic
134D109—81no—
135#3136—1091nomaximal, Sylow
136C109#313611yescharacteristic, complemented, Sylow
137#544—1091no—
138#535—1091no—
139#490—1092no—
140#489—1092no—
141#488—1092no—
142#486—1092no—
143#418D10911yescharacteristic, complemented
144#417—1091no—
145#416—1091no—
146C2×#102—1091no—
147#405—1091no—
148C2×#41—1091no—
149C22×D4C2×D10911yescharacteristic
150C2×M16C2×D10911yescharacteristic
151C22×C8—1091no—
152C4⋊Q8—1092no—
153C4⋊Q8—2181no—
154#127—1092no—
155#125—1092no—
156#125—2181no—
157#124—1092no—
158C22⋊Q8—2182no—
159#121—2181no—
160#121—2181no—
161#120—2182no—
162#117—1091no—
163#117C2×D10911yescharacteristic
164C2×C4⋊C4—1091no—
165#103—1092no—
166#103—1091no—
167#103—1091no—
168#102—1092no—
169#102—1092no—
170#102—1092no—
171#102—1091no—
172#102—1091no—
173#102C2×D10912yes—
174#102C2×D10912yes—
175#97—1092no—
176C4×C8—1092no—
177C24—21no—
178#41—2182no—
179#41—2182no—
180C2×Q8—1091no—
181C2×Q8—2182no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—1091no—
186C2×D4—2182no—
187C2×D4—2182no—
188C2×D4—2181no—
189C2×D4C22×D10912yes—
190C22×C4—2181no—
191C22×C4—2181no—
192C22×C4C22×D10911yescharacteristic
193M16—22no—
194C2×C8—1091no—
195C2×C8—1091no—
196C2×C8—2182no—
197C2×C8C22×D10912yes—
198C4⋊C4—1091no—
199C4⋊C4—1091no—
200C4⋊C4—1092no—
201C4⋊C4—2182no—
202C4⋊C4—2182no—
203C4⋊C4—2182no—
204C4⋊C4—2181no—
205C4⋊C4C22×D10911yescharacteristic
206C4⋊C4C22×D10911yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—1092no—
209C22⋊C4—2182no—
210C22⋊C4—2182no—
211C42—21no—
212C42—1092no—
213C23—22no—
214C23—42no—
215C23—81no—
216C23—81no—
217C23—2181no—
218C23D4×D10911yescharacteristic
219Q8—8722no—
220Q8—2181no—
221Q8—2181no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—8722no—
226D4—8722no—
227D4—4362no—
228D4—2181no—
229D4—2181no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—8721no—
233C2×C4—1092no—
234C2×C4—1092no—
235C2×C4—4362no—
236C2×C4—2181no—
237C2×C4—2182no—
238C2×C4D4×D10911yescharacteristic
239C2×C4#1318202812yes—
240C2×C4#1318202912yes—
241C2×C4C23×D10911yescharacteristic
242C8—22no—
243C8—2181no—
244C8—2181no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22—4361no—
252C22—4361no—
253C22—4361no—
254C22#1681268512yes—
255C22C2×D4×D10911yescharacteristic
256C4—41no—
257C4—41no—
258C4—4361no—
259C4—4362no—
260C4—2182no—
261C4—2182no—
262C4#1681268512yes—
263C4C2×#1318202811yescharacteristic
264C4C2×#1318202911yescharacteristic
265C2—22no—
266C2—82no—
267C2—8721no—
268C2C2×#1681268511yescharacteristic
269C2#2029583011yescharacteristic
270C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#418×C109C2
socleC22×C109C22×D4
prime radical#20295593C2
hypercenter#418D109
hypocenterC109#3136
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #20295654C2semidirect
3. #20295389C2semidirect
4. #20295384C2semidirect
5. #20295379C2semidirect
6. #20295371C2semidirect
7. #20295028C2semidirect
8. #20295020C2semidirect
9. C2×#16812620C2semidirect
10. #20294984C2semidirect
11. #418×C109C2semidirect
15. #16812698C22semidirect
18. #16812690C22semidirect
26. #16812624C22semidirect
27. #16812622C22semidirect
28. #16812622C22semidirect
29. #16812621C22semidirect
32. #16812619C22semidirect
33. #16812619C22semidirect
34. #16812609C22semidirect
35. C8×Dic109C22semidirect
136. C109#3136semidirect
143. #418D109semidirect

Maximal subgroups

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

subgroupindex
#202956542
#202955932
#202953892
#202953842
#202953792
#202953712
#202950282
#202950202
C2×#168126202
#202949842
#418×C1092
#3136109