Finite Group Databasebeta

← #28373417

Subgroups — #28373417

Order 13952.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20295581C211yesmaximal, characteristic, complemented
2#124×D109C211yesmaximal, characteristic
3#20295388C211yesmaximal, characteristic, complemented
4#20295386C211yesmaximal, characteristic, complemented
5#20295385C211yesmaximal, characteristic, complemented
6#20295382C212yesmaximal, complemented
7#20295379C212yesmaximal, complemented
8#20295033C212yesmaximal, complemented
9#20295029C212yesmaximal
10#20294978C211yesmaximal, characteristic, complemented
11#489×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2211yescharacteristic
13C2×D4×D109C2211yescharacteristic
14C2×#13182012C2212yes—
15#16812699C2211yescharacteristic, complemented
16#16812690C2212yescomplemented
17#16812688C2211yescharacteristic
18#16812687C2211yescharacteristic
19#16812686C2211yescharacteristic
20#16812685—21no—
21#16812685—21no—
22#16812679—21no—
23#16812675—21no—
24Q8×Dic109—21no—
25#16812664—21no—
26#16812657C2212yes—
27C22⋊C4×D109—21no—
28C22⋊C4×D109—21no—
29C42×D109C2211yescharacteristic
30#16812624C2212yescomplemented
31#16812623C2212yescomplemented
32#16812622C2212yescomplemented
33#16812621C2212yes—
34#16812620C2212yes—
35#16812619C2212yes—
36#16812610C2211yescharacteristic, complemented
37C8×Dic109C2212yescomplemented
38C4⋊Q8×C109C2211yescharacteristic, complemented
39#124×C109C2211yescharacteristic
40#103×C109C2212yescomplemented
41#102×C109C2212yes—
42#97×C109C2211yescharacteristic, complemented
43C23×D109—21no—
44C2×Dic218—21no—
45C2×Dic218—21no—
46C2×Dic218C2311yescharacteristic
47Q8×D109—24no—
48C2×D436C2311yescharacteristic
49C2×C22⋊D109—21no—
50D4×D109—24no—
51C2×C4×D109—21no—
52C2×C4×D109D411yescharacteristic
53C2×C4×D109D411yescharacteristic
54C2×C4×D109C2311yescharacteristic
55#13182012—24no—
56C2×#11724784C2312yes—
57C4⋊Dic109—21no—
58C4⋊Dic109—22no—
59C4⋊Dic109C2312yes—
60#13182008—22no—
61C22⋊Dic109—21no—
62C22⋊Dic109—21no—
63#13182006—22no—
64#13182006—21no—
65#13182006—21no—
66#13182006—21no—
67#13182006—21no—
68C4×Dic109—22no—
69C4×Dic109D411yescharacteristic
70C4×Dic109D411yescharacteristic
71C4×Dic109C2311yescharacteristic
72C2×Q8×C109—21no—
73C2×Q8×C109C2311yescharacteristic
74C2×D4×C109C2311yescharacteristic
75C2×C8×C109C2312yes—
76C4⋊C4×C109—21no—
77C4⋊C4×C109C2312yes—
78C22⋊C4×C109—21no—
79C22⋊C4×C109—21no—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C22×D109—21no—
83C22×D109—42no—
84C22×D109—42no—
85C22×D109—42no—
86C22×D109C2×D411yescharacteristic
87Q8×C109—22no—
88Q8×C109—42no—
89D4×C109—22no—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2411yescharacteristic
95C22⋊D109—42no—
96C2×Dic109—22no—
97C2×Dic109—21no—
98C2×Dic109C2×D411yescharacteristic
99C2×Dic109C2×D411yescharacteristic
100C2×Dic109C2×D411yescharacteristic
101D436—22no—
102C4×D109—41no—
103C4×D109—41no—
104C4×D109—42no—
105C4×D109#4114yes—
106Dic218—22no—
107Dic218—42no—
108Dic218—42no—
109C8×C109—22no—
110#11724784—22no—
111C22×C109—41no—
112C22×C109—41no—
113C22×C109—41no—
114C22×C109C22×D411yescharacteristic
115C2×D109—21no—
116C2×D109—21no—
117C2×D109—41no—
118C2×D109—41no—
119C2×D109—41no—
120C2×D109—84no—
121C2×D109#12412yes—
122C4×C109—21no—
123C4×C109—21no—
124C4×C109—41no—
125C4×C109—42no—
126C4×C109C2×#4112yes—
127Dic109—21no—
128Dic109—21no—
129Dic109—41no—
130Dic109—42no—
131Dic109#12412yes—
132C2×C109—81no—
133C2×C109C2×#12411yescharacteristic
134C2×C109#57511yescharacteristic
135C2×C109#57511yescharacteristic
136D109—22no—
137D109—81no—
138#3140—1091nomaximal, Sylow
139C109#314011yescharacteristic, complemented, Sylow
140#533—1091no—
141C2×#124—1091no—
142#489—1091no—
143#489—1091no—
144#489—1091no—
145#489D10911yescharacteristic, complemented
146#488—1092no—
147#488—1092no—
148#419—1092no—
149#418—1092no—
150C4×M16—1091no—
151C22×Q8—1091no—
152C22×D4—1091no—
153C2×M16—1092no—
154C4⋊Q8—1091no—
155C4⋊Q8C2×D10911yescharacteristic, complemented
156#125—1092no—
157#124—1091no—
158#124—1091no—
159#124—1091no—
160#124—2181no—
161#124—2181no—
162#124C2×D10911yescharacteristic
163C22⋊Q8—2181no—
164C22⋊Q8—2181no—
165C4×Q8—2181no—
166C4×Q8—2181no—
167#117—1092no—
168C2×C22⋊C4—2181no—
169C2×C22⋊C4—2181no—
170C2×C42—1091no—
171#103—1092no—
172#103—1092no—
173#103—1092no—
174#103C2×D10912yescomplemented
175#102—1092no—
176#102—1092no—
177#102—1092no—
178#102C2×D10912yes—
179#97—1091no—
180#97C2×D10911yescharacteristic, complemented
181C4×C8—1092no—
182C24—2181no—
183C2×Q8—21no—
184C2×Q8—1091no—
185C2×Q8—2181no—
186C2×Q8—2184no—
187C2×Q8—2181no—
188C2×Q8C22×D10911yescharacteristic
189C2×D4—1091no—
190C2×D4—2184no—
191C2×D4—2181no—
192C2×D4C22×D10911yescharacteristic
193C22×C4—1091no—
194C22×C4—1091no—
195C22×C4—1091no—
196C22×C4—2181no—
197M16—2184no—
198C2×C8—1092no—
199C2×C8C22×D10912yes—
200C4⋊C4—21no—
201C4⋊C4—1092no—
202C4⋊C4—2182no—
203C4⋊C4—2182no—
204C4⋊C4—2181no—
205C4⋊C4C22×D10912yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—2181no—
209C22⋊C4—2181no—
210C22⋊C4—2182no—
211C22⋊C4—2181no—
212C22⋊C4—2181no—
213C22⋊C4—2181no—
214C22⋊C4—2181no—
215C42—1091no—
216C42—1091no—
217C42—1091no—
218C42—2182no—
219C42C22×D10911yescharacteristic
220C23—21no—
221C23—1091no—
222C23—4362no—
223C23—4362no—
224C23—4362no—
225C23—2181no—
226Q8—22no—
227Q8—42no—
228Q8—4362no—
229Q8—4362no—
230Q8—2182no—
231D4—22no—
232D4—4362no—
233D4—2182no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—1091no—
237C2×C4—1094no—
238C2×C4—1091no—
239C2×C4—1091no—
240C2×C4—4361no—
241C2×C4—4361no—
242C2×C4—4362no—
243C2×C4—2182no—
244C2×C4—2181no—
245C2×C4D4×D10911yescharacteristic
246C2×C4D4×D10911yescharacteristic
247C2×C4C23×D10911yescharacteristic
248C8—22no—
249C8—2182no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—8724no—
254C22—1092no—
255C22—4361no—
256C22—4361no—
257C22—4361no—
258C22—2181no—
259C22—2181no—
260C22C2×D4×D10911yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—1092no—
266C4—4361no—
267C4—4362no—
268C4—2181no—
269C4—2181no—
270C4#41×D10912yes—
271C2—81no—
272C2—8721no—
273C2—2182no—
274C2#124×D10911yescharacteristic
275C2#2029582211yescharacteristic
276C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#489×C109C2
socleC22×C109C22×D4
prime radicalC2×D4×D109C22
hypercenter#489D109
hypocenterC109#3140
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20295581C2semidirect
3. #20295388C2semidirect
4. #20295386C2semidirect
5. #20295385C2semidirect
6. #20295382C2semidirect
7. #20295379C2semidirect
8. #20295033C2semidirect
10. #20294978C2semidirect
11. #489×C109C2semidirect
15. #16812699C22semidirect
16. #16812690C22semidirect
30. #16812624C22semidirect
31. #16812623C22semidirect
32. #16812622C22semidirect
36. #16812610C22semidirect
37. C8×Dic109C22semidirect
38. C4⋊Q8×C109C22semidirect
40. #103×C109C22semidirect
42. #97×C109C22semidirect
139. C109#3140semidirect
145. #489D109semidirect
155. C4⋊Q8C2×D109semidirect
174. #103C2×D109semidirect
180. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955812
#124×D1092
#202953882
#202953862
#202953852
#202953822
#202953792
#202950332
#202950292
#202949782
#489×C1092
#3140109