Finite Group Databasebeta

← #28374856

Subgroups — #28374856

Order 13952.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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#20295733C211yesmaximal, characteristic, complemented
2#20295372C211yesmaximal, characteristic, complemented
3#20295342C212yesmaximal, complemented
4#20295316C211yesmaximal, characteristic, complemented
5#20295253C212yesmaximal, complemented
6#20295223C211yesmaximal, characteristic, complemented
7#20295177C212yesmaximal, complemented
8C4×#13182017C211yesmaximal, characteristic, complemented
9D4×#11724784C211yesmaximal, characteristic, complemented
10C2×#16812620C212yesmaximal, complemented
11D42×C109C211yesmaximal, characteristic
12C2×#13182017C2211yescharacteristic
13C2×#13182017C2212yescomplemented
14C22×#11724784C2212yescomplemented
15#16812699C2211yescharacteristic, complemented
16#16812690—21no—
17#16812678C2212yescomplemented
18#16812676—22no—
19C4×Dic218C2211yescharacteristic, complemented
20D4×Dic109—21no—
21C2×C4⋊Dic109C2212yescomplemented
22#16812630C2212yescomplemented
23#16812630C2211yescharacteristic, complemented
24C4⋊#11724784C2211yescharacteristic, complemented
25#16812622C2211yescharacteristic, complemented
26#16812620C2212yes—
27#16812620C2211yescharacteristic
28#16812620C2212yescomplemented
29#16812620C2212yescomplemented
30#16812620C2212yescomplemented
31C22⋊#11724784C2212yescomplemented
32C4×#11724784C2211yescharacteristic, complemented
33C22×D4×C109—21no—
34C22×D4×C109C2212yes—
35#127×C109C2211yescharacteristic
36#121×C109—21no—
37#121×C109C2212yes—
38#120×C109—22no—
39C4×D4×C109C2211yescharacteristic
40C4×D4×C109C2211yescharacteristic
41C2×Dic218—21no—
42C2×Dic218C2311yescharacteristic
43C22×Dic109—22no—
44#13182017—22no—
45#13182017—42no—
46C2×#11724784—42no—
47C2×#11724784C2311yescharacteristic
48C2×#11724784C2311yescharacteristic
49C2×#11724784C2312yes—
50C4⋊Dic109—22no—
51C4⋊Dic109—21no—
52C4⋊Dic109C2311yescharacteristic
53C4⋊Dic109C2312yes—
54#13182008—21no—
55#13182008—22no—
56C22⋊Dic109—22no—
57C4×Dic109—21no—
58C24×C109—22no—
59C2×D4×C109—24no—
60C2×D4×C109—22no—
61C2×D4×C109—22no—
62C2×D4×C109—21no—
63C2×D4×C109—21no—
64C2×D4×C109—42no—
65C2×D4×C109—44no—
66C2×D4×C109D411yescharacteristic
67C2×D4×C109C2312yes—
68C2×D4×C109C2311yescharacteristic
69C22×C4×C109—21no—
70C22×C4×C109C2312yes—
71C4⋊C4×C109D411yescharacteristic
72C4⋊C4×C109C2311yescharacteristic
73C22⋊C4×C109—21no—
74C22⋊C4×C109—22no—
75C22⋊C4×C109D412yes—
76C42×C109C2311yescharacteristic
77C23×C109—21no—
78C23×C109—22no—
79C23×C109—44no—
80C23×C109—44no—
81C23×C109—82no—
82C23×C109C2×D412yes—
83D4×C109—22no—
84D4×C109—21no—
85D4×C109—22no—
86D4×C109—44no—
87D4×C109—44no—
88D4×C109—42no—
89D4×C109—81no—
90D4×C109SD1614yescomplemented
91D4×C109#4112yes—
92C2×C4×C109—22no—
93C2×C4×C109—22no—
94C2×C4×C109—21no—
95C2×C4×C109—42no—
96C2×C4×C109C2×D411yescharacteristic
97C2×C4×C109C2×D412yes—
98C2×C4×C109C2×D411yescharacteristic
99C2×C4×C109C2411yescharacteristic
100C2×Dic109—21no—
101C2×Dic109—21no—
102C2×Dic109—22no—
103C2×Dic109—82no—
104Dic218—21no—
105Dic218—21no—
106Dic218—81no—
107#11724784—21no—
108#11724784—21no—
109#11724784—22no—
110C22×C109—22no—
111C22×C109—22no—
112C22×C109—22no—
113C22×C109—41no—
114C22×C109—41no—
115C22×C109—42no—
116C22×C109—42no—
117C22×C109—42no—
118C22×C109—88no—
119C22×C109C2×SD1614yes—
120C22×C109C22×D411yescharacteristic
121C4×C109—22no—
122C4×C109—21no—
123C4×C109—41no—
124C4×C109C2×SD1612yes—
125C4×C109C2×#4111yescharacteristic
126C4×C109H(5,𝔽2)11yescharacteristic
127Dic109—41no—
128Dic109—42no—
129Dic109—41no—
130C2×C109—24no—
131C2×C109—42no—
132C2×C109—82no—
133C2×C109#54611yescharacteristic
134C2×C109C22×SD1611yescharacteristic
135C2×C109#57611yescharacteristic
136#3295—1091nomaximal, Sylow
137C109#329511yescharacteristic, complemented, Sylow
138#554—1091no—
139D42D10911yescharacteristic
140#486—1091no—
141#481—1092no—
142#476—1091no—
143#465—1092no—
144#460—1091no—
145#450—1092no—
146C4×SD16—1091no—
147C8×D4—1091no—
148C2×#102—1092no—
149C22×D4—21no—
150C22×D4C2×D10912yes—
151C2×SD16—1092no—
152C2×SD16—1091no—
153C22×C8—1092no—
154C4⋊Q8—1091no—
155#127C2×D10911yescharacteristic
156#125—2181no—
157C22⋊Q8—1092no—
158C22⋊Q8—2182no—
159#121—21no—
160#121C2×D10912yes—
161#120—22no—
162C4×Q8—1091no—
163C4×D4—2181no—
164C4×D4C2×D10911yescharacteristic
165C4×D4C2×D10911yescharacteristic
166C2×C4⋊C4—1092no—
167#106—1091no—
168#106—1092no—
169C4⋊C8—1091no—
170#103—1091no—
171#102—1091no—
172#102—1092no—
173#102—1092no—
174#102—1092no—
175#102—1092no—
176C22⋊C8—1092no—
177C4×C8—1091no—
178C24—22no—
179C2×Q8—1091no—
180C2×Q8—2181no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D10911yescharacteristic
189C2×D4C22×D10912yes—
190C2×D4C22×D10911yescharacteristic
191C22×C4—21no—
192C22×C4—2182no—
193C22×C4C22×D10912yes—
194SD16—4362no—
195SD16—2182no—
196C2×C8—1091no—
197C2×C8—1091no—
198C2×C8—1092no—
199C2×C8—4362no—
200C4⋊C4—1091no—
201C4⋊C4—1092no—
202C4⋊C4—2181no—
203C4⋊C4—2182no—
204C4⋊C4—2182no—
205C4⋊C4—2181no—
206C4⋊C4C22⋊D10911yescharacteristic
207C4⋊C4C22×D10911yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—2182no—
211C22⋊C4C22⋊D10912yes—
212C42—2181no—
213C42C22×D10911yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D10912yes—
220Q8—8721no—
221Q8—2181no—
222Q8—2181no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1318201714yescomplemented
231D4#1318202812yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—8722no—
237C2×C4—2181no—
238C2×C4—2181no—
239C2×C4—2182no—
240C2×C4C2×C22⋊D10911yescharacteristic
241C2×C4C2×C22⋊D10912yes—
242C2×C4C2×C22⋊D10911yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—2181no—
245C8—2181no—
246C8—2182no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1318201714yes—
257C22C22×C22⋊D10911yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—4361no—
262C4—4362no—
263C4—4361no—
264C4C2×#1318201712yes—
265C4C2×#1318202811yescharacteristic
266C4#1681274411yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2029567011yescharacteristic
271C2C22×#1318201711yescharacteristic
272C2#2029582911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting, prime radicalD42×C109C2
socleC22×C109C22×D4
hypercenterD42D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20295733C2semidirect
2. #20295372C2semidirect
3. #20295342C2semidirect
4. #20295316C2semidirect
5. #20295253C2semidirect
6. #20295223C2semidirect
7. #20295177C2semidirect
8. C4×#13182017C2semidirect
9. D4×#11724784C2semidirect
10. C2×#16812620C2semidirect
13. C2×#13182017C22semidirect
14. C22×#11724784C22semidirect
15. #16812699C22semidirect
17. #16812678C22semidirect
19. C4×Dic218C22semidirect
21. C2×C4⋊Dic109C22semidirect
22. #16812630C22semidirect
23. #16812630C22semidirect
24. C4⋊#11724784C22semidirect
25. #16812622C22semidirect
28. #16812620C22semidirect
29. #16812620C22semidirect
30. #16812620C22semidirect
31. C22⋊#11724784C22semidirect
32. C4×#11724784C22semidirect
90. D4×C109SD16semidirect
137. C109#3295semidirect
230. D4#13182017semidirect

Maximal subgroups

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

subgroupindex
#202957332
#202953722
#202953422
#202953162
#202952532
#202952232
#202951772
C4×#131820172
D4×#117247842
C2×#168126202
D42×C1092
#3295109