Finite Group Databasebeta

← #28374827

Subgroups — #28374827

Order 13952.

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

normal subgroups only

This group has 460 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#20295837C212yesmaximal, complemented
2#20295836C212yesmaximal, complemented
3#20295398C211yesmaximal, characteristic, complemented
4#20295288C212yesmaximal, complemented
5#20295287C212yesmaximal, complemented
6#20295210C212yesmaximal, complemented
7#20295193C211yesmaximal, characteristic, complemented
8#20295140C212yesmaximal, complemented
9#453×C109C211yesmaximal, characteristic, complemented
10#16812746C2212yes—
11#16812744C2211yescharacteristic, complemented
12#16812732C2212yescomplemented
13#16812729—22no—
14#16812729C2212yescomplemented
15#16812728—22no—
16#16812725—22no—
17#16812725C2212yes—
18#16812723—22no—
19#16812719—22no—
20#16812719C2212yes—
21#16812718—22no—
22C2×#13182017—22no—
23C2×#13182017C2212yescomplemented
24C2×#13182017C2212yescomplemented
25SD16×D109—22no—
26#16812707C2212yescomplemented
27#16812706C2212yescomplemented
28#16812699C2211yescharacteristic, complemented
29#16812634C2212yescomplemented
30#16812627C2212yescomplemented
31#16812617C2212yescomplemented
32#16812616C2211yescharacteristic, complemented
33C4×#11724784C2211yescharacteristic, complemented
34H(5,𝔽2)×C109C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C109C2212yes—
36#127×C109C2211yescharacteristic
37#104×C109C2212yescomplemented
38#100×C109C2211yescharacteristic, complemented
39#13182029—22no—
40#13182029C2312yes—
41#13182028—22no—
42#13182028—22no—
43#13182028—42no—
44C2×Dic218—22no—
45C2×Dic218C2312yes—
46Q8×D109—22no—
47C2×C22⋊D109—41no—
48D4×D109—22no—
49Dic436—22no—
50#13182019—22no—
51#13182019D412yescomplemented
52#13182018—22no—
53#13182017—22no—
54#13182017—22no—
55#13182017—22no—
56#13182017—22no—
57#13182017—22no—
58#13182017—22no—
59#13182017D412yescomplemented
60#13182016D412yescomplemented
61#13182014—22no—
62#13182014D412yescomplemented
63#13182013—22no—
64#13182013C2312yes—
65#13182012—22no—
66C2×#11724784—22no—
67C2×#11724784C2312yes—
68C8×D109—22no—
69C4⋊Dic109—41no—
70#41×C109—22no—
71#41×C109C2312yes—
72C2×D4×C109—22no—
73C2×D4×C109—41no—
74C2×D4×C109—41no—
75C2×D4×C109C2311yescharacteristic
76C2×D4×C109C2311yescharacteristic
77SD16×C109—22no—
78D8×C109—22no—
79M16×C109C2312yes—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C23×C109—21no—
83C23×C109—41no—
84C22×D109—41no—
85Q8×C109C2×D412yes—
86D4×C109—22no—
87D4×C109—22no—
88D4×C109—22no—
89D4×C109—42no—
90D4×C109—82no—
91D4×C109C2×D412yes—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109—41no—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—22no—
97C22⋊D109—42no—
98C22⋊D109—42no—
99C2×Dic109—22no—
100C2×Dic109—41no—
101C2×Dic109—42no—
102D436C2×D412yes—
103C4×D109—22no—
104C4×D109—42no—
105Dic218—22no—
106Dic218—22no—
107Dic218—42no—
108Dic218C2×D412yes—
109C8×C109—22no—
110#11724784—22no—
111#11724784C2×D412yes—
112#11724784C2×D412yes—
113C22×C109—22no—
114C22×C109—41no—
115C22×C109—41no—
116C22×C109—81no—
117C22×C109—81no—
118C22×C109—161no—
119C22×C109H(5,𝔽2)11yescharacteristic
120C2×D109—22no—
121C2×D109—161no—
122C4×C109—22no—
123C4×C109—22no—
124C4×C109C22×D412yes—
125Dic109—22no—
126Dic109—42no—
127C2×C109—21no—
128C2×C109—42no—
129C2×C109—81no—
130C2×C109—81no—
131C2×C109D4211yescharacteristic
132D109—42no—
133#3291—1091nomaximal, Sylow
134C109#329111yescharacteristic, complemented, Sylow
135#577—1092no—
136#577—1092no—
137#492—1091no—
138#471—1092no—
139#471—1092no—
140#456—1092no—
141#453—1091no—
142#453D10911yescharacteristic, complemented
143#443—1092no—
1442−1+4—1092no—
145H(5,𝔽2)—1091no—
146H(5,𝔽2)C2×D10911yescharacteristic, complemented
147#137—1092no—
148#137—1092no—
149#137—2182no—
150#137—2182no—
151AGL(1,ℤ/8)—1092no—
152AGL(1,ℤ/8)—2182no—
153AGL(1,ℤ/8)—2182no—
154AGL(1,ℤ/8)C2×D10912yes—
155#135—1092no—
156#135—2182no—
157#135—2182no—
158C2×SD16—1092no—
159C2×SD16—1092no—
160C2×SD16—2182no—
161C2×SD16—2182no—
162#131—1092no—
163#131—1092no—
164C4⋊Q8—1091no—
165#127C2×D10911yescharacteristic
166#108—1092no—
167#104—1092no—
168#104C2×D10912yescomplemented
169#101—1092no—
170#100—1091no—
171#100C2×D10911yescharacteristic, complemented
172C4×C8—1091no—
173#41—22no—
174#41—1092no—
175#41—4362no—
176#41—2182no—
177#41—2182no—
178#41—2182no—
179#41C22×D10912yes—
180C2×Q8—1092no—
181C2×Q8—2182no—
182C2×Q8—2182no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—4361no—
187C2×D4—2182no—
188C2×D4C22×D10911yescharacteristic
189C2×D4C22×D10911yescharacteristic
190Q16—1092no—
191Q16—2182no—
192Q16—2182no—
193SD16—22no—
194SD16—1092no—
195SD16—1092no—
196SD16—2182no—
197SD16—2182no—
198SD16—2182no—
199SD16—2182no—
200SD16—2182no—
201SD16—2182no—
202SD16—2182no—
203D8—22no—
204D8—1092no—
205D8—2182no—
206M16—1092no—
207M16—2182no—
208M16—2182no—
209M16C22×D10912yes—
210C2×C8—1092no—
211C2×C8—2182no—
212C2×C8—2182no—
213C4⋊C4—4361no—
214C42C22×D10911yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—4361no—
219Q8—1092no—
220Q8—4362no—
221Q8—2182no—
222Q8—2182no—
223Q8D4×D10912yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—1092no—
230D4—4362no—
231D4—4362no—
232D4—2182no—
233D4D4×D10912yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—4362no—
238C2×C4—4362no—
239C2×C4—4361no—
240C2×C4—2182no—
241C2×C4—2182no—
242C2×C4C23×D10911yescharacteristic
243C8—22no—
244C8—1092no—
245C8—1092no—
246C8—2182no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—17441no—
254C22—2182no—
255C22#1681274411yescharacteristic
256C4—22no—
257C4—22no—
258C4—4362no—
259C4—2182no—
260C4C2×D4×D10912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—4362no—
266C2#2029565811yescharacteristic
subgroupG/H
centerC2#20295658
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#453×C109C2
socleC2×C109D42
hypercenter#453D109
hypocenterC109#3291
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20295837C2semidirect
2. #20295836C2semidirect
3. #20295398C2semidirect
4. #20295288C2semidirect
5. #20295287C2semidirect
6. #20295210C2semidirect
7. #20295193C2semidirect
8. #20295140C2semidirect
9. #453×C109C2semidirect
11. #16812744C22semidirect
12. #16812732C22semidirect
14. #16812729C22semidirect
23. C2×#13182017C22semidirect
24. C2×#13182017C22semidirect
26. #16812707C22semidirect
27. #16812706C22semidirect
28. #16812699C22semidirect
29. #16812634C22semidirect
30. #16812627C22semidirect
31. #16812617C22semidirect
32. #16812616C22semidirect
33. C4×#11724784C22semidirect
34. H(5,𝔽2)×C109C22semidirect
37. #104×C109C22semidirect
38. #100×C109C22semidirect
51. #13182019D4semidirect
59. #13182017D4semidirect
60. #13182016D4semidirect
62. #13182014D4semidirect
134. C109#3291semidirect
142. #453D109semidirect
146. H(5,𝔽2)C2×D109semidirect
168. #104C2×D109semidirect
171. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958372
#202958362
#202953982
#202952882
#202952872
#202952102
#202951932
#202951402
#453×C1092
#3291109