Finite Group Databasebeta

← #28374822

Subgroups — #28374822

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#20295835C212yesmaximal, complemented
2#20295834C212yesmaximal, complemented
3#20295400C211yesmaximal, characteristic, complemented
4#20295292C212yesmaximal, complemented
5#20295291C212yesmaximal, complemented
6#20295208C211yesmaximal, characteristic, complemented
7#20295195C212yesmaximal, complemented
8#20295140C212yesmaximal, complemented
9#456×C109C211yesmaximal, characteristic, complemented
10#16812747C2211yescharacteristic
11#16812745C2212yescomplemented
12#16812731—22no—
13#16812730—22no—
14#16812730C2212yescomplemented
15#16812727C2212yes—
16#16812726—22no—
17#16812724—22no—
18#16812724C2212yes—
19#16812722—22no—
20#16812719—22no—
21#16812719C2212yes—
22C2×#13182016—22no—
23C2×#13182016C2212yescomplemented
24C2×#13182016C2212yescomplemented
25SD16×D109—22no—
26#16812707C2212yescomplemented
27#16812706C2212yescomplemented
28#16812696C2211yescharacteristic
29#16812634C2212yescomplemented
30#16812627C2212yescomplemented
31#16812618C2211yescharacteristic, complemented
32#16812615C2212yescomplemented
33C4×#11724784C2211yescharacteristic, complemented
342−1+4×C109C2211yescharacteristic
35#137×C109C2212yescomplemented
36C4⋊Q8×C109C2211yescharacteristic, complemented
37#104×C109C2212yescomplemented
38#101×C109C2211yescharacteristic, complemented
39#13182030—22no—
40#13182030—22no—
41#13182029—22no—
42#13182029—41no—
43#13182029C2312yes—
44Q8×D109—22no—
45C2×D436—22no—
46C2×D436—41no—
47C2×D436C2312yes—
48D4×D109—22no—
49D4×D109—42no—
50#13182019—22no—
51#13182019D412yescomplemented
52#13182018—22no—
53#13182017D412yescomplemented
54#13182016—22no—
55#13182016—22no—
56#13182016—22no—
57#13182016—22no—
58#13182016—22no—
59#13182016—22no—
60#13182016D412yescomplemented
61D872—22no—
62#13182014—22no—
63#13182014D412yescomplemented
64#13182013—22no—
65#13182013C2312yes—
66#13182012—22no—
67C2×#11724784—22no—
68C2×#11724784C2312yes—
69C8×D109—22no—
70#41×C109—22no—
71#41×C109—41no—
72#41×C109C2312yes—
73C2×Q8×C109—22no—
74C2×Q8×C109C2311yescharacteristic
75C2×Q8×C109C2311yescharacteristic
76Q16×C109—22no—
77SD16×C109—22no—
78M16×C109C2312yes—
79C4⋊C4×C109—41no—
80C42×C109C2311yescharacteristic
81C22×D109—22no—
82C22×D109—42no—
83Q8×C109—22no—
84Q8×C109—22no—
85Q8×C109—41no—
86Q8×C109C2×D412yes—
87D4×C109—22no—
88D4×C109—41no—
89D4×C109C2×D412yes—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109—21no—
93C2×C4×C109—21no—
94C2×C4×C109—42no—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—22no—
97C22⋊D109—42no—
98D436—22no—
99D436—22no—
100D436—42no—
101D436—41no—
102D436—82no—
103D436C2×D412yes—
104C4×D109—22no—
105C4×D109—42no—
106C4×D109—42no—
107Dic218—41no—
108Dic218C2×D412yes—
109C8×C109—22no—
110#11724784—22no—
111#11724784C2×D412yes—
112#11724784C2×D412yes—
113C22×C109—22no—
114C22×C109H(5,𝔽2)11yescharacteristic
115C2×D109—22no—
116C2×D109—42no—
117C2×D109—82no—
118C2×D109—162no—
119C4×C109—22no—
120C4×C109—22no—
121C4×C109—41no—
122C4×C109—41no—
123C4×C109C22×D412yes—
124Dic109—22no—
125C2×C109—21no—
126C2×C109—42no—
127C2×C109D4211yescharacteristic
128D109—42no—
129D109—82no—
130#3291—1091nomaximal, Sylow
131C109#329111yescharacteristic, complemented, Sylow
132#577—1092no—
133#577—1092no—
134#492—1091no—
135#471—1092no—
136#471—1092no—
137#456—1091no—
138#456D10911yescharacteristic, complemented
139#453—1092no—
140#443—1092no—
1412−1+4—1091no—
1422−1+4C2×D10911yescharacteristic
143H(5,𝔽2)—1092no—
144#137—1092no—
145#137—2182no—
146#137—2182no—
147#137C2×D10912yescomplemented
148AGL(1,ℤ/8)—1092no—
149AGL(1,ℤ/8)—1092no—
150AGL(1,ℤ/8)—2182no—
151AGL(1,ℤ/8)—2182no—
152#135—1092no—
153#135—2182no—
154#135—2182no—
155C2×SD16—1092no—
156C2×SD16—1092no—
157C2×SD16—2182no—
158C2×SD16—2182no—
159#131—1092no—
160#131—1092no—
161C4⋊Q8C2×D10911yescharacteristic, complemented
162#127—1091no—
163#108—1092no—
164#104—1092no—
165#104C2×D10912yescomplemented
166#101—1091no—
167#101C2×D10911yescharacteristic, complemented
168#100—1092no—
169C4×C8—1091no—
170#41—22no—
171#41—41no—
172#41—1092no—
173#41—4361no—
174#41—2182no—
175#41—2182no—
176#41—2182no—
177#41C22×D10912yes—
178C2×Q8—22no—
179C2×Q8—2182no—
180C2×Q8C22×D10911yescharacteristic
181C2×Q8C22×D10911yescharacteristic
182C2×D4—1092no—
183C2×D4—4361no—
184C2×D4—4362no—
185C2×D4—2182no—
186C2×D4—2182no—
187Q16—22no—
188Q16—1092no—
189Q16—2182no—
190SD16—22no—
191SD16—1092no—
192SD16—1092no—
193SD16—2182no—
194SD16—2182no—
195SD16—2182no—
196SD16—2182no—
197SD16—2182no—
198SD16—2182no—
199SD16—2182no—
200D8—1092no—
201D8—2182no—
202D8—2182no—
203M16—1092no—
204M16—2182no—
205M16—2182no—
206M16C22×D10912yes—
207C2×C8—1092no—
208C2×C8—2182no—
209C2×C8—2182no—
210C4⋊C4—41no—
211C42C22×D10911yescharacteristic
212C23—4362no—
213C23—2182no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—1092no—
218Q8—4361no—
219Q8D4×D10912yes—
220D4—22no—
221D4—41no—
222D4—8722no—
223D4—1092no—
224D4—4362no—
225D4—4361no—
226D4—4362no—
227D4—2182no—
228D4—2182no—
229D4—2182no—
230D4D4×D10912yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—4362no—
237C2×C4—4362no—
238C2×C4—2182no—
239C2×C4C23×D10911yescharacteristic
240C8—22no—
241C8—1092no—
242C8—1092no—
243C8—2182no—
244C22—22no—
245C22—8722no—
246C22—4362no—
247C22—17442no—
248C22—2182no—
249C22#1681274411yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2182no—
255C4C2×D4×D10912yes—
256C2—21no—
257C2—42no—
258C2—8722no—
259C2—4362no—
260C2#2029565811yescharacteristic
subgroupG/H
centerC2#20295658
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#456×C109C2
socleC2×C109D42
hypercenter#456D109
hypocenterC109#3291
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20295835C2semidirect
2. #20295834C2semidirect
3. #20295400C2semidirect
4. #20295292C2semidirect
5. #20295291C2semidirect
6. #20295208C2semidirect
7. #20295195C2semidirect
8. #20295140C2semidirect
9. #456×C109C2semidirect
11. #16812745C22semidirect
14. #16812730C22semidirect
23. C2×#13182016C22semidirect
24. C2×#13182016C22semidirect
26. #16812707C22semidirect
27. #16812706C22semidirect
29. #16812634C22semidirect
30. #16812627C22semidirect
31. #16812618C22semidirect
32. #16812615C22semidirect
33. C4×#11724784C22semidirect
35. #137×C109C22semidirect
36. C4⋊Q8×C109C22semidirect
37. #104×C109C22semidirect
38. #101×C109C22semidirect
51. #13182019D4semidirect
53. #13182017D4semidirect
60. #13182016D4semidirect
63. #13182014D4semidirect
131. C109#3291semidirect
138. #456D109semidirect
147. #137C2×D109semidirect
161. C4⋊Q8C2×D109semidirect
165. #104C2×D109semidirect
167. #101C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958352
#202958342
#202954002
#202952922
#202952912
#202952082
#202951952
#202951402
#456×C1092
#3291109