Finite Group Databasebeta

← #28374826

Subgroups — #28374826

Order 13952.

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

normal subgroups only

This group has 460 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#20295838C212yesmaximal, complemented
2#20295833C212yesmaximal, complemented
3#20295399C211yesmaximal, characteristic, complemented
4#20295290C212yesmaximal, complemented
5#20295289C212yesmaximal, complemented
6#20295210C212yesmaximal, complemented
7#20295195C212yesmaximal, complemented
8#20295144C211yesmaximal, characteristic, complemented
9#443×C109C211yesmaximal, characteristic, complemented
10#16812746C2212yes—
11#16812745C2212yescomplemented
12#16812732—22no—
13#16812732C2212yescomplemented
14#16812731—22no—
15#16812729C2212yescomplemented
16#16812727—22no—
17#16812727C2212yes—
18#16812724C2212yes—
19#16812723—22no—
20#16812721—22no—
21#16812721C2211yescharacteristic
22#16812720—22no—
23C2×#13182018—22no—
24C2×#13182018C2212yescomplemented
25C2×#13182018C2212yescomplemented
26SD16×D109—22no—
27#16812705C2212yescomplemented
28#16812699C2211yescharacteristic, complemented
29#16812696C2211yescharacteristic
30#16812635C2211yescharacteristic, complemented
31#16812627C2212yescomplemented
32#16812617C2212yescomplemented
33#16812615C2212yescomplemented
34#135×C109C2211yescharacteristic
35#131×C109C2212yescomplemented
36#108×C109C2211yescharacteristic, complemented
37#104×C109C2212yescomplemented
38C4×C8×C109C2211yescharacteristic, complemented
39#13182030—22no—
40#13182029—22no—
41#13182029—22no—
42#13182029C2312yes—
43#13182028—22no—
44#13182028—42no—
45C2×Dic218—22no—
46C2×Dic218C2312yes—
47Q8×D109—22no—
48C2×D436—22no—
49C2×D436—41no—
50C2×D436C2312yes—
51D4×D109—22no—
52D4×D109—42no—
53Dic436—22no—
54Dic436D411yescharacteristic, complemented
55#13182019—22no—
56#13182018—22no—
57#13182018—22no—
58#13182018—22no—
59#13182018—22no—
60#13182018—22no—
61#13182018—22no—
62#13182018D412yescomplemented
63#13182017—22no—
64#13182016—22no—
65D872—22no—
66D872D411yescharacteristic, complemented
67#13182014—22no—
68#13182013C2312yes—
69#13182012—22no—
70C8×D109—22no—
71C4⋊Dic109—41no—
72#41×C109C2312yes—
73Q16×C109D411yescharacteristic, complemented
74SD16×C109D412yescomplemented
75D8×C109D411yescharacteristic, complemented
76M16×C109—22no—
77M16×C109C2312yes—
78C2×C8×C109—22no—
79C2×C8×C109C2311yescharacteristic
80C2×C8×C109C2311yescharacteristic
81C42×C109C2311yescharacteristic
82C22×D109—22no—
83C22×D109—42no—
84Q8×C109C2×D412yes—
85D4×C109—22no—
86D4×C109C2×D412yes—
87C2×C4×C109—21no—
88C2×C4×C109—22no—
89C2×C4×C109C2411yescharacteristic
90C22⋊D109—22no—
91C22⋊D109—42no—
92C22⋊D109—42no—
93C2×Dic109—22no—
94C2×Dic109—42no—
95D436—22no—
96D436—22no—
97D436—42no—
98D436—82no—
99D436C2×D412yes—
100C4×D109—22no—
101C4×D109—42no—
102C4×D109—42no—
103Dic218—22no—
104Dic218—22no—
105Dic218—42no—
106Dic218C2×D412yes—
107C8×C109—22no—
108C8×C109C2×D411yescharacteristic
109C8×C109C2×D411yescharacteristic
110C8×C109C2×D411yescharacteristic
111C8×C109C2×D411yescharacteristic
112#11724784—22no—
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×C109C22×D411yescharacteristic
122C4×C109C22×D411yescharacteristic
123Dic109—22no—
124Dic109—42no—
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—1092no—
138#453—1092no—
139#443—1091no—
140#443D10911yescharacteristic, complemented
1412−1+4—1092no—
142H(5,𝔽2)—1092no—
143#137—1092no—
144#137—1092no—
145#137—2182no—
146#137—2182no—
147AGL(1,ℤ/8)—1092no—
148AGL(1,ℤ/8)—1092no—
149AGL(1,ℤ/8)—2182no—
150AGL(1,ℤ/8)—2182no—
151#135—1091no—
152#135—2182no—
153#135—2182no—
154#135C2×D10911yescharacteristic
155C2×SD16—1092no—
156C2×SD16—1092no—
157C2×SD16—2182no—
158C2×SD16—2182no—
159#131—1092no—
160#131C2×D10912yescomplemented
161C4⋊Q8—1091no—
162#127—1091no—
163#108—1091no—
164#108C2×D10911yescharacteristic, complemented
165#104—1092no—
166#104C2×D10912yescomplemented
167#101—1092no—
168#100—1092no—
169C4×C8C2×D10911yescharacteristic, complemented
170#41—1092no—
171#41—4362no—
172#41—2182no—
173#41—2182no—
174#41—2182no—
175#41—2182no—
176#41C22×D10912yes—
177C2×Q8—1092no—
178C2×Q8—2182no—
179C2×Q8—2182no—
180C2×D4—1092no—
181C2×D4—4361no—
182C2×D4—4362no—
183C2×D4—2182no—
184C2×D4—2182no—
185Q16—1091no—
186Q16—2182no—
187Q16—2182no—
188Q16D43611yescharacteristic, complemented
189SD16—1092no—
190SD16—2182no—
191SD16—2182no—
192SD16—2182no—
193SD16—2182no—
194SD16—2182no—
195SD16—2182no—
196SD16—2182no—
197SD16—2182no—
198SD16D43612yescomplemented
199D8—1091no—
200D8—2182no—
201D8—2182no—
202D8D43611yescharacteristic, complemented
203M16—22no—
204M16—1092no—
205M16—2182no—
206M16C22×D10912yes—
207C2×C8—22no—
208C2×C8—2182no—
209C2×C8C22×D10911yescharacteristic
210C2×C8C22×D10911yescharacteristic
211C4⋊C4—4361no—
212C42C22×D10911yescharacteristic
213C23—4362no—
214C23—2182no—
215Q8—1092no—
216Q8—4362no—
217Q8—2182no—
218Q8—2182no—
219Q8C2×D43612yes—
220D4—22no—
221D4—8722no—
222D4—1092no—
223D4—4362no—
224D4—4362no—
225D4—4362no—
226D4—2182no—
227D4—2182no—
228D4—2182no—
229D4C2×D43612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—4362no—
233C2×C4—4362no—
234C2×C4—4362no—
235C2×C4—2182no—
236C2×C4—2182no—
237C2×C4C23×D10911yescharacteristic
238C8—22no—
239C8—2182no—
240C8D4×D10911yescharacteristic
241C8D4×D10911yescharacteristic
242C8C2×D43611yescharacteristic
243C8C2×D43611yescharacteristic
244C22—22no—
245C22—8722no—
246C22—4362no—
247C22—17442no—
248C22—2182no—
249C22#1681274511yescharacteristic
250C4—22no—
251C4—22no—
252C4—4362no—
253C4—2182no—
254C4C2×D4×D10911yescharacteristic
255C4C22×D43611yescharacteristic
256C2—21no—
257C2—42no—
258C2—8722no—
259C2—4362no—
260C2D4×D43611yescharacteristic
subgroupG/H
centerC2D4×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#443×C109C2
socleC2×C109D42
hypercenter#443D109
hypocenterC109#3291
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20295838C2semidirect
2. #20295833C2semidirect
3. #20295399C2semidirect
4. #20295290C2semidirect
5. #20295289C2semidirect
6. #20295210C2semidirect
7. #20295195C2semidirect
8. #20295144C2semidirect
9. #443×C109C2semidirect
11. #16812745C22semidirect
13. #16812732C22semidirect
15. #16812729C22semidirect
24. C2×#13182018C22semidirect
25. C2×#13182018C22semidirect
27. #16812705C22semidirect
28. #16812699C22semidirect
30. #16812635C22semidirect
31. #16812627C22semidirect
32. #16812617C22semidirect
33. #16812615C22semidirect
35. #131×C109C22semidirect
36. #108×C109C22semidirect
37. #104×C109C22semidirect
38. C4×C8×C109C22semidirect
54. Dic436D4semidirect
62. #13182018D4semidirect
66. D872D4semidirect
73. Q16×C109D4semidirect
74. SD16×C109D4semidirect
75. D8×C109D4semidirect
131. C109#3291semidirect
140. #443D109semidirect
160. #131C2×D109semidirect
164. #108C2×D109semidirect
166. #104C2×D109semidirect
169. C4×C8C2×D109semidirect
188. Q16D436semidirect
198. SD16D436semidirect
202. D8D436semidirect

Maximal subgroups

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

subgroupindex
#202958382
#202958332
#202953992
#202952902
#202952892
#202952102
#202951952
#202951442
#443×C1092
#3291109