Finite Group Databasebeta

← #28377812

Subgroups — #28377812

Order 13952.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#20295862C211yesmaximal, characteristic, complemented
2#20295837C218yesmaximal, complemented
3#20295829C218yesmaximal, complemented
4#20295824C216yesmaximal, complemented
5C2×#16812725C216yesmaximal, complemented
6#20295793C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C109C211yesmaximal, characteristic, complemented
8#16812747—21no—
9#16812746C2214yescomplemented
10#16812745C2214yescomplemented
11#16812744—23no—
12C2×#13182029C2216yescomplemented
13#41×D109—24no—
14#16812730C2214yescomplemented
15#16812729C22112yescomplemented
16#16812725C22112yescomplemented
17#16812725C22124yescomplemented
18#16812724C22112yescomplemented
19#16812719C22124yescomplemented
20C2×#13182017C22112yescomplemented
21C2×#13182014C22112yescomplemented
22#16812706C2218yescomplemented
23C2×#13182013C2216yescomplemented
24H(5,𝔽2)×C109—24no—
25H(5,𝔽2)×C109C2218yescomplemented
26C2×#41×C109—21no—
27C2×#41×C109C2211yescharacteristic
28C2×#41×C109C2213yescomplemented
29C22×D4×C109—23no—
30C22×D4×C109C2213yescomplemented
31#13182030—24no—
32#13182029—26no—
33#13182029—26no—
34#13182029C2314yes—
35#13182029C23112yescomplemented
36#13182028—212no—
37C2×Dic218C2316yescomplemented
38Q8×D109—24no—
39C2×D436C2316yescomplemented
40C2×C22⋊D109—26no—
41D4×D109—212no—
42C2×C4×D109—26no—
43#13182019C2318yescomplemented
44#13182017C23124yescomplemented
45#13182016C2318yescomplemented
46#13182014C23124yescomplemented
47#13182013C2314yes—
48#13182013C23112yescomplemented
49C2×#11724784C23112yescomplemented
50C24×C109—23no—
51#41×C109—28no—
52#41×C109—26no—
53#41×C109D414yes—
54#41×C109C23112yes—
55#41×C109C2314yes—
56C2×Q8×C109D411yescharacteristic
57C2×Q8×C109C2311yescharacteristic
58C2×D4×C109—23no—
59C2×D4×C109—224no—
60C2×D4×C109—23no—
61C2×D4×C109—26no—
62C2×D4×C109D413yes—
63C2×D4×C109C2313yes—
64C2×D4×C109C23112yes—
65C22×C4×C109—23no—
66C22×C4×C109C2313yes—
67C23×C109—23no—
68C23×C109—212no—
69C23×C109—86no—
70C23×C109C2×D413yes—
71C22×D109—26no—
72Q8×C109—21no—
73Q8×C109C2×D414yes—
74Q8×C109C2412yes—
75D4×C109—212no—
76D4×C109—212no—
77D4×C109—23no—
78D4×C109C2×D4112yes—
79D4×C109C2416yes—
80C2×C4×C109—21no—
81C2×C4×C109—212no—
82C2×C4×C109C2×D413yes—
83C2×C4×C109C2×D416yes—
84C2×C4×C109C2411yescharacteristic
85C2×C4×C109C2416yes—
86C22⋊D109—212no—
87C22⋊D109—24no—
88C2×Dic109—26no—
89D436—26no—
90D436C2414yes—
91C4×D109—24no—
92C4×D109—212no—
93Dic218—26no—
94Dic218C2414yes—
95#11724784C2418yes—
96C22×C109—26no—
97C22×C109—43no—
98C22×C109—46no—
99C22×C109—812no—
100C22×C109C22×D416yes—
101C22×C109C22×D411yescharacteristic
102C2×D109—24no—
103C2×D109—86no—
104C4×C109—22no—
105C4×C109C22×D416yes—
106C4×C109C22×D411yescharacteristic
107C4×C109C2511yescharacteristic
108Dic109—24no—
109C2×C109—26no—
110C2×C109—21no—
111C2×C109—46no—
112C2×C109C23×D411yescharacteristic
113D109—44no—
114#3585—1091nomaximal, Sylow
115C109#358511yescharacteristic, complemented, Sylow
116#585—1091no—
117C2×H(5,𝔽2)D10911yescharacteristic, complemented
118#577—1098no—
119#576—1098no—
120#575—1096no—
121C2×AGL(1,ℤ/8)—1096no—
122#568—1091no—
1232−1+4—1094no—
1242−1+4—2181no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—1094no—
127H(5,𝔽2)—2183no—
128H(5,𝔽2)C2×D10918yescomplemented
129C2×#41—21no—
130C2×#41—1096no—
131C2×#41—2184no—
132C2×#41C2×D10911yescharacteristic
133C2×#41C2×D10913yescomplemented
134C22×D4—23no—
135C22×D4C2×D10913yescomplemented
136#137—10912no—
137#137—1094no—
138AGL(1,ℤ/8)—10912no—
139AGL(1,ℤ/8)—10912no—
140AGL(1,ℤ/8)—10924no—
141#135—10924no—
142C2×SD16—10912no—
143C2×D8—10912no—
144#131—1098no—
145C2×M16—1096no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—1094no—
150#41—10912no—
151#41—2184no—
152#41—21812no—
153#41—2186no—
154#41—2186no—
155#41C22⋊D10914yes—
156#41C22×D109112yes—
157#41C22×D10914yes—
158C2×Q8—1096no—
159C2×Q8—2184no—
160C2×Q8C22⋊D10911yescharacteristic
161C2×Q8C22×D10911yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—1096no—
167C2×D4—2186no—
168C2×D4—21812no—
169C2×D4C22⋊D10913yes—
170C2×D4C22×D109112yes—
171C2×D4C22×D10913yes—
172C22×C4—23no—
173C22×C4—2186no—
174C22×C4C22×D10913yes—
175Q16—1098no—
176SD16—1098no—
177SD16—10924no—
178D8—10924no—
179M16—1094no—
180M16—10912no—
181C2×C8—10912no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—2186no—
186C23C2×C22⋊D10913yes—
187Q8—21no—
188Q8—1094no—
189Q8—2186no—
190Q8C2×C22⋊D10914yes—
191Q8C23×D10912yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—1094no—
196D4—2184no—
197D4—21812no—
198D4—2186no—
199D4C2×C22⋊D109112yes—
200D4C23×D10916yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—2186no—
204C2×C4—2184no—
205C2×C4—21812no—
206C2×C4C2×C22⋊D10913yes—
207C2×C4C2×C22⋊D10916yes—
208C2×C4C23×D10916yes—
209C2×C4C23×D10911yescharacteristic
210C8—1098no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—8726no—
216C22—2184no—
217C22C22×C22⋊D10911yescharacteristic
218C22C22×C22⋊D10916yes—
219C4—22no—
220C4—2184no—
221C4C22×C22⋊D10916yes—
222C4C22×C22⋊D10911yescharacteristic
223C4C24×D10911yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—4364no—
228C2C23×C22⋊D10911yescharacteristic
subgroupG/H
centerC2C23×C22⋊D109
commutator, cosocleC4×C109C25
FrattiniC4C24×D109
FittingC2×H(5,𝔽2)×C109C2
socleC2×C109C23×D4
hypercenterC2×H(5,𝔽2)D109
hypocenterC109#3585
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. #20295837C2semidirect
3. #20295829C2semidirect
4. #20295824C2semidirect
5. C2×#16812725C2semidirect
6. #20295793C2semidirect
7. C2×H(5,𝔽2)×C109C2semidirect
9. #16812746C22semidirect
10. #16812745C22semidirect
12. C2×#13182029C22semidirect
14. #16812730C22semidirect
15. #16812729C22semidirect
16. #16812725C22semidirect
17. #16812725C22semidirect
18. #16812724C22semidirect
19. #16812719C22semidirect
20. C2×#13182017C22semidirect
21. C2×#13182014C22semidirect
22. #16812706C22semidirect
23. C2×#13182013C22semidirect
25. H(5,𝔽2)×C109C22semidirect
28. C2×#41×C109C22semidirect
30. C22×D4×C109C22semidirect
35. #13182029C23semidirect
37. C2×Dic218C23semidirect
39. C2×D436C23semidirect
43. #13182019C23semidirect
44. #13182017C23semidirect
45. #13182016C23semidirect
46. #13182014C23semidirect
48. #13182013C23semidirect
49. C2×#11724784C23semidirect
115. C109#3585semidirect
117. C2×H(5,𝔽2)D109semidirect
128. H(5,𝔽2)C2×D109semidirect
133. C2×#41C2×D109semidirect
135. C22×D4C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
#202958372
#202958292
#202958242
C2×#168127252
#202957932
C2×H(5,𝔽2)×C1092
#3585109