Finite Group Databasebeta

← #28377813

Subgroups — #28377813

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
2C2×#16812745C211yesmaximal, characteristic, complemented
3#20295838C218yesmaximal, complemented
4#20295831C218yesmaximal, complemented
5#20295826C216yesmaximal, complemented
6C2×#16812727C216yesmaximal, complemented
7#568×C109C211yesmaximal, characteristic, complemented
8#16812747—21no—
9#16812746C2214yescomplemented
10#16812745—24no—
11#16812745C2218yescomplemented
12#16812745C2214yescomplemented
13#16812744—23no—
14C2×#13182030—21no—
15C2×#13182029C2213yescomplemented
16C2×#13182029C2216yescomplemented
17#41×D109—24no—
18C22×D436C2213yescomplemented
19C2×D4×D109—23no—
20#16812732C2214yescomplemented
21#16812732C22112yescomplemented
22#16812727C22112yescomplemented
23#16812727C22124yescomplemented
24#16812727C22112yescomplemented
25#16812721C22124yescomplemented
26C2×#13182018C22112yescomplemented
27C2×D872C22112yescomplemented
28C2×#41×C109C2211yescharacteristic
29#131×C109C2218yescomplemented
30C2×M16×C109C2216yescomplemented
31C23×D109—23no—
32#13182030—24no—
33#13182030—28no—
34#13182029—26no—
35#13182029—26no—
36#13182029—26no—
37#13182029C23112yes—
38#13182029C2314yes—
39#13182029C23112yescomplemented
40#13182028—212no—
41C2×Dic218C2311yescharacteristic
42C2×Dic218C2316yescomplemented
43Q8×D109—24no—
44C2×D436—23no—
45C2×D436—26no—
46C2×D436C23112yes—
47C2×D436C2313yes—
48C2×D436C2316yescomplemented
49C2×C22⋊D109—23no—
50C2×C22⋊D109—26no—
51D4×D109—224no—
52D4×D109—212no—
53C2×C4×D109—23no—
54C2×C4×D109—26no—
55Dic436C2318yescomplemented
56#13182018C23124yescomplemented
57#13182018C2318yescomplemented
58D872C23124yescomplemented
59#41×C109D414yes—
60#41×C109C2314yes—
61C2×Q8×C109D411yescharacteristic
62C2×D4×C109D413yes—
63C22×C4×C109C2313yes—
64M16×C109C2314yes—
65M16×C109C23112yescomplemented
66C2×C8×C109C23112yescomplemented
67C23×C109C2×D413yes—
68C22×D109—23no—
69C22×D109—26no—
70C22×D109—212no—
71C22×D109—86no—
72Q8×C109C2×D414yes—
73D4×C109C2×D4112yes—
74C2×C4×C109C2×D413yes—
75C2×C4×C109C2×D416yes—
76C2×C4×C109C2416yes—
77C2×C4×C109C2411yescharacteristic
78C22⋊D109—212no—
79C22⋊D109—212no—
80C22⋊D109—24no—
81C2×Dic109—21no—
82C2×Dic109—26no—
83D436—212no—
84D436—26no—
85D436—23no—
86D436C2414yes—
87D436C2416yes—
88C4×D109—24no—
89C4×D109—212no—
90C4×D109—212no—
91Dic218—26no—
92Dic218—21no—
93Dic218C2414yes—
94Dic218C2412yes—
95C8×C109C2418yes—
96C22×C109—43no—
97C22×C109C22×D411yescharacteristic
98C22×C109C22×D416yes—
99C2×D109—24no—
100C2×D109—26no—
101C2×D109—46no—
102C2×D109—86no—
103C2×D109—812no—
104C4×C109C22×D411yescharacteristic
105C4×C109C22×D416yes—
106C4×C109C2511yescharacteristic
107Dic109—24no—
108Dic109—22no—
109C2×C109—21no—
110C2×C109—26no—
111C2×C109C23×D411yescharacteristic
112D109—46no—
113D109—44no—
114#3585—1091nomaximal, Sylow
115C109#358511yescharacteristic, complemented, Sylow
116#585—1091no—
117C2×H(5,𝔽2)—1091no—
118#577—1098no—
119#576—1098no—
120#575—1096no—
121C2×AGL(1,ℤ/8)—1096no—
122#568D10911yescharacteristic, complemented
1232−1+4—1094no—
1242−1+4—2181no—
125H(5,𝔽2)—1094no—
126H(5,𝔽2)—1098no—
127H(5,𝔽2)—2184no—
128H(5,𝔽2)—2183no—
129C2×#41—1096no—
130C2×#41—1093no—
131C2×#41—2184no—
132C2×#41—2181no—
133C2×#41C2×D10911yescharacteristic
134C22×D4—1093no—
135C22×D4—2183no—
136#137—1094no—
137#137—10912no—
138AGL(1,ℤ/8)—10912no—
139AGL(1,ℤ/8)—10924no—
140AGL(1,ℤ/8)—10912no—
141#135—10924no—
142C2×SD16—10912no—
143C2×D8—10912no—
144#131C2×D10918yescomplemented
145C2×M16C2×D10916yescomplemented
146C24—2183no—
147#41—10912no—
148#41—1094no—
149#41—10912no—
150#41—2188no—
151#41—2184no—
152#41—21812no—
153#41—2186no—
154#41—2186no—
155#41—2186no—
156#41D43614yes—
157#41C22×D10914yes—
158C2×Q8—1091no—
159C2×Q8—1096no—
160C2×Q8—2184no—
161C2×Q8D43611yescharacteristic
162C2×D4—10912no—
163C2×D4—1093no—
164C2×D4—1096no—
165C2×D4—2183no—
166C2×D4—2186no—
167C2×D4—21824no—
168C2×D4—21812no—
169C2×D4—2183no—
170C2×D4—2186no—
171C2×D4D43613yes—
172C22×C4—2183no—
173C22×C4—2186no—
174C22×C4C22×D10913yes—
175Q16—1098no—
176SD16—10924no—
177SD16—1098no—
178D8—10924no—
179M16C22×D10914yes—
180M16C22×D109112yescomplemented
181C2×C8C22×D109112yescomplemented
182C23—8726no—
183C23—2186no—
184C23—21812no—
185C23—2183no—
186C23C2×D43613yes—
187Q8—1094no—
188Q8—1092no—
189Q8—2186no—
190Q8—2181no—
191Q8C2×D43614yes—
192D4—1094no—
193D4—1096no—
194D4—21812no—
195D4—2184no—
196D4—21812no—
197D4—2183no—
198D4—21812no—
199D4—2186no—
200D4C2×D436112yes—
201C2×C4—2181no—
202C2×C4—2186no—
203C2×C4—2184no—
204C2×C4—21812no—
205C2×C4—21812no—
206C2×C4C2×D43613yes—
207C2×C4C2×D43616yes—
208C2×C4C23×D10916yes—
209C2×C4C23×D10911yescharacteristic
210C8C23×D10918yes—
211C22—43no—
212C22—87212no—
213C22—8726no—
214C22—4366no—
215C22—2184no—
216C22—2186no—
217C22C22×D43611yescharacteristic
218C22C22×D43616yes—
219C4—2182no—
220C4—2184no—
221C4C22×D43611yescharacteristic
222C4C22×D43616yes—
223C4C24×D10911yescharacteristic
224C2—21no—
225C2—26no—
226C2—4366no—
227C2—4364no—
228C2C23×D43611yescharacteristic
subgroupG/H
centerC2C23×D436
commutator, cosocleC4×C109C25
FrattiniC4C24×D109
Fitting#568×C109C2
socleC2×C109C23×D4
hypercenter#568D109
hypocenterC109#3585
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. C2×#16812745C2semidirect
3. #20295838C2semidirect
4. #20295831C2semidirect
5. #20295826C2semidirect
6. C2×#16812727C2semidirect
7. #568×C109C2semidirect
9. #16812746C22semidirect
11. #16812745C22semidirect
12. #16812745C22semidirect
15. C2×#13182029C22semidirect
16. C2×#13182029C22semidirect
18. C22×D436C22semidirect
20. #16812732C22semidirect
21. #16812732C22semidirect
22. #16812727C22semidirect
23. #16812727C22semidirect
24. #16812727C22semidirect
25. #16812721C22semidirect
26. C2×#13182018C22semidirect
27. C2×D872C22semidirect
29. #131×C109C22semidirect
30. C2×M16×C109C22semidirect
39. #13182029C23semidirect
42. C2×Dic218C23semidirect
48. C2×D436C23semidirect
55. Dic436C23semidirect
56. #13182018C23semidirect
57. #13182018C23semidirect
58. D872C23semidirect
65. M16×C109C23semidirect
66. C2×C8×C109C23semidirect
115. C109#3585semidirect
122. #568D109semidirect
144. #131C2×D109semidirect
145. C2×M16C2×D109semidirect
180. M16C22×D109semidirect
181. C2×C8C22×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
C2×#168127452
#202958382
#202958312
#202958262
C2×#168127272
#568×C1092
#3585109