Finite Group Databasebeta

← #28372274

Subgroups — #28372274

Order 13952.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#16812730C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic, complemented
3#20295537C211yesmaximal, characteristic, complemented
4#20295188C214yesmaximal, complemented
5#20295171C214yesmaximal, complemented
6#20295035C211yesmaximal, characteristic, complemented
7#20295031C211yesmaximal, characteristic, complemented
8#20294990C211yesmaximal, characteristic, complemented
9C22×#41×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11#16812730—24no—
12#16812725—24no—
13C2×#13182019C2212yescomplemented
14C2×#13182017C2212yescomplemented
15C2×#13182016C2212yescomplemented
16C2×#13182014C2212yescomplemented
17C2×#13182013C2211yescharacteristic
18C2×#13182013C2211yescharacteristic
19#16812682—21no—
20#16812678C2212yescomplemented
21#16812673C2212yescomplemented
22#16812668—21no—
23C4×C22⋊D109—22no—
24#16812658C2211yescharacteristic
25#16812623C2214yescomplemented
26#16812620C2214yescomplemented
27C22⋊#11724784C2214yescomplemented
28C2×#41×C109—24no—
29C2×#41×C109—24no—
30C2×#41×C109—22no—
31C2×#41×C109C2214yes—
32C22×Q8×C109C2211yescharacteristic
33C22×D4×C109—21no—
34C22×D4×C109C2211yescharacteristic
35C23×C4×C109—21no—
36C23×C4×C109C2211yescharacteristic
37#13182029—22no—
38#13182029—22no—
39C2×Dic218C2311yescharacteristic
40C2×D436C2311yescharacteristic
41C2×C22⋊D109—22no—
42C2×C22⋊D109—21no—
43C2×C4×D109—21no—
44#13182019—24no—
45#13182017—24no—
46#13182016—24no—
47#13182014—24no—
48#13182013—22no—
49#13182013—22no—
50C2×#11724784C2312yes—
51C2×#11724784C2312yes—
52C4⋊Dic109C2312yes—
53#13182008—22no—
54C22⋊Dic109—21no—
55C22⋊Dic109—22no—
56#13182006—22no—
57C4×Dic109—21no—
58C24×C109—21no—
59C24×C109D411yescharacteristic
60#41×C109—216no—
61#41×C109—28no—
62#41×C109—28no—
63C2×Q8×C109—22no—
64C2×Q8×C109—21no—
65C2×Q8×C109D414yescomplemented
66C2×Q8×C109C2312yes—
67C2×D4×C109—28no—
68C2×D4×C109—24no—
69C2×D4×C109—21no—
70C2×D4×C109—22no—
71C2×D4×C109D414yescomplemented
72C2×D4×C109C2312yes—
73C22×C4×C109—28no—
74C22×C4×C109—21no—
75C22×C4×C109—24no—
76C22×C4×C109—22no—
77C22×C4×C109—22no—
78C22×C4×C109D411yescharacteristic
79C22×C4×C109D412yes—
80C22×C4×C109C2312yes—
81C22×C4×C109C2311yescharacteristic
82C23×C109—22no—
83C23×C109—24no—
84C23×C109—22no—
85C23×C109—42no—
86C23×C109—44no—
87C23×C109C2×D412yes—
88C23×C109C2×D411yescharacteristic
89C22×D109—21no—
90Q8×C109—24no—
91Q8×C109—22no—
92Q8×C109C2×D414yes—
93D4×C109—216no—
94D4×C109—24no—
95D4×C109—22no—
96D4×C109C2×D414yes—
97C2×C4×C109—216no—
98C2×C4×C109—22no—
99C2×C4×C109—24no—
100C2×C4×C109—22no—
101C2×C4×C109—24no—
102C2×C4×C109—22no—
103C2×C4×C109—22no—
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D411yescharacteristic
106C2×C4×C109C2×D412yes—
107C2×C4×C109C2×D412yes—
108C2×C4×C109C2411yescharacteristic
109C22⋊D109—44no—
110C22⋊D109—42no—
111C2×Dic109—22no—
112C2×Dic109—21no—
113D436—21no—
114D436—21no—
115C4×D109—42no—
116Dic218—21no—
117Dic218—21no—
118#11724784—22no—
119#11724784—22no—
120C22×C109—22no—
121C22×C109—22no—
122C22×C109—24no—
123C22×C109—22no—
124C22×C109—22no—
125C22×C109—44no—
126C22×C109—48no—
127C22×C109—44no—
128C22×C109#12012yes—
129C22×C109C22×D411yescharacteristic
130C2×D109—41no—
131C2×D109—42no—
132C4×C109—22no—
133C4×C109—24no—
134C4×C109#12012yes—
135C4×C109C22×D411yescharacteristic
136C4×C109C22×D411yescharacteristic
137Dic109—41no—
138Dic109—42no—
139C2×C109—22no—
140C2×C109—42no—
141C2×C109—44no—
142C2×C109C2×#12011yescharacteristic
143C2×C109#57512yes—
144D109—81no—
145#3005—1091nomaximal, Sylow
146C109#300511yescharacteristic, complemented, Sylow
147C22×#41D10911yescharacteristic, complemented
148C2×#137—1091no—
149C2×AGL(1,ℤ/8)—1091no—
150#525—1091no—
151#452—1094no—
152#449—1094no—
153#419—1091no—
154#418—1091no—
155#407—1091no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1091no—
160C2×#41C2×D10914yes—
161C22×Q8C2×D10911yescharacteristic
162C22×D4—21no—
163C22×D4C2×D10911yescharacteristic
164C23×C4—21no—
165C23×C4C2×D10911yescharacteristic
166#137—2184no—
167AGL(1,ℤ/8)—2184no—
168C2×Q16—1092no—
169C2×SD16—1092no—
170C2×SD16—1092no—
171C2×D8—1092no—
172C2×M16—1091no—
173C2×M16—1091no—
174#123—2181no—
175C22⋊Q8—1092no—
176#121—1092no—
177#120—2181no—
178C4×D4—2182no—
179#117—1091no—
180#103—1094no—
181#102—1094no—
182C22⋊C8—1094no—
183C24—21no—
184C24C22⋊D10911yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—2182no—
189#41—2182no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1091no—
193C2×Q8C22⋊D10914yescomplemented
194C2×Q8C22×D10912yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—1091no—
200C2×D4—2182no—
201C2×D4—2181no—
202C2×D4C22⋊D10914yescomplemented
203C2×D4C22×D10912yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2181no—
210C22×C4C22⋊D10912yes—
211C22×C4C22⋊D10911yescharacteristic
212C22×C4C22×D10912yes—
213C22×C4C22×D10911yescharacteristic
214Q16—2184no—
215SD16—2184no—
216SD16—2184no—
217D8—2184no—
218M16—2182no—
219M16—2182no—
220C2×C8—1092no—
221C2×C8—1092no—
222C4⋊C4—1092no—
223C4⋊C4—2182no—
224C22⋊C4—2181no—
225C22⋊C4—2182no—
226C22⋊C4—2182no—
227C42—2181no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—2181no—
234C23C2×C22⋊D10912yes—
235C23C2×C22⋊D10911yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—2181no—
239Q8—2181no—
240Q8C2×C22⋊D10914yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—4364no—
245D4—4362no—
246D4—2181no—
247D4—2181no—
248D4C2×C22⋊D10914yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—4362no—
257C2×C4—2182no—
258C2×C4—2181no—
259C2×C4C2×C22⋊D10912yes—
260C2×C4C2×C22⋊D10911yescharacteristic
261C2×C4C2×C22⋊D10912yes—
262C2×C4C2×C22⋊D10912yes—
263C2×C4C23×D10911yescharacteristic
264C8—2182no—
265C8—2182no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—4361no—
275C22—4362no—
276C22#1681266812yes—
277C22C22×C22⋊D10911yescharacteristic
278C4—22no—
279C4—24no—
280C4—4361no—
281C4—4362no—
282C4#1681266812yes—
283C4C22×C22⋊D10911yescharacteristic
284C4C22×C22⋊D10911yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—8721no—
289C2C2×#1681266811yescharacteristic
290C2#2029582412yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×#41×C109C2
socleC22×C109C22×D4
hypercenterC22×#41D109
hypocenterC109#3005
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
2. C2×#16812725C2semidirect
3. #20295537C2semidirect
4. #20295188C2semidirect
5. #20295171C2semidirect
6. #20295035C2semidirect
7. #20295031C2semidirect
8. #20294990C2semidirect
9. C22×#41×C109C2semidirect
13. C2×#13182019C22semidirect
14. C2×#13182017C22semidirect
15. C2×#13182016C22semidirect
16. C2×#13182014C22semidirect
20. #16812678C22semidirect
21. #16812673C22semidirect
25. #16812623C22semidirect
26. #16812620C22semidirect
27. C22⋊#11724784C22semidirect
65. C2×Q8×C109D4semidirect
71. C2×D4×C109D4semidirect
146. C109#3005semidirect
147. C22×#41D109semidirect
193. C2×Q8C22⋊D109semidirect
202. C2×D4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
C2×#168127252
#202955372
#202951882
#202951712
#202950352
#202950312
#202949902
C22×#41×C1092
#3005109