Finite Group Databasebeta

← #28372277

Subgroups — #28372277

Order 13952.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#13182029C211yesmaximal, characteristic, complemented
2C2×#16812732C211yesmaximal, characteristic, complemented
3C2×#16812727C211yesmaximal, characteristic, complemented
4#20295537C211yesmaximal, characteristic, complemented
5#20295190C214yesmaximal, complemented
6#20295173C214yesmaximal, complemented
7#20295036C211yesmaximal, characteristic, complemented
8#20295032C211yesmaximal, characteristic, complemented
9#407×C109C211yesmaximal, characteristic, complemented
10C2×#13182029—24no—
11C2×#13182029—24no—
12C2×#13182029—22no—
13C2×#13182029C2214yes—
14C2×#13182029C2211yescharacteristic
15C22×Dic218C2211yescharacteristic
16C22×D436C2211yescharacteristic
17C22×C22⋊D109—21no—
18C22×C4×D109—21no—
19#16812732—24no—
20#16812727—24no—
21C2×Dic436C2212yescomplemented
22C2×#13182018C2212yescomplemented
23C2×#13182018C2212yescomplemented
24C2×D872C2212yescomplemented
25#16812682—21no—
26#16812678C2212yescomplemented
27#16812673C2212yescomplemented
28#16812668—21no—
29C4×C22⋊D109—22no—
30#16812658C2211yescharacteristic
31#16812624C2214yescomplemented
32#16812621C2214yescomplemented
33C23×C4×C109C2211yescharacteristic
34C2×M16×C109C2211yescharacteristic
35C2×M16×C109C2211yescharacteristic
36C22⋊C8×C109C2214yescomplemented
37C23×D109—21no—
38#13182029—216no—
39#13182029—28no—
40#13182029—22no—
41#13182029—22no—
42#13182029—28no—
43C2×Dic218—22no—
44C2×Dic218—21no—
45C2×Dic218D414yescomplemented
46C2×Dic218C2311yescharacteristic
47C2×Dic218C2312yes—
48C2×D436—22no—
49C2×D436—21no—
50C2×D436D414yescomplemented
51C2×D436C2311yescharacteristic
52C2×D436C2312yes—
53C2×C22⋊D109—28no—
54C2×C22⋊D109—24no—
55C2×C22⋊D109—22no—
56C2×C22⋊D109—21no—
57C22×Dic109—21no—
58C2×C4×D109—28no—
59C2×C4×D109—24no—
60C2×C4×D109—21no—
61Dic436—24no—
62#13182018—24no—
63#13182018—24no—
64D872—24no—
65C4⋊Dic109C2312yes—
66#13182008—22no—
67C22⋊Dic109—21no—
68C22⋊Dic109—22no—
69#13182006—22no—
70C4×Dic109—21no—
71C24×C109D411yescharacteristic
72C22×C4×C109—22no—
73C22×C4×C109—22no—
74C22×C4×C109D412yes—
75C22×C4×C109D411yescharacteristic
76C22×C4×C109C2312yes—
77C22×C4×C109C2311yescharacteristic
78M16×C109—22no—
79M16×C109—22no—
80C2×C8×C109C2312yes—
81C2×C8×C109C2312yes—
82C23×C109—22no—
83C23×C109—42no—
84C23×C109C2×D412yes—
85C23×C109C2×D411yescharacteristic
86C22×D109—24no—
87C22×D109—22no—
88C22×D109—21no—
89C22×D109—44no—
90C2×C4×C109—22no—
91C2×C4×C109—24no—
92C2×C4×C109—22no—
93C2×C4×C109—22no—
94C2×C4×C109C2×D412yes—
95C2×C4×C109C2×D412yes—
96C2×C4×C109C2×D412yes—
97C2×C4×C109C2×D411yescharacteristic
98C2×C4×C109C2411yescharacteristic
99C22⋊D109—216no—
100C22⋊D109—44no—
101C22⋊D109—42no—
102C2×Dic109—24no—
103C2×Dic109—22no—
104C2×Dic109—22no—
105C2×Dic109—21no—
106D436—24no—
107D436—22no—
108D436—21no—
109D436—21no—
110D436C2×D414yes—
111C4×D109—216no—
112C4×D109—42no—
113Dic218—24no—
114Dic218—22no—
115Dic218—21no—
116Dic218—21no—
117Dic218C2×D414yes—
118C8×C109—22no—
119C8×C109—22no—
120C22×C109—22no—
121C22×C109—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—44no—
125C22×C109#12012yes—
126C22×C109C22×D411yescharacteristic
127C2×D109—24no—
128C2×D109—48no—
129C2×D109—44no—
130C2×D109—41no—
131C2×D109—42no—
132C4×C109—22no—
133C4×C109#12012yes—
134C4×C109C22×D411yescharacteristic
135C4×C109C22×D411yescharacteristic
136Dic109—24no—
137Dic109—41no—
138Dic109—42no—
139C2×C109—22no—
140C2×C109—42no—
141C2×C109C2×#12011yescharacteristic
142C2×C109#57512yes—
143D109—44no—
144D109—81no—
145#3005—1091nomaximal, Sylow
146C109#300511yescharacteristic, complemented, Sylow
147C22×#41—1091no—
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#407D10911yescharacteristic, complemented
156C2×#41—1091no—
157C2×#41—1094no—
158C2×#41—2184no—
159C2×#41—2184no—
160C2×#41—2182no—
161C22×Q8—1091no—
162C22×D4—1091no—
163C22×D4—2181no—
164C23×C4—2181no—
165C23×C4C2×D10911yescharacteristic
166#137—2184no—
167AGL(1,ℤ/8)—2184no—
168C2×Q16—1092no—
169C2×SD16—1092no—
170C2×SD16—1092no—
171C2×D8—1092no—
172C2×M16C2×D10911yescharacteristic
173C2×M16C2×D10911yescharacteristic
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⋊C8C2×D10914yescomplemented
183C24—2181no—
184C24D43611yescharacteristic
185#41—2188no—
186#41—2188no—
187#41—21816no—
188#41—2182no—
189#41—2182no—
190C2×Q8—1094no—
191C2×Q8—1092no—
192C2×Q8—1091no—
193C2×Q8—2182no—
194C2×Q8—2181no—
195C2×D4—1094no—
196C2×D4—1092no—
197C2×D4—1091no—
198C2×D4—2182no—
199C2×D4—2181no—
200C2×D4—2188no—
201C2×D4—2184no—
202C2×D4—2182no—
203C2×D4—2181no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2188no—
207C22×C4—2181no—
208C22×C4—2181no—
209C22×C4—2184no—
210C22×C4D43612yes—
211C22×C4D43611yescharacteristic
212C22×C4C22×D10912yes—
213C22×C4C22×D10911yescharacteristic
214Q16—2184no—
215SD16—2184no—
216SD16—2184no—
217D8—2184no—
218M16—22no—
219M16—22no—
220C2×C8C22×D10912yes—
221C2×C8C22×D10912yes—
222C4⋊C4—1092no—
223C4⋊C4—2182no—
224C22⋊C4—2181no—
225C22⋊C4—2182no—
226C22⋊C4—2182no—
227C42—2181no—
228C23—22no—
229C23—42no—
230C23—4364no—
231C23—2181no—
232C23—2184no—
233C23—2182no—
234C23C2×D43612yes—
235C23C2×D43611yescharacteristic
236Q8—1094no—
237Q8—2181no—
238Q8—2181no—
239Q8—2184no—
240Q8—2182no—
241D4—1094no—
242D4—4364no—
243D4—4362no—
244D4—21816no—
245D4—2184no—
246D4—2181no—
247D4—2181no—
248D4—2182no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—4362no—
254C2×C4—21816no—
255C2×C4—2182no—
256C2×C4—2181no—
257C2×C4—2184no—
258C2×C4—2182no—
259C2×C4D4×D10912yes—
260C2×C4D4×D10912yes—
261C2×C4C2×D43612yes—
262C2×C4C2×D43611yescharacteristic
263C2×C4C23×D10911yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—4368no—
272C22—4361no—
273C22—4362no—
274C22—4364no—
275C22—2184no—
276C22#1681266912yes—
277C22C22×D43611yescharacteristic
278C4—22no—
279C4—4361no—
280C4—4362no—
281C4—2184no—
282C4#1681266912yes—
283C4C2×D4×D10911yescharacteristic
284C4C2×D4×D10911yescharacteristic
285C2—22no—
286C2—42no—
287C2—8721no—
288C2—4364no—
289C2C2×#1681266911yescharacteristic
290C2#2029582612yes—
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#407×C109C2
socleC22×C109C22×D4
hypercenter#407D109
hypocenterC109#3005
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13182029C2semidirect
2. C2×#16812732C2semidirect
3. C2×#16812727C2semidirect
4. #20295537C2semidirect
5. #20295190C2semidirect
6. #20295173C2semidirect
7. #20295036C2semidirect
8. #20295032C2semidirect
9. #407×C109C2semidirect
21. C2×Dic436C22semidirect
22. C2×#13182018C22semidirect
23. C2×#13182018C22semidirect
24. C2×D872C22semidirect
26. #16812678C22semidirect
27. #16812673C22semidirect
31. #16812624C22semidirect
32. #16812621C22semidirect
36. C22⋊C8×C109C22semidirect
45. C2×Dic218D4semidirect
50. C2×D436D4semidirect
146. C109#3005semidirect
155. #407D109semidirect
182. C22⋊C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
C22×#131820292
C2×#168127322
C2×#168127272
#202955372
#202951902
#202951732
#202950362
#202950322
#407×C1092
#3005109