Finite Group Databasebeta

← #28374855

Subgroups — #28374855

Order 13952.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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#20295737C211yesmaximal, characteristic, complemented
2#20295661C211yesmaximal, characteristic
3#20295343C212yesmaximal, complemented
4#20295313C211yesmaximal, characteristic, complemented
5#20295254C212yesmaximal, complemented
6#20295223C211yesmaximal, characteristic, complemented
7#20295178C212yesmaximal, complemented
8C4×#13182018C211yesmaximal, characteristic, complemented
9C8×D436C211yesmaximal, characteristic, complemented
10#102×D109C212yesmaximal, complemented
11#486×C109C211yesmaximal, characteristic, complemented
12C2×D4×D109—21no—
13C2×D4×D109C2212yes—
14C2×#13182018C2211yescharacteristic
15C2×#13182017C2212yescomplemented
16C2×C8×D109C2212yescomplemented
17#16812690—21no—
18#16812679C2212yescomplemented
19#16812675—22no—
20#16812670—21no—
21#16812670C2212yes—
22#16812667—22no—
23C4×Dic218C2211yescharacteristic, complemented
24C4×D436C2211yescharacteristic
25C4×D436C2211yescharacteristic
26#16812661—21no—
27C4⋊C4×D109C2212yescomplemented
28#16812631C2211yescharacteristic, complemented
29#16812630C2212yescomplemented
30C4⋊#11724784C2211yescharacteristic, complemented
31#16812624C2211yescharacteristic, complemented
32#16812621C2211yescharacteristic
33#16812620C2212yescomplemented
34#16812619C2212yes—
35#16812611C2212yescomplemented
36C4⋊Q8×C109C2211yescharacteristic, complemented
37#127×C109C2211yescharacteristic
38#102×C109C2212yescomplemented
39#102×C109C2212yescomplemented
40C4×C8×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42C2×Dic218C2311yescharacteristic
43C2×D436D411yescharacteristic
44C2×D436C2311yescharacteristic
45C2×C22⋊D109—22no—
46C2×C22⋊D109—22no—
47D4×D109—24no—
48D4×D109—42no—
49D4×D109—44no—
50C2×C4×D109—21no—
51C2×C4×D109—22no—
52C2×C4×D109C2312yes—
53#13182018—22no—
54#13182017—42no—
55C2×#11724784C2312yes—
56C8×D109—42no—
57C4⋊Dic109—22no—
58C4⋊Dic109D411yescharacteristic
59C4⋊Dic109C2311yescharacteristic
60C4⋊Dic109C2311yescharacteristic
61#13182008—21no—
62#13182008—22no—
63C22⋊Dic109—22no—
64#13182006—22no—
65#13182006—21no—
66#13182006D412yes—
67C4×Dic109—21no—
68C2×Q8×C109—21no—
69C2×D4×C109—21no—
70C2×D4×C109—21no—
71C2×D4×C109C2312yes—
72C2×C8×C109C2311yescharacteristic
73C2×C8×C109C2311yescharacteristic
74C4⋊C4×C109—21no—
75C4⋊C4×C109C2312yes—
76C42×C109C2311yescharacteristic
77C23×C109—22no—
78C22×D109—21no—
79C22×D109—44no—
80C22×D109—44no—
81C22×D109—82no—
82C22×D109C2×D412yes—
83Q8×C109—81no—
84D4×C109—22no—
85D4×C109—22no—
86D4×C109—42no—
87D4×C109—81no—
88C2×C4×C109—22no—
89C2×C4×C109C2×D411yescharacteristic
90C2×C4×C109C2×D411yescharacteristic
91C2×C4×C109C2411yescharacteristic
92C22⋊D109—44no—
93C22⋊D109—44no—
94C2×Dic109—21no—
95C2×Dic109—21no—
96C2×Dic109—21no—
97C2×Dic109C2×D412yes—
98D436—21no—
99D436SD1614yescomplemented
100D436#4112yes—
101C4×D109—22no—
102C4×D109—22no—
103C4×D109—42no—
104C4×D109—82no—
105Dic218—21no—
106Dic218—21no—
107C8×C109—21no—
108C8×C109—21no—
109#11724784—22no—
110C22×C109—42no—
111C22×C109—42no—
112C22×C109—42no—
113C22×C109C22×D411yescharacteristic
114C2×D109—22no—
115C2×D109—22no—
116C2×D109—22no—
117C2×D109—41no—
118C2×D109—41no—
119C2×D109—88no—
120C2×D109C2×SD1614yes—
121C4×C109—21no—
122C4×C109—42no—
123C4×C109C2×SD1612yes—
124C4×C109C2×#4111yescharacteristic
125C4×C109H(5,𝔽2)11yescharacteristic
126Dic109—22no—
127Dic109—41no—
128Dic109—41no—
129Dic109—41no—
130C2×C109—82no—
131C2×C109#54611yescharacteristic
132C2×C109C22×SD1611yescharacteristic
133C2×C109#57611yescharacteristic
134D109—24no—
135D109—42no—
136#3295—1091nomaximal, Sylow
137C109#329511yescharacteristic, complemented, Sylow
138#554—1091no—
139D42—1091no—
140#486D10911yescharacteristic, complemented
141#481—1092no—
142#476—1091no—
143#465—1092no—
144#460—1091no—
145#450—1092no—
146C4×SD16—1091no—
147C8×D4—1091no—
148C2×#102—1092no—
149C22×D4—1092no—
150C22×D4—2181no—
151C2×SD16—1092no—
152C2×SD16—1091no—
153C22×C8—1092no—
154C4⋊Q8C2×D10911yescharacteristic, complemented
155#127C2×D10911yescharacteristic
156#125—2181no—
157C22⋊Q8—1092no—
158C22⋊Q8—2182no—
159#121—1092no—
160#121—2181no—
161#120—2182no—
162C4×Q8—1091no—
163C4×D4—1091no—
164C4×D4—1091no—
165C4×D4—2181no—
166C2×C4⋊C4—1092no—
167#106—1091no—
168#106—1092no—
169C4⋊C8—1091no—
170#103—1091no—
171#102—1092no—
172#102—1091no—
173#102—1092no—
174#102C2×D10912yescomplemented
175#102C2×D10912yescomplemented
176C22⋊C8—1092no—
177C4×C8C2×D10911yescharacteristic, complemented
178C24—2182no—
179C2×Q8—21no—
180C2×Q8—1091no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—1091no—
184C2×D4—1091no—
185C2×D4—4364no—
186C2×D4—4362no—
187C2×D4—2182no—
188C2×D4—2184no—
189C2×D4—2182no—
190C2×D4C22×D10912yes—
191C22×C4—1092no—
192C22×C4—2182no—
193C22×C4—2181no—
194SD16—4362no—
195SD16—2182no—
196C2×C8—1092no—
197C2×C8—4362no—
198C2×C8C22×D10911yescharacteristic
199C2×C8C22×D10911yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1091no—
202C4⋊C4—1091no—
203C4⋊C4—1091no—
204C4⋊C4—2182no—
205C4⋊C4—2182no—
206C4⋊C4—2181no—
207C4⋊C4C22×D10912yes—
208C22⋊C4—1092no—
209C22⋊C4—2182no—
210C22⋊C4—2182no—
211C22⋊C4—2181no—
212C42—2181no—
213C42C22×D10911yescharacteristic
214C23—22no—
215C23—8722no—
216C23—1092no—
217C23—4364no—
218C23—4364no—
219C23—2181no—
220Q8—81no—
221Q8—2181no—
222Q8—2181no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—1094no—
228D4—1092no—
229D4—4364no—
230D4—4364no—
231D4—2181no—
232C2×C4—22no—
233C2×C4—8722no—
234C2×C4—1092no—
235C2×C4—4362no—
236C2×C4—2181no—
237C2×C4—2181no—
238C2×C4—2181no—
239C2×C4—2182no—
240C2×C4—2182no—
241C2×C4D4×D10911yescharacteristic
242C2×C4D4×D10911yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—21no—
245C8—21no—
246C8—2182no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—8728no—
251C22—1094no—
252C22—4361no—
253C22—4361no—
254C22—2182no—
255C22—2182no—
256C22—2182no—
257C22C2×D4×D10911yescharacteristic
258C4—21no—
259C4—42no—
260C4—4361no—
261C4—4361no—
262C4—4361no—
263C4—2182no—
264C4SD16×D10912yes—
265C4#41×D10911yescharacteristic
266C4#1681274511yescharacteristic
267C2—82no—
268C2—4362no—
269C2—2184no—
270C2#2029566911yescharacteristic
271C2C2×SD16×D10911yescharacteristic
272C2#2029582711yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#486×C109C2
socleC22×C109C22×D4
prime radical#20295661C2
hypercenter#486D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20295737C2semidirect
3. #20295343C2semidirect
4. #20295313C2semidirect
5. #20295254C2semidirect
6. #20295223C2semidirect
7. #20295178C2semidirect
8. C4×#13182018C2semidirect
9. C8×D436C2semidirect
10. #102×D109C2semidirect
11. #486×C109C2semidirect
15. C2×#13182017C22semidirect
16. C2×C8×D109C22semidirect
18. #16812679C22semidirect
23. C4×Dic218C22semidirect
27. C4⋊C4×D109C22semidirect
28. #16812631C22semidirect
29. #16812630C22semidirect
30. C4⋊#11724784C22semidirect
31. #16812624C22semidirect
33. #16812620C22semidirect
35. #16812611C22semidirect
36. C4⋊Q8×C109C22semidirect
38. #102×C109C22semidirect
39. #102×C109C22semidirect
40. C4×C8×C109C22semidirect
99. D436SD16semidirect
137. C109#3295semidirect
140. #486D109semidirect
154. C4⋊Q8C2×D109semidirect
174. #102C2×D109semidirect
175. #102C2×D109semidirect
177. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957372
#202956612
#202953432
#202953132
#202952542
#202952232
#202951782
C4×#131820182
C8×D4362
#102×D1092
#486×C1092
#3295109