Finite Group Databasebeta

← #28374821

Subgroups — #28374821

Order 13952.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#20295839C211yesmaximal, characteristic, complemented
2#20295835C212yesmaximal, complemented
3#20295831C211yesmaximal, characteristic, complemented
4#20295411C211yesmaximal, characteristic, complemented
5#20295297C211yesmaximal, characteristic, complemented
6#20295292C212yesmaximal, complemented
7#20295202C212yesmaximal, complemented
8#20295201C212yesmaximal, complemented
9#20295142C211yesmaximal, characteristic, complemented
10#20295141C211yesmaximal, characteristic, complemented
11#473×C109C211yesmaximal, characteristic, complemented
12#16812747C2212yescomplemented
13#16812745C2212yes—
14#16812731—22no—
15#16812731—22no—
16#16812730C2212yescomplemented
17#16812727—22no—
18#16812727C2212yes—
19#16812727C2212yescomplemented
20#16812726—22no—
21#16812722—22no—
22#16812722—21no—
23#16812721—21no—
24#16812721C2211yescharacteristic
25#16812719C2211yescharacteristic
26Q16×D109—21no—
27C2×#13182016C2212yescomplemented
28SD16×D109—22no—
29C2×D872—21no—
30C2×D872C2211yescharacteristic
31#16812707C2212yescomplemented
32#16812705C2211yescharacteristic, complemented
33#16812688C2212yescomplemented
34#16812634C2211yescharacteristic, complemented
35#16812626C2211yescharacteristic
36#16812625C2211yescharacteristic, complemented
37#16812625C2212yescomplemented
38#16812615C2212yescomplemented
39C8×Dic109C2211yescharacteristic, complemented
40#137×C109C2212yes—
41C2×Q16×C109C2211yescharacteristic, complemented
42#131×C109C2211yescharacteristic
43#108×C109C2211yescharacteristic, complemented
44#101×C109C2212yescomplemented
45#13182030—22no—
46#13182030—22no—
47#13182030—22no—
48#13182029—22no—
49#13182029—42no—
50#13182029C2311yescharacteristic
51#13182029C2312yes—
52Q8×D109—22no—
53Q8×D109—22no—
54C2×D436—22no—
55C2×D436C2312yes—
56D4×D109—22no—
57D4×D109—42no—
58Dic436D411yescharacteristic
59#13182019—22no—
60#13182019D411yescharacteristic
61#13182018—22no—
62#13182018—22no—
63#13182018D412yes—
64#13182017D411yescharacteristic
65#13182016—22no—
66#13182016—22no—
67#13182016—22no—
68#13182016D411yescharacteristic
69D872—22no—
70D872—22no—
71D872—21no—
72D872—21no—
73D872D411yescharacteristic
74#13182014D411yescharacteristic
75#13182013C2311yescharacteristic
76#13182012—21no—
77#13182012—22no—
78C2×#11724784C2311yescharacteristic
79C8×D109—21no—
80C8×D109—22no—
81#13182006—42no—
82C4×Dic109C2311yescharacteristic
83#41×C109C2311yescharacteristic
84C2×Q8×C109C2312yes—
85Q16×C109—22no—
86Q16×C109—21no—
87Q16×C109—21no—
88SD16×C109—22no—
89M16×C109—21no—
90M16×C109C2311yescharacteristic
91M16×C109C2312yes—
92C2×C8×C109—21no—
93C2×C8×C109C2311yescharacteristic
94C22×D109—22no—
95C22×D109—42no—
96Q8×C109—22no—
97Q8×C109—22no—
98Q8×C109C2×D411yescharacteristic
99D4×C109—21no—
100D4×C109C2×D411yescharacteristic
101C2×C4×C109—21no—
102C2×C4×C109—22no—
103C2×C4×C109C2411yescharacteristic
104C22⋊D109—22no—
105C22⋊D109—21no—
106C22⋊D109—42no—
107C2×Dic109—21no—
108D436—22no—
109D436—22no—
110D436—42no—
111D436—42no—
112D436C2×D411yescharacteristic
113D436C2×D412yes—
114C4×D109—22no—
115C4×D109—21no—
116C4×D109—42no—
117C4×D109—42no—
118C4×D109—42no—
119Dic218—42no—
120Dic218C2×D411yescharacteristic
121Dic218C2×D412yes—
122C8×C109—21no—
123C8×C109—22no—
124C8×C109C2×D411yescharacteristic
125C8×C109C2×D411yescharacteristic
126#11724784—21no—
127#11724784C2×D411yescharacteristic
128#11724784C2×D411yescharacteristic
129C22×C109—21no—
130C22×C109H(5,𝔽2)11yescharacteristic
131C2×D109—21no—
132C2×D109—22no—
133C2×D109—42no—
134C2×D109—82no—
135C2×D109—162no—
136C4×C109—21no—
137C4×C109—42no—
138C4×C109C22×D411yescharacteristic
139C4×C109C22×D411yescharacteristic
140Dic109—21no—
141Dic109—22no—
142Dic109—41no—
143C2×C109—21no—
144C2×C109—41no—
145C2×C109D4211yescharacteristic
146D109—41no—
147D109—42no—
148D109—82no—
149#3290—1091nomaximal, Sylow
150C109#329011yescharacteristic, complemented, Sylow
151#578—1091no—
152#577—1092no—
153#576—1091no—
154#495—1091no—
155#473D10911yescharacteristic, complemented
156#472—1091no—
157#471—1092no—
158#455—1092no—
159#454—1092no—
160#443—1091no—
161#443—1091no—
1622−1+4—1092no—
163H(5,𝔽2)—1092no—
164#137—1092no—
165#137—2182no—
166#137—2182no—
167#137C2×D10912yes—
168AGL(1,ℤ/8)—1092no—
169AGL(1,ℤ/8)—1092no—
170AGL(1,ℤ/8)—2182no—
171AGL(1,ℤ/8)—2182no—
172#135—1091no—
173#135—1091no—
174#135—2181no—
175#135—2181no—
176#135—2182no—
177C2×Q16—2181no—
178C2×Q16C2×D10911yescharacteristic, complemented
179C2×SD16—1092no—
180C2×SD16—2182no—
181C2×D8—1091no—
182C2×D8—2181no—
183#131—1092no—
184#131—1091no—
185#131C2×D10911yescharacteristic
186#124—1092no—
187#108—1091no—
188#108C2×D10911yescharacteristic, complemented
189#104—1091no—
190#104—1091no—
191#104—1092no—
192#101C2×D10912yescomplemented
193#100—1092no—
194C4×C8—1091no—
195#41—1091no—
196#41—1092no—
197#41—4362no—
198#41—2182no—
199#41—2182no—
200#41—2182no—
201#41—2182no—
202#41C22×D10911yescharacteristic
203C2×Q8—2182no—
204C2×Q8—2182no—
205C2×Q8C22×D10912yes—
206C2×D4—1092no—
207C2×D4—4362no—
208C2×D4—2182no—
209C2×D4—2182no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—1091no—
214Q16—1091no—
215Q16—2182no—
216SD16—22no—
217SD16—1091no—
218SD16—1091no—
219SD16—1092no—
220SD16—2182no—
221SD16—2182no—
222SD16—2182no—
223SD16—2182no—
224SD16—2182no—
225D8—1091no—
226D8—1091no—
227D8—2182no—
228D8—2182no—
229D8—2181no—
230D8—2181no—
231M16—21no—
232M16—1091no—
233M16—2182no—
234M16—2181no—
235M16C22×D10912yes—
236M16C22×D10911yescharacteristic
237C2×C8—21no—
238C2×C8—1091no—
239C2×C8—2182no—
240C2×C8—2181no—
241C2×C8C22×D10911yescharacteristic
242C22⋊C4—4362no—
243C42—1091no—
244C23—4362no—
245C23—2182no—
246Q8—22no—
247Q8—22no—
248Q8—1091no—
249Q8—1092no—
250Q8—4362no—
251Q8D4×D10911yescharacteristic
252D4—21no—
253D4—1091no—
254D4—1092no—
255D4—4362no—
256D4—4362no—
257D4—4362no—
258D4—2182no—
259D4—2181no—
260D4—2182no—
261D4—2182no—
262D4D4×D10911yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—4362no—
266C2×C4—4362no—
267C2×C4—4362no—
268C2×C4—2181no—
269C2×C4—2182no—
270C2×C4—2181no—
271C2×C4C23×D10911yescharacteristic
272C8—21no—
273C8—22no—
274C8—1091no—
275C8—1091no—
276C8—2181no—
277C8D4×D10911yescharacteristic
278C8D4×D10911yescharacteristic
279C22—21no—
280C22—8722no—
281C22—4362no—
282C22—17442no—
283C22—2181no—
284C22—2182no—
285C22#1681274511yescharacteristic
286C4—21no—
287C4—42no—
288C4—4361no—
289C4—2182no—
290C4—2181no—
291C4C2×D4×D10911yescharacteristic
292C4C2×D4×D10911yescharacteristic
293C2—21no—
294C2—41no—
295C2—8722no—
296C2—4362no—
297C2—4361no—
298C2#2029566011yescharacteristic
subgroupG/H
centerC2#20295660
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#473×C109C2
socleC2×C109D42
hypercenter#473D109
hypocenterC109#3290
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #20295839C2semidirect
2. #20295835C2semidirect
3. #20295831C2semidirect
4. #20295411C2semidirect
5. #20295297C2semidirect
6. #20295292C2semidirect
7. #20295202C2semidirect
8. #20295201C2semidirect
9. #20295142C2semidirect
10. #20295141C2semidirect
11. #473×C109C2semidirect
12. #16812747C22semidirect
16. #16812730C22semidirect
19. #16812727C22semidirect
27. C2×#13182016C22semidirect
31. #16812707C22semidirect
32. #16812705C22semidirect
33. #16812688C22semidirect
34. #16812634C22semidirect
36. #16812625C22semidirect
37. #16812625C22semidirect
38. #16812615C22semidirect
39. C8×Dic109C22semidirect
41. C2×Q16×C109C22semidirect
43. #108×C109C22semidirect
44. #101×C109C22semidirect
150. C109#3290semidirect
155. #473D109semidirect
178. C2×Q16C2×D109semidirect
188. #108C2×D109semidirect
192. #101C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958392
#202958352
#202958312
#202954112
#202952972
#202952922
#202952022
#202952012
#202951422
#202951412
#473×C1092
#3290109