Finite Group Databasebeta

← #28376172

Subgroups — #28376172

Order 13952.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#20295828C211yesmaximal, characteristic, complemented
2#20295827C211yesmaximal, characteristic, complemented
3#20295464C211yesmaximal, characteristic, complemented
4#20295463C212yesmaximal, complemented
5#20295460C212yesmaximal, complemented
6#20295457C212yesmaximal, complemented
7#20295456C211yesmaximal, characteristic, complemented
8C2×#16812640C211yesmaximal, characteristic, complemented
9D16×D109C212yesmaximal, complemented
10#20295444C211yesmaximal, characteristic, complemented
11#509×C109C211yesmaximal, characteristic, complemented
12#16812745—21no—
13#16812744—21no—
14#16812726—22no—
15#16812725—21no—
16#16812724—21no—
17#16812723—22no—
18#16812722C2212yescomplemented
19#16812721C2211yescharacteristic, complemented
20#16812718C2212yescomplemented
21C2×D872C2211yescharacteristic, complemented
22D8×D109C2212yescomplemented
23D8×D109C2212yescomplemented
24#16812705C2211yescharacteristic
25#16812645C2211yescharacteristic, complemented
26#16812644C2212yescomplemented
27#16812643C2211yescharacteristic, complemented
28#16812642C2211yescharacteristic, complemented
29D1744C2212yescomplemented
30#16812640C2211yescharacteristic, complemented
31#16812640C2212yescomplemented
32#16812640C2212yescomplemented
33#16812638C2212yescomplemented
34C2×#13182004C2211yescharacteristic, complemented
35C16×D109C2212yescomplemented
36#135×C109C2211yescharacteristic, complemented
37C2×D8×C109C2211yescharacteristic, complemented
38SD32×C109C2212yescomplemented
39D16×C109C2212yescomplemented
40M32×C109C2211yescharacteristic, complemented
41#13182030—22no—
42#13182029—21no—
43#13182029—41no—
44#13182029D411yescharacteristic
45#13182028—22no—
46#13182028—41no—
47C2×D436—21no—
48C2×D436—41no—
49C2×C22⋊D109—42no—
50D4×D109—22no—
51D4×D109—22no—
52D4×D109—42no—
53D4×D109—41no—
54Dic436C2311yescharacteristic
55#13182018—21no—
56#13182017—22no—
57#13182016—22no—
58D872—21no—
59D872C2312yes—
60D872C2311yescharacteristic
61#13182014—22no—
62#13182014—22no—
63#13182013D411yescharacteristic
64#13182012D412yes—
65C8×D109C2312yes—
66#13182004C2311yescharacteristic
67#13182004C2311yescharacteristic
68#41×C109—21no—
69C2×D4×C109—21no—
70Q16×C109C2311yescharacteristic
71SD16×C109—21no—
72D8×C109—21no—
73D8×C109C2311yescharacteristic
74D8×C109C2312yes—
75C2×C8×C109C2311yescharacteristic
76C16×C109C2312yes—
77C23×C109—41no—
78C22×D109—41no—
79C22×D109—42no—
80C22×D109—42no—
81Q8×C109—21no—
82D4×C109—22no—
83D4×C109—21no—
84D4×C109—41no—
85D4×C109—41no—
86C2×C4×C109—41no—
87C2×C4×C109C2×D411yescharacteristic
88C22⋊D109—41no—
89C22⋊D109—42no—
90C22⋊D109—42no—
91C22⋊D109—42no—
92C22⋊D109D812yes—
93C2×Dic109—42no—
94D436—21no—
95D436—22no—
96D436—42no—
97D436—41no—
98D436D811yescharacteristic
99C4×D109—41no—
100C4×D109—42no—
101C4×D109C2×D412yes—
102Dic218—21no—
103Dic218D811yescharacteristic
104C8×C109C2×D411yescharacteristic
105C8×C109C2411yescharacteristic
106#11724784C2×D412yes—
107C22×C109—42no—
108C22×C109—41no—
109C22×C109—161no—
110C22×C109C2×D811yescharacteristic
111C2×D109—42no—
112C2×D109—41no—
113C2×D109—162no—
114C2×D109—162no—
115C2×D109—161no—
116C2×D109C2×D812yes—
117C4×C109—41no—
118C4×C109C2×D811yescharacteristic
119C4×C109C22×D411yescharacteristic
120Dic109—41no—
121Dic109C2×D812yes—
122C2×C109—21no—
123C2×C109—81no—
124C2×C109—82no—
125C2×C109C22×D811yescharacteristic
126D109—22no—
127D109—82no—
128D109—81no—
129#3415—1091nomaximal, Sylow
130C109#341511yescharacteristic, complemented, Sylow
131#576—1091no—
132#576—1091no—
133#509—1092no—
134#509—1092no—
135#509—1091no—
136#509D10911yescharacteristic, complemented
137#508—1091no—
138#508—1092no—
139C2×D16—1091no—
140C2×D16—1092no—
141#504—1091no—
142H(5,𝔽2)—2181no—
143H(5,𝔽2)—2181no—
144AGL(1,ℤ/8)—2182no—
145AGL(1,ℤ/8)—2182no—
146AGL(1,ℤ/8)—2181no—
147AGL(1,ℤ/8)—2181no—
148#135—1092no—
149#135—1092no—
150#135—1091no—
151#135C2×D10911yescharacteristic, complemented
152C2×D8—1092no—
153C2×D8—1092no—
154C2×D8—1091no—
155C2×D8C2×D10911yescharacteristic, complemented
156#131—1091no—
157Q32—1091no—
158SD32—1091no—
159SD32—1091no—
160SD32—1092no—
161SD32C2×D10912yescomplemented
162D16—1092no—
163D16—1091no—
164D16—1092no—
165D16—1092no—
166D16C2×D10912yescomplemented
167M32—1092no—
168M32C2×D10911yescharacteristic, complemented
169C2×C16—1091no—
170C2×C16—1092no—
171#41—21no—
172#41—1091no—
173#41—4361no—
174#41—4361no—
175#41—2182no—
176#41—2182no—
177#41—2181no—
178C2×D4—21no—
179C2×D4—4361no—
180C2×D4—4362no—
181C2×D4—4361no—
182C2×D4—4362no—
183C2×D4—2182no—
184C2×D4—2182no—
185C2×D4—2181no—
186Q16—1091no—
187Q16C22×D10911yescharacteristic
188SD16—21no—
189SD16—2182no—
190SD16—2182no—
191SD16—2181no—
192D8—21no—
193D8—1092no—
194D8—1091no—
195D8—2182no—
196D8—2182no—
197D8—2181no—
198D8C22×D10912yes—
199D8C22×D10911yescharacteristic
200M16—1091no—
201M16—1092no—
202C2×C8—1092no—
203C2×C8C22×D10911yescharacteristic
204C16—1091no—
205C16—1091no—
206C16C22×D10912yes—
207C23—41no—
208C23—4362no—
209C23—4362no—
210C23—4361no—
211Q8—21no—
212Q8—1091no—
213Q8—2181no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—1092no—
219D4—1091no—
220D4—4362no—
221D4—4362no—
222D4—4362no—
223D4—4362no—
224D4—4361no—
225D4—4361no—
226D4—2181no—
227D4—2182no—
228C2×C4—41no—
229C2×C4—1092no—
230C2×C4—4362no—
231C2×C4—4362no—
232C2×C4—4361no—
233C2×C4D4×D10911yescharacteristic
234C8—1092no—
235C8D4×D10911yescharacteristic
236C8C23×D10911yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—1092no—
241C22—4362no—
242C22—4361no—
243C22—17441no—
244C22—17442no—
245C22—17442no—
246C22D8×D10911yescharacteristic
247C4—41no—
248C4—1092no—
249C4—4361no—
250C4D8×D10911yescharacteristic
251C4C2×D4×D10911yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—8722no—
256C2—8721no—
257C2—2182no—
258C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#509×C109C2
socleC2×C109C22×D8
hypercenter#509D109
hypocenterC109#3415
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20295828C2semidirect
2. #20295827C2semidirect
3. #20295464C2semidirect
4. #20295463C2semidirect
5. #20295460C2semidirect
6. #20295457C2semidirect
7. #20295456C2semidirect
8. C2×#16812640C2semidirect
9. D16×D109C2semidirect
10. #20295444C2semidirect
11. #509×C109C2semidirect
18. #16812722C22semidirect
19. #16812721C22semidirect
20. #16812718C22semidirect
21. C2×D872C22semidirect
22. D8×D109C22semidirect
23. D8×D109C22semidirect
25. #16812645C22semidirect
26. #16812644C22semidirect
27. #16812643C22semidirect
28. #16812642C22semidirect
29. D1744C22semidirect
30. #16812640C22semidirect
31. #16812640C22semidirect
32. #16812640C22semidirect
33. #16812638C22semidirect
34. C2×#13182004C22semidirect
35. C16×D109C22semidirect
36. #135×C109C22semidirect
37. C2×D8×C109C22semidirect
38. SD32×C109C22semidirect
39. D16×C109C22semidirect
40. M32×C109C22semidirect
130. C109#3415semidirect
136. #509D109semidirect
151. #135C2×D109semidirect
155. C2×D8C2×D109semidirect
161. SD32C2×D109semidirect
166. D16C2×D109semidirect
168. M32C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958282
#202958272
#202954642
#202954632
#202954602
#202954572
#202954562
C2×#168126402
D16×D1092
#202954442
#509×C1092
#3415109