Finite Group Databasebeta

← #28376178

Subgroups — #28376178

Order 13952.

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

normal subgroups only

This group has 360 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#20295840C211yesmaximal, characteristic, complemented
2#20295827C211yesmaximal, characteristic, complemented
3#20295469C211yesmaximal, characteristic, complemented
4#20295465C212yesmaximal, complemented
5#20295460C212yesmaximal, complemented
6#20295456C211yesmaximal, characteristic, complemented
7#20295455C212yesmaximal, complemented
8C2×#16812643C211yesmaximal, characteristic, complemented
9SD32×D109C212yesmaximal, complemented
10#20295444C211yesmaximal, characteristic, complemented
11#509×C109C211yesmaximal, characteristic, complemented
12#16812746—21no—
13#16812744—21no—
14#16812729—21no—
15#16812728—22no—
16#16812725—21no—
17#16812723—22no—
18#16812721C2211yescharacteristic
19#16812718C2212yes—
20#16812718C2212yescomplemented
21C2×Dic436C2211yescharacteristic, complemented
22Q16×D109C2212yescomplemented
23D8×D109C2212yes—
24#16812705C2211yescharacteristic
25Dic872C2212yescomplemented
26#16812645C2211yescharacteristic, complemented
27#16812644C2212yescomplemented
28#16812643C2212yescomplemented
29#16812643C2212yescomplemented
30#16812643C2211yescharacteristic, complemented
31#16812642C2211yescharacteristic, complemented
32#16812640C2211yescharacteristic, complemented
33#16812638C2212yescomplemented
34C2×#13182004C2211yescharacteristic, complemented
35C16×D109C2212yescomplemented
36#135×C109C2211yescharacteristic, complemented
37C2×D8×C109C2211yescharacteristic
38SD32×C109C2212yescomplemented
39D16×C109C2212yescomplemented
40M32×C109C2211yescharacteristic, complemented
41#13182029—21no—
42#13182029—41no—
43#13182029D411yescharacteristic
44#13182028—22no—
45#13182028—22no—
46#13182028—42no—
47#13182028—41no—
48C2×Dic218—21no—
49C2×Dic218—41no—
50Q8×D109—22no—
51C2×C22⋊D109—42no—
52D4×D109—22no—
53D4×D109—41no—
54Dic436—21no—
55Dic436C2312yes—
56Dic436C2311yescharacteristic
57#13182019—22no—
58#13182018—21no—
59#13182017—22no—
60#13182017—22no—
61D872C2311yescharacteristic
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—42no—
79Q8×C109—21no—
80D4×C109—22no—
81D4×C109—21no—
82D4×C109—41no—
83D4×C109—41no—
84C2×C4×C109—41no—
85C2×C4×C109C2×D411yescharacteristic
86C22⋊D109—41no—
87C22⋊D109—42no—
88C22⋊D109—42no—
89C22⋊D109—42no—
90C22⋊D109D812yes—
91C2×Dic109—41no—
92C2×Dic109—42no—
93C2×Dic109—42no—
94D436—21no—
95D436D811yescharacteristic
96C4×D109—42no—
97C4×D109—41no—
98C4×D109C2×D412yes—
99Dic218—21no—
100Dic218—22no—
101Dic218—41no—
102Dic218—42no—
103Dic218D811yescharacteristic
104C8×C109C2×D411yescharacteristic
105C8×C109C2411yescharacteristic
106#11724784C2×D412yes—
107C22×C109—41no—
108C22×C109—42no—
109C22×C109—161no—
110C22×C109C2×D811yescharacteristic
111C2×D109—41no—
112C2×D109—162no—
113C2×D109C2×D812yes—
114C4×C109—41no—
115C4×C109C2×D811yescharacteristic
116C4×C109C22×D411yescharacteristic
117Dic109—42no—
118Dic109—41no—
119Dic109C2×D812yes—
120C2×C109—21no—
121C2×C109—82no—
122C2×C109—81no—
123C2×C109C22×D811yescharacteristic
124D109—22no—
125D109—81no—
126#3416—1091nomaximal, Sylow
127C109#341611yescharacteristic, complemented, Sylow
128#578—1091no—
129#576—1091no—
130#510—1092no—
131#510—1091no—
132#509—1092no—
133#509D10911yescharacteristic, complemented
134#508—1091no—
135#508—1092no—
136C2×SD32—1091no—
137C2×SD32—1092no—
138#504—1091no—
1392−1+4—2181no—
140H(5,𝔽2)—2181no—
141#137—2182no—
142#137—2181no—
143AGL(1,ℤ/8)—2182no—
144AGL(1,ℤ/8)—2181no—
145#135—1092no—
146#135—1092no—
147#135—1091no—
148#135C2×D10911yescharacteristic, complemented
149C2×Q16—1092no—
150C2×Q16—1091no—
151C2×D8—1092no—
152C2×D8C2×D10911yescharacteristic
153#131—1091no—
154Q32—1091no—
155Q32—1092no—
156SD32—1091no—
157SD32—1092no—
158SD32—1092no—
159SD32—1091no—
160SD32—1092no—
161SD32C2×D10912yescomplemented
162D16—1091no—
163D16C2×D10912yescomplemented
164M32—1092no—
165M32C2×D10911yescharacteristic, complemented
166C2×C16—1091no—
167C2×C16—1092no—
168#41—21no—
169#41—1091no—
170#41—4361no—
171#41—4362no—
172#41—4361no—
173#41—2182no—
174#41—2182no—
175#41—2181no—
176C2×Q8—4361no—
177C2×Q8—2182no—
178C2×Q8—2181no—
179C2×D4—21no—
180C2×D4—4361no—
181C2×D4—4362no—
182C2×D4—2182no—
183Q16—1092no—
184Q16—1091no—
185Q16—2182no—
186Q16—2181no—
187Q16C22×D10911yescharacteristic
188SD16—21no—
189SD16—2182no—
190SD16—2182no—
191SD16—2181no—
192D8—21no—
193D8—1091no—
194D8—2182no—
195D8C22×D10911yescharacteristic
196D8C22×D10912yes—
197M16—1091no—
198M16—1092no—
199C2×C8—1092no—
200C2×C8C22×D10911yescharacteristic
201C16—1091no—
202C16—1091no—
203C16C22×D10912yes—
204C23—41no—
205C23—4362no—
206Q8—21no—
207Q8—1091no—
208Q8—4362no—
209Q8—4361no—
210Q8—2181no—
211Q8—2182no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—1092no—
217D4—1091no—
218D4—4362no—
219D4—4362no—
220D4—4362no—
221D4—4361no—
222D4—2181no—
223C2×C4—41no—
224C2×C4—1092no—
225C2×C4—4362no—
226C2×C4—4362no—
227C2×C4—4362no—
228C2×C4—4361no—
229C2×C4—4361no—
230C2×C4D4×D10911yescharacteristic
231C8—1092no—
232C8D4×D10911yescharacteristic
233C8C23×D10911yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—1092no—
238C22—4361no—
239C22—17442no—
240C22D8×D10911yescharacteristic
241C4—41no—
242C4—1092no—
243C4—4361no—
244C4—4362no—
245C4D8×D10911yescharacteristic
246C4C2×D4×D10911yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—8721no—
251C2—2182no—
252C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#509×C109C2
socleC2×C109C22×D8
hypercenter#509D109
hypocenterC109#3416
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. #20295840C2semidirect
2. #20295827C2semidirect
3. #20295469C2semidirect
4. #20295465C2semidirect
5. #20295460C2semidirect
6. #20295456C2semidirect
7. #20295455C2semidirect
8. C2×#16812643C2semidirect
9. SD32×D109C2semidirect
10. #20295444C2semidirect
11. #509×C109C2semidirect
20. #16812718C22semidirect
21. C2×Dic436C22semidirect
22. Q16×D109C22semidirect
25. Dic872C22semidirect
26. #16812645C22semidirect
27. #16812644C22semidirect
28. #16812643C22semidirect
29. #16812643C22semidirect
30. #16812643C22semidirect
31. #16812642C22semidirect
32. #16812640C22semidirect
33. #16812638C22semidirect
34. C2×#13182004C22semidirect
35. C16×D109C22semidirect
36. #135×C109C22semidirect
38. SD32×C109C22semidirect
39. D16×C109C22semidirect
40. M32×C109C22semidirect
127. C109#3416semidirect
133. #509D109semidirect
148. #135C2×D109semidirect
161. SD32C2×D109semidirect
163. D16C2×D109semidirect
165. M32C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958402
#202958272
#202954692
#202954652
#202954602
#202954562
#202954552
C2×#168126432
SD32×D1092
#202954442
#509×C1092
#3416109