Finite Group Databasebeta

← #28376177

Subgroups — #28376177

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#20295839C211yesmaximal, characteristic, complemented
2#20295828C211yesmaximal, characteristic, complemented
3#20295468C212yesmaximal, complemented
4#20295464C211yesmaximal, characteristic, complemented
5#20295463C212yesmaximal, complemented
6#20295459C212yesmaximal, complemented
7#20295456C211yesmaximal, characteristic, complemented
8C2×#16812642C211yesmaximal, characteristic, complemented
9SD32×D109C212yesmaximal, complemented
10#20295444C211yesmaximal, characteristic, complemented
11#510×C109C211yesmaximal, characteristic, complemented
12#16812747—21no—
13#16812745—21no—
14#16812731—22no—
15#16812730—21no—
16#16812726—22no—
17#16812724—21no—
18#16812722C2212yes—
19#16812722C2212yescomplemented
20#16812721C2211yescharacteristic, complemented
21Q16×D109C2212yescomplemented
22C2×D872C2211yescharacteristic
23D8×D109C2212yes—
24#16812705C2211yescharacteristic
25#16812645C2211yescharacteristic, complemented
26#16812644C2212yescomplemented
27#16812643C2211yescharacteristic, complemented
28#16812642C2212yescomplemented
29#16812642C2212yescomplemented
30#16812642C2211yescharacteristic, complemented
31D1744C2212yescomplemented
32#16812640C2211yescharacteristic, complemented
33#16812638C2212yescomplemented
34C2×#13182004C2211yescharacteristic, complemented
35C16×D109C2212yescomplemented
36#135×C109C2211yescharacteristic
37C2×Q16×C109C2211yescharacteristic, complemented
38Q32×C109C2212yescomplemented
39SD32×C109C2212yescomplemented
40M32×C109C2211yescharacteristic, complemented
41#13182030—22no—
42#13182030—22no—
43#13182030—41no—
44#13182029—21no—
45#13182029—42no—
46#13182029—41no—
47#13182029D411yescharacteristic
48Q8×D109—22no—
49Q8×D109—41no—
50C2×D436—21no—
51C2×D436—41no—
52D4×D109—22no—
53D4×D109—42no—
54Dic436C2311yescharacteristic
55#13182019—22no—
56#13182018—21no—
57#13182016—22no—
58#13182016—22no—
59D872—21no—
60D872C2311yescharacteristic
61D872C2312yes—
62#13182014—22no—
63#13182013D411yescharacteristic
64#13182012D412yes—
65C8×D109C2312yes—
66#13182004C2311yescharacteristic
67#13182004C2311yescharacteristic
68#41×C109—21no—
69C2×Q8×C109—21no—
70Q16×C109—21no—
71Q16×C109C2312yes—
72Q16×C109C2311yescharacteristic
73SD16×C109—21no—
74D8×C109C2311yescharacteristic
75C2×C8×C109C2311yescharacteristic
76C16×C109C2312yes—
77C22×D109—41no—
78C22×D109—42no—
79Q8×C109—21no—
80Q8×C109—22no—
81Q8×C109—41no—
82D4×C109—21no—
83D4×C109—41no—
84C2×C4×C109—41no—
85C2×C4×C109—41no—
86C2×C4×C109C2×D411yescharacteristic
87C22⋊D109—41no—
88C22⋊D109—42no—
89C22⋊D109D812yes—
90D436—22no—
91D436—21no—
92D436—42no—
93D436—42no—
94D436—41no—
95D436D811yescharacteristic
96C4×D109—42no—
97C4×D109—41no—
98C4×D109—42no—
99C4×D109—42no—
100C4×D109C2×D412yes—
101Dic218—21no—
102Dic218—42no—
103Dic218D811yescharacteristic
104C8×C109C2×D411yescharacteristic
105C8×C109C2411yescharacteristic
106#11724784C2×D412yes—
107C22×C109—41no—
108C22×C109C2×D811yescharacteristic
109C2×D109—41no—
110C2×D109—42no—
111C2×D109—162no—
112C2×D109—161no—
113C2×D109C2×D812yes—
114C4×C109—42no—
115C4×C109—41no—
116C4×C109C2×D811yescharacteristic
117C4×C109C22×D411yescharacteristic
118Dic109—41no—
119Dic109C2×D812yes—
120C2×C109—21no—
121C2×C109—81no—
122C2×C109C22×D811yescharacteristic
123D109—22no—
124D109—81no—
125D109—82no—
126#3416—1091nomaximal, Sylow
127C109#341611yescharacteristic, complemented, Sylow
128#578—1091no—
129#576—1091no—
130#510—1092no—
131#510D10911yescharacteristic, complemented
132#509—1092no—
133#509—1091no—
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
149C2×Q16—1092no—
150C2×Q16C2×D10911yescharacteristic, complemented
151C2×D8—1092no—
152C2×D8—1091no—
153#131—1091no—
154Q32—1091no—
155Q32C2×D10912yescomplemented
156SD32—1092no—
157SD32—1092no—
158SD32—1091no—
159SD32—1091no—
160SD32—1092no—
161SD32C2×D10912yescomplemented
162D16—1091no—
163D16—1092no—
164M32—1092no—
165M32C2×D10911yescharacteristic, complemented
166C2×C16—1091no—
167C2×C16—1092no—
168#41—21no—
169#41—1091no—
170#41—4362no—
171#41—4361no—
172#41—4361no—
173#41—2182no—
174#41—2182no—
175#41—2181no—
176C2×Q8—21no—
177C2×Q8—4361no—
178C2×Q8—2182no—
179C2×D4—4361no—
180C2×D4—4362no—
181C2×D4—2182no—
182C2×D4—2181no—
183Q16—21no—
184Q16—1091no—
185Q16—2182no—
186Q16C22×D10912yes—
187Q16C22×D10911yescharacteristic
188SD16—21no—
189SD16—2182no—
190SD16—2182no—
191SD16—2181no—
192D8—1091no—
193D8—1092no—
194D8—2182no—
195D8—2181no—
196D8C22×D10911yescharacteristic
197M16—1091no—
198M16—1092no—
199C2×C8—1092no—
200C2×C8C22×D10911yescharacteristic
201C16—1091no—
202C16—1091no—
203C16C22×D10912yes—
204C23—4362no—
205C23—4361no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—1091no—
210Q8—4362no—
211Q8—2181no—
212D4—21no—
213D4—41no—
214D4—1092no—
215D4—1091no—
216D4—4362no—
217D4—4362no—
218D4—4362no—
219D4—4361no—
220D4—4361no—
221D4—2182no—
222D4—2181no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1092no—
226C2×C4—4362no—
227C2×C4—4362no—
228C2×C4—4362no—
229C2×C4—4361no—
230C2×C4D4×D10911yescharacteristic
231C8—1092no—
232C8D4×D10911yescharacteristic
233C8C23×D10911yescharacteristic
234C22—41no—
235C22—1092no—
236C22—4361no—
237C22—4362no—
238C22—17441no—
239C22—17442no—
240C22D8×D10911yescharacteristic
241C4—41no—
242C4—42no—
243C4—1092no—
244C4—4361no—
245C4D8×D10911yescharacteristic
246C4C2×D4×D10911yescharacteristic
247C2—21no—
248C2—81no—
249C2—8722no—
250C2—8721no—
251C2—2182no—
252C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#510×C109C2
socleC2×C109C22×D8
hypercenter#510D109
hypocenterC109#3416
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #20295839C2semidirect
2. #20295828C2semidirect
3. #20295468C2semidirect
4. #20295464C2semidirect
5. #20295463C2semidirect
6. #20295459C2semidirect
7. #20295456C2semidirect
8. C2×#16812642C2semidirect
9. SD32×D109C2semidirect
10. #20295444C2semidirect
11. #510×C109C2semidirect
19. #16812722C22semidirect
20. #16812721C22semidirect
21. Q16×D109C22semidirect
25. #16812645C22semidirect
26. #16812644C22semidirect
27. #16812643C22semidirect
28. #16812642C22semidirect
29. #16812642C22semidirect
30. #16812642C22semidirect
31. D1744C22semidirect
32. #16812640C22semidirect
33. #16812638C22semidirect
34. C2×#13182004C22semidirect
35. C16×D109C22semidirect
37. C2×Q16×C109C22semidirect
38. Q32×C109C22semidirect
39. SD32×C109C22semidirect
40. M32×C109C22semidirect
127. C109#3416semidirect
131. #510D109semidirect
150. C2×Q16C2×D109semidirect
155. Q32C2×D109semidirect
161. SD32C2×D109semidirect
165. M32C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958392
#202958282
#202954682
#202954642
#202954632
#202954592
#202954562
C2×#168126422
SD32×D1092
#202954442
#510×C1092
#3416109