Finite Group Databasebeta

← #28376165

Subgroups — #28376165

Order 13952.

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

normal subgroups only

This group has 364 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#135×D109C212yesmaximal, complemented
2#20295468C212yesmaximal, complemented
3#20295465C212yesmaximal, complemented
4#20295463C212yesmaximal, complemented
5#20295460C212yesmaximal, complemented
6#20295458C211yesmaximal, characteristic, complemented
7#20295456C212yesmaximal, complemented
8C2×#16812638C211yesmaximal, characteristic, complemented
9#508×C109C211yesmaximal, characteristic, complemented
10#41×D109—22no—
11#16812722C2212yes—
12#16812721C2212yescomplemented
13#16812720—22no—
14#16812719—22no—
15#16812718C2212yes—
16Q16×D109C2212yescomplemented
17SD16×D109—22no—
18D8×D109C2212yescomplemented
19C2×C8×D109C2211yescharacteristic
20Dic872C2211yescharacteristic, complemented
21#16812645C2212yescomplemented
22#16812644C2212yescomplemented
23#16812643C2212yescomplemented
24#16812642C2212yescomplemented
25D1744C2211yescharacteristic, complemented
26#16812640C2212yescomplemented
27#16812638C2212yescomplemented
28#16812638C2212yescomplemented
29C2×#13182004C2211yescharacteristic, complemented
30#135×C109C2212yescomplemented
31Q32×C109C2211yescharacteristic, complemented
32SD32×C109C2212yescomplemented
33D16×C109C2211yescharacteristic, complemented
34C2×C16×C109C2211yescharacteristic, complemented
35#13182030—22no—
36#13182029—22no—
37#13182029—42no—
38#13182028—22no—
39#13182028—42no—
40Q8×D109—22no—
41D4×D109—22no—
42D4×D109—42no—
43C2×C4×D109—42no—
44C2×C4×D109D411yescharacteristic
45Dic436C2312yes—
46#13182019—22no—
47#13182018—22no—
48#13182017—22no—
49#13182016—22no—
50D872C2312yes—
51#13182014—22no—
52C2×#11724784D411yescharacteristic
53C8×D109D412yes—
54C8×D109C2312yes—
55#13182004C2312yes—
56#41×C109—22no—
57Q16×C109C2312yes—
58SD16×C109—22no—
59D8×C109C2312yes—
60C2×C8×C109C2311yescharacteristic
61C16×C109C2311yescharacteristic
62C16×C109C2311yescharacteristic
63C22×D109—42no—
64C22×D109D811yescharacteristic
65Q8×C109—22no—
66D4×C109—22no—
67D4×C109—42no—
68C2×C4×C109—42no—
69C2×C4×C109C2×D411yescharacteristic
70C22⋊D109—42no—
71C22⋊D109—42no—
72C22⋊D109—42no—
73C2×Dic109—42no—
74C2×Dic109D811yescharacteristic
75D436—22no—
76D436—42no—
77C4×D109—42no—
78C4×D109—42no—
79C4×D109—42no—
80C4×D109D812yes—
81C4×D109C2×D412yes—
82Dic218—22no—
83Dic218—42no—
84C8×C109C2×D411yescharacteristic
85C8×C109C2411yescharacteristic
86#11724784C2×D412yes—
87C22×C109—42no—
88C22×C109C2×D811yescharacteristic
89C2×D109—41no—
90C2×D109—42no—
91C2×D109—84no—
92C2×D109C2×D812yes—
93C4×C109—42no—
94C4×C109C2×D811yescharacteristic
95C4×C109C22×D411yescharacteristic
96Dic109—42no—
97Dic109C2×D812yes—
98C2×C109—21no—
99C2×C109—82no—
100C2×C109C22×D811yescharacteristic
101D109—22no—
102D109—82no—
103#3414—1091nomaximal, Sylow
104C109#341411yescharacteristic, complemented, Sylow
105C2×#135—1092no—
106#510—1092no—
107#510—1092no—
108#509—1092no—
109#509—1092no—
110#508—1092no—
111#508—1091no—
112#508D10911yescharacteristic, complemented
113C2×M32—1091no—
114C2×#41—2182no—
115#135—1092no—
116#135—1092no—
117#135—1092no—
118#135—2182no—
119#135—2182no—
120#135C2×D10912yescomplemented
121C2×Q16—1092no—
122C2×SD16—2182no—
123C2×D8—1092no—
124C22×C8—1091no—
125Q32—1092no—
126Q32—1091no—
127Q32C2×D10911yescharacteristic, complemented
128SD32—1092no—
129SD32—1092no—
130SD32—1092no—
131SD32C2×D10912yescomplemented
132D16—1092no—
133D16—1091no—
134D16C2×D10911yescharacteristic, complemented
135M32—1092no—
136M32—1092no—
137C2×C16—1091no—
138C2×C16C2×D10911yescharacteristic, complemented
139#41—22no—
140#41—4362no—
141#41—4362no—
142#41—2182no—
143#41—2182no—
144#41—2182no—
145C2×Q8—2182no—
146C2×D4—4362no—
147C2×D4—2182no—
148C22×C4—1091no—
149C22×C4—4362no—
150Q16—1092no—
151Q16—2182no—
152Q16C22×D10912yes—
153SD16—22no—
154SD16—2182no—
155SD16—2182no—
156SD16—2182no—
157D8—1092no—
158D8—2182no—
159D8C22×D10912yes—
160C2×C8—1091no—
161C2×C8—1092no—
162C2×C8—1092no—
163C2×C8C22×D10911yescharacteristic
164C16—1092no—
165C16C22×D10911yescharacteristic
166C16C22×D10911yescharacteristic
167C23—1091no—
168C23—4362no—
169Q8—22no—
170Q8—4362no—
171Q8—2182no—
172D4—22no—
173D4—42no—
174D4—4362no—
175D4—4362no—
176D4—4362no—
177D4—4362no—
178D4—2182no—
179C2×C4—42no—
180C2×C4—1091no—
181C2×C4—1092no—
182C2×C4—1092no—
183C2×C4—4362no—
184C2×C4—4362no—
185C2×C4—4362no—
186C2×C4—4362no—
187C2×C4D4×D10911yescharacteristic
188C8—1092no—
189C8D4×D10911yescharacteristic
190C8C23×D10911yescharacteristic
191C22—42no—
192C22—8724no—
193C22—1092no—
194C22—4361no—
195C22—4362no—
196C22D8×D10911yescharacteristic
197C4—42no—
198C4—1092no—
199C4—4362no—
200C4D8×D10911yescharacteristic
201C4C2×D4×D10911yescharacteristic
202C2—21no—
203C2—82no—
204C2—8722no—
205C2—2182no—
206C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC4D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#508×C109C2
socleC2×C109C22×D8
hypercenter#508D109
hypocenterC109#3414
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D109C2semidirect
2. #20295468C2semidirect
3. #20295465C2semidirect
4. #20295463C2semidirect
5. #20295460C2semidirect
6. #20295458C2semidirect
7. #20295456C2semidirect
8. C2×#16812638C2semidirect
9. #508×C109C2semidirect
12. #16812721C22semidirect
16. Q16×D109C22semidirect
18. D8×D109C22semidirect
20. Dic872C22semidirect
21. #16812645C22semidirect
22. #16812644C22semidirect
23. #16812643C22semidirect
24. #16812642C22semidirect
25. D1744C22semidirect
26. #16812640C22semidirect
27. #16812638C22semidirect
28. #16812638C22semidirect
29. C2×#13182004C22semidirect
30. #135×C109C22semidirect
31. Q32×C109C22semidirect
32. SD32×C109C22semidirect
33. D16×C109C22semidirect
34. C2×C16×C109C22semidirect
104. C109#3414semidirect
112. #508D109semidirect
120. #135C2×D109semidirect
127. Q32C2×D109semidirect
131. SD32C2×D109semidirect
134. D16C2×D109semidirect
138. C2×C16C2×D109semidirect

Maximal subgroups

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

subgroupindex
#135×D1092
#202954682
#202954652
#202954632
#202954602
#202954582
#202954562
C2×#168126382
#508×C1092
#3414109