Finite Group Databasebeta

← #508×D109 #28376146

Subgroups — #508×D109 #28376146

Order 13952.

368 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 115 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#20295459C211yesmaximal, characteristic, complemented
3#20295458C211yesmaximal, characteristic, complemented
4#20295457C212yesmaximal, complemented
5#20295456C212yesmaximal, complemented
6#20295455C211yesmaximal, characteristic, complemented
7Q32×D109C211yesmaximal, characteristic, complemented
8SD32×D109C212yesmaximal, complemented
9D16×D109C211yesmaximal, characteristic, complemented
10C2×C16×D109C211yesmaximal, characteristic, complemented
11#508×C109C211yesmaximal, characteristic, complemented
12#41×D109—22no—
13#16812722C2212yescomplemented
14#16812721C2212yescomplemented
15#16812720—22no—
16#16812719—22no—
17#16812718C2212yescomplemented
18Q16×D109C2212yes—
19SD16×D109—22no—
20D8×D109C2212yes—
21C2×C8×D109C2211yescharacteristic
22Dic872C2211yescharacteristic, complemented
23#16812645C2212yescomplemented
24#16812644C2212yescomplemented
25#16812643C2212yescomplemented
26#16812642C2212yescomplemented
27D1744C2211yescharacteristic, complemented
28#16812640C2212yescomplemented
29C2×#13182004C2211yescharacteristic, complemented
30C16×D109C2211yescharacteristic
31C16×D109C2211yescharacteristic
32C16×D109C2211yescharacteristic, complemented
33C16×D109C2211yescharacteristic, complemented
34#135×C109C2212yescomplemented
35Q32×C109C2211yescharacteristic, complemented
36SD32×C109C2212yescomplemented
37D16×C109C2211yescharacteristic, complemented
38C2×C16×C109C2211yescharacteristic, complemented
39#13182030—22no—
40#13182029—22no—
41#13182029—42no—
42#13182028—22no—
43#13182028—42no—
44Q8×D109—22no—
45D4×D109—22no—
46D4×D109—42no—
47C2×C4×D109—42no—
48C2×C4×D109D411yescharacteristic
49Dic436C2312yes—
50#13182019—22no—
51#13182018—22no—
52#13182017—22no—
53#13182016—22no—
54D872C2312yes—
55#13182014—22no—
56C2×#11724784D411yescharacteristic
57C8×D109D411yescharacteristic
58C8×D109D411yescharacteristic
59C8×D109C2311yescharacteristic
60C8×D109C2311yescharacteristic
61#13182004C2311yescharacteristic
62#13182004C2311yescharacteristic
63#41×C109—22no—
64Q16×C109C2312yes—
65SD16×C109—22no—
66D8×C109C2312yes—
67C2×C8×C109C2311yescharacteristic
68C16×C109C2311yescharacteristic
69C16×C109C2311yescharacteristic
70C22×D109—42no—
71C22×D109D811yescharacteristic
72Q8×C109—22no—
73D4×C109—22no—
74D4×C109—42no—
75C2×C4×C109—42no—
76C2×C4×C109C2×D411yescharacteristic
77C22⋊D109—42no—
78C22⋊D109—42no—
79C22⋊D109—42no—
80C2×Dic109—42no—
81C2×Dic109D811yescharacteristic
82D436—22no—
83D436—42no—
84C4×D109—42no—
85C4×D109—42no—
86C4×D109—42no—
87C4×D109D811yescharacteristic
88C4×D109D811yescharacteristic
89C4×D109C2×D411yescharacteristic
90C4×D109C2×D411yescharacteristic
91Dic218—22no—
92Dic218—42no—
93C8×C109C2×D411yescharacteristic
94C8×C109C2411yescharacteristic
95#11724784C2×D411yescharacteristic
96#11724784C2×D411yescharacteristic
97C22×C109—42no—
98C22×C109C2×D811yescharacteristic
99C2×D109—22no—
100C2×D109—42no—
101C2×D109—84no—
102C2×D109C2×D811yescharacteristic
103C2×D109C2×D811yescharacteristic
104C4×C109—42no—
105C4×C109C2×D811yescharacteristic
106C4×C109C22×D411yescharacteristic
107Dic109—42no—
108Dic109C2×D811yescharacteristic
109Dic109C2×D811yescharacteristic
110C2×C109—21no—
111C2×C109—82no—
112C2×C109C22×D811yescharacteristic
113D109—21no—
114D109—82no—
115D109#50812yesdirect product factor
116C2×#508—1091nomaximal, Sylow
117C109C2×#50811yescharacteristic, complemented, Sylow
118C2×#135—1092no—
119#508—1092no—
120#508—1091no—
121#508—1092no—
122#508—1091no—
123#508—1091no—
124#508D10911yescharacteristic, direct product factor
125C2×Q32—1091no—
126C2×SD32—1092no—
127C2×D16—1091no—
128C22×C16—1091no—
129C2×#41—2182no—
130#135—1092no—
131#135—1092no—
132#135—1092no—
133#135—2182no—
134#135—2182no—
135#135C2×D10912yescomplemented
136C2×Q16—1092no—
137C2×SD16—2182no—
138C2×D8—1092no—
139C22×C8—1091no—
140Q32—1092no—
141Q32—1091no—
142Q32C2×D10911yescharacteristic, complemented
143SD32—1092no—
144SD32—1092no—
145SD32—1092no—
146SD32C2×D10912yescomplemented
147D16—1092no—
148D16—1091no—
149D16C2×D10911yescharacteristic, complemented
150C2×C16—1091no—
151C2×C16—1091no—
152C2×C16—1091no—
153C2×C16—1091no—
154C2×C16—1091no—
155C2×C16C2×D10911yescharacteristic, complemented
156#41—22no—
157#41—4362no—
158#41—4362no—
159#41—2182no—
160#41—2182no—
161#41—2182no—
162C2×Q8—2182no—
163C2×D4—4362no—
164C2×D4—2182no—
165C22×C4—1091no—
166C22×C4—4362no—
167Q16—1092no—
168Q16—2182no—
169Q16C22×D10912yes—
170SD16—22no—
171SD16—2182no—
172SD16—2182no—
173SD16—2182no—
174D8—1092no—
175D8—2182no—
176D8C22×D10912yes—
177C2×C8—1091no—
178C2×C8—1091no—
179C2×C8—1091no—
180C2×C8—1091no—
181C2×C8—1091no—
182C2×C8C22×D10911yescharacteristic
183C16—1091no—
184C16—1091no—
185C16C22×D10911yescharacteristic
186C16C22×D10911yescharacteristic
187C23—1091no—
188C23—4362no—
189Q8—22no—
190Q8—4362no—
191Q8—2182no—
192D4—22no—
193D4—42no—
194D4—4362no—
195D4—4362no—
196D4—4362no—
197D4—4362no—
198D4—2182no—
199C2×C4—42no—
200C2×C4—1091no—
201C2×C4—1091no—
202C2×C4—1091no—
203C2×C4—1091no—
204C2×C4—1091no—
205C2×C4—4362no—
206C2×C4—4362no—
207C2×C4—4362no—
208C2×C4—4362no—
209C2×C4D4×D10911yescharacteristic
210C8—1091no—
211C8—1091no—
212C8D4×D10911yescharacteristic
213C8C23×D10911yescharacteristic
214C22—42no—
215C22—8724no—
216C22—1091no—
217C22—1091no—
218C22—4362no—
219C22—2182no—
220C22D8×D10911yescharacteristic
221C4—42no—
222C4—1091no—
223C4—1091no—
224C4—4362no—
225C4D8×D10911yescharacteristic
226C4C2×D4×D10911yescharacteristic
227C2—21no—
228C2—82no—
229C2—8722no—
230C2—1092no—
231C2—2181no—
232C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC4D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#508×C109C2
socleC2×C109C22×D8
hypercenter#508D109
hypocenterC109C2×#508
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. #20295459C2semidirect
3. #20295458C2semidirect
4. #20295457C2semidirect
5. #20295456C2semidirect
6. #20295455C2semidirect
7. Q32×D109C2semidirect
8. SD32×D109C2semidirect
9. D16×D109C2semidirect
10. C2×C16×D109C2semidirect
11. #508×C109C2semidirect
13. #16812722C22semidirect
14. #16812721C22semidirect
17. #16812718C22semidirect
22. Dic872C22semidirect
23. #16812645C22semidirect
24. #16812644C22semidirect
25. #16812643C22semidirect
26. #16812642C22semidirect
27. D1744C22semidirect
28. #16812640C22semidirect
29. C2×#13182004C22semidirect
32. C16×D109C22semidirect
33. C16×D109C22semidirect
34. #135×C109C22semidirect
35. Q32×C109C22semidirect
36. SD32×C109C22semidirect
37. D16×C109C22semidirect
38. C2×C16×C109C22semidirect
115. D109#508direct (N × H)
117. C109C2×#508semidirect
124. #508D109direct (N × H)
135. #135C2×D109semidirect
142. Q32C2×D109semidirect
146. SD32C2×D109semidirect
149. D16C2×D109semidirect
155. 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
#202954592
#202954582
#202954572
#202954562
#202954552
Q32×D1092
SD32×D1092
D16×D1092
C2×C16×D1092
#508×C1092
C2×#508109