Finite Group Databasebeta

← #508×D107 #28349575

Subgroups — #508×D107 #28349575

Order 13696.

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×D107C212yesmaximal, complemented
2#20293060C211yesmaximal, characteristic, complemented
3#20293059C211yesmaximal, characteristic, complemented
4#20293058C212yesmaximal, complemented
5#20293057C212yesmaximal, complemented
6#20293056C211yesmaximal, characteristic, complemented
7Q32×D107C211yesmaximal, characteristic, complemented
8SD32×D107C212yesmaximal, complemented
9D16×D107C211yesmaximal, characteristic, complemented
10C2×C16×D107C211yesmaximal, characteristic, complemented
11#508×C107C211yesmaximal, characteristic, complemented
12#41×D107—22no—
13#16812132C2212yescomplemented
14#16812131C2212yescomplemented
15#16812130—22no—
16#16812129—22no—
17#16812128C2212yescomplemented
18Q16×D107C2212yes—
19SD16×D107—22no—
20D8×D107C2212yes—
21C2×C8×D107C2211yescharacteristic
22Dic856C2211yescharacteristic, complemented
23#16812055C2212yescomplemented
24#16812054C2212yescomplemented
25#16812053C2212yescomplemented
26#16812052C2212yescomplemented
27D1712C2211yescharacteristic, complemented
28#16812050C2212yescomplemented
29C2×C107⋊C16C2211yescharacteristic, complemented
30C16×D107C2211yescharacteristic
31C16×D107C2211yescharacteristic
32C16×D107C2211yescharacteristic, complemented
33C16×D107C2211yescharacteristic, complemented
34#135×C107C2212yescomplemented
35Q32×C107C2211yescharacteristic, complemented
36SD32×C107C2212yescomplemented
37D16×C107C2211yescharacteristic, complemented
38C2×C16×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133818—22no—
41#13133818—42no—
42#13133817—22no—
43#13133817—42no—
44Q8×D107—22no—
45D4×D107—22no—
46D4×D107—42no—
47C2×C4×D107—42no—
48C2×C4×D107D411yescharacteristic
49Dic428C2312yes—
50#13133808—22no—
51#13133807—22no—
52#13133806—22no—
53#13133805—22no—
54D856C2312yes—
55#13133803—22no—
56C2×C107⋊C8D411yescharacteristic
57C8×D107D411yescharacteristic
58C8×D107D411yescharacteristic
59C8×D107C2311yescharacteristic
60C8×D107C2311yescharacteristic
61C107⋊C16C2311yescharacteristic
62C107⋊C16C2311yescharacteristic
63#41×C107—22no—
64Q16×C107C2312yes—
65SD16×C107—22no—
66D8×C107C2312yes—
67C2×C8×C107C2311yescharacteristic
68C16×C107C2311yescharacteristic
69C16×C107C2311yescharacteristic
70C22×D107—42no—
71C22×D107D811yescharacteristic
72Q8×C107—22no—
73D4×C107—22no—
74D4×C107—42no—
75C2×C4×C107—42no—
76C2×C4×C107C2×D411yescharacteristic
77C22⋊D107—42no—
78C22⋊D107—42no—
79C22⋊D107—42no—
80C2×Dic107—42no—
81C2×Dic107D811yescharacteristic
82D428—22no—
83D428—42no—
84C4×D107—42no—
85C4×D107—42no—
86C4×D107—42no—
87C4×D107D811yescharacteristic
88C4×D107D811yescharacteristic
89C4×D107C2×D411yescharacteristic
90C4×D107C2×D411yescharacteristic
91Dic214—22no—
92Dic214—42no—
93C8×C107C2×D411yescharacteristic
94C8×C107C2411yescharacteristic
95C107⋊C8C2×D411yescharacteristic
96C107⋊C8C2×D411yescharacteristic
97C22×C107—42no—
98C22×C107C2×D811yescharacteristic
99C2×D107—22no—
100C2×D107—42no—
101C2×D107—84no—
102C2×D107C2×D811yescharacteristic
103C2×D107C2×D811yescharacteristic
104C4×C107—42no—
105C4×C107C2×D811yescharacteristic
106C4×C107C22×D411yescharacteristic
107Dic107—42no—
108Dic107C2×D811yescharacteristic
109Dic107C2×D811yescharacteristic
110C2×C107—21no—
111C2×C107—82no—
112C2×C107C22×D811yescharacteristic
113D107—21no—
114D107—82no—
115D107#50812yesdirect product factor
116C2×#508—1071nomaximal, Sylow
117C107C2×#50811yescharacteristic, complemented, Sylow
118C2×#135—1072no—
119#508—1072no—
120#508—1071no—
121#508—1072no—
122#508—1071no—
123#508—1071no—
124#508D10711yescharacteristic, direct product factor
125C2×Q32—1071no—
126C2×SD32—1072no—
127C2×D16—1071no—
128C22×C16—1071no—
129C2×#41—2142no—
130#135—1072no—
131#135—1072no—
132#135—1072no—
133#135—2142no—
134#135—2142no—
135#135C2×D10712yescomplemented
136C2×Q16—1072no—
137C2×SD16—2142no—
138C2×D8—1072no—
139C22×C8—1071no—
140Q32—1072no—
141Q32—1071no—
142Q32C2×D10711yescharacteristic, complemented
143SD32—1072no—
144SD32—1072no—
145SD32—1072no—
146SD32C2×D10712yescomplemented
147D16—1072no—
148D16—1071no—
149D16C2×D10711yescharacteristic, complemented
150C2×C16—1071no—
151C2×C16—1071no—
152C2×C16—1071no—
153C2×C16—1071no—
154C2×C16—1071no—
155C2×C16C2×D10711yescharacteristic, complemented
156#41—22no—
157#41—4282no—
158#41—4282no—
159#41—2142no—
160#41—2142no—
161#41—2142no—
162C2×Q8—2142no—
163C2×D4—4282no—
164C2×D4—2142no—
165C22×C4—1071no—
166C22×C4—4282no—
167Q16—1072no—
168Q16—2142no—
169Q16C22×D10712yes—
170SD16—22no—
171SD16—2142no—
172SD16—2142no—
173SD16—2142no—
174D8—1072no—
175D8—2142no—
176D8C22×D10712yes—
177C2×C8—1071no—
178C2×C8—1071no—
179C2×C8—1071no—
180C2×C8—1071no—
181C2×C8—1071no—
182C2×C8C22×D10711yescharacteristic
183C16—1071no—
184C16—1071no—
185C16C22×D10711yescharacteristic
186C16C22×D10711yescharacteristic
187C23—1071no—
188C23—4282no—
189Q8—22no—
190Q8—4282no—
191Q8—2142no—
192D4—22no—
193D4—42no—
194D4—4282no—
195D4—4282no—
196D4—4282no—
197D4—4282no—
198D4—2142no—
199C2×C4—42no—
200C2×C4—1071no—
201C2×C4—1071no—
202C2×C4—1071no—
203C2×C4—1071no—
204C2×C4—1071no—
205C2×C4—4282no—
206C2×C4—4282no—
207C2×C4—4282no—
208C2×C4—4282no—
209C2×C4D4×D10711yescharacteristic
210C8—1071no—
211C8—1071no—
212C8D4×D10711yescharacteristic
213C8C23×D10711yescharacteristic
214C22—42no—
215C22—8564no—
216C22—1071no—
217C22—1071no—
218C22—4282no—
219C22—2142no—
220C22D8×D10711yescharacteristic
221C4—42no—
222C4—1071no—
223C4—1071no—
224C4—4282no—
225C4D8×D10711yescharacteristic
226C4C2×D4×D10711yescharacteristic
227C2—21no—
228C2—82no—
229C2—8562no—
230C2—1072no—
231C2—2141no—
232C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC4D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#508×C107C2
socleC2×C107C22×D8
hypercenter#508D107
hypocenterC107C2×#508
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D107C2semidirect
2. #20293060C2semidirect
3. #20293059C2semidirect
4. #20293058C2semidirect
5. #20293057C2semidirect
6. #20293056C2semidirect
7. Q32×D107C2semidirect
8. SD32×D107C2semidirect
9. D16×D107C2semidirect
10. C2×C16×D107C2semidirect
11. #508×C107C2semidirect
13. #16812132C22semidirect
14. #16812131C22semidirect
17. #16812128C22semidirect
22. Dic856C22semidirect
23. #16812055C22semidirect
24. #16812054C22semidirect
25. #16812053C22semidirect
26. #16812052C22semidirect
27. D1712C22semidirect
28. #16812050C22semidirect
29. C2×C107⋊C16C22semidirect
32. C16×D107C22semidirect
33. C16×D107C22semidirect
34. #135×C107C22semidirect
35. Q32×C107C22semidirect
36. SD32×C107C22semidirect
37. D16×C107C22semidirect
38. C2×C16×C107C22semidirect
115. D107#508direct (N × H)
117. C107C2×#508semidirect
124. #508D107direct (N × H)
135. #135C2×D107semidirect
142. Q32C2×D107semidirect
146. SD32C2×D107semidirect
149. D16C2×D107semidirect
155. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
#135×D1072
#202930602
#202930592
#202930582
#202930572
#202930562
Q32×D1072
SD32×D1072
D16×D1072
C2×C16×D1072
#508×C1072
C2×#508107