Finite Group Databasebeta

← #28349594

Subgroups — #28349594

Order 13696.

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×D107C212yesmaximal, complemented
2#20293069C212yesmaximal, complemented
3#20293066C212yesmaximal, complemented
4#20293064C212yesmaximal, complemented
5#20293061C212yesmaximal, complemented
6#20293059C211yesmaximal, characteristic, complemented
7#20293057C212yesmaximal, complemented
8C2×#16812048C211yesmaximal, characteristic, complemented
9#508×C107C211yesmaximal, characteristic, complemented
10#41×D107—22no—
11#16812132C2212yes—
12#16812131C2212yescomplemented
13#16812130—22no—
14#16812129—22no—
15#16812128C2212yes—
16Q16×D107C2212yescomplemented
17SD16×D107—22no—
18D8×D107C2212yescomplemented
19C2×C8×D107C2211yescharacteristic
20Dic856C2211yescharacteristic, complemented
21#16812055C2212yescomplemented
22#16812054C2212yescomplemented
23#16812053C2212yescomplemented
24#16812052C2212yescomplemented
25D1712C2211yescharacteristic, complemented
26#16812050C2212yescomplemented
27#16812048C2212yescomplemented
28#16812048C2212yescomplemented
29C2×C107⋊C16C2211yescharacteristic, complemented
30#135×C107C2212yescomplemented
31Q32×C107C2211yescharacteristic, complemented
32SD32×C107C2212yescomplemented
33D16×C107C2211yescharacteristic, complemented
34C2×C16×C107C2211yescharacteristic, complemented
35#13133819—22no—
36#13133818—22no—
37#13133818—42no—
38#13133817—22no—
39#13133817—42no—
40Q8×D107—22no—
41D4×D107—22no—
42D4×D107—42no—
43C2×C4×D107—42no—
44C2×C4×D107D411yescharacteristic
45Dic428C2312yes—
46#13133808—22no—
47#13133807—22no—
48#13133806—22no—
49#13133805—22no—
50D856C2312yes—
51#13133803—22no—
52C2×C107⋊C8D411yescharacteristic
53C8×D107D412yes—
54C8×D107C2312yes—
55C107⋊C16C2312yes—
56#41×C107—22no—
57Q16×C107C2312yes—
58SD16×C107—22no—
59D8×C107C2312yes—
60C2×C8×C107C2311yescharacteristic
61C16×C107C2311yescharacteristic
62C16×C107C2311yescharacteristic
63C22×D107—42no—
64C22×D107D811yescharacteristic
65Q8×C107—22no—
66D4×C107—22no—
67D4×C107—42no—
68C2×C4×C107—42no—
69C2×C4×C107C2×D411yescharacteristic
70C22⋊D107—42no—
71C22⋊D107—42no—
72C22⋊D107—42no—
73C2×Dic107—42no—
74C2×Dic107D811yescharacteristic
75D428—22no—
76D428—42no—
77C4×D107—42no—
78C4×D107—42no—
79C4×D107—42no—
80C4×D107D812yes—
81C4×D107C2×D412yes—
82Dic214—22no—
83Dic214—42no—
84C8×C107C2×D411yescharacteristic
85C8×C107C2411yescharacteristic
86C107⋊C8C2×D412yes—
87C22×C107—42no—
88C22×C107C2×D811yescharacteristic
89C2×D107—41no—
90C2×D107—42no—
91C2×D107—84no—
92C2×D107C2×D812yes—
93C4×C107—42no—
94C4×C107C2×D811yescharacteristic
95C4×C107C22×D411yescharacteristic
96Dic107—42no—
97Dic107C2×D812yes—
98C2×C107—21no—
99C2×C107—82no—
100C2×C107C22×D811yescharacteristic
101D107—22no—
102D107—82no—
103#3414—1071nomaximal, Sylow
104C107#341411yescharacteristic, complemented, Sylow
105C2×#135—1072no—
106#510—1072no—
107#510—1072no—
108#509—1072no—
109#509—1072no—
110#508—1072no—
111#508—1071no—
112#508D10711yescharacteristic, complemented
113C2×M32—1071no—
114C2×#41—2142no—
115#135—1072no—
116#135—1072no—
117#135—1072no—
118#135—2142no—
119#135—2142no—
120#135C2×D10712yescomplemented
121C2×Q16—1072no—
122C2×SD16—2142no—
123C2×D8—1072no—
124C22×C8—1071no—
125Q32—1072no—
126Q32—1071no—
127Q32C2×D10711yescharacteristic, complemented
128SD32—1072no—
129SD32—1072no—
130SD32—1072no—
131SD32C2×D10712yescomplemented
132D16—1072no—
133D16—1071no—
134D16C2×D10711yescharacteristic, complemented
135M32—1072no—
136M32—1072no—
137C2×C16—1071no—
138C2×C16C2×D10711yescharacteristic, complemented
139#41—22no—
140#41—4282no—
141#41—4282no—
142#41—2142no—
143#41—2142no—
144#41—2142no—
145C2×Q8—2142no—
146C2×D4—4282no—
147C2×D4—2142no—
148C22×C4—1071no—
149C22×C4—4282no—
150Q16—1072no—
151Q16—2142no—
152Q16C22×D10712yes—
153SD16—22no—
154SD16—2142no—
155SD16—2142no—
156SD16—2142no—
157D8—1072no—
158D8—2142no—
159D8C22×D10712yes—
160C2×C8—1071no—
161C2×C8—1072no—
162C2×C8—1072no—
163C2×C8C22×D10711yescharacteristic
164C16—1072no—
165C16C22×D10711yescharacteristic
166C16C22×D10711yescharacteristic
167C23—1071no—
168C23—4282no—
169Q8—22no—
170Q8—4282no—
171Q8—2142no—
172D4—22no—
173D4—42no—
174D4—4282no—
175D4—4282no—
176D4—4282no—
177D4—4282no—
178D4—2142no—
179C2×C4—42no—
180C2×C4—1071no—
181C2×C4—1072no—
182C2×C4—1072no—
183C2×C4—4282no—
184C2×C4—4282no—
185C2×C4—4282no—
186C2×C4—4282no—
187C2×C4D4×D10711yescharacteristic
188C8—1072no—
189C8D4×D10711yescharacteristic
190C8C23×D10711yescharacteristic
191C22—42no—
192C22—8564no—
193C22—1072no—
194C22—4281no—
195C22—4282no—
196C22D8×D10711yescharacteristic
197C4—42no—
198C4—1072no—
199C4—4282no—
200C4D8×D10711yescharacteristic
201C4C2×D4×D10711yescharacteristic
202C2—21no—
203C2—82no—
204C2—8562no—
205C2—2142no—
206C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC4D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#508×C107C2
socleC2×C107C22×D8
hypercenter#508D107
hypocenterC107#3414
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. #20293069C2semidirect
3. #20293066C2semidirect
4. #20293064C2semidirect
5. #20293061C2semidirect
6. #20293059C2semidirect
7. #20293057C2semidirect
8. C2×#16812048C2semidirect
9. #508×C107C2semidirect
12. #16812131C22semidirect
16. Q16×D107C22semidirect
18. D8×D107C22semidirect
20. Dic856C22semidirect
21. #16812055C22semidirect
22. #16812054C22semidirect
23. #16812053C22semidirect
24. #16812052C22semidirect
25. D1712C22semidirect
26. #16812050C22semidirect
27. #16812048C22semidirect
28. #16812048C22semidirect
29. C2×C107⋊C16C22semidirect
30. #135×C107C22semidirect
31. Q32×C107C22semidirect
32. SD32×C107C22semidirect
33. D16×C107C22semidirect
34. C2×C16×C107C22semidirect
104. C107#3414semidirect
112. #508D107semidirect
120. #135C2×D107semidirect
127. Q32C2×D107semidirect
131. SD32C2×D107semidirect
134. D16C2×D107semidirect
138. 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
#202930692
#202930662
#202930642
#202930612
#202930592
#202930572
C2×#168120482
#508×C1072
#3414107