Finite Group Databasebeta

← #28346834

Subgroups — #28346834

Order 13696.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#20293183C211yesmaximal, characteristic, complemented
2#127×D107C211yesmaximal, characteristic
3#20292993C211yesmaximal, characteristic, complemented
4#20292992C211yesmaximal, characteristic, complemented
5#20292991C211yesmaximal, characteristic, complemented
6#20292972C214yesmaximal, complemented
7#20292630C214yesmaximal, complemented
8#20292579C211yesmaximal, characteristic, complemented
9#490×C107C211yesmaximal, characteristic, complemented
10C2×D4×D107—22no—
11C2×D4×D107C2212yes—
12C2×#13133801C2212yes—
13#16812108C2212yescomplemented
14#16812106C2211yescharacteristic
15#16812105—22no—
16#16812105C2212yes—
17#16812101C2211yescharacteristic, complemented
18#16812087—22no—
19#16812074—22no—
20#16812067C2212yes—
21C42×D107C2211yescharacteristic
22#16812031C2214yescomplemented
23#16812030C2214yescomplemented
24#16812029C2214yescomplemented
25#16812020C2211yescharacteristic, complemented
26C8×Dic107C2212yescomplemented
27#127×C107C2211yescharacteristic
28#125×C107C2211yescharacteristic, complemented
29#102×C107C2214yescomplemented
30#97×C107C2211yescharacteristic, complemented
31C23×D107—22no—
32C2×Dic214—22no—
33C2×D428—22no—
34C2×D428C2312yes—
35C2×C22⋊D107—22no—
36C2×C22⋊D107—24no—
37D4×D107—28no—
38D4×D107—48no—
39C2×C4×D107D412yes—
40C2×C4×D107C2311yescharacteristic
41#13133801—24no—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—22no—
45Dic107⋊C4—22no—
46(C2×D107)⋊C4—22no—
47C4×Dic107—22no—
48C4×Dic107D412yes—
49C4×Dic107C2311yescharacteristic
50C2×D4×C107—22no—
51C2×D4×C107C2312yes—
52C2×C8×C107C2312yes—
53C4⋊C4×C107—22no—
54C4⋊C4×C107C2312yes—
55C42×C107C2311yescharacteristic
56C23×C107—22no—
57C22×D107—22no—
58C22×D107—48no—
59C22×D107—44no—
60C22×D107C2×D411yescharacteristic
61D4×C107—24no—
62D4×C107—44no—
63C2×C4×C107—22no—
64C2×C4×C107C2×D412yes—
65C2×C4×C107C2411yescharacteristic
66C22⋊D107—44no—
67C22⋊D107—48no—
68C2×Dic107—22no—
69C2×Dic107C2×D411yescharacteristic
70C2×Dic107C2×D412yes—
71D428—24no—
72D428—44no—
73C4×D107—24no—
74C4×D107#4114yes—
75Dic214—44no—
76C8×C107—22no—
77C107⋊C8—22no—
78C22×C107—42no—
79C22×C107—44no—
80C22×C107C22×D411yescharacteristic
81C2×D107—22no—
82C2×D107—42no—
83C2×D107—44no—
84C2×D107—88no—
85C2×D107#12412yes—
86C4×C107—22no—
87C4×C107—42no—
88C4×C107C2×#4112yes—
89Dic107—42no—
90Dic107#12412yes—
91Dic107AGL(1,ℤ/8)14yes—
92C2×C107—82no—
93C2×C107C2×#12411yescharacteristic
94C2×C107C2×AGL(1,ℤ/8)12yes—
95D107—22no—
96D107—82no—
97#3138—1071nomaximal, Sylow
98C107#313811yescharacteristic, complemented, Sylow
99#533—1071no—
100C2×#127—1071no—
101#490—1071no—
102#490—1071no—
103#490—1071no—
104#490D10711yescharacteristic, complemented
105#486—1074no—
106#418—1074no—
107C4×M16—1071no—
108C22×D4—1072no—
109C22×D4—2142no—
110C2×M16—1072no—
111C4⋊Q8—1072no—
112#127—1072no—
113#127—1071no—
114#127—2142no—
115#127C2×D10711yescharacteristic
116#125—1071no—
117#125C2×D10711yescharacteristic, complemented
118C22⋊Q8—2142no—
119C4×Q8—2142no—
120#117—1072no—
121C2×C42—1071no—
122#102—1074no—
123#102—1074no—
124#102—1074no—
125#102C2×D10714yescomplemented
126#97—1071no—
127#97C2×D10711yescharacteristic, complemented
128C4×C8—1072no—
129C24—2142no—
130C2×Q8—2142no—
131C2×D4—22no—
132C2×D4—1072no—
133C2×D4—4288no—
134C2×D4—2148no—
135C2×D4—2144no—
136C2×D4—2142no—
137C2×D4—2142no—
138C2×D4C22×D10712yes—
139C22×C4—1071no—
140C22×C4—1072no—
141M16—2144no—
142C2×C8—1072no—
143C2×C8C22×D10712yes—
144C4⋊C4—22no—
145C4⋊C4—1072no—
146C4⋊C4—2142no—
147C4⋊C4—2142no—
148C4⋊C4C22×D10712yes—
149C22⋊C4—2142no—
150C42—1072no—
151C42—1071no—
152C42—2142no—
153C42C22×D10711yescharacteristic
154C23—22no—
155C23—1071no—
156C23—4284no—
157C23—4288no—
158C23—2142no—
159Q8—4284no—
160D4—24no—
161D4—44no—
162D4—4284no—
163D4—4288no—
164D4—4284no—
165D4—2144no—
166C2×C4—22no—
167C2×C4—1071no—
168C2×C4—1072no—
169C2×C4—1074no—
170C2×C4—2144no—
171C2×C4—2142no—
172C2×C4D4×D10712yes—
173C2×C4C23×D10711yescharacteristic
174C8—22no—
175C8—2142no—
176C22—42no—
177C22—44no—
178C22—8568no—
179C22—1072no—
180C22—4282no—
181C22—4284no—
182C22—2142no—
183C22C2×D4×D10711yescharacteristic
184C4—22no—
185C4—42no—
186C4—1072no—
187C4—1074no—
188C4—4282no—
189C4#41×D10712yes—
190C2—82no—
191C2—8562no—
192C2—2142no—
193C2#124×D10711yescharacteristic
194C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#490×C107C2
socleC22×C107C22×D4
prime radical#127×D107C2
hypercenter#490D107
hypocenterC107#3138
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20293183C2semidirect
3. #20292993C2semidirect
4. #20292992C2semidirect
5. #20292991C2semidirect
6. #20292972C2semidirect
7. #20292630C2semidirect
8. #20292579C2semidirect
9. #490×C107C2semidirect
13. #16812108C22semidirect
17. #16812101C22semidirect
22. #16812031C22semidirect
23. #16812030C22semidirect
24. #16812029C22semidirect
25. #16812020C22semidirect
26. C8×Dic107C22semidirect
28. #125×C107C22semidirect
29. #102×C107C22semidirect
30. #97×C107C22semidirect
98. C107#3138semidirect
104. #490D107semidirect
117. #125C2×D107semidirect
125. #102C2×D107semidirect
127. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931832
#127×D1072
#202929932
#202929922
#202929912
#202929722
#202926302
#202925792
#490×C1072
#3138107