Finite Group Databasebeta

← #28346849

Subgroups — #28346849

Order 13696.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293180C211yesmaximal, characteristic, complemented
2C2×#16812098C211yesmaximal, characteristic
3#20292988C214yesmaximal, complemented
4#20292983C214yesmaximal, complemented
5#20292636C211yesmaximal, characteristic, complemented
6#20292633C211yesmaximal, characteristic
7#20292631C211yesmaximal, characteristic
8M16×Dic107C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133802C2211yescharacteristic
12#16812108C2212yescomplemented
13#16812100C2212yescomplemented
14#16812098—22no—
15#16812098C2212yes—
16#16812098C2212yes—
17#16812086—22no—
18C4×Dic214—22no—
19#16812068C2211yescharacteristic
20C2×(C2×D107)⋊C4—22no—
21C2×C4×Dic107C2211yescharacteristic
22#16812033C2214yescomplemented
23#16812031C2214yes—
24#16812029C2214yes—
25#16812019C2212yescomplemented
26C8×Dic107C2212yescomplemented
27C22×Q8×C107C2211yescharacteristic
28C2×M16×C107C2211yescharacteristic
29#117×C107C2211yescharacteristic
30#103×C107C2214yescomplemented
31C23×D107—21no—
32C2×Dic214—22no—
33C2×D428—22no—
34C2×D428—22no—
35C2×D428—21no—
36C2×D428C2312yes—
37C22×Dic107D412yes—
38#13133802—22no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107—22no—
41C4⋊Dic107C2312yes—
42Dic107⋊C4—22no—
43Dic107⋊C4—22no—
44C22⋊Dic107—21no—
45(C2×D107)⋊C4—24no—
46(C2×D107)⋊C4—24no—
47C4×Dic107—21no—
48C4×Dic107D412yes—
49C4×Dic107C2311yescharacteristic
50C4×Dic107C2311yescharacteristic
51C2×Q8×C107—21no—
52C2×Q8×C107—22no—
53C2×Q8×C107—22no—
54C2×Q8×C107C2312yes—
55C22×C4×C107—21no—
56C22×C4×C107C2311yescharacteristic
57M16×C107—22no—
58C2×C8×C107C2312yes—
59C4⋊C4×C107C2312yes—
60C22⋊C4×C107—21no—
61C42×C107—21no—
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64C22×D107—44no—
65C22×D107—42no—
66Q8×C107—22no—
67Q8×C107—22no—
68Q8×C107—42no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—42no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107#4112yes—
74C2×C4×C107#4112yes—
75C2×C4×C107C2411yescharacteristic
76C2×Dic107—22no—
77C2×Dic107—42no—
78C2×Dic107C2×D412yes—
79C2×Dic107C2×D412yes—
80D428—22no—
81D428—22no—
82D428—42no—
83Dic214—44no—
84C8×C107—22no—
85C107⋊C8—22no—
86C22×C107—22no—
87C22×C107#12412yes—
88C22×C107C22×D411yescharacteristic
89C2×D107—42no—
90C2×D107—44no—
91C2×D107—84no—
92C4×C107—42no—
93C4×C107—42no—
94C4×C107#12412yes—
95C4×C107C2×#4111yescharacteristic
96C4×C107C2×#4111yescharacteristic
97Dic107—22no—
98Dic107—22no—
99Dic107—42no—
100C2×C107—22no—
101C2×C107C2×#12411yescharacteristic
102C2×C107#57512yes—
103D107—82no—
104#3140—1071nomaximal, Sylow
105C107#314011yescharacteristic, complemented, Sylow
106#533—1071no—
107C2×#124—1071no—
108#489—1074no—
109#488—1074no—
110#419—1071no—
111#419D10711yescharacteristic, complemented
112#418—1071no—
113#418—1071no—
114C4×M16—1071no—
115C22×Q8C2×D10711yescharacteristic
116C22×D4—1071no—
117C2×M16—1071no—
118C2×M16C2×D10711yescharacteristic
119C4⋊Q8—1072no—
120#125—1072no—
121#124—1072no—
122#124—1072no—
123#124—2142no—
124C22⋊Q8—2142no—
125C4×Q8—2142no—
126#117—1071no—
127#117C2×D10711yescharacteristic
128C2×C22⋊C4—2142no—
129C2×C42—1071no—
130#103—1074no—
131#103C2×D10714yescomplemented
132#102—1074no—
133#102—1074no—
134#97—1072no—
135C4×C8—1072no—
136C24—2141no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—2142no—
141C2×Q8C22×D10712yes—
142C2×D4—1072no—
143C2×D4—2141no—
144C2×D4—2142no—
145C2×D4—2142no—
146C22×C4—21no—
147C22×C4—1072no—
148C22×C4C22×D10711yescharacteristic
149M16—22no—
150M16—2142no—
151C2×C8—1072no—
152C2×C8C22×D10712yes—
153C4⋊C4—1072no—
154C4⋊C4—2142no—
155C4⋊C4—2142no—
156C4⋊C4—2142no—
157C4⋊C4C22×D10712yes—
158C22⋊C4—21no—
159C22⋊C4—2144no—
160C22⋊C4—2144no—
161C22⋊C4—2141no—
162C42—21no—
163C42—1072no—
164C42—1071no—
165C42—1071no—
166C42—2141no—
167C23—4284no—
168C23—4282no—
169C23—2142no—
170C23D4×D10711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—4284no—
175D4—4282no—
176D4—2142no—
177D4—2142no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—1072no—
182C2×C4—1072no—
183C2×C4—4282no—
184C2×C4—2142no—
185C2×C4D4×D10711yescharacteristic
186C2×C4#1313381712yes—
187C2×C4#1313381812yes—
188C2×C4C23×D10711yescharacteristic
189C8—22no—
190C8—2142no—
191C22—22no—
192C22—8564no—
193C22—4282no—
194C22—4284no—
195C22#1681209512yes—
196C22C2×D4×D10711yescharacteristic
197C4—42no—
198C4—42no—
199C4—4282no—
200C4—2142no—
201C4—2142no—
202C4#1681209512yes—
203C4C2×#1313381711yescharacteristic
204C4C2×#1313381811yescharacteristic
205C2—22no—
206C2—8562no—
207C2C2×#1681209511yescharacteristic
208C2#2029342612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#419D107
hypocenterC107#3140
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #20293180C2semidirect
3. #20292988C2semidirect
4. #20292983C2semidirect
5. #20292636C2semidirect
8. M16×Dic107C2semidirect
9. #419×C107C2semidirect
12. #16812108C22semidirect
13. #16812100C22semidirect
22. #16812033C22semidirect
25. #16812019C22semidirect
26. C8×Dic107C22semidirect
30. #103×C107C22semidirect
105. C107#3140semidirect
111. #419D107semidirect
131. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×#168120982
#202929882
#202929832
#202926362
#202926332
#202926312
M16×Dic1072
#419×C1072
#3140107