Finite Group Databasebeta

← #28347032

Subgroups — #28347032

Order 13696.

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

normal subgroups only

This group has 380 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#20293163C212yesmaximal, complemented
2#20293028C211yesmaximal, characteristic, complemented
3#20293012C211yesmaximal, characteristic, complemented
4#20292983C212yesmaximal, complemented
5#20292863C211yesmaximal, characteristic
6#20292858C212yesmaximal, complemented
7Q16×Dic107C211yesmaximal, characteristic, complemented
8#20292712C212yesmaximal, complemented
9#20292709C211yesmaximal, characteristic, complemented
10C2×C8×Dic107C211yesmaximal, characteristic, complemented
11#467×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2212yes—
13C2×D856C2211yescharacteristic
14C22×C107⋊C8C2211yescharacteristic
15#16812104—22no—
16#16812100C2212yes—
17#16812098C2212yes—
18#16812093—22no—
19#16812083C2212yes—
20Q8×Dic107C2212yes—
21#16812071C2212yes—
22(C22⋊D107)⋊C4—22no—
23#16812067—22no—
24C2×C4×Dic107C2211yescharacteristic
25#16812043C2211yescharacteristic
26#16812040C2212yes—
27#16812033C2212yes—
28#16812031C2212yes—
29#16812029C2212yes—
30C8×Dic107C2211yescharacteristic
31C8×Dic107C2211yescharacteristic
32C8×Dic107C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34C2×Q16×C107C2211yescharacteristic
35C22×C8×C107C2211yescharacteristic
36C22⋊Q8×C107C2212yes—
37#107×C107C2211yescharacteristic
38#103×C107C2212yes—
39C2×D428C2312yes—
40C2×C22⋊D107—22no—
41C22×Dic107D411yescharacteristic
42C22×Dic107D411yescharacteristic
43C2×C4×D107—22no—
44#13133805—24no—
45D856—22no—
46C2×C107⋊C8—22no—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C4⋊Dic107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107C2312yes—
52Dic107⋊C4—22no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—22no—
57C4×Dic107—22no—
58C4×Dic107D411yescharacteristic
59C4×Dic107D411yescharacteristic
60C4×Dic107C2311yescharacteristic
61C4×Dic107C2311yescharacteristic
62C2×Q8×C107C2312yes—
63C22×C4×C107C2311yescharacteristic
64Q16×C107—22no—
65C2×C8×C107—22no—
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107C2312yes—
70C22⋊C4×C107—22no—
71C23×C107C2×D411yescharacteristic
72C22×D107—22no—
73Q8×C107—22no—
74Q8×C107—22no—
75C2×C4×C107—21no—
76C2×C4×C107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2411yescharacteristic
81C22⋊D107—44no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107C2×D411yescharacteristic
86C2×Dic107C2×D411yescharacteristic
87C2×Dic107C2×D411yescharacteristic
88C2×Dic107C2×D411yescharacteristic
89D428—22no—
90D428—22no—
91C4×D107—44no—
92C8×C107—21no—
93C8×C107#4112yes—
94C107⋊C8—21no—
95C107⋊C8#4112yes—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107#13512yes—
99C22×C107C22×D411yescharacteristic
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—42no—
103C4×C107—21no—
104C4×C107—42no—
105C4×C107—42no—
106C4×C107C2×#4111yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—21no—
109Dic107—21no—
110Dic107—21no—
111Dic107—42no—
112Dic107#13512yes—
113C2×C107—22no—
114C2×C107#53211yescharacteristic
115C2×C107C2×#13511yescharacteristic
116C2×C107C2×#13511yescharacteristic
117D107—82no—
118#3168—1071nomaximal, Sylow
119C107#316811yescharacteristic, complemented, Sylow
120#529—1072no—
121#499—1071no—
122#495—1071no—
123#488—1072no—
124#467D10711yescharacteristic, complemented
125#466—1071no—
126#465—1072no—
127C4×Q16—1071no—
128C4×SD16—1072no—
129C4×D8—1071no—
130C2×C4×C8—1071no—
131C2×Q16C2×D10711yescharacteristic
132C2×SD16—1072no—
133C2×D8—1071no—
134C22×C8—1071no—
135C22×C8C2×D10711yescharacteristic
136#126—2142no—
137#125—1072no—
138#124—1072no—
139#123—2142no—
140C22⋊Q8C2×D10712yes—
141#121—1072no—
142C4×Q8—1072no—
143C4×D4—1072no—
144C4×D4—2142no—
145#117—2142no—
146C2×C42—1071no—
147#107—1071no—
148#107C2×D10711yescharacteristic
149#106—1072no—
150#103—1072no—
151#103C2×D10712yes—
152#102—1072no—
153#102—1072no—
154C4×C8—1071no—
155C4×C8—1071no—
156C4×C8—1071no—
157C4×C8—1071no—
158C2×Q8C22×D10712yes—
159C2×D4—1072no—
160C2×D4—2142no—
161C22×C4—1071no—
162C22×C4—1071no—
163C22×C4—2142no—
164C22×C4C22×D10711yescharacteristic
165Q16—22no—
166SD16—2144no—
167D8—2142no—
168C2×C8—22no—
169C2×C8—1071no—
170C2×C8—1071no—
171C2×C8—2142no—
172C2×C8C22×D10711yescharacteristic
173C2×C8C22×D10711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—1072no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C4⋊C4—2142no—
179C4⋊C4C22×D10712yes—
180C22⋊C4—22no—
181C22⋊C4—2142no—
182C22⋊C4—2142no—
183C22⋊C4—2142no—
184C22⋊C4—2142no—
185C42—1071no—
186C42—1071no—
187C42—1071no—
188C42—1071no—
189C42—2142no—
190C23—2142no—
191C23D4×D10711yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—4284no—
195D4—2142no—
196D4—2142no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—1071no—
202C2×C4—1071no—
203C2×C4—1071no—
204C2×C4—1071no—
205C2×C4—4284no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4—2142no—
209C2×C4D4×D10711yescharacteristic
210C2×C4C23×D10711yescharacteristic
211C8—21no—
212C8—1072no—
213C8—2141no—
214C8#1313381712yes—
215C22—21no—
216C22—21no—
217C22—4282no—
218C22—4282no—
219C22—4282no—
220C22#1681213212yes—
221C22C2×D4×D10711yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—1072no—
226C4—4282no—
227C4—2141no—
228C4—2141no—
229C4—2141no—
230C4#41×D10711yescharacteristic
231C4C2×#1313381711yescharacteristic
232C2—22no—
233C2—8562no—
234C2#2029317211yescharacteristic
235C2#135×D10711yescharacteristic
236C2C2×#1681213211yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radical#20292863C2
hypercenter#467D107
hypocenterC107#3168
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20293163C2semidirect
2. #20293028C2semidirect
3. #20293012C2semidirect
4. #20292983C2semidirect
6. #20292858C2semidirect
7. Q16×Dic107C2semidirect
8. #20292712C2semidirect
9. #20292709C2semidirect
10. C2×C8×Dic107C2semidirect
11. #467×C107C2semidirect
119. C107#3168semidirect
124. #467D107semidirect

Maximal subgroups

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

subgroupindex
#202931632
#202930282
#202930122
#202929832
#202928632
#202928582
Q16×Dic1072
#202927122
#202927092
C2×C8×Dic1072
#467×C1072
#3168107