Finite Group Databasebeta

← #28347033

Subgroups — #28347033

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#20293162C212yesmaximal, complemented
2#20293011C211yesmaximal, characteristic, complemented
3#20292984C212yesmaximal, complemented
4#20292869C211yesmaximal, characteristic, complemented
5#20292864C211yesmaximal, characteristic
6#20292857C212yesmaximal, complemented
7#20292721C211yesmaximal, characteristic, complemented
8#20292714C212yesmaximal, complemented
9#20292709C211yesmaximal, characteristic, complemented
10C4×C8×D107C211yesmaximal, characteristic, complemented
11#499×C107C211yesmaximal, characteristic, complemented
12C2×Dic428C2211yescharacteristic
13C2×#13133807C2212yes—
14C2×D856C2211yescharacteristic
15C2×C8×D107C2211yescharacteristic
16C2×C8×D107C2211yescharacteristic
17#16812104—22no—
18#16812097C2212yes—
19#16812094—22no—
20#16812089C2212yes—
21#16812084C2212yes—
22#16812074C2212yes—
23#16812071—22no—
24#16812071C2212yes—
25#16812067—22no—
26C42×D107C2211yescharacteristic
27#16812032C2212yes—
28#16812032C2212yes—
29#16812029C2212yes—
30#16812029C2212yes—
31C4×C107⋊C8C2211yescharacteristic
32C8×Dic107C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34#125×C107C2212yes—
35#107×C107C2211yescharacteristic
36#107×C107C2211yescharacteristic
37#106×C107C2212yes—
38C4×C8×C107C2211yescharacteristic
39C2×Dic214C2312yes—
40C2×D428—22no—
41C2×D428C2312yes—
42C2×C4×D107—22no—
43C2×C4×D107D411yescharacteristic
44C2×C4×D107D411yescharacteristic
45C2×C4×D107C2311yescharacteristic
46Dic428—22no—
47#13133807—24no—
48D856—22no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C8×D107—22no—
52C8×D107—22no—
53C4⋊Dic107—22no—
54Dic107⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60C4×Dic107—22no—
61C4×Dic107D411yescharacteristic
62C4×Dic107D411yescharacteristic
63C4×Dic107C2311yescharacteristic
64C2×C8×C107C2311yescharacteristic
65C2×C8×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107C2312yes—
69C4⋊C4×C107C2312yes—
70C42×C107C2311yescharacteristic
71C22×D107—22no—
72C22×D107C2×D411yescharacteristic
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2411yescharacteristic
78C2×Dic107—22no—
79C2×Dic107C2×D411yescharacteristic
80C2×Dic107C2×D411yescharacteristic
81C2×Dic107C2×D411yescharacteristic
82D428—22no—
83D428—22no—
84D428—44no—
85C4×D107—22no—
86C4×D107—21no—
87C4×D107—21no—
88C4×D107—22no—
89C4×D107—44no—
90Dic214—22no—
91Dic214—22no—
92C8×C107#4112yes—
93C8×C107#4112yes—
94C107⋊C8—21no—
95C107⋊C8—21no—
96C22×C107C22×D411yescharacteristic
97C2×D107—21no—
98C2×D107—21no—
99C2×D107—42no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107#13512yes—
103C4×C107—21no—
104C4×C107—21no—
105C4×C107—42no—
106C4×C107—42no—
107C4×C107C2×#4111yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—21no—
110Dic107—21no—
111Dic107—42no—
112Dic107#13512yes—
113C2×C107#53211yescharacteristic
114C2×C107C2×#13511yescharacteristic
115C2×C107C2×#13511yescharacteristic
116D107—22no—
117D107—82no—
118#3168—1071nomaximal, Sylow
119C107#316811yescharacteristic, complemented, Sylow
120#529—1072no—
121#499D10711yescharacteristic, complemented
122#495—1071no—
123#488—1072no—
124#467—1071no—
125#466—1071no—
126#465—1072no—
127C4×Q16—1071no—
128C4×SD16—1072no—
129C4×D8—1071no—
130C2×C4×C8—1071no—
131C2×Q16—1071no—
132C2×SD16—1072no—
133C2×D8—1071no—
134C22×C8—1071no—
135C22×C8—1071no—
136#126—2142no—
137#125C2×D10712yes—
138#124—1072no—
139#123—2142no—
140C22⋊Q8—1072no—
141#121—1072no—
142C4×Q8—1072no—
143C4×D4—1072no—
144C4×D4—2142no—
145#117—2142no—
146C2×C42—1071no—
147#107C2×D10711yescharacteristic
148#107C2×D10711yescharacteristic
149#106C2×D10712yes—
150#103—1072no—
151#103—1072no—
152#102—1072no—
153#102—1072no—
154C4×C8—1071no—
155C4×C8—1071no—
156C4×C8—1071no—
157C4×C8C2×D10711yescharacteristic
158C2×Q8—1072no—
159C2×D4—1072no—
160C2×D4—2142no—
161C22×C4—1071no—
162C22×C4—1071no—
163C22×C4—1071no—
164C22×C4—2142no—
165Q16—2142no—
166SD16—2144no—
167D8—2142no—
168C2×C8—1071no—
169C2×C8—1071no—
170C2×C8—2142no—
171C2×C8—2142no—
172C2×C8C22×D10711yescharacteristic
173C2×C8C22×D10711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C4⋊C4C22×D10712yes—
179C4⋊C4C22×D10712yes—
180C22⋊C4—2142no—
181C22⋊C4—2142no—
182C22⋊C4—2142no—
183C22⋊C4—2142no—
184C22⋊C4—2142no—
185C42—1071no—
186C42—1071no—
187C42—1071no—
188C42—2142no—
189C42C22×D10711yescharacteristic
190C23—1071no—
191C23—2142no—
192Q8—2142no—
193Q8—2142no—
194D4—4284no—
195D4—2142no—
196D4—2142no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—1071no—
200C2×C4—1071no—
201C2×C4—1071no—
202C2×C4—4284no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2141no—
207C2×C4—2141no—
208C2×C4D4×D10711yescharacteristic
209C2×C4D4×D10711yescharacteristic
210C2×C4C23×D10711yescharacteristic
211C8—2141no—
212C8—2141no—
213C8#1313381912yes—
214C8#1313381912yes—
215C22—1072no—
216C22—4282no—
217C22—4282no—
218C22—4282no—
219C22—2141no—
220C22—2141no—
221C22C2×D4×D10711yescharacteristic
222C4—21no—
223C4—21no—
224C4—42no—
225C4—42no—
226C4—1072no—
227C4—4282no—
228C4—2141no—
229C4—2141no—
230C4C2×#1313381911yescharacteristic
231C4C2×#1313381911yescharacteristic
232C2—8562no—
233C2—2142no—
234C2#2029317811yescharacteristic
235C2#135×D10711yescharacteristic
236C2#135×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#499×C107C2
socleC22×C107C22×D4
prime radical#20292864C2
hypercenter#499D107
hypocenterC107#3168
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #20293162C2semidirect
2. #20293011C2semidirect
3. #20292984C2semidirect
4. #20292869C2semidirect
6. #20292857C2semidirect
7. #20292721C2semidirect
8. #20292714C2semidirect
9. #20292709C2semidirect
10. C4×C8×D107C2semidirect
11. #499×C107C2semidirect
119. C107#3168semidirect
121. #499D107semidirect

Maximal subgroups

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

subgroupindex
#202931622
#202930112
#202929842
#202928692
#202928642
#202928572
#202927212
#202927142
#202927092
C4×C8×D1072
#499×C1072
#3168107