Finite Group Databasebeta

← #28345522

Subgroups — #28345522

Order 13696.

402 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 157 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#20293139C211yesmaximal, characteristic
2#20292706C212yesmaximal, complemented
3#20292704C212yesmaximal, complemented
4#20292700C212yesmaximal, complemented
5#20292697C212yesmaximal, complemented
6#20292688C211yesmaximal, characteristic, complemented
7#20292592C212yesmaximal, complemented
8#20292589C212yesmaximal, complemented
9#433×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×C4×D107C2211yescharacteristic
12C2×#13133801C2212yes—
13C2×#13133801C2212yes—
14C2×#13133801C2212yescomplemented
15C2×C8×D107C2212yescomplemented
16#16812093C412yes—
17#16812089C412yes—
18#16812084C412yes—
19#16812079C412yes—
20C4×D428C2212yes—
21C4×D428C2212yes—
22C4⋊(C107⋊C8)C2212yescomplemented
23C22⋊(C107⋊C8)C2212yes—
24(C2×D107)⋊C8C2212yes—
25(C2×D107)⋊C8C2212yes—
26(C2×D107)⋊C8C2212yescomplemented
27(C2×D107)⋊C8C2212yescomplemented
28#16812020C2211yescharacteristic, complemented
29C4×C107⋊C8C2211yescharacteristic, complemented
30#117×C107C2211yescharacteristic
31C4⋊C8×C107C2212yescomplemented
32C22⋊C8×C107C2212yes—
33#97×C107C2211yescharacteristic, complemented
34C4×C8×C107C2211yescharacteristic, complemented
35C23×D107C2×C411yescharacteristic
36#13133818—42no—
37C2×Dic214C2×C411yescharacteristic
38C2×D428C2×C412yes—
39C2×D428C2×C412yes—
40C2×D428C2×C411yescharacteristic
41C2×C22⋊D107C2×C412yes—
42C22×Dic107C2×C411yescharacteristic
43C2×C4×D107—42no—
44C2×C4×D107C2×C412yes—
45C2×C4×D107C2312yes—
46C2×C4×D107C2312yes—
47#13133801—42no—
48#13133801—42no—
49#13133801—42no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52C8×D107—42no—
53C4⋊Dic107C2×C412yes—
54C4⋊Dic107C2×C412yes—
55(C2×D107)⋊C4C2×C412yes—
56(C2×D107)⋊C4C2×C412yes—
57(C2×D107)⋊C4C2×C412yes—
58(C2×D107)⋊C4C2×C412yes—
59C22×C4×C107C2311yescharacteristic
60C2×C8×C107C2312yes—
61C2×C8×C107C2312yes—
62C4⋊C4×C107C2×C412yes—
63C22⋊C4×C107C2×C412yes—
64C42×C107C2311yescharacteristic
65C42×C107C2311yescharacteristic
66C23×C107C22×C411yescharacteristic
67C22×D107—41no—
68C22×D107—42no—
69C22×D107C22×C412yes—
70C22×D107C22×C412yes—
71C2×C4×C107—22no—
72C2×C4×C107C22×C412yes—
73C2×C4×C107C22×C412yes—
74C2×C4×C107C22×C411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C22⋊D107—42no—
77C2×Dic107—41no—
78C2×Dic107C22×C412yes—
79C2×Dic107C22×C412yes—
80D428—42no—
81D428—42no—
82D428—41no—
83C4×D107—22no—
84C4×D107—22no—
85C4×D107—22no—
86C4×D107—22no—
87C4×D107—42no—
88Dic214—41no—
89C8×C107—22no—
90C8×C107—22no—
91C107⋊C8—22no—
92C107⋊C8—22no—
93C22×C107—21no—
94C22×C107—22no—
95C22×C107C23×C411yescharacteristic
96C2×D107—22no—
97C2×D107—22no—
98C2×D107—22no—
99C2×D107—22no—
100C2×D107—22no—
101C2×D107—22no—
102C2×D107—44no—
103C4×C107—21no—
104C4×C107—22no—
105C4×C107—22no—
106C4×C107H(5,𝔽2)12yes—
107Dic107—22no—
108Dic107—22no—
109C2×C107—41no—
110C2×C107#51811yescharacteristic
111C2×C107#56812yes—
112D107—42no—
113D107—42no—
114#2978—1071nomaximal, Sylow
115C107#297811yescharacteristic, complemented, Sylow
116#525—1071no—
117#436—1072no—
118#436—1072no—
119#435—1072no—
120#435—1072no—
121#433—1071no—
122#433D10711yescharacteristic, complemented
123#408—1072no—
124#407—1072no—
125C2×#41—1071no—
126C23×C4—1071no—
127C2×M16—1072no—
128C2×M16—1072no—
129C2×M16—1072no—
130C22×C8—1072no—
131#123—1072no—
132C22⋊Q8—1072no—
133#121—1072no—
134#120—1072no—
135C4×D4—1072no—
136C4×D4—1072no—
137#117C2×D10711yescharacteristic
138C4⋊C8—1072no—
139C4⋊C8C2×D10712yescomplemented
140C22⋊C8—1072no—
141C22⋊C8—1072no—
142C22⋊C8—1072no—
143C22⋊C8—1072no—
144C22⋊C8—1072no—
145C22⋊C8C2×D10712yes—
146#97—1071no—
147#97C2×D10711yescharacteristic, complemented
148C4×C8—1071no—
149C4×C8C2×D10711yescharacteristic, complemented
150C24—1071no—
151#41—4282no—
152C2×Q8—1071no—
153C2×D4—1072no—
154C2×D4—1072no—
155C2×D4—1072no—
156C2×D4—1071no—
157C22×C4—1071no—
158C22×C4—1072no—
159C22×C4—1072no—
160C22×C4—1072no—
161C22×C4—4282no—
162C22×C4C22×D10711yescharacteristic
163M16—4282no—
164M16—4282no—
165M16—4282no—
166C2×C8—1072no—
167C2×C8—1072no—
168C2×C8—4282no—
169C2×C8C22×D10712yes—
170C2×C8C22×D10712yes—
171C4⋊C4—1072no—
172C4⋊C4—1072no—
173C4⋊C4C4×D10712yes—
174C22⋊C4—1072no—
175C22⋊C4—1072no—
176C22⋊C4—1072no—
177C22⋊C4—1072no—
178C22⋊C4C4×D10712yes—
179C42C22×D10711yescharacteristic
180C42C22×D10711yescharacteristic
181C23—1072no—
182C23—1072no—
183C23—4281no—
184C23—4282no—
185C23C2×C4×D10711yescharacteristic
186Q8—4281no—
187D4—4281no—
188D4—4282no—
189D4—4282no—
190D4—4282no—
191C2×C4—22no—
192C2×C4—1072no—
193C2×C4—1072no—
194C2×C4—4282no—
195C2×C4—4281no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4C2×C4×D10712yes—
201C2×C4C2×C4×D10712yes—
202C2×C4C2×C4×D10711yescharacteristic
203C2×C4C23×D10711yescharacteristic
204C8—22no—
205C8—22no—
206C8—2142no—
207C8—2142no—
208C22—22no—
209C22—21no—
210C22—4284no—
211C22—2142no—
212C22—2142no—
213C22—2142no—
214C22—2142no—
215C22—2142no—
216C22—2142no—
217C22C22×C4×D10711yescharacteristic
218C4—22no—
219C4—22no—
220C4—21no—
221C4—2142no—
222C4—2142no—
223C4#1681215512yes—
224C2—41no—
225C2—4282no—
226C2—4282no—
227C2#2029310111yescharacteristic
228C2#2029339212yes—
subgroupG/H
center, FrattiniC2×C4C23×D107
commutator, socleC22×C107C23×C4
Fitting#433×C107C2
prime radical#20293139C2
cosocleC2×C4×C107C24
hypercenter#433D107
hypocenterC107#2978
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #20292706C2semidirect
3. #20292704C2semidirect
4. #20292700C2semidirect
5. #20292697C2semidirect
6. #20292688C2semidirect
7. #20292592C2semidirect
8. #20292589C2semidirect
9. #433×C107C2semidirect
14. C2×#13133801C22semidirect
15. C2×C8×D107C22semidirect
22. C4⋊(C107⋊C8)C22semidirect
26. (C2×D107)⋊C8C22semidirect
27. (C2×D107)⋊C8C22semidirect
28. #16812020C22semidirect
29. C4×C107⋊C8C22semidirect
31. C4⋊C8×C107C22semidirect
33. #97×C107C22semidirect
34. C4×C8×C107C22semidirect
115. C107#2978semidirect
122. #433D107semidirect
139. C4⋊C8C2×D107semidirect
147. #97C2×D107semidirect
149. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931392
#202927062
#202927042
#202927002
#202926972
#202926882
#202925922
#202925892
#433×C1072
#2978107