Finite Group Databasebeta

← #28346402

Subgroups — #28346402

Order 13696.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16812083C211yesmaximal, characteristic
2C2×#16812068C211yesmaximal, characteristic, complemented
3#20292952C214yesmaximal, complemented
4#20292946C212yesmaximal, complemented
5#20292941C212yesmaximal, complemented
6#20292669C212yesmaximal, complemented
7#20292633C212yesmaximal, complemented
8#407×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11#16812083—21no—
12#16812083—21no—
13#16812083C2212yes—
14#16812083C2212yes—
15#16812068—21no—
16#16812068—21no—
17#16812068C2212yes—
18#16812068C2212yescomplemented
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic, complemented
21C2×C4⋊Dic107C2211yescharacteristic, complemented
22C2×C22⋊Dic107—21no—
23C2×(C2×D107)⋊C4—21no—
24C2×C4×Dic107—21no—
25#16812043C2214yescomplemented
26#16812041C2214yescomplemented
27#16812031C2214yescomplemented
28#16812031C2214yescomplemented
29C23×C4×C107C2211yescharacteristic
30C2×M16×C107C2212yes—
31C22⋊C8×C107C2212yescomplemented
32C22⋊C8×C107C2212yescomplemented
33C23×D107—21no—
34C2×D428—24no—
35C2×D428—21no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C22⋊D107—22no—
39C2×C22⋊D107—44no—
40C22×Dic107—21no—
41C22×Dic107—21no—
42C22×Dic107—21no—
43C4⋊Dic107—21no—
44C4⋊Dic107—21no—
45C4⋊Dic107—21no—
46C4⋊Dic107C2312yes—
47C4⋊Dic107C2312yes—
48C4⋊Dic107C2312yes—
49C22⋊Dic107—22no—
50C22⋊Dic107—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—22no—
54C4×Dic107—22no—
55C24×C107D411yescharacteristic
56C22×C4×C107—24no—
57C22×C4×C107D411yescharacteristic
58C22×C4×C107D411yescharacteristic
59C22×C4×C107D411yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C22×C4×C107C2311yescharacteristic
62C22×C4×C107C2311yescharacteristic
63M16×C107—24no—
64C2×C8×C107C2314yes—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—41no—
68C23×C107C2×D411yescharacteristic
69C23×C107C2×D411yescharacteristic
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—42no—
73C22×D107—42no—
74C22×D107—42no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107#4114yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—44no—
86C22⋊D107—44no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—42no—
91C2×Dic107—42no—
92C2×Dic107—42no—
93D428—24no—
94D428—42no—
95C8×C107—24no—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107—21no—
99C22×C107—21no—
100C22×C107—21no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—44no—
104C22×C107#12312yes—
105C22×C107AGL(1,ℤ/8)12yes—
106C22×C107C22×D411yescharacteristic
107C2×D107—42no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—84no—
111C4×C107—21no—
112C4×C107—21no—
113C4×C107#12312yes—
114C4×C107C2×#4112yes—
115Dic107—42no—
116Dic107—42no—
117Dic107—42no—
118C2×C107—22no—
119C2×C107—22no—
120C2×C107—41no—
121C2×C107C2×#12311yescharacteristic
122C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C107#57511yescharacteristic
124D107—82no—
125#3094—1071nomaximal, Sylow
126C107#309411yescharacteristic, complemented, Sylow
127C2×#121—1071no—
128C2×#117—1071no—
129#482—1074no—
130#481—1072no—
131#480—1072no—
132#428—1072no—
133#418—1072no—
134#407D10711yescharacteristic, complemented
135C22×D4—1071no—
136C22×D4—2141no—
137C23×C4C2×D10711yescharacteristic
138C2×M16C2×D10712yes—
139#121—1072no—
140#121—1072no—
141#121—2141no—
142#121—2141no—
143#117—1072no—
144#117—1072no—
145#117—2141no—
146#117—2141no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4—1071no—
150C2×C22⋊C4—2141no—
151C2×C22⋊C4—2141no—
152C2×C42—2141no—
153#107—1074no—
154#106—1074no—
155#102—1074no—
156#102—1074no—
157C22⋊C8C2×D10712yescomplemented
158C22⋊C8C2×D10712yescomplemented
159C24—2141no—
160C24D42811yescharacteristic
161C2×D4—1072no—
162C2×D4—4284no—
163C2×D4—2142no—
164C2×D4—2142no—
165C2×D4—2141no—
166C2×D4—2144no—
167C22×C4—24no—
168C22×C4—2141no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4D42811yescharacteristic
172C22×C4D42811yescharacteristic
173C22×C4D42811yescharacteristic
174C22×C4C22×D10711yescharacteristic
175C22×C4C22×D10711yescharacteristic
176C22×C4C22×D10711yescharacteristic
177M16—24no—
178C2×C8C22×D10714yes—
179C4⋊C4—1072no—
180C4⋊C4—1072no—
181C4⋊C4—1072no—
182C4⋊C4—2141no—
183C4⋊C4—2141no—
184C4⋊C4—2141no—
185C22⋊C4—2142no—
186C22⋊C4—2142no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C42—2142no—
190C42—2142no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—4282no—
195C23—4282no—
196C23—4282no—
197C23—2142no—
198C23C2×D42811yescharacteristic
199C23C2×D42811yescharacteristic
200C23C2×D42811yescharacteristic
201D4—4284no—
202D4—4284no—
203D4—4282no—
204D4—2144no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—4282no—
211C2×C4—4282no—
212C2×C4—4282no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4C2×D42811yescharacteristic
217C2×C4C2×D42811yescharacteristic
218C2×C4C2×D42811yescharacteristic
219C2×C4#1313381714yes—
220C2×C4C23×D10711yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—8564no—
231C22—4282no—
232C22—4282no—
233C22—4282no—
234C22#1681209312yes—
235C22#1681213712yes—
236C22C22×D42811yescharacteristic
237C4—21no—
238C4—21no—
239C4—4282no—
240C4—4282no—
241C4—4282no—
242C4#1681209312yes—
243C4C2×#1313381712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—8562no—
248C2C2×#1681209311yescharacteristic
249C2C2×#1681213711yescharacteristic
250C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812083C2
hypercenter#407D107
hypocenterC107#3094
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16812068C2semidirect
3. #20292952C2semidirect
4. #20292946C2semidirect
5. #20292941C2semidirect
6. #20292669C2semidirect
7. #20292633C2semidirect
8. #407×C107C2semidirect
18. #16812068C22semidirect
20. C2×C4⋊Dic107C22semidirect
21. C2×C4⋊Dic107C22semidirect
25. #16812043C22semidirect
26. #16812041C22semidirect
27. #16812031C22semidirect
28. #16812031C22semidirect
31. C22⋊C8×C107C22semidirect
32. C22⋊C8×C107C22semidirect
126. C107#3094semidirect
134. #407D107semidirect
157. C22⋊C8C2×D107semidirect
158. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120682
#202929522
#202929462
#202929412
#202926692
#202926332
#407×C1072
#3094107