Finite Group Databasebeta

← #28344938

Subgroups — #28344938

Order 13696.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812155C211yesmaximal, characteristic
2#20293393C212yesmaximal, complemented
3#20292611C211yesmaximal, characteristic, complemented
4#20292610C214yesmaximal, complemented
5C2×#16812025C212yesmaximal, complemented
6#100×D107C214yesmaximal, complemented
7#413×C107C211yesmaximal, characteristic, complemented
8#16812155—24no—
9#16812155—24no—
10C2×#13133819C2212yes—
11C2×#13133818C412yes—
12C2×#13133818C2211yescharacteristic
13C22×D428C412yes—
14C22×D428C2211yescharacteristic
15C2×D4×D107C414yes—
16C2×D4×D107C2212yes—
17#16812117—24no—
18#16812115—24no—
19C2×#13133802C2212yescomplemented
20M16×D107C2218yescomplemented
21#16812028C2212yescomplemented
22#16812026C2212yescomplemented
23#16812025C2218yescomplemented
24C2×#41×C107C2211yescharacteristic
25C2×M16×C107C2212yescomplemented
26#101×C107C2212yescomplemented
27#100×C107C2212yescomplemented
28C23×D107C2×C412yes—
29C23×D107C2×C414yes—
30#13133819—28no—
31#13133818—24no—
32#13133818—24no—
33#13133818D414yes—
34#13133818D414yes—
35C2×Dic214C2×C411yescharacteristic
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—24no—
39C2×D428—24no—
40C2×D428C2×C414yes—
41C2×D428C2×C412yes—
42C2×D428C2×C411yescharacteristic
43C2×D428C2312yes—
44C2×C22⋊D107C2×C414yes—
45C2×C22⋊D107C2×C412yes—
46D4×D107—216no—
47D4×D107—28no—
48C2×C4×D107C2×C414yes—
49C2×C4×D107C2312yes—
50#13133802—22no—
51#13133802C2314yes—
52#13133801—28no—
53C2×C107⋊C8—22no—
54C8×D107—28no—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107C2311yescharacteristic
58C2×D4×C107C2×C412yes—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107C2×C412yes—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63M16×C107C2314yes—
64C2×C8×C107—22no—
65C23×C107C22×C412yes—
66C23×C107C22×C411yescharacteristic
67C22×D107—24no—
68C22×D107—28no—
69C22×D107—84no—
70C22×D107—82no—
71C22×D107C22×C414yes—
72C22×D107C22×C412yes—
73Q8×C107—22no—
74D4×C107—24no—
75D4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C22×C412yes—
79C2×C4×C107C22×C411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107C2×D412yes—
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—28no—
84C22⋊D107C22⋊C418yes—
85C2×Dic107C22×C412yes—
86D428—28no—
87D428—22no—
88D428—22no—
89D428—24no—
90D428C22⋊C412yes—
91D428C22⋊C412yes—
92C4×D107—28no—
93C4×D107C2×D418yes—
94Dic214C22⋊C412yes—
95Dic214C22⋊C412yes—
96C8×C107—24no—
97C107⋊C8—24no—
98C22×C107—22no—
99C22×C107—41no—
100C22×C107—42no—
101C22×C107C2×C22⋊C412yes—
102C22×C107C23×C411yescharacteristic
103C2×D107—24no—
104C2×D107—42no—
105C2×D107—44no—
106C2×D107—88no—
107C2×D107—84no—
108C2×D107C2×C22⋊C414yes—
109C4×C107—22no—
110C4×C107C2×C22⋊C411yescharacteristic
111C4×C107C2×C22⋊C411yescharacteristic
112C4×C107C22×D412yes—
113Dic107C2×C22⋊C414yes—
114C2×C107—21no—
115C2×C107—22no—
116C2×C107—42no—
117C2×C107C22×C22⋊C411yescharacteristic
118D107—24no—
119D107—44no—
120#2888—1071nomaximal, Sylow
121C107#288811yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1071no—
123#568—1072no—
124#413—1071no—
125#413—1074no—
126#413D10711yescharacteristic, complemented
127C2×#100—1072no—
128C2×#100—1074no—
129H(5,𝔽2)—2144no—
130H(5,𝔽2)—2144no—
131C2×#41—1072no—
132C2×#41—1072no—
133C2×#41—1071no—
134C2×#41C2×D10711yescharacteristic
135C22×D4—1074no—
136C22×D4—1072no—
137C22×D4—1072no—
138C22×D4—1071no—
139#131—2144no—
140#131—2144no—
141C2×M16—1072no—
142C2×M16—1078no—
143C2×M16C2×D10712yescomplemented
144#101—1072no—
145#101C2×D10712yescomplemented
146#100—1072no—
147#100—1078no—
148#100C2×D10712yescomplemented
149C24—1074no—
150C24—1072no—
151#41—22no—
152#41—22no—
153#41—1074no—
154#41—1074no—
155#41—2144no—
156#41—2144no—
157#41—2148no—
158C2×Q8—1071no—
159C2×Q8C22×D10711yescharacteristic
160C2×D4—1072no—
161C2×D4—1074no—
162C2×D4—1071no—
163C2×D4—1074no—
164C2×D4—1072no—
165C2×D4—1072no—
166C2×D4—2144no—
167C2×D4—2144no—
168C2×D4—2148no—
169C2×D4—21416no—
170C2×D4—2142no—
171C2×D4—2142no—
172C2×D4C4×D10712yes—
173C2×D4C22×D10711yescharacteristic
174C22×C4—1072no—
175C22×C4—1074no—
176C22×C4C4×D10712yes—
177C22×C4C22×D10711yescharacteristic
178M16—22no—
179M16—1074no—
180M16—2148no—
181M16—2142no—
182M16C22×D10714yes—
183C2×C8—22no—
184C2×C8—2148no—
185C2×C8—2142no—
186C23—8562no—
187C23—8564no—
188C23—1072no—
189C23—1074no—
190C23—2148no—
191C23—2144no—
192C23C2×C4×D10712yes—
193C23C2×C4×D10711yescharacteristic
194Q8—22no—
195Q8—1072no—
196Q8—1072no—
197D4—24no—
198D4—22no—
199D4—1078no—
200D4—1072no—
201D4—1072no—
202D4—2148no—
203D4—2148no—
204D4—2142no—
205D4—2142no—
206D4—2144no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—1072no—
210C2×C4—1078no—
211C2×C4—2148no—
212C2×C4C2×C4×D10712yes—
213C2×C4C2×C4×D10711yescharacteristic
214C2×C4D4×D10712yes—
215C2×C4D4×D10712yes—
216C2×C4C23×D10711yescharacteristic
217C8—24no—
218C8—2144no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—8564no—
223C22—8568no—
224C22—1074no—
225C22—4284no—
226C22—4282no—
227C22—2144no—
228C22C22⋊C4×D10712yes—
229C22C22×C4×D10711yescharacteristic
230C4—22no—
231C4—1074no—
232C4C22⋊C4×D10711yescharacteristic
233C4C22⋊C4×D10711yescharacteristic
234C4C2×D4×D10712yes—
235C2—21no—
236C2—22no—
237C2—42no—
238C2—4284no—
239C2—2144no—
240C2C2×C22⋊C4×D10711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D107
commutatorC22×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#413×C107C2
socleC2×C107C22×C22⋊C4
prime radicalC2×#16812155C2
cosocleC2×C4×C107C24
hypercenter#413D107
hypocenterC107#2888
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292611C2semidirect
4. #20292610C2semidirect
5. C2×#16812025C2semidirect
6. #100×D107C2semidirect
7. #413×C107C2semidirect
19. C2×#13133802C22semidirect
20. M16×D107C22semidirect
21. #16812028C22semidirect
22. #16812026C22semidirect
23. #16812025C22semidirect
25. C2×M16×C107C22semidirect
26. #101×C107C22semidirect
27. #100×C107C22semidirect
121. C107#2888semidirect
126. #413D107semidirect
143. C2×M16C2×D107semidirect
145. #101C2×D107semidirect
148. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
#202933932
#202926112
#202926102
C2×#168120252
#100×D1072
#413×C1072
#2888107