Finite Group Databasebeta

← #28344937

Subgroups — #28344937

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×#16812154C211yesmaximal, characteristic
2#20293393C212yesmaximal, complemented
3#20292609C212yesmaximal, complemented
4#20292608C214yesmaximal, complemented
5C2×#16812026C211yesmaximal, characteristic, complemented
6#100×D107C214yesmaximal, complemented
7C2×#100×C107C211yesmaximal, characteristic, complemented
8#16812154—24no—
9#16812154—24no—
10C2×#13133818C2211yescharacteristic
11C2×#13133818C2211yescharacteristic
12C2×#13133817C412yes—
13C2×#13133817C2212yes—
14C22×C22⋊D107C414yes—
15C2×D4×D107C412yes—
16C2×D4×D107C2212yes—
17#16812117—24no—
18#16812115—24no—
19C2×#13133802C2212yescomplemented
20M16×D107C2218yescomplemented
21#16812027C2214yescomplemented
22#16812026C2214yescomplemented
23#16812025C2214yescomplemented
24C22×D4×C107C2211yescharacteristic
25C2×M16×C107C2212yescomplemented
26#100×C107C2214yescomplemented
27C23×D107C2×C414yes—
28#13133818—22no—
29#13133818—22no—
30#13133818D414yes—
31#13133818D414yes—
32#13133817—28no—
33#13133817—28no—
34C2×Dic214C2×C411yescharacteristic
35C2×Dic214C2311yescharacteristic
36C2×D428C2×C411yescharacteristic
37C2×D428C2311yescharacteristic
38C2×C22⋊D107—28no—
39C2×C22⋊D107—28no—
40C2×C22⋊D107C2×C414yes—
41C2×C22⋊D107C2×C414yes—
42C2×C22⋊D107C2×C412yes—
43C2×C22⋊D107C2×C412yes—
44D4×D107—28no—
45D4×D107—28no—
46C22×Dic107C2×C414yes—
47C2×C4×D107C2×C412yes—
48C2×C4×D107C2312yes—
49#13133802—22no—
50#13133802C2314yes—
51#13133801—28no—
52C2×C107⋊C8—22no—
53C8×D107—28no—
54C24×C107C2×C412yes—
55C2×D4×C107—22no—
56C2×D4×C107—22no—
57C2×D4×C107C2×C412yes—
58C2×D4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60M16×C107—22no—
61M16×C107C2314yes—
62C2×C8×C107—22no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107—82no—
66C23×C107C22×C414yes—
67C23×C107C22×C411yescharacteristic
68C22×D107—28no—
69C22×D107—84no—
70C22×D107C22×C412yes—
71C22×D107C22×C412yes—
72D4×C107—24no—
73D4×C107—24no—
74C2×C4×C107C22×C411yescharacteristic
75C2×C4×C107C2×D412yes—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—22no—
79C22⋊D107—22no—
80C22⋊D107—28no—
81C22⋊D107—28no—
82C22⋊D107C22⋊C414yes—
83C22⋊D107C22⋊C414yes—
84C2×Dic107—28no—
85C2×Dic107C22×C412yes—
86C2×Dic107C22×C412yes—
87D428—22no—
88D428C22⋊C414yes—
89C4×D107—24no—
90C4×D107C2×D418yes—
91Dic214—22no—
92Dic214C22⋊C414yes—
93C8×C107—24no—
94C107⋊C8—24no—
95C22×C107—24no—
96C22×C107—41no—
97C22×C107—44no—
98C22×C107—82no—
99C22×C107—82no—
100C22×C107C2×C22⋊C411yescharacteristic
101C22×C107C2×C22⋊C411yescharacteristic
102C22×C107C23×C411yescharacteristic
103C2×D107—22no—
104C2×D107—42no—
105C2×D107—42no—
106C2×D107—88no—
107C2×D107C2×C22⋊C414yes—
108C4×C107C2×C22⋊C412yes—
109C4×C107C22×D412yes—
110Dic107—22no—
111Dic107C2×C22⋊C414yes—
112C2×C107—21no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—44no—
116C2×C107C22×C22⋊C411yescharacteristic
117D107—24no—
118D107—42no—
119#2888—1071nomaximal, Sylow
120C107#288811yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1071no—
122#568—1072no—
123#413—1072no—
124#413—1074no—
125C2×#100—1071no—
126C2×#100—1074no—
127C2×#100D10711yescharacteristic, complemented
128H(5,𝔽2)—2144no—
129H(5,𝔽2)—2144no—
130C2×#41—1072no—
131C2×#41—1072no—
132C2×#41—1071no—
133C2×#41—1071no—
134C22×D4—1074no—
135C22×D4—1072no—
136C22×D4—1072no—
137C22×D4C2×D10711yescharacteristic
138#131—2144no—
139#131—2144no—
140C2×M16—1072no—
141C2×M16—1078no—
142C2×M16C2×D10712yescomplemented
143#101—1074no—
144#100—1074no—
145#100—1074no—
146#100C2×D10714yescomplemented
147C24—1074no—
148C24C4×D10712yes—
149#41—1074no—
150#41—1074no—
151#41—2148no—
152#41—2148no—
153#41—2142no—
154#41—2142no—
155C2×Q8—1071no—
156C2×Q8—1071no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—1072no—
160C2×D4—1074no—
161C2×D4—1074no—
162C2×D4—1072no—
163C2×D4—1071no—
164C2×D4—1071no—
165C2×D4—2148no—
166C2×D4—2148no—
167C2×D4—2148no—
168C2×D4—2148no—
169C2×D4C4×D10712yes—
170C2×D4C22×D10712yes—
171C22×C4—1072no—
172C22×C4—1074no—
173C22×C4—1072no—
174C22×C4C22×D10711yescharacteristic
175M16—22no—
176M16—1074no—
177M16—2148no—
178M16—2142no—
179M16C22×D10714yes—
180C2×C8—22no—
181C2×C8—2148no—
182C2×C8—2142no—
183C23—22no—
184C23—22no—
185C23—82no—
186C23—8564no—
187C23—1072no—
188C23—1072no—
189C23—2148no—
190C23C2×C4×D10714yes—
191C23C2×C4×D10711yescharacteristic
192Q8—1074no—
193Q8—2142no—
194D4—24no—
195D4—24no—
196D4—1074no—
197D4—1074no—
198D4—1074no—
199D4—2148no—
200D4—2148no—
201D4—2142no—
202D4—2142no—
203D4—2142no—
204C2×C4—1072no—
205C2×C4—1078no—
206C2×C4—1072no—
207C2×C4—2148no—
208C2×C4—2144no—
209C2×C4C2×C4×D10711yescharacteristic
210C2×C4D4×D10712yes—
211C2×C4D4×D10712yes—
212C2×C4C23×D10711yescharacteristic
213C8—24no—
214C8—2144no—
215C22—24no—
216C22—41no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—8568no—
221C22—1074no—
222C22—4282no—
223C22—4282no—
224C22—2142no—
225C22C22⋊C4×D10711yescharacteristic
226C22C22⋊C4×D10711yescharacteristic
227C22C22×C4×D10711yescharacteristic
228C4—1074no—
229C4—2142no—
230C4C22⋊C4×D10712yes—
231C4C2×D4×D10712yes—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—4282no—
237C2—2144no—
238C2C2×C22⋊C4×D10711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D107
commutatorC22×C107C23×C4
FrattiniC2×C4C23×D107
FittingC2×#100×C107C2
socleC2×C107C22×C22⋊C4
prime radicalC2×#16812154C2
cosocleC2×C4×C107C24
hypercenterC2×#100D107
hypocenterC107#2888
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292609C2semidirect
4. #20292608C2semidirect
5. C2×#16812026C2semidirect
6. #100×D107C2semidirect
7. C2×#100×C107C2semidirect
19. C2×#13133802C22semidirect
20. M16×D107C22semidirect
21. #16812027C22semidirect
22. #16812026C22semidirect
23. #16812025C22semidirect
25. C2×M16×C107C22semidirect
26. #100×C107C22semidirect
120. C107#2888semidirect
127. C2×#100D107semidirect
142. C2×M16C2×D107semidirect
146. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
#202933932
#202926092
#202926082
C2×#168120262
#100×D1072
C2×#100×C1072
#2888107