Finite Group Databasebeta

← #28345954

Subgroups — #28345954

Order 13696.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic
3C2×#16812096C211yesmaximal, characteristic, complemented
4#20293095C211yesmaximal, characteristic, complemented
5#20292848C214yesmaximal, complemented
6#20292841C214yesmaximal, complemented
7#20292650C211yesmaximal, characteristic, complemented
8#20292635C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812142—24no—
13#16812135—24no—
14C2×Dic428C2212yescomplemented
15C2×#13133807C2212yescomplemented
16C2×#13133806C2212yes—
17C2×#13133803C2212yes—
18C2×#13133802C2211yescharacteristic
19#16812096—21no—
20#16812096—21no—
21#16812096C2214yes—
22#16812074C2212yescomplemented
23#16812071C2212yescomplemented
24(C22⋊D107)⋊C4—22no—
25#16812066—21no—
26C2×C22⋊Dic107—22no—
27C2×C4×Dic107C2211yescharacteristic
28C42×D107—21no—
29Dic107⋊C8C2214yescomplemented
30#16812032C2214yescomplemented
31C22×D4×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#102×C107C2214yes—
35#13133818—22no—
36#13133818D414yes—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214C2312yes—
41C2×Dic214C2311yescharacteristic
42C2×D428C2311yescharacteristic
43C2×C22⋊D107—21no—
44C22×Dic107—21no—
45C22×Dic107D412yes—
46C2×C4×D107—21no—
47C2×C4×D107—22no—
48Dic428—24no—
49#13133807—24no—
50#13133806—24no—
51#13133803—24no—
52#13133802—22no—
53C2×C107⋊C8C2312yes—
54Dic107⋊C4—22no—
55C22⋊Dic107—24no—
56C22⋊Dic107—24no—
57(C2×D107)⋊C4—22no—
58C4×Dic107—22no—
59C4×Dic107D411yescharacteristic
60C4×Dic107D411yescharacteristic
61C4×Dic107C2312yes—
62C24×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107—21no—
65C2×D4×C107—22no—
66C2×D4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68M16×C107—22no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107C2312yes—
71C22⋊C4×C107—21no—
72C42×C107—21no—
73C23×C107—22no—
74C23×C107—44no—
75C23×C107—42no—
76C23×C107C2×D411yescharacteristic
77C22×D107—21no—
78D4×C107—22no—
79D4×C107—22no—
80D4×C107—42no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107#4112yes—
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—24no—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—42no—
90C2×Dic107—42no—
91C2×Dic107C2×D412yes—
92C2×Dic107C2×D412yes—
93D428—21no—
94D428C2×D412yes—
95C4×D107—24no—
96C4×D107—44no—
97Dic214—21no—
98Dic214—22no—
99Dic214—22no—
100Dic214—42no—
101Dic214C2×D412yes—
102C8×C107—22no—
103C107⋊C8—22no—
104C22×C107—22no—
105C22×C107—42no—
106C22×C107—44no—
107C22×C107—84no—
108C22×C107#12112yes—
109C22×C107C22×D411yescharacteristic
110C2×D107—22no—
111C2×D107—42no—
112C4×C107—42no—
113C4×C107#12112yes—
114C4×C107C22×D411yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—22no—
117Dic107—22no—
118Dic107—22no—
119Dic107—42no—
120C2×C107—22no—
121C2×C107—82no—
122C2×C107C2×#12111yescharacteristic
123C2×C107#57512yes—
124D107—42no—
125#3040—1071nomaximal, Sylow
126C107#304011yescharacteristic, complemented, Sylow
127C2×#137—1071no—
128C2×AGL(1,ℤ/8)—1071no—
129C2×#124—1071no—
130C4×#41—1071no—
131#464—1074no—
132#463—1074no—
133#423—1071no—
134#419—1071no—
135#418D10711yescharacteristic, complemented
136C2×#41—1071no—
137C22×Q8—1071no—
138C22×D4C2×D10711yescharacteristic
139#137—2144no—
140AGL(1,ℤ/8)—2144no—
141C2×Q16—1072no—
142C2×SD16—1072no—
143C2×SD16—1072no—
144C2×D8—1072no—
145C2×M16—1071no—
146C2×M16C2×D10711yescharacteristic
147#124—1074no—
148#124—2141no—
149#124—2141no—
150C4×Q8—1072no—
151C4×D4—1072no—
152C4×D4—2142no—
153#117—2141no—
154#117C2×D10711yescharacteristic
155C2×C22⋊C4—2142no—
156C2×C42—1071no—
157C2×C42—2141no—
158C4⋊C8—1074no—
159#103—1074no—
160#102C2×D10714yes—
161C24—21no—
162#41—1074no—
163#41—2142no—
164C2×Q8—1071no—
165C2×Q8—1072no—
166C2×Q8—2142no—
167C2×Q8—2142no—
168C2×Q8—2141no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—1071no—
173C2×D4—2141no—
174C2×D4C22×D10712yes—
175C22×C4—1072no—
176C22×C4—2142no—
177C22×C4—2141no—
178C22×C4—2141no—
179C22×C4C22×D10711yescharacteristic
180Q16—2144no—
181SD16—2144no—
182SD16—2144no—
183D8—2144no—
184M16—22no—
185M16—2142no—
186C2×C8—1072no—
187C2×C8C22×D10712yes—
188C4⋊C4—2142no—
189C4⋊C4C22×D10712yes—
190C22⋊C4—21no—
191C22⋊C4—2142no—
192C22⋊C4—2144no—
193C22⋊C4—2144no—
194C42—21no—
195C42—1071no—
196C42—1071no—
197C42—1072no—
198C42—2142no—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2141no—
203C23D4×D10711yescharacteristic
204Q8—1072no—
205Q8—4282no—
206Q8—2142no—
207Q8—2142no—
208Q8—2141no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—1072no—
213D4—2144no—
214D4—2141no—
215C2×C4—22no—
216C2×C4—1072no—
217C2×C4—1072no—
218C2×C4—4284no—
219C2×C4—4282no—
220C2×C4—4282no—
221C2×C4—2142no—
222C2×C4—2141no—
223C2×C4—2144no—
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10712yes—
226C2×C4#1313381812yes—
227C2×C4C23×D10711yescharacteristic
228C8—22no—
229C8—2142no—
230C22—22no—
231C22—44no—
232C22—42no—
233C22—84no—
234C22—4282no—
235C22—2142no—
236C22#1681208212yes—
237C22C2×D4×D10711yescharacteristic
238C4—42no—
239C4—4282no—
240C4—2142no—
241C4—2142no—
242C4—2142no—
243C4#1681208212yes—
244C4C2×D4×D10711yescharacteristic
245C4C2×#1313381811yescharacteristic
246C2—22no—
247C2—82no—
248C2—4282no—
249C2C2×#1681208211yescharacteristic
250C2#2029342312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812135C2
hypercenter#418D107
hypocenterC107#3040
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
3. C2×#16812096C2semidirect
4. #20293095C2semidirect
5. #20292848C2semidirect
6. #20292841C2semidirect
7. #20292650C2semidirect
8. #20292635C2semidirect
9. #418×C107C2semidirect
14. C2×Dic428C22semidirect
15. C2×#13133807C22semidirect
22. #16812074C22semidirect
23. #16812071C22semidirect
29. Dic107⋊C8C22semidirect
30. #16812032C22semidirect
126. C107#3040semidirect
135. #418D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121352
C2×#168120962
#202930952
#202928482
#202928412
#202926502
#202926352
#418×C1072
#3040107