Finite Group Databasebeta

← #28345925

Subgroups — #28345925

Order 13696.

548 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812135C212yesmaximal, complemented
2#20293177C211yesmaximal, characteristic, complemented
3#20292842C214yesmaximal, complemented
4#20292832C212yesmaximal, complemented
5#20292822C212yesmaximal, complemented
6#20292651C211yesmaximal, characteristic, complemented
7#20292626C212yesmaximal, complemented
8C2×C4×D4×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2212yes—
10#16812135—28no—
11C2×#13133806C2214yes—
12C2×#13133803C2214yes—
13C2×#13133802C2212yes—
14#16812109C2211yescharacteristic, complemented
15#16812106C2211yescharacteristic, complemented
16#16812097C2212yescomplemented
17#16812083—21no—
18#16812083—21no—
19C4×D428—21no—
20C4×D428—21no—
21C4⋊(C107⋊C8)C2212yescomplemented
22C4⋊(C107⋊C8)C2212yescomplemented
23#16812032C2214yescomplemented
24#16812029C2214yescomplemented
25C22×D4×C107C2211yescharacteristic
26C23×C4×C107—21no—
27C4×D4×C107—21no—
28C4×D4×C107—21no—
29C4×D4×C107C2212yes—
30C4×D4×C107C2212yes—
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107—21no—
33C2×C42×C107C2211yescharacteristic
34#13133818—24no—
35#13133818—24no—
36C2×Dic214C2312yes—
37C2×D428—21no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—22no—
41C2×C4×D107—22no—
42#13133806—28no—
43#13133803—28no—
44#13133802—24no—
45C2×C107⋊C8C2314yes—
46C4⋊Dic107—21no—
47C4⋊Dic107—21no—
48(C2×D107)⋊C4—22no—
49(C2×D107)⋊C4—22no—
50C24×C107—21no—
51C2×D4×C107—21no—
52C2×D4×C107—22no—
53C2×D4×C107D414yescomplemented
54C2×D4×C107C2312yes—
55C22×C4×C107—21no—
56C22×C4×C107—22no—
57C22×C4×C107—22no—
58C22×C4×C107—44no—
59C22×C4×C107D411yescharacteristic
60C22×C4×C107D411yescharacteristic
61C22×C4×C107C2311yescharacteristic
62C4⋊C4×C107—21no—
63C4⋊C4×C107C2312yes—
64C22⋊C4×C107—22no—
65C22⋊C4×C107—22no—
66C42×C107D411yescharacteristic
67C42×C107D411yescharacteristic
68C42×C107C2311yescharacteristic
69C42×C107C2311yescharacteristic
70C23×C107—24no—
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—42no—
74C23×C107C2×D411yescharacteristic
75C22×D107—22no—
76D4×C107—24no—
77D4×C107—22no—
78D4×C107C2×D414yes—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—41no—
82C2×C4×C107—44no—
83C2×C4×C107—42no—
84C2×C4×C107—44no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D412yes—
91C2×C4×C107#4112yes—
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—44no—
94C2×Dic107—22no—
95D428—22no—
96D428—22no—
97D428—42no—
98D428—42no—
99C4×D107—44no—
100Dic214—22no—
101Dic214—22no—
102C107⋊C8—24no—
103C22×C107—21no—
104C22×C107—21no—
105C22×C107—24no—
106C22×C107—48no—
107C22×C107—42no—
108C22×C107—42no—
109C22×C107#12112yes—
110C22×C107C22×D411yescharacteristic
111C2×D107—42no—
112C2×D107—42no—
113C2×D107—42no—
114C4×C107—21no—
115C4×C107—21no—
116C4×C107—21no—
117C4×C107—42no—
118C4×C107#12112yes—
119C4×C107AGL(1,ℤ/8)12yescomplemented
120C4×C107C22×D411yescharacteristic
121C4×C107C2×#4111yescharacteristic
122Dic107—42no—
123C2×C107—22no—
124C2×C107—44no—
125C2×C107C2×#12111yescharacteristic
126C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C107#57511yescharacteristic
128D107—82no—
129#3036—1071nomaximal, Sylow
130C107#303611yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1072no—
132#532—1071no—
133C2×C4×D4D10711yescharacteristic, complemented
134#463—1074no—
135#461—1072no—
136#459—1072no—
137#423—1071no—
138#417—1072no—
139C2×#41—1072no—
140C22×D4C2×D10711yescharacteristic
141C23×C4—21no—
142AGL(1,ℤ/8)—2148no—
143C2×SD16—1074no—
144C2×D8—1074no—
145C2×M16—1072no—
146C4⋊Q8—1071no—
147#127—1071no—
148#124—1072no—
149#121—2141no—
150#121—2141no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—2141no—
154C4×D4—2141no—
155C4×D4C2×D10712yes—
156C4×D4C2×D10712yes—
157C2×C4⋊C4C2×D10711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C42C2×D10711yescharacteristic
160C4⋊C8—1072no—
161C4⋊C8—1072no—
162#103—1074no—
163#102—1074no—
164C24—21no—
165#41—2144no—
166#41—2144no—
167C2×Q8—1072no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—1072no—
171C2×D4—2142no—
172C2×D4—2141no—
173C2×D4—2141no—
174C2×D4D42814yescomplemented
175C2×D4C22×D10712yes—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—2142no—
181C22×C4C22⋊D10711yescharacteristic
182C22×C4C22⋊D10711yescharacteristic
183C22×C4C22×D10711yescharacteristic
184SD16—2148no—
185D8—2148no—
186M16—2144no—
187C2×C8—1074no—
188C4⋊C4—21no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4C22×D10712yes—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—2142no—
195C22⋊C4—2142no—
196C42C22⋊D10711yescharacteristic
197C42C22⋊D10711yescharacteristic
198C42C22×D10711yescharacteristic
199C42C22×D10711yescharacteristic
200C23—24no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—2142no—
205C23C2×C22⋊D10711yescharacteristic
206Q8—2142no—
207Q8—2142no—
208D4—24no—
209D4—22no—
210D4—4282no—
211D4—4284no—
212D4—4282no—
213D4—2142no—
214D4—2142no—
215D4C2×D42814yes—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—4284no—
223C2×C4—2142no—
224C2×C4C2×C22⋊D10711yescharacteristic
225C2×C4C2×C22⋊D10711yescharacteristic
226C2×C4C2×C22⋊D10711yescharacteristic
227C2×C4C2×C22⋊D10711yescharacteristic
228C2×C4C2×C22⋊D10711yescharacteristic
229C2×C4C2×D42812yes—
230C2×C4#1313381812yes—
231C2×C4C23×D10711yescharacteristic
232C8—2144no—
233C22—21no—
234C22—21no—
235C22—24no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—4282no—
240C22—4282no—
241C22—4282no—
242C22#1681208312yes—
243C22C22×C22⋊D10711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—4282no—
249C4#1681208312yes—
250C4#1681213512yescomplemented
251C4C22×D42811yescharacteristic
252C4C2×#1313381811yescharacteristic
253C2—22no—
254C2—44no—
255C2—8562no—
256C2C2×#1681208311yescharacteristic
257C2C2×#1681213511yescharacteristic
258C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4×D4×C107C2
socleC22×C107C22×D4
hypercenterC2×C4×D4D107
hypocenterC107#3036
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293177C2semidirect
3. #20292842C2semidirect
4. #20292832C2semidirect
5. #20292822C2semidirect
6. #20292651C2semidirect
7. #20292626C2semidirect
8. C2×C4×D4×C107C2semidirect
14. #16812109C22semidirect
15. #16812106C22semidirect
16. #16812097C22semidirect
21. C4⋊(C107⋊C8)C22semidirect
22. C4⋊(C107⋊C8)C22semidirect
23. #16812032C22semidirect
24. #16812029C22semidirect
53. C2×D4×C107D4semidirect
119. C4×C107AGL(1,ℤ/8)semidirect
130. C107#3036semidirect
133. C2×C4×D4D107semidirect
174. C2×D4D428semidirect
250. C4#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202931772
#202928422
#202928322
#202928222
#202926512
#202926262
C2×C4×D4×C1072
#3036107