Finite Group Databasebeta

← #28346106

Subgroups — #28346106

Order 13696.

524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13133803C211yesmaximal, characteristic, complemented
2C2×#16812116C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292878C214yesmaximal, complemented
5#20292861C214yesmaximal, complemented
6#20292663C211yesmaximal, characteristic, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9#535×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133803—24no—
12C2×#13133803C2212yes—
13C2×#13133803C2212yes—
14#16812116—24no—
15C2×#13133802C2211yescharacteristic
16C2×#13133802C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C22×C107⋊C8C2212yescomplemented
19#16812105—21no—
20#16812098—21no—
21#16812083C2214yescomplemented
22#16812077—22no—
23#16812068C2211yescharacteristic
24#16812042C2212yescomplemented
25#16812040C2212yescomplemented
26#16812030C2214yescomplemented
27#16812029C2214yescomplemented
28C2×#41×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30#127×C107—21no—
31#124×C107—21no—
32#121×C107C2214yescomplemented
33#120×C107—22no—
34#117×C107C2211yescharacteristic
35C23×D107—21no—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—24no—
41#13133803—24no—
42#13133803—24no—
43#13133802—22no—
44#13133802—24no—
45C2×C107⋊C8—24no—
46C2×C107⋊C8D414yescomplemented
47C2×C107⋊C8C2312yes—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107C2312yes—
51C22⋊Dic107—21no—
52(C2×D107)⋊C4—24no—
53C4×Dic107—21no—
54C24×C107—21no—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107D411yescharacteristic
58C2×D4×C107—24no—
59C2×D4×C107—22no—
60C2×D4×C107—21no—
61C2×D4×C107—22no—
62C2×D4×C107D412yes—
63C2×D4×C107D411yescharacteristic
64C2×D4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107C2312yes—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107—21no—
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—82no—
74C23×C107C2×D412yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—22no—
77C22×D107—42no—
78C22×D107—82no—
79Q8×C107—22no—
80D4×C107—24no—
81D4×C107—22no—
82D4×C107—22no—
83D4×C107—22no—
84D4×C107—42no—
85D4×C107—84no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107#4112yes—
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—84no—
95C2×Dic107—22no—
96D428—22no—
97D428—22no—
98D428—42no—
99C107⋊C8—22no—
100C107⋊C8C2×D412yes—
101C107⋊C8C2×D412yes—
102C22×C107—22no—
103C22×C107—22no—
104C22×C107—24no—
105C22×C107—42no—
106C22×C107—44no—
107C22×C107—84no—
108C22×C107#12112yes—
109C22×C107C22×D411yescharacteristic
110C2×D107—42no—
111C2×D107—44no—
112C2×D107—84no—
113C4×C107—22no—
114C4×C107—42no—
115C4×C107#12112yes—
116C4×C107C22×D411yescharacteristic
117C4×C107C2×#4111yescharacteristic
118Dic107—42no—
119C2×C107—22no—
120C2×C107—42no—
121C2×C107—82no—
122C2×C107C2×#12111yescharacteristic
123C2×C107#57612yes—
124D107—82no—
125#3058—1071nomaximal, Sylow
126C107#305811yescharacteristic, complemented, Sylow
127C22×D8—1071no—
128C2×#131—1071no—
129#535—1071no—
130#535D10711yescharacteristic, complemented
131#469—1074no—
132#466—1074no—
133#427—1071no—
134#418—1071no—
135#418—1071no—
136C2×#41C2×D10711yescharacteristic
137C22×D4—1071no—
138C22×D4C2×D10711yescharacteristic
139C2×D8—1072no—
140C2×D8—1072no—
141C2×D8—2144no—
142#131—2144no—
143C2×M16—1071no—
144C2×M16—1072no—
145C22×C8—1071no—
146C22×C8—1072no—
147#127—21no—
148#127—2141no—
149#124—21no—
150#124—2141no—
151#121—1074no—
152#121C2×D10714yescomplemented
153#120—22no—
154#120—2142no—
155#117—1071no—
156#117C2×D10711yescharacteristic
157#107—1072no—
158#106—1072no—
159#102—1074no—
160#102—1074no—
161C24—21no—
162C24—2141no—
163#41—22no—
164#41—22no—
165C2×Q8C22⋊D10711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—1072no—
171C2×D4—2144no—
172C2×D4—2142no—
173C2×D4—2141no—
174C2×D4—2142no—
175C2×D4C22⋊D10712yes—
176C2×D4C22⋊D10711yescharacteristic
177C2×D4C22×D10712yes—
178C22×C4C22×D10712yes—
179C22×C4C22×D10711yescharacteristic
180D8—2144no—
181D8—2144no—
182M16—2144no—
183M16—2142no—
184C2×C8—1074no—
185C2×C8—1071no—
186C2×C8—1071no—
187C2×C8—1072no—
188C2×C8—2144no—
189C4⋊C4—1072no—
190C4⋊C4C22×D10712yes—
191C22⋊C4—21no—
192C22⋊C4—24no—
193C22⋊C4—2141no—
194C22⋊C4—2144no—
195C42—21no—
196C42—2141no—
197C23—22no—
198C23—42no—
199C23—82no—
200C23—8562no—
201C23—4282no—
202C23—2142no—
203C23C2×C22⋊D10712yes—
204C23C2×C22⋊D10711yescharacteristic
205Q8—22no—
206D4—24no—
207D4—22no—
208D4—22no—
209D4—22no—
210D4—42no—
211D4—84no—
212D4—8564no—
213D4—4282no—
214D4—2142no—
215D4—2142no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—2142no—
220C2×C4D4×D10712yes—
221C2×C4C2×C22⋊D10712yes—
222C2×C4C2×C22⋊D10711yescharacteristic
223C2×C4#1313381712yes—
224C2×C4C23×D10711yescharacteristic
225C8—1072no—
226C8—1072no—
227C8—2142no—
228C22—22no—
229C22—24no—
230C22—22no—
231C22—44no—
232C22—42no—
233C22—84no—
234C22—8564no—
235C22—4284no—
236C22—4282no—
237C22#1681208112yes—
238C22C22×C22⋊D10711yescharacteristic
239C4—22no—
240C4—42no—
241C4—4282no—
242C4#1681208112yes—
243C4C2×D4×D10711yescharacteristic
244C4C2×#1313381711yescharacteristic
245C2—22no—
246C2—42no—
247C2—82no—
248C2—8562no—
249C2C2×#1681208111yescharacteristic
250C2#2029343012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3058
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C22×#13133803C2semidirect
2. C2×#16812116C2semidirect
3. #20293191C2semidirect
4. #20292878C2semidirect
5. #20292861C2semidirect
6. #20292663C2semidirect
7. #20292632C2semidirect
8. #20292631C2semidirect
9. #535×C107C2semidirect
16. C2×#13133802C22semidirect
18. C22×C107⋊C8C22semidirect
21. #16812083C22semidirect
24. #16812042C22semidirect
25. #16812040C22semidirect
26. #16812030C22semidirect
27. #16812029C22semidirect
32. #121×C107C22semidirect
46. C2×C107⋊C8D4semidirect
126. C107#3058semidirect
130. #535D107semidirect
152. #121C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338032
C2×#168121162
#202931912
#202928782
#202928612
#202926632
#202926322
#202926312
#535×C1072
#3058107