Finite Group Databasebeta

← #28340081

Subgroups — #28340081

Order 13696.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812105C211yesmaximal, characteristic, complemented
3C2×#16812096C211yesmaximal, characteristic, complemented
4#20292535C214yesmaximal, complemented
5#20292532C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292503C214yesmaximal, complemented
8C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
9C2×#121×C107C211yesmaximal, characteristic, complemented
10C2×#13133818—24no—
11C2×#13133818—24no—
12C2×#13133818—24no—
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C22×C22⋊D107C2214yes—
17C22×C4×D107C2212yes—
18#16812105—24no—
19#16812096—24no—
20C2×C22⋊Dic107C2214yes—
21C2×C22⋊Dic107C2212yes—
22C2×(C2×D107)⋊C4C2214yes—
23C2×C4×Dic107C2211yescharacteristic
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2214yes—
27C22×D4×C107C2211yescharacteristic
28C22×D4×C107C2212yes—
29C23×C4×C107C2211yescharacteristic
30#121×C107—24no—
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107C2212yes—
33C23×D107C2312yes—
34#13133818—232no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214D414yescomplemented
38C2×D428—22no—
39C2×D428—22no—
40C2×D428D414yescomplemented
41C2×C22⋊D107—24no—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—28no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—28no—
46C2×C22⋊D107—44no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2312yes—
50C2×C4×D107—24no—
51C2×C4×D107—24no—
52C2×C4×D107—24no—
53C22⋊Dic107—28no—
54C22⋊Dic107—24no—
55(C2×D107)⋊C4—44no—
56C4×Dic107—22no—
57C4×Dic107—22no—
58C24×C107C2312yes—
59C24×C107C2311yescharacteristic
60C2×D4×C107—24no—
61C2×D4×C107—22no—
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107—44no—
65C22×C4×C107—22no—
66C22×C4×C107—22no—
67C22×C4×C107D414yescomplemented
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—22no—
72C22⋊C4×C107—24no—
73C23×C107—21no—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—24no—
81C23×C107—24no—
82C23×C107C2×D412yes—
83C23×C107C2411yescharacteristic
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—24no—
87C22×D107—24no—
88C22×D107C2×D414yes—
89D4×C107—48no—
90D4×C107—44no—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—28no—
94C2×C4×C107—22no—
95C2×C4×C107—24no—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D412yes—
100C22⋊D107—216no—
101C22⋊D107—416no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107C2×D414yes—
110C2×Dic107#4114yescomplemented
111D428—28no—
112C4×D107—216no—
113Dic214—28no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—44no—
123C22×C107—44no—
124C22×C107C22×D411yescharacteristic
125C22×C107C22×D411yescharacteristic
126C22×C107C22×D411yescharacteristic
127C22×C107C2×#4112yes—
128C22×C107H(5,𝔽2)12yes—
129C2×D107—24no—
130C2×D107—28no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—44no—
134C4×C107—22no—
135C4×C107—42no—
136C4×C107#12014yes—
137Dic107—24no—
138Dic107—24no—
139Dic107—42no—
140C2×C107—42no—
141C2×C107—82no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#53211yescharacteristic
144C2×C107#53511yescharacteristic
145C2×C107#54612yes—
146C2×C107#55012yes—
147D107—44no—
148#2431—1071nomaximal, Sylow
149C107#243111yescharacteristic, complemented, Sylow
150C22×#41—1071no—
151C2×#127—1071no—
152C2×#124—1071no—
153C2×#121D10711yescharacteristic, complemented
154#392—1074no—
155#391—1071no—
156#390—1071no—
157#386—1074no—
158C4×C22⋊C4—1071no—
159C2×#41—2144no—
160C2×#41—2144no—
161C2×#41—2144no—
162C22×Q8—1071no—
163C22×D4—1074no—
164C22×D4—1072no—
165C22×D4—1071no—
166C22×D4C2×D10711yescharacteristic
167C22×D4C2×D10712yes—
168C23×C4—1072no—
169C23×C4C2×D10711yescharacteristic
170#127—2144no—
171#124—2144no—
172#121—24no—
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C22⋊C4—1074no—
175C2×C22⋊C4—1074no—
176C2×C22⋊C4—1072no—
177C2×C22⋊C4C2×D10712yes—
178C2×C42—1071no—
179C2×C42—1071no—
180#95—1072no—
181#95—1074no—
182C24—1072no—
183C24C22×D10712yes—
184C24C22×D10711yescharacteristic
185#41—21432no—
186C2×Q8—1074no—
187C2×Q8—2142no—
188C2×Q8—2142no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—44no—
194C2×D4—1074no—
195C2×D4—4284no—
196C2×D4—2148no—
197C2×D4—2144no—
198C2×D4—2148no—
199C2×D4—2144no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4—2142no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—1072no—
206C22×C4—1072no—
207C22×C4—1072no—
208C22×C4—2144no—
209C22×C4—2144no—
210C22×C4—2144no—
211C22×C4C22⋊D10714yescomplemented
212C22×C4C22×D10712yes—
213C22×C4C22×D10711yescharacteristic
214C22×C4C22×D10711yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—24no—
217C22⋊C4—4284no—
218C22⋊C4—2148no—
219C22⋊C4—2144no—
220C42—2142no—
221C42—2142no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—1074no—
232C23—2142no—
233C23—2142no—
234C23—2144no—
235C23—2144no—
236C23C2×C22⋊D10712yes—
237C23C23×D10711yescharacteristic
238Q8—2148no—
239D4—48no—
240D4—44no—
241D4—42816no—
242D4—21416no—
243D4—2148no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—1074no—
250C2×C4—1074no—
251C2×C4—21416no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4D4×D10712yes—
260C2×C4D4×D10712yes—
261C2×C4C2×C22⋊D10712yes—
262C2×C4C2×C22⋊D10712yes—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—44no—
271C22—44no—
272C22—42no—
273C22—4282no—
274C22—4282no—
275C22—4284no—
276C22—2148no—
277C22—2144no—
278C22C2×D4×D10711yescharacteristic
279C22C2×D4×D10711yescharacteristic
280C22C22×C22⋊D10711yescharacteristic
281C22#41×D10712yes—
282C22#1681215412yes—
283C4—22no—
284C4—42no—
285C4—4282no—
286C4—2144no—
287C4—2144no—
288C4#1681207714yes—
289C2—42no—
290C2—82no—
291C2—4284no—
292C2C2×#1681207711yescharacteristic
293C2#2029317311yescharacteristic
294C2#2029319411yescharacteristic
295C2#2029327412yes—
296C2#2029329512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#121×C107C2
hypercenterC2×#121D107
hypocenterC107#2431
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812105C2semidirect
3. C2×#16812096C2semidirect
4. #20292535C2semidirect
5. #20292532C2semidirect
6. #20292528C2semidirect
7. #20292503C2semidirect
8. C4×C22⋊Dic107C2semidirect
9. C2×#121×C107C2semidirect
37. C2×Dic214D4semidirect
40. C2×D428D4semidirect
67. C22×C4×C107D4semidirect
110. C2×Dic107#41semidirect
149. C107#2431semidirect
153. C2×#121D107semidirect
211. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121052
C2×#168120962
#202925352
#202925322
#202925282
#202925032
C4×C22⋊Dic1072
C2×#121×C1072
#2431107