Finite Group Databasebeta

← #28340082

Subgroups — #28340082

Order 13696.

952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133819C211yesmaximal, characteristic, complemented
2C2×#16812106C211yesmaximal, characteristic, complemented
3C2×#16812098C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5#20292537C214yesmaximal, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292505C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C2×#13133819—24no—
11C2×#13133819—24no—
12C2×#13133819—24no—
13C22×D428C2211yescharacteristic
14C22×D428C2212yes—
15C22×D428C2214yes—
16C22×D428C2211yescharacteristic
17C22×D428C2212yes—
18C22×C4×D107C2212yes—
19C22×C4×D107C2211yescharacteristic
20#16812106—24no—
21#16812098—24no—
22#16812084—24no—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2214yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2214yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2212yes—
29C22×Q8×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C42×C107C2211yescharacteristic
32#95×C107C2214yes—
33C23×D107C2312yes—
34C23×D107C2311yescharacteristic
35C23×D107C2312yes—
36#13133819—232no—
37C2×D428—24no—
38C2×D428—24no—
39C2×D428—22no—
40C2×D428—24no—
41C2×D428—22no—
42C2×D428—24no—
43C2×D428—28no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—28no—
48C2×D428—44no—
49C2×D428—44no—
50C2×D428D414yescomplemented
51C22×Dic107C2312yes—
52C22×Dic107C2311yescharacteristic
53C2×C4×D107—24no—
54C2×C4×D107—22no—
55C2×C4×D107—24no—
56C2×C4×D107—22no—
57C2×C4×D107—24no—
58C2×C4×D107D414yescomplemented
59(C2×D107)⋊C4—28no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—24no—
62(C2×D107)⋊C4—44no—
63C4×Dic107—22no—
64C2×Q8×C107—22no—
65C2×Q8×C107—22no—
66C2×Q8×C107D414yescomplemented
67C22×C4×C107C2312yes—
68C22×C4×C107C2312yes—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—22no—
72C42×C107—22no—
73C23×C107C2411yescharacteristic
74C22×D107—22no—
75C22×D107—21no—
76C22×D107—22no—
77C22×D107—21no—
78C22×D107—22no—
79C22×D107—24no—
80C22×D107—24no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—24no—
86C22×D107—24no—
87C22×D107C2×D414yes—
88C22×D107C2×D412yes—
89Q8×C107—28no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—24no—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D414yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107#4114yescomplemented
101C2×Dic107—21no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107C2×D412yes—
107D428—216no—
108D428—28no—
109D428—416no—
110D428—48no—
111D428—44no—
112C4×D107—216no—
113C4×D107—28no—
114C22×C107C22×D411yescharacteristic
115C22×C107C22×D411yescharacteristic
116C22×C107C22×D411yescharacteristic
117C22×C107C2×#4112yes—
118C22×C107H(5,𝔽2)12yes—
119C2×D107—24no—
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—22no—
123C2×D107—28no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C2×D107—44no—
129C2×D107—42no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—44no—
133C2×D107—44no—
134C4×C107—24no—
135C4×C107—24no—
136C4×C107—42no—
137C4×C107#12014yes—
138Dic107—22no—
139Dic107—42no—
140C2×C107C2×#12011yescharacteristic
141C2×C107#53211yescharacteristic
142C2×C107#53511yescharacteristic
143C2×C107#54612yes—
144C2×C107#55012yes—
145D107—42no—
146D107—44no—
147D107—82no—
148#2431—1071nomaximal, Sylow
149C107#243111yescharacteristic, complemented, Sylow
150C22×#41—1071no—
151C2×#127—1071no—
152C2×#124—1071no—
153C2×#121—1071no—
154#392—1074no—
155#391D10711yescharacteristic, complemented
156#390—1071no—
157#386—1074no—
158C4×C22⋊C4—1071no—
159C2×#41—2144no—
160C2×#41—2144no—
161C2×#41—2144no—
162C22×Q8C2×D10711yescharacteristic
163C22×D4—1071no—
164C22×D4—1072no—
165C22×D4—1074no—
166C22×D4—1071no—
167C22×D4—1072no—
168C23×C4—1071no—
169C23×C4—1072no—
170#127—2144no—
171#124—2144no—
172#121—2144no—
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C22⋊C4—1074no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1074no—
177C2×C22⋊C4—1072no—
178C2×C42—1071no—
179C2×C42C2×D10711yescharacteristic
180#95—1072no—
181#95C2×D10714yes—
182C24—1071no—
183C24—1072no—
184C24—1072no—
185#41—21432no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8D42814yescomplemented
189C2×D4—1074no—
190C2×D4—4284no—
191C2×D4—4284no—
192C2×D4—2144no—
193C2×D4—2148no—
194C2×D4—2144no—
195C2×D4—2144no—
196C2×D4—2142no—
197C2×D4—2142no—
198C2×D4—2142no—
199C2×D4—2142no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4—2148no—
203C22×C4—1071no—
204C22×C4—1074no—
205C22×C4—1072no—
206C22×C4—2144no—
207C22×C4—2144no—
208C22×C4—2142no—
209C22×C4—2144no—
210C22×C4—2142no—
211C22×C4C22×D10712yes—
212C22×C4C22×D10712yes—
213C22×C4C22×D10712yes—
214C22×C4C22×D10711yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—4284no—
217C22⋊C4—2144no—
218C22⋊C4—2148no—
219C22⋊C4—2144no—
220C42—22no—
221C42—2142no—
222C23—1072no—
223C23—1074no—
224C23—2142no—
225C23—2142no—
226C23—2142no—
227C23—2144no—
228C23—2144no—
229C23—2142no—
230C23—2141no—
231C23—2141no—
232C23—2142no—
233C23—2142no—
234C23—2144no—
235C23—2144no—
236C23—2142no—
237C23C23×D10711yescharacteristic
238Q8—28no—
239D4—42816no—
240D4—4288no—
241D4—4284no—
242D4—21416no—
243D4—2148no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—1072no—
251C2×C4—21416no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2141no—
256C2×C4—2141no—
257C2×C4—2148no—
258C2×C4D4×D10712yes—
259C2×C4D4×D10712yes—
260C2×C4C2×D42814yes—
261C2×C4C2×D42812yes—
262C2×C4#1313381914yescomplemented
263C22—4282no—
264C22—4282no—
265C22—4284no—
266C22—4284no—
267C22—4282no—
268C22—4282no—
269C22—4282no—
270C22—4282no—
271C22—4282no—
272C22—4284no—
273C22—2142no—
274C22—2142no—
275C22—2148no—
276C22—2144no—
277C22—2142no—
278C22C2×D4×D10711yescharacteristic
279C22C2×D4×D10711yescharacteristic
280C22C22×D42811yescharacteristic
281C22C2×#1313381912yes—
282C22#1681215512yes—
283C4—24no—
284C4—24no—
285C4—42no—
286C4—4282no—
287C4—2142no—
288C4#1681207914yes—
289C2—8562no—
290C2—4284no—
291C2—4282no—
292C2C2×#1681207911yescharacteristic
293C2#2029317811yescharacteristic
294C2#2029319511yescharacteristic
295C2#2029326812yes—
296C2#2029329912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
hypercenter#391D107
hypocenterC107#2431
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#13133819C2semidirect
2. C2×#16812106C2semidirect
3. C2×#16812098C2semidirect
4. C2×#16812084C2semidirect
5. #20292537C2semidirect
6. #20292528C2semidirect
7. #20292505C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #391×C107C2semidirect
50. C2×D428D4semidirect
58. C2×C4×D107D4semidirect
66. C2×Q8×C107D4semidirect
100. C2×C4×C107#41semidirect
149. C107#2431semidirect
155. #391D107semidirect
188. C2×Q8D428semidirect
262. C2×C4#13133819semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168121062
C2×#168120982
C2×#168120842
#202925372
#202925282
#202925052
C4×(C2×D107)⋊C42
#391×C1072
#2431107