Finite Group Databasebeta

← #28340080

Subgroups — #28340080

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×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812106C211yesmaximal, characteristic, complemented
3C2×#16812097C211yesmaximal, characteristic, complemented
4C2×#16812083C211yesmaximal, characteristic, complemented
5#20292536C214yesmaximal, complemented
6#20292533C211yesmaximal, characteristic, complemented
7#20292529C211yesmaximal, characteristic, complemented
8#20292505C214yesmaximal, complemented
9C4×C22⋊C4×C107C211yesmaximal, characteristic, complemented
10C2×#13133818—24no—
11C2×#13133818—24no—
12C2×#13133818—24no—
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×D428C2214yes—
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2212yes—
18C22×C22⋊D107C2212yes—
19C22×C4×D107C2212yes—
20#16812106—24no—
21#16812097—24no—
22#16812083—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2214yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2214yes—
27C107⋊#95C2214yes—
28C23×C4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2212yes—
30C2×C42×C107C2211yescharacteristic
31C2×C42×C107C2211yescharacteristic
32#95×C107C2212yes—
33C23×D107C2312yes—
34C23×D107C2312yes—
35#13133818—232no—
36C2×Dic214—22no—
37C2×Dic214—22no—
38C2×Dic214D414yescomplemented
39C2×D428—22no—
40C2×D428—22no—
41C2×D428—28no—
42C2×D428—22no—
43C2×D428—22no—
44C2×D428—22no—
45C2×D428—28no—
46C2×D428—44no—
47C2×D428D414yescomplemented
48C2×C22⋊D107—24no—
49C2×C22⋊D107—24no—
50C2×C22⋊D107—24no—
51C2×C22⋊D107—24no—
52C2×C22⋊D107—44no—
53C22×Dic107C2312yes—
54C22×Dic107C2312yes—
55C2×C4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107—24no—
58C4⋊Dic107—22no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—28no—
61(C2×D107)⋊C4—44no—
62C24×C107C2311yescharacteristic
63C22×C4×C107—22no—
64C22×C4×C107—22no—
65C22×C4×C107D414yescomplemented
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22⋊C4×C107—24no—
71C42×C107—22no—
72C42×C107—22no—
73C23×C107—21no—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107C2×D412yes—
78C23×C107C2411yescharacteristic
79C22×D107—22no—
80C22×D107—22no—
81C22×D107—24no—
82C22×D107—24no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—24no—
87C22×D107—24no—
88C22×D107C2×D414yes—
89C2×C4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—28no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107#4114yescomplemented
102C22⋊D107—216no—
103C22⋊D107—48no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—24no—
108C2×Dic107C2×D414yes—
109D428—28no—
110D428—416no—
111D428—44no—
112C4×D107—216no—
113Dic214—28no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107C22×D411yescharacteristic
120C22×C107C22×D411yescharacteristic
121C22×C107C22×D411yescharacteristic
122C22×C107C2×#4112yes—
123C22×C107H(5,𝔽2)12yes—
124C2×D107—24no—
125C2×D107—28no—
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—22no—
135C4×C107—24no—
136C4×C107—42no—
137C4×C107#12014yes—
138Dic107—24no—
139Dic107—42no—
140C2×C107—42no—
141C2×C107C2×#12011yescharacteristic
142C2×C107#53211yescharacteristic
143C2×C107#53511yescharacteristic
144C2×C107#54612yes—
145C2×C107#55012yes—
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#391—1071no—
156#390—1071no—
157#386—1074no—
158C4×C22⋊C4D10711yescharacteristic, complemented
159C2×#41—2144no—
160C2×#41—2144no—
161C2×#41—2144no—
162C22×Q8—1071no—
163C22×D4—1072no—
164C22×D4—1072no—
165C22×D4—1071no—
166C22×D4—1074no—
167C22×D4—1071no—
168C23×C4—1072no—
169C23×C4C2×D10711yescharacteristic
170#127—2144no—
171#124—2144no—
172#121—2144no—
173C2×C4⋊C4—1071no—
174C2×C22⋊C4—1074no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1074no—
177C2×C22⋊C4C2×D10712yes—
178C2×C42C2×D10711yescharacteristic
179C2×C42C2×D10711yescharacteristic
180#95—1074no—
181#95C2×D10712yes—
182C24—1072no—
183C24—1072no—
184C24C22×D10711yescharacteristic
185#41—21432no—
186C2×Q8—1074no—
187C2×Q8—2142no—
188C2×Q8—2142no—
189C2×D4—1074no—
190C2×D4—4284no—
191C2×D4—4284no—
192C2×D4—2148no—
193C2×D4—2144no—
194C2×D4—2144no—
195C2×D4—2144no—
196C2×D4—2144no—
197C2×D4—2142no—
198C2×D4—2142no—
199C2×D4—2142no—
200C2×D4—2142no—
201C2×D4—2142no—
202C2×D4—2148no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—1072no—
206C22×C4—1072no—
207C22×C4—2144no—
208C22×C4—2144no—
209C22×C4—2144no—
210C22×C4D42814yescomplemented
211C22×C4C22×D10712yes—
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10712yes—
214C22×C4C22×D10711yescharacteristic
215C4⋊C4—2142no—
216C22⋊C4—24no—
217C22⋊C4—4284no—
218C22⋊C4—2148no—
219C22⋊C4—2144no—
220C42—22no—
221C42—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—1074no—
227C23—2142no—
228C23—2142no—
229C23—2142no—
230C23—2144no—
231C23—2144no—
232C23—2142no—
233C23—2142no—
234C23—2144no—
235C23—2144no—
236C23C2×D42812yes—
237C23C23×D10711yescharacteristic
238Q8—2148no—
239D4—42816no—
240D4—4288no—
241D4—4284no—
242D4—21416no—
243D4—2148no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—1074no—
253C2×C4—21416no—
254C2×C4—2142no—
255C2×C4—2144no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4D4×D10712yes—
259C2×C4D4×D10712yes—
260C2×C4C2×D42812yes—
261C2×C4C2×D42812yes—
262C2×C4#1313381814yescomplemented
263C22—22no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—4282no—
269C22—4282no—
270C22—4284no—
271C22—4284no—
272C22—4282no—
273C22—4282no—
274C22—4282no—
275C22—4284no—
276C22—2148no—
277C22—2144no—
278C22C2×D4×D10711yescharacteristic
279C22C2×D4×D10711yescharacteristic
280C22C22×D42811yescharacteristic
281C22C2×#1313381812yes—
282C22#1681215512yes—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—4282no—
287C4—2144no—
288C4#1681207914yes—
289C2—42no—
290C2—8562no—
291C2—4284no—
292C2C2×#1681207911yescharacteristic
293C2#2029317711yescharacteristic
294C2#2029319611yescharacteristic
295C2#2029327312yes—
296C2#2029329812yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC4×C22⋊C4×C107C2
hypercenterC4×C22⋊C4D107
hypocenterC107#2431
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812106C2semidirect
3. C2×#16812097C2semidirect
4. C2×#16812083C2semidirect
5. #20292536C2semidirect
6. #20292533C2semidirect
7. #20292529C2semidirect
8. #20292505C2semidirect
9. C4×C22⋊C4×C107C2semidirect
38. C2×Dic214D4semidirect
47. C2×D428D4semidirect
65. C22×C4×C107D4semidirect
101. C2×C4×C107#41semidirect
149. C107#2431semidirect
158. C4×C22⋊C4D107semidirect
210. C22×C4D428semidirect
262. C2×C4#13133818semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121062
C2×#168120972
C2×#168120832
#202925362
#202925332
#202925292
#202925052
C4×C22⋊C4×C1072
#2431107