Finite Group Databasebeta

← #28340079

Subgroups — #28340079

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×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812105C211yesmaximal, characteristic, complemented
3C2×#16812097C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5#20292536C214yesmaximal, complemented
6#20292532C211yesmaximal, characteristic, complemented
7#20292504C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9#390×C107C211yesmaximal, characteristic, complemented
10C2×#13133817—24no—
11C2×#13133817—24no—
12C2×#13133817—24no—
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×D428C2212yes—
16C22×C22⋊D107C2212yes—
17C22×C22⋊D107C2214yes—
18C23×Dic107C2212yes—
19C22×C4×D107C2211yescharacteristic
20#16812105—24no—
21#16812097—24no—
22#16812084—24no—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2214yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2212yes—
28C107⋊#95C2214yes—
29C22×D4×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C22⋊C4×C107C2214yes—
32C2×C42×C107C2211yescharacteristic
33C23×D107C2312yes—
34C23×D107C2311yescharacteristic
35#13133817—232no—
36C2×Dic214—22no—
37C2×Dic214—22no—
38C2×Dic214D414yescomplemented
39C2×D428—24no—
40C2×D428—22no—
41C2×D428—22no—
42C2×D428—22no—
43C2×D428—44no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—28no—
47C2×C22⋊D107—24no—
48C2×C22⋊D107—28no—
49C2×C22⋊D107—44no—
50C22×Dic107—24no—
51C22×Dic107—24no—
52C22×Dic107—24no—
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2312yes—
55C22×Dic107C2312yes—
56C2×C4×D107—22no—
57C2×C4×D107—22no—
58C2×C4×D107D414yescomplemented
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—28no—
61(C2×D107)⋊C4—24no—
62C4×Dic107—22no—
63C24×C107C2312yes—
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107D414yescomplemented
67C22×C4×C107C2312yes—
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C4⋊C4×C107—22no—
71C22⋊C4×C107—44no—
72C42×C107—22no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107—24no—
76C23×C107—24no—
77C23×C107C2×D414yes—
78C23×C107C2411yescharacteristic
79C22×D107—21no—
80C22×D107—21no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—24no—
87C22×D107—24no—
88C22×D107C2×D412yes—
89D4×C107—28no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—24no—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D412yes—
98C22⋊D107—216no—
99C22⋊D107—416no—
100C2×Dic107—216no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107C2×D412yes—
108C2×Dic107C2×D414yes—
109C2×Dic107#4114yescomplemented
110D428—48no—
111D428—44no—
112C4×D107—28no—
113Dic214—28no—
114C22×C107—24no—
115C22×C107—28no—
116C22×C107—42no—
117C22×C107—42no—
118C22×C107—44no—
119C22×C107C22×D411yescharacteristic
120C22×C107C22×D411yescharacteristic
121C22×C107C22×D411yescharacteristic
122C22×C107C2×#4112yes—
123C22×C107H(5,𝔽2)12yes—
124C2×D107—22no—
125C2×D107—22no—
126C2×D107—22no—
127C2×D107—42no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—44no—
133C2×D107—44no—
134C4×C107—42no—
135C4×C107—42no—
136C4×C107#12014yes—
137Dic107—24no—
138Dic107—22no—
139Dic107—24no—
140C2×C107—44no—
141C2×C107C2×#12011yescharacteristic
142C2×C107#53211yescharacteristic
143C2×C107#53511yescharacteristic
144C2×C107#54612yes—
145C2×C107#55012yes—
146D107—42no—
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#390D10711yescharacteristic, complemented
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—1072no—
166C22×D4—1071no—
167C22×D4C2×D10711yescharacteristic
168C23×C4—1071no—
169C23×C4—1072no—
170#127—2144no—
171#124—2144no—
172#121—2144no—
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1074no—
176C2×C22⋊C4—1072no—
177C2×C22⋊C4C2×D10714yes—
178C2×C42—1071no—
179C2×C42C2×D10711yescharacteristic
180#95—1072no—
181#95—1074no—
182C24—1071no—
183C24—1072no—
184C24C22×D10712yes—
185#41—21432no—
186C2×Q8—1074no—
187C2×Q8—2142no—
188C2×Q8—2142no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—4284no—
192C2×D4—4284no—
193C2×D4—2144no—
194C2×D4—2148no—
195C2×D4—2144no—
196C2×D4—2148no—
197C2×D4—2144no—
198C2×D4—2144no—
199C2×D4—2142no—
200C2×D4—2142no—
201C2×D4—2142no—
202C2×D4D42814yescomplemented
203C22×C4—1072no—
204C22×C4—1071no—
205C22×C4—1072no—
206C22×C4—1074no—
207C22×C4—2144no—
208C22×C4—2144no—
209C22×C4—2144no—
210C22×C4—2142no—
211C22×C4—2142no—
212C22×C4C22×D10712yes—
213C22×C4C22×D10712yes—
214C22×C4C22×D10711yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—44no—
217C22⋊C4—2144no—
218C22⋊C4—2148no—
219C22⋊C4—2144no—
220C42—22no—
221C42—2142no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—1072no—
227C23—2142no—
228C23—2142no—
229C23—2142no—
230C23—2144no—
231C23—2144no—
232C23—2141no—
233C23—2141no—
234C23—2142no—
235C23—2142no—
236C23C2×D42814yes—
237C23C23×D10711yescharacteristic
238Q8—2148no—
239D4—28no—
240D4—4288no—
241D4—42816no—
242D4—4284no—
243D4—21416no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—1072no—
250C2×C4—1074no—
251C2×C4—1074no—
252C2×C4—21416no—
253C2×C4—2141no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2141no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2148no—
260C2×C4D4×D10712yes—
261C2×C4D4×D10712yes—
262C2×C4C2×D42812yes—
263C22—28no—
264C22—24no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—4282no—
269C22—4282no—
270C22—4284no—
271C22—4284no—
272C22—4282no—
273C22—4282no—
274C22—4282no—
275C22—2142no—
276C22—2142no—
277C22—2142no—
278C22C2×D4×D10711yescharacteristic
279C22C2×D4×D10711yescharacteristic
280C22C22×D42811yescharacteristic
281C22#41×D10712yes—
282C22#1681215412yes—
283C4—42no—
284C4—42no—
285C4—2144no—
286C4—2142no—
287C4—2144no—
288C4#1681207914yes—
289C2—44no—
290C2—8562no—
291C2—4282no—
292C2C2×#1681207911yescharacteristic
293C2#2029317611yescharacteristic
294C2#2029319011yescharacteristic
295C2#2029327212yes—
296C2#2029329512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
hypercenter#390D107
hypocenterC107#2431
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812105C2semidirect
3. C2×#16812097C2semidirect
4. C2×#16812084C2semidirect
5. #20292536C2semidirect
6. #20292532C2semidirect
7. #20292504C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #390×C107C2semidirect
38. C2×Dic214D4semidirect
58. C2×C4×D107D4semidirect
66. C2×D4×C107D4semidirect
109. C2×Dic107#41semidirect
149. C107#2431semidirect
156. #390D107semidirect
202. C2×D4D428semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168121052
C2×#168120972
C2×#168120842
#202925362
#202925322
#202925042
C4×(C2×D107)⋊C42
#390×C1072
#2431107