Finite Group Databasebeta

← #28345922

Subgroups — #28345922

Order 13696.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16812135C211yesmaximal, characteristic, complemented
2#20293173C211yesmaximal, characteristic, complemented
3C2×D4×Dic107C211yesmaximal, characteristic, complemented
4#20292844C212yesmaximal, complemented
5#20292840C212yesmaximal, complemented
6#20292830C212yesmaximal, complemented
7#20292820C212yesmaximal, complemented
8#20292650C211yesmaximal, characteristic, complemented
9#20292629C211yesmaximal, characteristic, complemented
10#20292628C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C23×Dic107—21no—
14#16812135—24no—
15C2×#13133806C2212yes—
16C2×#13133803C2212yes—
17C2×#13133802C2211yescharacteristic
18#16812107C2211yescharacteristic, complemented
19#16812105C2211yescharacteristic, complemented
20#16812098C2211yescharacteristic, complemented
21#16812096C2211yescharacteristic, complemented
22#16812081—21no—
23#16812081—21no—
24D4×Dic107—21no—
25D4×Dic107—21no—
26D4×Dic107C2212yes—
27D4×Dic107C2212yes—
28C4×C22⋊D107—21no—
29C4×C22⋊D107—21no—
30C2×C4⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107—21no—
32C2×C4×Dic107C2211yescharacteristic
33Dic107⋊C8C2212yescomplemented
34Dic107⋊C8C2212yescomplemented
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812031C2212yescomplemented
38#16812030C2212yescomplemented
39C2×#41×C107C2211yescharacteristic
40C22×D4×C107C2211yescharacteristic
41AGL(1,ℤ/8)×C107—24no—
42C2×SD16×C107C2212yes—
43C2×D8×C107C2212yes—
44C2×M16×C107C2211yescharacteristic
45#13133818—22no—
46#13133818—22no—
47C2×Dic214C2311yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—21no—
52C22×Dic107—21no—
53C22×Dic107—22no—
54C22×Dic107—22no—
55C22×Dic107—44no—
56C22×Dic107D411yescharacteristic
57C22×Dic107D411yescharacteristic
58C2×C4×D107—21no—
59#13133806—24no—
60#13133803—24no—
61#13133802—22no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107—21no—
64C4⋊Dic107C2312yes—
65Dic107⋊C4—21no—
66Dic107⋊C4—21no—
67C22⋊Dic107—22no—
68C22⋊Dic107—22no—
69C22⋊Dic107—21no—
70C22⋊Dic107—21no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—21no—
73C4×Dic107D411yescharacteristic
74C4×Dic107D411yescharacteristic
75C4×Dic107C2311yescharacteristic
76C4×Dic107C2311yescharacteristic
77C24×C107—21no—
78#41×C107—22no—
79#41×C107—22no—
80C2×Q8×C107C2311yescharacteristic
81C2×D4×C107—21no—
82C2×D4×C107—21no—
83C2×D4×C107—22no—
84C2×D4×C107D414yescomplemented
85C2×D4×C107C2311yescharacteristic
86C2×D4×C107C2312yes—
87C22×C4×C107—21no—
88C22×C4×C107C2311yescharacteristic
89SD16×C107—24no—
90D8×C107—24no—
91M16×C107—22no—
92C2×C8×C107C2312yes—
93C23×C107—24no—
94C23×C107—22no—
95C23×C107—21no—
96C23×C107—42no—
97C23×C107—42no—
98C23×C107C2×D411yescharacteristic
99C22×D107—21no—
100Q8×C107—21no—
101Q8×C107—21no—
102D4×C107—24no—
103D4×C107—21no—
104D4×C107—22no—
105D4×C107—21no—
106D4×C107—42no—
107D4×C107C2×D414yes—
108C2×C4×C107—21no—
109C2×C4×C107—42no—
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D412yes—
112C2×C4×C107#4112yes—
113C2×C4×C107C2411yescharacteristic
114C22⋊D107—42no—
115C22⋊D107—42no—
116C22⋊D107—42no—
117C2×Dic107—22no—
118C2×Dic107—21no—
119C2×Dic107—22no—
120C2×Dic107—41no—
121C2×Dic107—44no—
122C2×Dic107—42no—
123C2×Dic107—44no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128D428—21no—
129D428—21no—
130C4×D107—42no—
131Dic214—21no—
132Dic214—21no—
133C8×C107—22no—
134C107⋊C8—22no—
135C22×C107—21no—
136C22×C107—21no—
137C22×C107—24no—
138C22×C107—48no—
139C22×C107—42no—
140C22×C107—42no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107#12112yes—
145C22×C107C22×D411yescharacteristic
146C2×D107—41no—
147C2×D107—41no—
148C2×D107—41no—
149C4×C107—41no—
150C4×C107#12112yes—
151C4×C107C22×D411yescharacteristic
152C4×C107C2×#4111yescharacteristic
153Dic107—21no—
154Dic107—21no—
155Dic107—21no—
156Dic107—41no—
157Dic107—42no—
158Dic107AGL(1,ℤ/8)12yescomplemented
159C2×C107—22no—
160C2×C107—44no—
161C2×C107—81no—
162C2×C107C2×#12111yescharacteristic
163C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C107#57511yescharacteristic
165D107—81no—
166#3036—1071nomaximal, Sylow
167C107#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1071no—
169C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
170#532—1071no—
171C2×C4×D4—1071no—
172#463—1072no—
173#463—1072no—
174#461—1072no—
175#459—1072no—
176#423—1071no—
177#417—1071no—
178#417—1071no—
179C2×#41—1071no—
180C2×#41C2×D10711yescharacteristic
181C22×D4C2×D10711yescharacteristic
182C23×C4—2141no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—2144no—
185C2×SD16—1072no—
186C2×SD16C2×D10712yes—
187C2×D8—1072no—
188C2×D8C2×D10712yes—
189C2×M16—1071no—
190C2×M16C2×D10711yescharacteristic
191C4⋊Q8—1071no—
192#127—1071no—
193#124—1071no—
194#124—1071no—
195#121—2141no—
196#121—2141no—
197C4×D4—1072no—
198C4×D4—1072no—
199C4×D4—2141no—
200C4×D4—2141no—
201C4×D4—2141no—
202C4×D4—2141no—
203C2×C4⋊C4—1071no—
204C2×C22⋊C4—2141no—
205C2×C42—1071no—
206C4⋊C8—1072no—
207C4⋊C8—1072no—
208#103—1072no—
209#103—1072no—
210#102—1072no—
211#102—1072no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—2142no—
216#41—2142no—
217C2×Q8—1071no—
218C2×Q8C22×D10711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—1071no—
223C2×D4—2141no—
224C2×D4—2141no—
225C2×D4—2141no—
226C2×D4C22⋊D10714yescomplemented
227C2×D4C22×D10711yescharacteristic
228C2×D4C22×D10712yes—
229C22×C4—21no—
230C22×C4—1071no—
231C22×C4—1071no—
232C22×C4—4284no—
233C22×C4—2142no—
234C22×C4—2142no—
235C22×C4—2141no—
236C22×C4—2141no—
237C22×C4C22×D10711yescharacteristic
238SD16—24no—
239SD16—2144no—
240D8—24no—
241D8—2144no—
242M16—22no—
243M16—2142no—
244C2×C8—1072no—
245C2×C8C22×D10712yes—
246C4⋊C4—1072no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C22⋊C4—2141no—
251C22⋊C4—2141no—
252C22⋊C4—2142no—
253C22⋊C4—2142no—
254C22⋊C4—2141no—
255C22⋊C4—2141no—
256C42—1071no—
257C42—1071no—
258C42—1071no—
259C42—1071no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—2141no—
266C23D4×D10711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2141no—
270Q8—2141no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—4282no—
277D4—4282no—
278D4—4282no—
279D4—2141no—
280D4—2141no—
281D4C2×C22⋊D10714yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—1071no—
285C2×C4—1071no—
286C2×C4—1071no—
287C2×C4—1071no—
288C2×C4—4284no—
289C2×C4—4282no—
290C2×C4—4284no—
291C2×C4—4282no—
292C2×C4—4281no—
293C2×C4—2142no—
294C2×C4—2142no—
295C2×C4—2141no—
296C2×C4D4×D10711yescharacteristic
297C2×C4C2×C22⋊D10712yes—
298C2×C4#1313381712yes—
299C2×C4C23×D10711yescharacteristic
300C8—22no—
301C8—2142no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—4281no—
312C22—4281no—
313C22—4281no—
314C22#1681208112yes—
315C22C2×D4×D10711yescharacteristic
316C4—41no—
317C4—1072no—
318C4—4282no—
319C4—4281no—
320C4—2141no—
321C4—2141no—
322C4—2141no—
323C4#1681208112yes—
324C4C22×C22⋊D10711yescharacteristic
325C4C2×#1313381711yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—8561no—
330C2C2×#1681208111yescharacteristic
331C2AGL(1,ℤ/8)×D10711yescharacteristic
332C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3036
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293173C2semidirect
3. C2×D4×Dic107C2semidirect
4. #20292844C2semidirect
5. #20292840C2semidirect
6. #20292830C2semidirect
7. #20292820C2semidirect
8. #20292650C2semidirect
9. #20292629C2semidirect
10. #20292628C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
18. #16812107C22semidirect
19. #16812105C22semidirect
20. #16812098C22semidirect
21. #16812096C22semidirect
33. Dic107⋊C8C22semidirect
34. Dic107⋊C8C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812031C22semidirect
38. #16812030C22semidirect
84. C2×D4×C107D4semidirect
158. Dic107AGL(1,ℤ/8)semidirect
167. C107#3036semidirect
169. C2×AGL(1,ℤ/8)D107semidirect
226. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202931732
C2×D4×Dic1072
#202928442
#202928402
#202928302
#202928202
#202926502
#202926292
#202926282
C2×AGL(1,ℤ/8)×C1072
#3036107