Finite Group Databasebeta

← #28346083

Subgroups — #28346083

Order 13696.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16812134C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293134C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5#20292879C212yesmaximal, complemented
6#20292876C212yesmaximal, complemented
7#20292862C212yesmaximal, complemented
8#20292859C212yesmaximal, complemented
9#20292667C211yesmaximal, characteristic, complemented
10#20292633C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C22×D428—21no—
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812134—24no—
16C2×#13133805C2212yescomplemented
17C2×#13133803C2212yescomplemented
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2212yescomplemented
20M16×D107—24no—
21C2×C8×D107C2212yescomplemented
22#16812090—21no—
23#16812085C2211yescharacteristic, complemented
24#16812085C2211yescharacteristic, complemented
25#16812084—21no—
26#16812084—21no—
27#16812084C2212yes—
28#16812084C2212yes—
29#16812080C2211yescharacteristic, complemented
30#16812080C2211yescharacteristic, complemented
31#16812077—21no—
32C4×C22⋊D107—21no—
33C4×C22⋊D107—21no—
34#16812068C2211yescharacteristic
35C2×(C2×D107)⋊C4—21no—
36#16812042C2212yescomplemented
37#16812040C2212yescomplemented
38#16812031C2212yescomplemented
39#16812031C2212yescomplemented
40C2×#41×C107C2211yescharacteristic
41C2×M16×C107C2211yescharacteristic
42C2×C4⋊C4×C107C2211yescharacteristic
43#103×C107C2212yescomplemented
44#102×C107C2212yescomplemented
45C23×D107—21no—
46C23×D107D411yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—22no—
50C2×D428—22no—
51C2×D428—21no—
52C2×D428—44no—
53C2×D428C2312yes—
54C2×C22⋊D107—21no—
55C2×C22⋊D107—21no—
56C22×Dic107D411yescharacteristic
57C2×C4×D107—22no—
58C2×C4×D107—22no—
59C2×C4×D107D411yescharacteristic
60C2×C4×D107D411yescharacteristic
61C2×C4×D107C2311yescharacteristic
62C2×C4×D107C2311yescharacteristic
63#13133805—24no—
64#13133803—24no—
65#13133802D414yescomplemented
66#13133801—24no—
67C2×C107⋊C8C2312yes—
68C8×D107—24no—
69C4⋊Dic107C2311yescharacteristic
70C4⋊Dic107C2311yescharacteristic
71Dic107⋊C4—21no—
72Dic107⋊C4—21no—
73C22⋊Dic107—21no—
74C22⋊Dic107—21no—
75C22⋊Dic107—21no—
76(C2×D107)⋊C4—21no—
77(C2×D107)⋊C4—21no—
78(C2×D107)⋊C4—22no—
79(C2×D107)⋊C4—22no—
80C4×Dic107—21no—
81#41×C107—22no—
82#41×C107—22no—
83C2×Q8×C107C2311yescharacteristic
84C2×D4×C107—21no—
85C2×D4×C107C2311yescharacteristic
86C22×C4×C107—21no—
87C22×C4×C107—21no—
88C22×C4×C107C2311yescharacteristic
89M16×C107—22no—
90C2×C8×C107C2312yes—
91C4⋊C4×C107—21no—
92C4⋊C4×C107C2312yes—
93C23×C107—21no—
94C23×C107C2×D411yescharacteristic
95C22×D107—22no—
96C22×D107—22no—
97C22×D107—22no—
98C22×D107—41no—
99C22×D107—42no—
100C22×D107—42no—
101C22×D107—42no—
102C22×D107C2×D411yescharacteristic
103C22×D107C2×D411yescharacteristic
104Q8×C107—21no—
105Q8×C107—21no—
106D4×C107—21no—
107D4×C107—21no—
108D4×C107—42no—
109C2×C4×C107—22no—
110C2×C4×C107—21no—
111C2×C4×C107—42no—
112C2×C4×C107—42no—
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2×D412yes—
115C2×C4×C107#4112yes—
116C2×C4×C107C2411yescharacteristic
117C22⋊D107—42no—
118C22⋊D107—42no—
119C2×Dic107—22no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124D428—22no—
125D428—22no—
126D428—44no—
127D428—44no—
128D428—42no—
129C4×D107—22no—
130C4×D107—22no—
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—21no—
134C4×D107—21no—
135C8×C107—22no—
136C107⋊C8C2×D414yescomplemented
137C22×C107—21no—
138C22×C107—21no—
139C22×C107—41no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107#12112yes—
143C22×C107C22×D411yescharacteristic
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—21no—
148C2×D107—21no—
149C2×D107—44no—
150C2×D107—42no—
151C2×D107—42no—
152C2×D107—42no—
153C2×D107—84no—
154C2×D107AGL(1,ℤ/8)12yescomplemented
155C4×C107—41no—
156C4×C107—42no—
157C4×C107#12112yes—
158C4×C107C22×D411yescharacteristic
159C4×C107C2×#4111yescharacteristic
160Dic107—21no—
161Dic107—21no—
162Dic107—41no—
163Dic107—41no—
164C2×C107—22no—
165C2×C107—81no—
166C2×C107C2×#12111yescharacteristic
167C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C107#57511yescharacteristic
169D107—22no—
170D107—41no—
171D107—82no—
172#3055—1071nomaximal, Sylow
173C107#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1071no—
175C22×M16—1071no—
176#525—1071no—
177C2×#121—1071no—
178#469—1072no—
179#468—1072no—
180#466—1072no—
181#465—1072no—
182#428—1071no—
183#418—1071no—
184#417D10711yescharacteristic, complemented
185C2×#41C2×D10711yescharacteristic
186C22×D4—1071no—
187C22×D4—2141no—
188C23×C4—1071no—
189AGL(1,ℤ/8)—2144no—
190C2×SD16—1072no—
191C2×D8—1072no—
192C2×M16—1071no—
193C2×M16—1072no—
194C2×M16—2144no—
195C2×M16C2×D10711yescharacteristic
196C22×C8—1072no—
197#123—2141no—
198C22⋊Q8—1071no—
199C22⋊Q8—1071no—
200#121—1071no—
201#121—1071no—
202#121—1072no—
203#121—1072no—
204#121—2141no—
205#121—2141no—
206#120—2141no—
207C4×D4—2141no—
208C4×D4—2141no—
209#117—1071no—
210C2×C4⋊C4C2×D10711yescharacteristic
211C2×C22⋊C4—2141no—
212#107—1072no—
213#106—1072no—
214#103C2×D10712yescomplemented
215#102—1072no—
216#102—1072no—
217#102C2×D10712yescomplemented
218C24—1071no—
219C24—2141no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D10711yescharacteristic
223C2×D4—21no—
224C2×D4—1072no—
225C2×D4—4284no—
226C2×D4—2142no—
227C2×D4—2142no—
228C2×D4—2141no—
229C2×D4—2141no—
230C2×D4—2141no—
231C2×D4—2142no—
232C2×D4—2142no—
233C2×D4C22×D10711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1071no—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—2142no—
242C22×C4—2142no—
243C22×C4C22×D10711yescharacteristic
244SD16—2144no—
245D8—2144no—
246M16—22no—
247M16—1074no—
248M16—2144no—
249C2×C8—1072no—
250C2×C8—2144no—
251C2×C8C22×D10712yes—
252C4⋊C4—21no—
253C4⋊C4—1071no—
254C4⋊C4—1071no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4C22×D10712yes—
258C22⋊C4—2142no—
259C22⋊C4—2142no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C42—2141no—
266C23—21no—
267C23—1071no—
268C23—1071no—
269C23—4282no—
270C23—4282no—
271C23—4281no—
272C23—4282no—
273C23—2142no—
274C23—2142no—
275C23—2142no—
276C23D4×D10711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—4284no—
283D4—4284no—
284D4—4282no—
285D4—4282no—
286D4—4282no—
287D4—2142no—
288D4—2142no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1071no—
294C2×C4—1071no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2142no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4—2141no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4D4×D10711yescharacteristic
305C2×C4D4×D10712yes—
306C2×C4#1313381812yes—
307C2×C4C23×D10711yescharacteristic
308C8—22no—
309C8—1074no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—8564no—
316C22—1072no—
317C22—4284no—
318C22—4282no—
319C22—4282no—
320C22—4282no—
321C22—2141no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22—2141no—
326C22#1681208212yes—
327C22C2×D4×D10711yescharacteristic
328C4—42no—
329C4—41no—
330C4—4281no—
331C4—4281no—
332C4—2141no—
333C4—2141no—
334C4#1681208212yes—
335C4C2×D4×D10711yescharacteristic
336C4C2×#1313381811yescharacteristic
337C2—22no—
338C2—81no—
339C2—8562no—
340C2—4281no—
341C2—2142no—
342C2C2×#1681208211yescharacteristic
343C2AGL(1,ℤ/8)×D10711yescharacteristic
344C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3055
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×M16×D107C2semidirect
3. #20293134C2semidirect
4. C2×#16812084C2semidirect
5. #20292879C2semidirect
6. #20292876C2semidirect
7. #20292862C2semidirect
8. #20292859C2semidirect
9. #20292667C2semidirect
10. #20292633C2semidirect
11. #417×C107C2semidirect
16. C2×#13133805C22semidirect
17. C2×#13133803C22semidirect
19. C2×#13133801C22semidirect
21. C2×C8×D107C22semidirect
23. #16812085C22semidirect
24. #16812085C22semidirect
29. #16812080C22semidirect
30. #16812080C22semidirect
36. #16812042C22semidirect
37. #16812040C22semidirect
38. #16812031C22semidirect
39. #16812031C22semidirect
43. #103×C107C22semidirect
44. #102×C107C22semidirect
65. #13133802D4semidirect
136. C107⋊C8C2×D4semidirect
154. C2×D107AGL(1,ℤ/8)semidirect
173. C107#3055semidirect
184. #417D107semidirect
214. #103C2×D107semidirect
217. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×M16×D1072
#202931342
C2×#168120842
#202928792
#202928762
#202928622
#202928592
#202926672
#202926332
#417×C1072
#3055107