Finite Group Databasebeta

← #28345764

Subgroups — #28345764

Order 13696.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812154C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293205C211yesmaximal, characteristic, complemented
4#20292786C212yesmaximal, complemented
5#20292776C212yesmaximal, complemented
6#20292769C212yesmaximal, complemented
7#20292759C212yesmaximal, complemented
8#20292629C211yesmaximal, characteristic, complemented
9C2×#16812030C211yesmaximal, characteristic, complemented
10#20292595C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12#16812154—24no—
13#16812154—24no—
14C2×#13133818C2211yescharacteristic
15C2×#13133818C2211yescharacteristic
16C2×#13133817—21no—
17C2×#13133817C2212yes—
18C22×C22⋊D107—22no—
19C2×D4×D107—21no—
20C2×D4×D107C2212yes—
21#16812129—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic, complemented
25C2×#13133803C2211yescharacteristic, complemented
26C22×C107⋊C8C2211yescharacteristic, complemented
27#16812088—21no—
28#16812085C2212yescomplemented
29#16812083—21no—
30#16812081—22no—
31#16812080C2212yescomplemented
32C2×C4⋊Dic107C2211yescharacteristic, complemented
33#16812034C2212yescomplemented
34#16812031C2212yescomplemented
35#16812030C2212yescomplemented
36#16812030C2212yescomplemented
37(C2×D107)⋊C8C2212yescomplemented
38(C2×D107)⋊C8C2212yescomplemented
39C2×#41×C107C2211yescharacteristic, complemented
40C22×D4×C107C2211yescharacteristic
41AGL(1,ℤ/8)×C107—24no—
42C2×SD16×C107C2212yescomplemented
43C2×D8×C107C2212yescomplemented
44C2×M16×C107C2211yescharacteristic, complemented
45C23×D107—22no—
46#13133818—22no—
47#13133818—22no—
48#13133818—22no—
49#13133818D414yescomplemented
50#13133817—28no—
51#13133817—24no—
52#13133817—24no—
53C2×Dic214D411yescharacteristic
54C2×Dic214C2311yescharacteristic
55C2×D428D411yescharacteristic
56C2×D428C2311yescharacteristic
57C2×C22⋊D107—216no—
58C2×C22⋊D107—21no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107D412yes—
63D4×D107—28no—
64D4×D107—24no—
65D4×D107—24no—
66C22×Dic107—22no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107C2312yes—
70#13133808—22no—
71#13133806—22no—
72#13133805—22no—
73#13133803—22no—
74C2×C107⋊C8—22no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107—21no—
78C4⋊Dic107C2312yes—
79Dic107⋊C4—22no—
80C22⋊Dic107—22no—
81(C2×D107)⋊C4—22no—
82C24×C107—21no—
83#41×C107—22no—
84#41×C107—22no—
85C2×Q8×C107C2311yescharacteristic
86C2×D4×C107—21no—
87C2×D4×C107—21no—
88C2×D4×C107—22no—
89C2×D4×C107D414yescomplemented
90C2×D4×C107C2311yescharacteristic
91C2×D4×C107C2312yes—
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94SD16×C107—24no—
95D8×C107—24no—
96M16×C107—22no—
97C2×C8×C107C2312yes—
98C23×C107—22no—
99C23×C107—24no—
100C23×C107—21no—
101C23×C107—81no—
102C23×C107—81no—
103C23×C107C2×D411yescharacteristic
104C22×D107—21no—
105C22×D107—28no—
106C22×D107—82no—
107C22×D107—82no—
108C22×D107C2×D412yes—
109Q8×C107—21no—
110Q8×C107—21no—
111D4×C107—24no—
112D4×C107—22no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—42no—
116D4×C107C2×D414yes—
117C2×C4×C107—21no—
118C2×C4×C107—42no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D412yes—
121C2×C4×C107C2×D412yes—
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—28no—
124C22⋊D107—28no—
125C22⋊D107—24no—
126C22⋊D107—24no—
127C22⋊D107—82no—
128C2×Dic107—28no—
129C2×Dic107—21no—
130C2×Dic107—22no—
131C2×Dic107—81no—
132C2×Dic107C2×D412yes—
133D428—22no—
134D428—21no—
135D428C2×D412yes—
136C4×D107—24no—
137C4×D107—22no—
138C4×D107—22no—
139Dic214—22no—
140Dic214—21no—
141Dic214C2×D412yes—
142C8×C107—22no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—24no—
148C22×C107—42no—
149C22×C107—42no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—84no—
154C22×C107#12012yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—22no—
157C2×D107—22no—
158C2×D107—22no—
159C2×D107—22no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—88no—
163C4×C107—41no—
164C4×C107#12012yes—
165C4×C107C22×D411yescharacteristic
166C4×C107C22×D411yescharacteristic
167Dic107—22no—
168Dic107—22no—
169Dic107—42no—
170C2×C107—22no—
171C2×C107—44no—
172C2×C107—81no—
173C2×C107C2×#12011yescharacteristic
174C2×C107#57611yescharacteristic
175C2×C107#57711yescharacteristic
176D107—42no—
177D107—42no—
178#3012—1071nomaximal, Sylow
179C107#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1071no—
181C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
182C2×#135—1071no—
183#537—1071no—
184#452—1072no—
185#450—1072no—
186#449—1072no—
187#447—1072no—
188#417—1071no—
189C2×#102—1071no—
190#408—1071no—
191H(5,𝔽2)—2144no—
192H(5,𝔽2)—2144no—
193C2×#41—1071no—
194C2×#41—1072no—
195C2×#41—1071no—
196C2×#41—2141no—
197C2×#41C2×D10711yescharacteristic, complemented
198C22×D4—1072no—
199C22×D4—2141no—
200C22×D4—2142no—
201C22×D4C2×D10711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2144no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C2×SD16C2×D10712yescomplemented
208C2×D8—1071no—
209C2×D8C2×D10712yescomplemented
210C2×M16C2×D10711yescharacteristic, complemented
211C22×C8—1071no—
212C22⋊Q8—1072no—
213C22⋊Q8—2141no—
214#121—1072no—
215#121—2141no—
216#121—2142no—
217C2×C4⋊C4—1071no—
218#103—1072no—
219#102—1072no—
220#102—1072no—
221#102—1072no—
222C22⋊C8—1072no—
223C22⋊C8—1072no—
224C24—21no—
225C24—2142no—
226#41—22no—
227#41—22no—
228#41—1074no—
229#41—2144no—
230#41—2148no—
231#41—2144no—
232#41—2142no—
233#41—2142no—
234#41—2142no—
235C2×Q8—1071no—
236C2×Q8—1071no—
237C2×Q8C22×D10711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—1072no—
242C2×D4—1071no—
243C2×D4—1071no—
244C2×D4—2142no—
245C2×D4—2144no—
246C2×D4—2148no—
247C2×D4—2142no—
248C2×D4—2142no—
249C2×D4—21416no—
250C2×D4—2141no—
251C2×D4—2144no—
252C2×D4C22⋊D10714yescomplemented
253C2×D4C22×D10711yescharacteristic
254C2×D4C22×D10712yes—
255C22×C4—21no—
256C22×C4—1072no—
257C22×C4—2141no—
258C22×C4—2142no—
259C22×C4—2141no—
260C22×C4C22×D10711yescharacteristic
261Q16—2142no—
262SD16—24no—
263SD16—2142no—
264SD16—2142no—
265D8—24no—
266D8—2142no—
267M16—22no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—2142no—
271C2×C8C22×D10712yes—
272C4⋊C4—1072no—
273C4⋊C4—2142no—
274C4⋊C4—2141no—
275C22⋊C4—2142no—
276C22⋊C4—2142no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—8562no—
283C23—8562no—
284C23—1072no—
285C23—2148no—
286C23—2141no—
287C23D4×D10711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—1072no—
291Q8—2142no—
292Q8—2141no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—8562no—
299D4—1072no—
300D4—2148no—
301D4—2148no—
302D4—2144no—
303D4—2144no—
304D4—2142no—
305D4—2141no—
306D4C2×C22⋊D10714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—8561no—
310C2×C4—1072no—
311C2×C4—2142no—
312C2×C4—2148no—
313C2×C4—2141no—
314C2×C4—2144no—
315C2×C4—2142no—
316C2×C4—2142no—
317C2×C4D4×D10711yescharacteristic
318C2×C4D4×D10712yes—
319C2×C4C2×C22⋊D10712yes—
320C2×C4C23×D10711yescharacteristic
321C8—22no—
322C8—2141no—
323C8—2141no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—8568no—
334C22—4281no—
335C22—4281no—
336C22—2142no—
337C22—2142no—
338C22—2142no—
339C22—2142no—
340C22#1681207712yes—
341C22C2×D4×D10711yescharacteristic
342C4—41no—
343C4—4282no—
344C4—2142no—
345C4—2142no—
346C4#1681207712yes—
347C4C2×D4×D10711yescharacteristic
348C4C22×C22⋊D10711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—4282no—
353C2—4282no—
354C2C2×#1681207711yescharacteristic
355C2#2029343111yescharacteristic
356C2#2029343711yescharacteristic
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#3012
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×#16812154C2semidirect
2. C2×#16812129C2semidirect
3. #20293205C2semidirect
4. #20292786C2semidirect
5. #20292776C2semidirect
6. #20292769C2semidirect
7. #20292759C2semidirect
8. #20292629C2semidirect
9. C2×#16812030C2semidirect
10. #20292595C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
22. C2×#13133808C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133805C22semidirect
25. C2×#13133803C22semidirect
26. C22×C107⋊C8C22semidirect
28. #16812085C22semidirect
31. #16812080C22semidirect
32. C2×C4⋊Dic107C22semidirect
33. #16812034C22semidirect
34. #16812031C22semidirect
35. #16812030C22semidirect
36. #16812030C22semidirect
37. (C2×D107)⋊C8C22semidirect
38. (C2×D107)⋊C8C22semidirect
39. C2×#41×C107C22semidirect
42. C2×SD16×C107C22semidirect
43. C2×D8×C107C22semidirect
44. C2×M16×C107C22semidirect
49. #13133818D4semidirect
89. C2×D4×C107D4semidirect
179. C107#3012semidirect
181. C2×AGL(1,ℤ/8)D107semidirect
197. C2×#41C2×D107semidirect
207. C2×SD16C2×D107semidirect
209. C2×D8C2×D107semidirect
210. C2×M16C2×D107semidirect
252. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
C2×#168121292
#202932052
#202927862
#202927762
#202927692
#202927592
#202926292
C2×#168120302
#202925952
C2×AGL(1,ℤ/8)×C1072
#3012107