Finite Group Databasebeta

← #28348206

Subgroups — #28348206

Order 13696.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133808C211yesmaximal, characteristic, complemented
2C2×Q16×D107C211yesmaximal, characteristic, complemented
3Q8×C22⋊D107C211yesmaximal, characteristic
4#20293290C211yesmaximal, characteristic
5#20293005C211yesmaximal, characteristic, complemented
6#20292867C211yesmaximal, characteristic, complemented
7#20292866C211yesmaximal, characteristic, complemented
8#20292836C211yesmaximal, characteristic, complemented
9#20292835C211yesmaximal, characteristic, complemented
10#20292785C211yesmaximal, characteristic, complemented
11#20292784C211yesmaximal, characteristic, complemented
12#20292781C211yesmaximal, characteristic, complemented
13#20292719C211yesmaximal, characteristic, complemented
14(C22⋊D107)⋊C8C211yesmaximal, characteristic
15#451×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19C2×Dic428C2211yescharacteristic
20C2×Dic428C2211yescharacteristic
21C2×#13133808—24no—
22C2×#13133808C2211yescharacteristic
23C2×#13133808C2211yescharacteristic
24C2×#13133808C2211yescharacteristic
25C2×#13133808C2211yescharacteristic
26C2×#13133808C2211yescharacteristic
27Q16×D107—24no—
28C22×C107⋊C8C2211yescharacteristic
29C2×C8×D107C2211yescharacteristic
30#16812108—21no—
31#16812107—21no—
32#16812107C2211yescharacteristic
33#16812107C2211yescharacteristic
34#16812089C2211yescharacteristic
35#16812088—21no—
36#16812088C2211yescharacteristic
37#16812087—21no—
38#16812086—21no—
39#16812085—21no—
40#16812085C2211yescharacteristic
41Q8×Dic107C2211yescharacteristic
42#16812074C2211yescharacteristic
43C4×C22⋊D107—21no—
44C4×C22⋊D107C2211yescharacteristic
45(C22⋊D107)⋊C4—21no—
46#16812042C2211yescharacteristic
47Dic107⋊C8C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812034C2211yescharacteristic
50#16812033C2211yescharacteristic
51#16812032C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53C8×Dic107C2211yescharacteristic
54C22×Q8×C107C2211yescharacteristic
55C2×Q16×C107C2211yescharacteristic
56C2×Q16×C107C2211yescharacteristic
57C22⋊Q8×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59#103×C107C2211yescharacteristic
60C22⋊C8×C107C2211yescharacteristic
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—21no—
64C2×Dic214—21no—
65C2×Dic214—41no—
66C2×Dic214C2311yescharacteristic
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69Q8×D107—22no—
70Q8×D107—22no—
71Q8×D107—41no—
72Q8×D107—41no—
73C2×C22⋊D107D411yescharacteristic
74C22×Dic107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107C2311yescharacteristic
78Dic428—22no—
79Dic428—41no—
80#13133808—22no—
81#13133808—22no—
82#13133808—22no—
83#13133808—22no—
84#13133808D414yescomplemented
85C2×C107⋊C8—22no—
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C8×D107—22no—
89C4⋊Dic107—21no—
90C4⋊Dic107—21no—
91C4⋊Dic107C2311yescharacteristic
92Dic107⋊C4—21no—
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4—21no—
98Dic107⋊C4—21no—
99Dic107⋊C4D411yescharacteristic
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4D411yescharacteristic
105C4×Dic107—21no—
106C4×Dic107—21no—
107C4×Dic107C2311yescharacteristic
108C2×Q8×C107—21no—
109C2×Q8×C107—22no—
110C2×Q8×C107—41no—
111C2×Q8×C107C2311yescharacteristic
112C2×Q8×C107C2311yescharacteristic
113C2×Q8×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—22no—
117Q16×C107—41no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107—21no—
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107C2×D411yescharacteristic
124C22×D107C2×D411yescharacteristic
125Q8×C107—21no—
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—42no—
131Q8×C107C2×D412yes—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—42no—
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2411yescharacteristic
141C22⋊D107Q1614yescomplemented
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—42no—
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149C2×Dic107C2×D411yescharacteristic
150C4×D107—21no—
151C4×D107—21no—
152C4×D107—42no—
153C4×D107—42no—
154Dic214—21no—
155Dic214—21no—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—42no—
160Dic214—42no—
161Dic214—42no—
162Dic214C2×D412yes—
163C8×C107—21no—
164C8×C107—21no—
165C107⋊C8—21no—
166C107⋊C8C2×D412yes—
167C22×C107—21no—
168C22×C107—21no—
169C22×C107C2×Q1612yescomplemented
170C22×C107C22×D411yescharacteristic
171C2×D107—21no—
172C2×D107—21no—
173C2×D107C2×Q1612yescomplemented
174C4×C107—21no—
175C4×C107—22no—
176C4×C107—41no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107C22×D411yescharacteristic
180C4×C107H(5,𝔽2)11yescharacteristic
181Dic107—22no—
182Dic107—21no—
183Dic107—21no—
184Dic107—41no—
185Dic107—41no—
186Dic107—41no—
187Dic107C2×Q1612yescomplemented
188C2×C107—22no—
189C2×C107D4211yescharacteristic
190C2×C107C22×Q1611yescharacteristic
191C2×C107#57811yescharacteristic
192D107—22no—
193D4×Q16—1071nomaximal, Sylow
194C107D4×Q1611yescharacteristic, complemented, Sylow
195C22×Q16—1071no—
196C22×Q16—1071no—
197D4×Q8—1071no—
198D4×Q8—1071no—
199#494—1071no—
200#467—1071no—
201#467—1071no—
202#462—1071no—
203#462—1071no—
204#451—1071no—
205#451—1071no—
206#451—1071no—
207#451D10711yescharacteristic, complemented
208C4×Q16—1071no—
209C8×D4—1071no—
210C22×Q8—1071no—
211C22×Q8—1071no—
212C22×Q8—1071no—
213C22×Q8C2×D10711yescharacteristic
214C2×Q16—1071no—
215C2×Q16—1071no—
216C2×Q16—1071no—
217C2×Q16—1071no—
218C2×Q16—1071no—
219C2×Q16—1071no—
220C2×Q16—1071no—
221C2×Q16—2144no—
222C2×Q16—2144no—
223C2×Q16C2×D10711yescharacteristic
224C2×Q16C2×D10711yescharacteristic
225C22×C8—1071no—
226C22×C8—1071no—
227C4⋊Q8—1071no—
228C4⋊Q8—1071no—
229C4⋊Q8—2141no—
230C4⋊Q8—2141no—
231C22⋊Q8—1071no—
232C22⋊Q8—1071no—
233C22⋊Q8—1071no—
234C22⋊Q8—2141no—
235C22⋊Q8—2141no—
236C22⋊Q8—2141no—
237C22⋊Q8—2141no—
238C22⋊Q8C2×D10711yescharacteristic
239C4×Q8—1071no—
240C4×Q8—1071no—
241C4×D4—1071no—
242C4×D4—2141no—
243C4×D4—2141no—
244#107—1071no—
245C4⋊C8—1071no—
246#103—1071no—
247#103—1071no—
248#103—1071no—
249#103—1071no—
250#103C2×D10711yescharacteristic
251#103C2×D10711yescharacteristic
252C22⋊C8—1071no—
253C22⋊C8C2×D10711yescharacteristic
254C4×C8—1071no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—1071no—
259C2×Q8—1071no—
260C2×Q8—1071no—
261C2×Q8—4281no—
262C2×Q8—4281no—
263C2×Q8—4281no—
264C2×Q8—2141no—
265C2×Q8—2141no—
266C2×Q8—2142no—
267C2×Q8—2142no—
268C2×Q8—2142no—
269C2×Q8—2141no—
270C2×Q8C22×D10711yescharacteristic
271C2×Q8C22×D10711yescharacteristic
272C2×Q8C22×D10711yescharacteristic
273C2×D4—1071no—
274C22×C4—21no—
275C22×C4—1071no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4—2141no—
279C22×C4C22×D10711yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—1074no—
283Q16—4281no—
284Q16—2142no—
285Q16—2142no—
286Q16—2142no—
287Q16—2142no—
288Q16—2142no—
289C2×C8—1071no—
290C2×C8—1071no—
291C2×C8—2142no—
292C2×C8—2142no—
293C2×C8C22×D10711yescharacteristic
294C2×C8C22×D10711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—1071no—
297C4⋊C4—1071no—
298C4⋊C4—2141no—
299C4⋊C4—2141no—
300C4⋊C4—2141no—
301C4⋊C4—2141no—
302C4⋊C4—2141no—
303C4⋊C4—2141no—
304C4⋊C4—2141no—
305C4⋊C4—2141no—
306C4⋊C4—2141no—
307C4⋊C4C22×D10711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—1071no—
310C22⋊C4—1071no—
311C22⋊C4—2141no—
312C22⋊C4—2141no—
313C22⋊C4—2141no—
314C42—1071no—
315C42—2141no—
316C42—2141no—
317C23—1071no—
318C23D4×D10711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—1072no—
326Q8—4282no—
327Q8—4282no—
328Q8—4282no—
329Q8—2141no—
330Q8—2141no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334Q8D4×D10712yes—
335D4—1074no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1071no—
344C2×C4—1071no—
345C2×C4—1071no—
346C2×C4—4282no—
347C2×C4—4282no—
348C2×C4—4282no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4D4×D10711yescharacteristic
356C2×C4C23×D10711yescharacteristic
357C8—21no—
358C8—21no—
359C8—1072no—
360C8—2141no—
361C22—21no—
362C22—21no—
363C22—1072no—
364C22—2141no—
365C22—2141no—
366C22Q16×D10712yescomplemented
367C22C2×D4×D10711yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—1072no—
374C4—4281no—
375C4—4281no—
376C4—4281no—
377C4—2141no—
378C4—2141no—
379C4—2142no—
380C4C2×D4×D10711yescharacteristic
381C4#1681215411yescharacteristic
382C2—22no—
383C2—2142no—
384C2#2029325911yescharacteristic
385C2C2×Q16×D10711yescharacteristic
386C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#451×C107C2
socleC22×C107C22×D4
prime radicalC2×C22⋊D107D4
hypercenter#451D107
hypocenterC107D4×Q16
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#13133808C2semidirect
2. C2×Q16×D107C2semidirect
5. #20293005C2semidirect
6. #20292867C2semidirect
7. #20292866C2semidirect
8. #20292836C2semidirect
9. #20292835C2semidirect
10. #20292785C2semidirect
11. #20292784C2semidirect
12. #20292781C2semidirect
13. #20292719C2semidirect
15. #451×C107C2semidirect
84. #13133808D4semidirect
141. C22⋊D107Q16semidirect
169. C22×C107C2×Q16semidirect
173. C2×D107C2×Q16semidirect
187. Dic107C2×Q16semidirect
194. C107D4×Q16semidirect
207. #451D107semidirect
366. C22Q16×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×Q16×D1072
Q8×C22⋊D1072
#202932902
#202930052
#202928672
#202928662
#202928362
#202928352
#202927852
#202927842
#202927812
#202927192
(C22⋊D107)⋊C82
#451×C1072
D4×Q16107