Finite Group Databasebeta

← #28349163

Subgroups — #28349163

Order 13696.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293336C211yesmaximal, characteristic, complemented
2#20293297C211yesmaximal, characteristic, complemented
3#20292991C211yesmaximal, characteristic, complemented
4#20292990C211yesmaximal, characteristic, complemented
5#20292972C211yesmaximal, characteristic, complemented
6#20292928C211yesmaximal, characteristic, complemented
7#20292925C211yesmaximal, characteristic, complemented
8#20292915C211yesmaximal, characteristic, complemented
9#20292820C211yesmaximal, characteristic, complemented
10#20292820C211yesmaximal, characteristic, complemented
11#20292756C211yesmaximal, characteristic, complemented
12#20292739C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14D8×Dic107C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133803C2211yescharacteristic
18#16812108C2211yescharacteristic
19#16812105C2211yescharacteristic
20#16812105C2211yescharacteristic
21#16812101C2211yescharacteristic
22#16812100C2211yescharacteristic
23#16812088—21no—
24#16812086—21no—
25#16812086—21no—
26#16812081—21no—
27#16812081—21no—
28#16812080—21no—
29C4×Dic214C2211yescharacteristic
30C4×D428C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35C2×Dic107⋊C4—21no—
36#16812043C2211yescharacteristic
37#16812041C2211yescharacteristic
38#16812040C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812031C2211yescharacteristic
42#16812031C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812020C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×D8×C107C2211yescharacteristic
53C2×D8×C107C2211yescharacteristic
54#127×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×Dic214D411yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C22×Dic107—21no—
68C22×Dic107—21no—
69C22×Dic107—21no—
70C22×Dic107—21no—
71C2×C4×D107—21no—
72#13133803—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C4⋊Dic107—21no—
76C4⋊Dic107D411yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4D411yescharacteristic
85Dic107⋊C4D411yescharacteristic
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93C2×D4×C107—21no—
94C2×D4×C107—21no—
95C2×D4×C107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99D8×C107—22no—
100D8×C107—22no—
101C2×C8×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C42×C107C2311yescharacteristic
106C23×C107—21no—
107C23×C107—21no—
108C23×C107—21no—
109C22×D107—21no—
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—42no—
116D4×C107—81no—
117D4×C107#4112yescomplemented
118C2×C4×C107—21no—
119C2×C4×C107—81no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—42no—
124C22⋊D107—42no—
125C22⋊D107—81no—
126C22⋊D107—81no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—42no—
131C2×Dic107—42no—
132C2×Dic107—42no—
133C2×Dic107—81no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—21no—
139D428—21no—
140C4×D107—42no—
141Dic214—22no—
142Dic214—81no—
143C8×C107—21no—
144C8×C107—21no—
145C107⋊C8—21no—
146C107⋊C8—21no—
147C22×C107—22no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107—41no—
156C22×C107C22×D411yescharacteristic
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107—41no—
163C4×C107C2×#4111yescharacteristic
164C4×C1072−1+411yescharacteristic
165Dic107—21no—
166Dic107—21no—
167Dic107—21no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171Dic107AGL(1,ℤ/8)12yes—
172C2×C107—42no—
173C2×C107—81no—
174C2×C107—81no—
175C2×C107#54811yescharacteristic
176C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C107#57611yescharacteristic
178D107—81no—
179#3366—1071nomaximal, Sylow
180C107#336611yescharacteristic, complemented, Sylow
181#554—1071no—
182#550—1071no—
183#490—1071no—
184#490—1071no—
185#486—1071no—
186#478—1071no—
187#478—1071no—
188#476—1071no—
189#459—1071no—
190#459—1071no—
191#459D10711yescharacteristic, complemented
192#446—1071no—
193#442—1071no—
194#442—1071no—
195C4×D8—1071no—
196C2×#41—2141no—
197C2×D8—1071no—
198C2×D8C2×D10711yescharacteristic
199C2×D8C2×D10711yescharacteristic
200C4⋊Q8—1071no—
201#127—1071no—
202#127—1071no—
203#127C2×D10711yescharacteristic
204#125—1071no—
205#125—1071no—
206C22⋊Q8—2141no—
207C22⋊Q8—2141no—
208C22⋊Q8—2141no—
209#121—2141no—
210#121—2141no—
211#121—2141no—
212C4×Q8—1071no—
213C4×D4—1071no—
214C4×D4—1071no—
215C4×D4—1071no—
216C4×D4—1071no—
217C4×D4—1071no—
218C4×D4C2×D10711yescharacteristic
219C2×C4⋊C4—2141no—
220#107—1071no—
221#106—1071no—
222#106—1071no—
223C4⋊C8—1071no—
224C4⋊C8—1071no—
225C4⋊C8C2×D10711yescharacteristic
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#102C2×D10711yescharacteristic
235#102C2×D10711yescharacteristic
236#97—1071no—
237#97—1071no—
238C4×C8—1071no—
239#41—4284no—
240C2×Q8—1071no—
241C2×Q8—2141no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1071no—
245C2×D4—2141no—
246C2×D4—2141no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4C22×D10711yescharacteristic
250C2×D4C22×D10711yescharacteristic
251C2×D4C22×D10711yescharacteristic
252C22×C4—21no—
253C22×C4—2141no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4—2141no—
257C22×C4—2141no—
258D8—22no—
259D8—22no—
260D8—2142no—
261C2×C8—1071no—
262C2×C8—1071no—
263C2×C8C22×D10711yescharacteristic
264C2×C8C22×D10711yescharacteristic
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4—2141no—
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C42—1071no—
284C42—1071no—
285C42C22×D10711yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—2141no—
290Q8—8561no—
291Q8—2142no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—8561no—
300D4—8561no—
301D4—4282no—
302D4—4282no—
303D4—2141no—
304D4—2141no—
305D4#1313381712yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—8561no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—4282no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—4282no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4D4×D10711yescharacteristic
321C2×C4D4×D10711yescharacteristic
322C2×C4C23×D10711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2141no—
326C8—2141no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—4281no—
337C22—4281no—
338C22—4281no—
339C22C2×D4×D10711yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—1072no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4—2141no—
348C4—2141no—
349C4—2141no—
350C4C2×#1313381711yescharacteristic
351C4#1681215611yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—8561no—
356C2#2029328311yescharacteristic
357C2AGL(1,ℤ/8)×D10711yescharacteristic
358C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3366
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293336C2semidirect
2. #20293297C2semidirect
3. #20292991C2semidirect
4. #20292990C2semidirect
5. #20292972C2semidirect
6. #20292928C2semidirect
7. #20292925C2semidirect
8. #20292915C2semidirect
9. #20292820C2semidirect
10. #20292820C2semidirect
11. #20292756C2semidirect
12. #20292739C2semidirect
13. #20292737C2semidirect
14. D8×Dic107C2semidirect
15. #459×C107C2semidirect
117. D4×C107#41semidirect
180. C107#3366semidirect
191. #459D107semidirect
305. D4#13133817semidirect

Maximal subgroups

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

subgroupindex
#202933362
#202932972
#202929912
#202929902
#202929722
#202929282
#202929252
#202929152
#202928202
#202927562
#202927392
#202927372
D8×Dic1072
#459×C1072
#3366107