Finite Group Databasebeta

← #28349051

Subgroups — #28349051

Order 13696.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293341C211yesmaximal, characteristic, complemented
2#20293277C211yesmaximal, characteristic, complemented
3#20293024C211yesmaximal, characteristic, complemented
4#20292949C211yesmaximal, characteristic, complemented
5#20292948C211yesmaximal, characteristic, complemented
6#20292944C211yesmaximal, characteristic, complemented
7#20292943C211yesmaximal, characteristic, complemented
8#20292915C211yesmaximal, characteristic, complemented
9#20292905C211yesmaximal, characteristic, complemented
10#20292877C211yesmaximal, characteristic, complemented
11D8×Dic107C211yesmaximal, characteristic, complemented
12#20292703C211yesmaximal, characteristic, complemented
13#20292669C211yesmaximal, characteristic, complemented
14#20292666C211yesmaximal, characteristic, complemented
15#469×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133801C2211yescharacteristic
18#16812108C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812096—21no—
21#16812095—21no—
22#16812091—21no—
23#16812090—21no—
24#16812086—21no—
25#16812081—21no—
26#16812080C2211yescharacteristic
27#16812080C2211yescharacteristic
28D4×Dic107C2211yescharacteristic
29D4×Dic107C2211yescharacteristic
30C4×C22⋊D107C2211yescharacteristic
31#16812068C2211yescharacteristic
32#16812067C2211yescharacteristic
33#16812065—21no—
34C2×C4⋊Dic107C2211yescharacteristic
35C4⋊C4×D107C2211yescharacteristic
36#16812043C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812040C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×D8×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54#121×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107D411yescharacteristic
65C22×Dic107—21no—
66C22×Dic107—21no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107C2311yescharacteristic
70#13133801—22no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107D411yescharacteristic
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4D411yescharacteristic
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C2×D4×C107—21no—
94C2×D4×C107—21no—
95C2×D4×C107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98D8×C107—22no—
99M16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C23×C107—21no—
107C23×C107—21no—
108C23×C107C2×D411yescharacteristic
109C22×D107C2×D411yescharacteristic
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—21no—
113D4×C107—21no—
114D4×C107—42no—
115D4×C107—81no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C22⋊D107—22no—
123C22⋊D107—42no—
124C22⋊D107—81no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—42no—
130C2×Dic107—42no—
131C2×Dic107—42no—
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—42no—
138Dic214—42no—
139Dic214—81no—
140C8×C107—21no—
141C8×C107#4112yescomplemented
142C107⋊C8—21no—
143C107⋊C8—21no—
144C22×C107—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107C22×D411yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107C2×#4111yescharacteristic
161C4×C1072−1+411yescharacteristic
162Dic107—21no—
163Dic107—21no—
164Dic107—41no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168Dic107AGL(1,ℤ/8)12yes—
169C2×C107—41no—
170C2×C107—81no—
171C2×C107—81no—
172C2×C107#54711yescharacteristic
173C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C107#57711yescharacteristic
175D107—41no—
176#3354—1071nomaximal, Sylow
177C107#335411yescharacteristic, complemented, Sylow
178#555—1071no—
179#547—1071no—
180#498—1071no—
181#482—1071no—
182#482—1071no—
183#481—1071no—
184#481—1071no—
185#476—1071no—
186#474—1071no—
187#469—1071no—
188#469D10711yescharacteristic, complemented
189C4×D8—1071no—
190#436—1071no—
191#428—1071no—
192#428—1071no—
193C2×#41—2141no—
194C2×D8C2×D10711yescharacteristic
195C2×M16—1071no—
196C2×M16C2×D10711yescharacteristic
197C4⋊Q8—1071no—
198C4⋊Q8—1071no—
199#124—2141no—
200#124—2141no—
201#123—2141no—
202#123—2141no—
203C22⋊Q8—2141no—
204#121—1071no—
205#121—1071no—
206#121—2141no—
207#121C2×D10711yescharacteristic
208#121C2×D10711yescharacteristic
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—1071no—
212#117—1071no—
213#117—1071no—
214#117—2141no—
215C2×C4⋊C4—1071no—
216C2×C4⋊C4—1071no—
217#107—1071no—
218#107—1071no—
219#107—1071no—
220#106—1071no—
221#106—1071no—
222#106—1071no—
223#106—1071no—
224#106—1071no—
225#106C2×D10711yescharacteristic
226C4⋊C8—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102C2×D10711yescharacteristic
232#102C2×D10711yescharacteristic
233C22⋊C8—1071no—
234C22⋊C8—1071no—
235C4×C8—1071no—
236#41—4284no—
237C2×Q8—2141no—
238C2×Q8—2141no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—1071no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4C22×D10711yescharacteristic
245C2×D4C22×D10711yescharacteristic
246C22×C4—1071no—
247C22×C4—2141no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252D8—22no—
253M16—22no—
254M16—2142no—
255C2×C8—1071no—
256C2×C8—1071no—
257C2×C8C22×D10711yescharacteristic
258C2×C8C22×D10711yescharacteristic
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—1071no—
263C4⋊C4—1071no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1071no—
273C22⋊C4—1071no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C42—1071no—
281C42—2141no—
282C42—2141no—
283C23—21no—
284C23—21no—
285C23—1071no—
286C23D4×D10711yescharacteristic
287Q8—8561no—
288Q8—4282no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—8561no—
296D4—4282no—
297D4—2142no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1071no—
303C2×C4—1071no—
304C2×C4—1071no—
305C2×C4—4282no—
306C2×C4—4282no—
307C2×C4—4282no—
308C2×C4—4282no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—2141no—
319C8—2141no—
320C8#1313381712yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—2141no—
331C22—2141no—
332C22—2141no—
333C22C2×D4×D10711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—1072no—
338C4—4281no—
339C4—4281no—
340C4—4281no—
341C4—4281no—
342C4—2141no—
343C4—2141no—
344C4C2×#1313381711yescharacteristic
345C4#1681215611yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—4281no—
350C2#2029327711yescharacteristic
351C2AGL(1,ℤ/8)×D10711yescharacteristic
352C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107#3354
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #20293341C2semidirect
2. #20293277C2semidirect
3. #20293024C2semidirect
4. #20292949C2semidirect
5. #20292948C2semidirect
6. #20292944C2semidirect
7. #20292943C2semidirect
8. #20292915C2semidirect
9. #20292905C2semidirect
10. #20292877C2semidirect
11. D8×Dic107C2semidirect
12. #20292703C2semidirect
13. #20292669C2semidirect
14. #20292666C2semidirect
15. #469×C107C2semidirect
141. C8×C107#41semidirect
177. C107#3354semidirect
188. #469D107semidirect
320. C8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202933412
#202932772
#202930242
#202929492
#202929482
#202929442
#202929432
#202929152
#202929052
#202928772
D8×Dic1072
#202927032
#202926692
#202926662
#469×C1072
#3354107