Finite Group Databasebeta

← #28347929

Subgroups — #28347929

Order 13696.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293249C211yesmaximal, characteristic, complemented
2#20293246C211yesmaximal, characteristic, complemented
3#20292918C211yesmaximal, characteristic, complemented
4#20292915C211yesmaximal, characteristic, complemented
5#20292912C211yesmaximal, characteristic, complemented
6#20292910C211yesmaximal, characteristic, complemented
7#20292882C211yesmaximal, characteristic, complemented
8#20292879C211yesmaximal, characteristic
9#20292859C211yesmaximal, characteristic, complemented
10#20292853C211yesmaximal, characteristic, complemented
11#20292851C211yesmaximal, characteristic, complemented
12#20292844C211yesmaximal, characteristic, complemented
13#20292839C211yesmaximal, characteristic, complemented
14#20292652C211yesmaximal, characteristic, complemented
15#464×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×C8×D107C2211yescharacteristic
20#16812109C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812103—21no—
23#16812102—21no—
24#16812098C2211yescharacteristic
25#16812097—21no—
26#16812095—21no—
27#16812094—21no—
28#16812093—21no—
29#16812087—21no—
30#16812086—21no—
31#16812085C2211yescharacteristic
32#16812085C2211yescharacteristic
33#16812084C2211yescharacteristic
34#16812080C2211yescharacteristic
35Q8×Dic107C2211yescharacteristic
36C4×D428C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38#16812066C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812040C2211yescharacteristic
43C4⋊(C107⋊C8)C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812033C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812030C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C2×C4×D107—21no—
66C2×C4×D107C2311yescharacteristic
67#13133805—41no—
68#13133803—41no—
69#13133801—41no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C8×D107—41no—
73C4⋊Dic107—21no—
74C4⋊Dic107—21no—
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98Q16×C107—41no—
99SD16×C107—41no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107—21no—
110C22×D107C2×D411yescharacteristic
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115D4×C107—21no—
116D4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—81no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—81no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131D428—21no—
132D428—21no—
133D428—81no—
134C4×D107—21no—
135C4×D107—21no—
136C4×D107—81no—
137Dic214—81no—
138Dic214—81no—
139C8×C107—21no—
140C8×C107—21no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107C22×D411yescharacteristic
147C2×D107—21no—
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C4×C107—21no—
154C4×C107—21no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107—41no—
158C4×C107H(5,𝔽2)11yescharacteristic
159C4×C1072−1+411yescharacteristic
160Dic107—21no—
161Dic107—21no—
162Dic107—21no—
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—81no—
167C2×C107#53711yescharacteristic
168C2×C107#57711yescharacteristic
169C2×C107#57711yescharacteristic
170D107—41no—
171D107—81no—
172#3263—1071nomaximal, Sylow
173C107#326311yescharacteristic, complemented, Sylow
174#542—1071no—
175#542—1071no—
176#476—1071no—
177#476—1071no—
178#475—1071no—
179#475—1071no—
180#470—1071no—
181#469—1071no—
182#465—1071no—
183#465—1071no—
184#464—1071no—
185#464D10711yescharacteristic, complemented
186#463—1071no—
187#463—1071no—
188#424—1071no—
189C2×Q16C2×D10711yescharacteristic
190C2×SD16—1071no—
191C2×SD16C2×D10711yescharacteristic
192C2×D8—1071no—
193C2×M16—1071no—
194C22×C8—1071no—
195C4⋊Q8—1071no—
196C4⋊Q8—1071no—
197#126—2141no—
198#126—2141no—
199#124—1071no—
200#124—2141no—
201#124—2141no—
202#124C2×D10711yescharacteristic
203#123—2141no—
204#123—2141no—
205C22⋊Q8—1071no—
206C22⋊Q8—1071no—
207C22⋊Q8—2141no—
208C22⋊Q8—2141no—
209#121—1071no—
210#121—1071no—
211C4×Q8—1071no—
212C4×Q8C2×D10711yescharacteristic
213C4×D4—1071no—
214C4×D4—1071no—
215#117—1071no—
216#106—1071no—
217#106—1071no—
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223C4⋊C8C2×D10711yescharacteristic
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103C2×D10711yescharacteristic
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102C2×D10711yescharacteristic
232C2×Q8—2141no—
233C2×Q8—2141no—
234C2×Q8C22×D10711yescharacteristic
235C2×Q8C22×D10711yescharacteristic
236C2×D4—1071no—
237C2×D4—2141no—
238C2×D4—2141no—
239C2×D4C22×D10711yescharacteristic
240C22×C4—1071no—
241C22×C4—2141no—
242C22×C4—2141no—
243Q16—41no—
244SD16—41no—
245SD16—4281no—
246D8—4281no—
247M16—4281no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8—4281no—
251C2×C8C22×D10711yescharacteristic
252C2×C8C22×D10711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
256C4⋊C4—1071no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—1071no—
268C22⋊C4—1071no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C42—21no—
278C42—1071no—
279C42—2141no—
280C42C22×D10711yescharacteristic
281C23—21no—
282C23—1071no—
283C23—2141no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—8561no—
289Q8—8561no—
290D4—21no—
291D4—21no—
292D4—8561no—
293D4—8561no—
294D4—2141no—
295D4—2141no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—8561no—
300C2×C4—8561no—
301C2×C4—1071no—
302C2×C4—1071no—
303C2×C4—1071no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4D4×D10711yescharacteristic
310C2×C4D4×D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—21no—
313C8—21no—
314C8—2141no—
315C8—2141no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—4281no—
320C22—4281no—
321C22—4281no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22C2×D4×D10711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—4281no—
332C4—4281no—
333C4—4281no—
334C4—2141no—
335C4—2141no—
336C4—2141no—
337C4#1681215411yescharacteristic
338C4#1681215611yescharacteristic
339C2—81no—
340C2—8561no—
341C2—4281no—
342C2#2029320611yescharacteristic
343C2#2029343411yescharacteristic
344C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radical#20292879C2
hypercenter#464D107
hypocenterC107#3263
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #20293249C2semidirect
2. #20293246C2semidirect
3. #20292918C2semidirect
4. #20292915C2semidirect
5. #20292912C2semidirect
6. #20292910C2semidirect
7. #20292882C2semidirect
9. #20292859C2semidirect
10. #20292853C2semidirect
11. #20292851C2semidirect
12. #20292844C2semidirect
13. #20292839C2semidirect
14. #20292652C2semidirect
15. #464×C107C2semidirect
173. C107#3263semidirect
185. #464D107semidirect

Maximal subgroups

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

subgroupindex
#202932492
#202932462
#202929182
#202929152
#202929122
#202929102
#202928822
#202928792
#202928592
#202928532
#202928512
#202928442
#202928392
#202926522
#464×C1072
#3263107