Finite Group Databasebeta

← #28347926

Subgroups — #28347926

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#20293248C211yesmaximal, characteristic, complemented
2#20292917C211yesmaximal, characteristic, complemented
3#20292914C211yesmaximal, characteristic, complemented
4#20292912C211yesmaximal, characteristic, complemented
5#20292909C211yesmaximal, characteristic, complemented
6#20292883C211yesmaximal, characteristic, complemented
7#20292878C211yesmaximal, characteristic
8#20292858C211yesmaximal, characteristic, complemented
9#20292854C211yesmaximal, characteristic, complemented
10#20292849C211yesmaximal, characteristic, complemented
11#20292846C211yesmaximal, characteristic, complemented
12#20292842C211yesmaximal, characteristic, complemented
13#20292839C211yesmaximal, characteristic, complemented
14#20292654C211yesmaximal, characteristic, complemented
15#542×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133802C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22#16812109C2211yescharacteristic
23#16812104—21no—
24#16812097C2211yescharacteristic
25#16812095—21no—
26#16812091—21no—
27#16812088C2211yescharacteristic
28#16812087—21no—
29#16812083C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×D428C2211yescharacteristic
32#16812040C2211yescharacteristic
33#16812040C2211yescharacteristic
34#16812040C2211yescharacteristic
35#16812040C2211yescharacteristic
36C4⋊(C107⋊C8)C2211yescharacteristic
37C4⋊(C107⋊C8)C2211yescharacteristic
38C4⋊(C107⋊C8)C2211yescharacteristic
39C4⋊(C107⋊C8)C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812032C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48C4⋊Q8×C107C2211yescharacteristic
49#126×C107—21no—
50#124×C107—21no—
51#124×C107C2211yescharacteristic
52#123×C107—21no—
53C22⋊Q8×C107—21no—
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic
58#117×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C4×D107—21no—
64#13133808—41no—
65#13133806—41no—
66#13133805—41no—
67#13133803—41no—
68#13133802—41no—
69C2×C107⋊C8—41no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84C4×Dic107—21no—
85C2×Q8×C107—21no—
86C2×Q8×C107C2311yescharacteristic
87C2×D4×C107—21no—
88C2×D4×C107C2311yescharacteristic
89C22×C4×C107—21no—
90C22×C4×C107C2311yescharacteristic
91C4⋊C4×C107—21no—
92C4⋊C4×C107—21no—
93C4⋊C4×C107—21no—
94C4⋊C4×C107D411yescharacteristic
95C4⋊C4×C107D411yescharacteristic
96C4⋊C4×C107C2311yescharacteristic
97C4⋊C4×C107C2311yescharacteristic
98C22⋊C4×C107—21no—
99C22⋊C4×C107—21no—
100C22⋊C4×C107—21no—
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C22⋊C4×C107D411yescharacteristic
104C22⋊C4×C107D411yescharacteristic
105C42×C107—21no—
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—81no—
114D4×C107—21no—
115D4×C107—21no—
116D4×C107—81no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—81no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—81no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133D428—21no—
134D428—21no—
135C4×D107—81no—
136Dic214—21no—
137Dic214—21no—
138Dic214—81no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C22×C107—21no—
144C22×C107—21no—
145C22×C107—21no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107C22×D411yescharacteristic
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C4×C107—21no—
154C4×C107—21no—
155C4×C107—21no—
156C4×C107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107H(5,𝔽2)11yescharacteristic
162C4×C1072−1+411yescharacteristic
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—41no—
167C2×C107—81no—
168C2×C107#53711yescharacteristic
169C2×C107#57711yescharacteristic
170C2×C107#57711yescharacteristic
171D107—81no—
172#3263—1071nomaximal, Sylow
173C107#326311yescharacteristic, complemented, Sylow
174#542—1071no—
175#542D10711yescharacteristic, complemented
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#464—1071no—
186#463—1071no—
187#463—1071no—
188#424—1071no—
189C2×Q16—1071no—
190C2×SD16—1071no—
191C2×SD16—1071no—
192C2×D8—1071no—
193C2×M16—1071no—
194C22×C8—1071no—
195C4⋊Q8—1071no—
196C4⋊Q8C2×D10711yescharacteristic
197#126—21no—
198#126—2141no—
199#124—21no—
200#124—1071no—
201#124—2141no—
202#124C2×D10711yescharacteristic
203#123—21no—
204#123—2141no—
205C22⋊Q8—21no—
206C22⋊Q8—1071no—
207C22⋊Q8—2141no—
208C22⋊Q8C2×D10711yescharacteristic
209#121—1071no—
210#121C2×D10711yescharacteristic
211C4×Q8—1071no—
212C4×Q8C2×D10711yescharacteristic
213C4×D4—1071no—
214C4×D4C2×D10711yescharacteristic
215#117C2×D10711yescharacteristic
216#106—1071no—
217#106—1071no—
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223C4⋊C8—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102—1071no—
232C2×Q8—21no—
233C2×Q8—1071no—
234C2×Q8—2141no—
235C2×Q8C22×D10711yescharacteristic
236C2×D4—21no—
237C2×D4—1071no—
238C2×D4—2141no—
239C2×D4C22×D10711yescharacteristic
240C22×C4—21no—
241C22×C4—2141no—
242C22×C4C22×D10711yescharacteristic
243Q16—4281no—
244SD16—4281no—
245SD16—4281no—
246D8—4281no—
247M16—4281no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8—1071no—
251C2×C8—1071no—
252C2×C8—4281no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—1071no—
257C4⋊C4—1071no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4C22⋊D10711yescharacteristic
262C4⋊C4C22⋊D10711yescharacteristic
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4C22⋊D10711yescharacteristic
276C22⋊C4C22⋊D10711yescharacteristic
277C42—21no—
278C42—2141no—
279C42C22×D10711yescharacteristic
280C42C22×D10711yescharacteristic
281C23—21no—
282C23—2141no—
283C23C2×C22⋊D10711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—8561no—
288Q8—2141no—
289Q8—2141no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—8561no—
294D4—2141no—
295D4—2141no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—8561no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4C2×C22⋊D10711yescharacteristic
307C2×C4C2×C22⋊D10711yescharacteristic
308C2×C4C2×C22⋊D10711yescharacteristic
309C2×C4C2×C22⋊D10711yescharacteristic
310C2×C4C2×C22⋊D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—2141no—
313C8—2141no—
314C8—2141no—
315C8—2141no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—4281no—
323C22—4281no—
324C22—4281no—
325C22C22×C22⋊D10711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—4281no—
335C4—4281no—
336C4—4281no—
337C4#1681215511yescharacteristic
338C4#1681215611yescharacteristic
339C2—41no—
340C2—81no—
341C2—8561no—
342C2#2029320511yescharacteristic
343C2#2029343511yescharacteristic
344C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#542×C107C2
socleC22×C107C22×D4
prime radical#20292878C2
hypercenter#542D107
hypocenterC107#3263
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20293248C2semidirect
2. #20292917C2semidirect
3. #20292914C2semidirect
4. #20292912C2semidirect
5. #20292909C2semidirect
6. #20292883C2semidirect
8. #20292858C2semidirect
9. #20292854C2semidirect
10. #20292849C2semidirect
11. #20292846C2semidirect
12. #20292842C2semidirect
13. #20292839C2semidirect
14. #20292654C2semidirect
15. #542×C107C2semidirect
173. C107#3263semidirect
175. #542D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202929172
#202929142
#202929122
#202929092
#202928832
#202928782
#202928582
#202928542
#202928492
#202928462
#202928422
#202928392
#202926542
#542×C1072
#3263107