Finite Group Databasebeta

← #28346746

Subgroups — #28346746

Order 13696.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293230C211yesmaximal, characteristic
2#20293105C211yesmaximal, characteristic, complemented
3#20292952C211yesmaximal, characteristic, complemented
4#20292948C211yesmaximal, characteristic, complemented
5#20292936C211yesmaximal, characteristic, complemented
6#20292932C211yesmaximal, characteristic, complemented
7#20292825C211yesmaximal, characteristic, complemented
8#20292825C211yesmaximal, characteristic, complemented
9#20292766C211yesmaximal, characteristic, complemented
10#20292730C211yesmaximal, characteristic, complemented
11#20292728C211yesmaximal, characteristic, complemented
12SD16×Dic107C211yesmaximal, characteristic, complemented
13#20292712C211yesmaximal, characteristic, complemented
14#20292685C211yesmaximal, characteristic, complemented
15#448×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18#16812105C2211yescharacteristic
19#16812100C2211yescharacteristic
20#16812093—21no—
21#16812091—21no—
22#16812083C2211yescharacteristic
23#16812081—21no—
24#16812080—21no—
25Q8×Dic107—21no—
26Q8×Dic107—21no—
27Q8×Dic107C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29Q8×Dic107C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32#16812071C2211yescharacteristic
33#16812068—21no—
34#16812068—21no—
35#16812068C2211yescharacteristic
36#16812068C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22×Q8×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C22×Dic107—21no—
64C2×C4×D107—21no—
65#13133805—22no—
66#13133805—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107D411yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80C22⋊Dic107—21no—
81C22⋊Dic107—21no—
82C22⋊Dic107D411yescharacteristic
83C22⋊Dic107D411yescharacteristic
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86C4×Dic107—21no—
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93C2×Q8×C107—22no—
94C2×Q8×C107—22no—
95C2×Q8×C107—21no—
96C2×Q8×C107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98C2×D4×C107—21no—
99C2×D4×C107C2311yescharacteristic
100C22×C4×C107—21no—
101C22×C4×C107C2311yescharacteristic
102SD16×C107—22no—
103SD16×C107—22no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—24no—
112Q8×C107—21no—
113Q8×C107—21no—
114Q8×C107#4112yes—
115Q8×C107#4112yes—
116D4×C107—21no—
117D4×C107—21no—
118D4×C107—81no—
119C2×C4×C107—24no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—81no—
128C22⋊D107—81no—
129C22⋊D107—81no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—81no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—21no—
139D428—21no—
140C4×D107—81no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—21no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C4×C107—22no—
156C4×C107—22no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C107C2×#4111yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—21no—
164Dic107—21no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168C2×C107—41no—
169C2×C107—81no—
170C2×C107#52511yescharacteristic
171C2×C107#57711yescharacteristic
172C2×C107#57711yescharacteristic
173D107—81no—
174#3126—1071nomaximal, Sylow
175C107#312611yescharacteristic, complemented, Sylow
176#540—1071no—
177#519—1071no—
178#482—1071no—
179#482—1071no—
180#479—1071no—
181#479—1071no—
182#460—1071no—
183#460—1071no—
184#448—1071no—
185#448D10711yescharacteristic, complemented
186#440—1071no—
187#440—1071no—
188C4×SD16—1071no—
189C4×SD16—1071no—
190#433—1071no—
191C22×Q8C2×D10711yescharacteristic
192C2×SD16—1071no—
193C2×SD16—1071no—
194C2×SD16C2×D10711yescharacteristic
195C2×SD16C2×D10711yescharacteristic
196#127—1071no—
197#125—1071no—
198#123—2141no—
199#123—2141no—
200#121—1071no—
201#121—2141no—
202#121—2141no—
203#121C2×D10711yescharacteristic
204C4×Q8—1071no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×Q8—2141no—
209C4×Q8—2141no—
210C4×D4—1071no—
211C4×D4—1071no—
212#117—1071no—
213#117—1071no—
214#117—2141no—
215#117—2141no—
216#107—1071no—
217#107—1071no—
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222#103—1071no—
223#103—1071no—
224#103C2×D10711yescharacteristic
225#103C2×D10711yescharacteristic
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230C22⋊C8—1071no—
231C22⋊C8C2×D10711yescharacteristic
232#97—1071no—
233C4×C8—1071no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D10711yescharacteristic
238C2×Q8C22×D10711yescharacteristic
239C2×D4—21no—
240C2×D4—1071no—
241C2×D4—2141no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4C22×D10711yescharacteristic
245C22×C4—21no—
246C22×C4—2141no—
247C22×C4—2141no—
248C22×C4C22×D10711yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—2142no—
252SD16—2142no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8C22×D10711yescharacteristic
256C2×C8C22×D10711yescharacteristic
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1071no—
271C22⋊C4—1071no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C42—1071no—
277C42—1071no—
278C42—2141no—
279C42—2141no—
280C42—2141no—
281C42—2141no—
282C42—2141no—
283C23—21no—
284C23—2141no—
285C23D4×D10711yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1313381712yes—
290Q8#1313381712yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—8561no—
295D4—8561no—
296D4—8561no—
297D4—2141no—
298D4—2141no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—8561no—
306C2×C4—8561no—
307C2×C4—1071no—
308C2×C4—1071no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4D4×D10711yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2141no—
319C8—2141no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22C2×D4×D10711yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—4281no—
335C4—4281no—
336C4—4281no—
337C4—2141no—
338C4—2141no—
339C4—2141no—
340C4—2141no—
341C4C2×#1313381711yescharacteristic
342C4C2×#1313381711yescharacteristic
343C2—41no—
344C2—81no—
345C2—8561no—
346C2#2029313311yescharacteristic
347C2#2029343311yescharacteristic
348C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#20293230C2
hypercenter#448D107
hypocenterC107#3126
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #20293105C2semidirect
3. #20292952C2semidirect
4. #20292948C2semidirect
5. #20292936C2semidirect
6. #20292932C2semidirect
7. #20292825C2semidirect
8. #20292825C2semidirect
9. #20292766C2semidirect
10. #20292730C2semidirect
11. #20292728C2semidirect
12. SD16×Dic107C2semidirect
13. #20292712C2semidirect
14. #20292685C2semidirect
15. #448×C107C2semidirect
175. C107#3126semidirect
185. #448D107semidirect

Maximal subgroups

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

subgroupindex
#202932302
#202931052
#202929522
#202929482
#202929362
#202929322
#202928252
#202927662
#202927302
#202927282
SD16×Dic1072
#202927122
#202926852
#448×C1072
#3126107