Finite Group Databasebeta

← #28349517

Subgroups — #28349517

Order 13696.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293359C211yesmaximal, characteristic
2#20293355C211yesmaximal, characteristic, complemented
3#20293022C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292998C211yesmaximal, characteristic
6#20292852C211yesmaximal, characteristic, complemented
7#20292835C211yesmaximal, characteristic, complemented
8#20292833C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292826C211yesmaximal, characteristic
11#20292756C211yesmaximal, characteristic, complemented
12#20292725C211yesmaximal, characteristic, complemented
13#20292724C211yesmaximal, characteristic, complemented
14#20292719C211yesmaximal, characteristic, complemented
15#464×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133807C2211yescharacteristic
20C2×#13133806C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133805C2211yescharacteristic
23#16812109C2211yescharacteristic
24#16812107C2211yescharacteristic
25#16812107C2211yescharacteristic
26#16812105C2211yescharacteristic
27#16812101—21no—
28#16812099—21no—
29#16812099—21no—
30#16812098C2211yescharacteristic
31#16812095—21no—
32#16812093—21no—
33#16812091—21no—
34Q8×Dic107C2211yescharacteristic
35C4×Dic214—21no—
36C4×Dic214C2211yescharacteristic
37#16812074C2211yescharacteristic
38#16812074C2211yescharacteristic
39C4×D428C2211yescharacteristic
40D4×Dic107C2211yescharacteristic
41(C22⋊D107)⋊C4—21no—
42#16812042C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812020C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214D411yescharacteristic
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C22×Dic107—21no—
66C2×C4×D107—21no—
67#13133808—22no—
68#13133808—41no—
69#13133807—41no—
70#13133807—41no—
71#13133806—22no—
72#13133806—41no—
73#13133805—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107D411yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4D411yescharacteristic
88Dic107⋊C4D411yescharacteristic
89C22⋊Dic107—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107C2311yescharacteristic
101Q16×C107—41no—
102SD16×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—42no—
126C22⋊D107—81no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—81no—
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135D428—21no—
136D428—21no—
137C4×D107—81no—
138Dic214—22no—
139Dic214—21no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143C8×C107—21no—
144C8×C107—21no—
145C107⋊C8—21no—
146C107⋊C8#4112yes—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C22×D411yescharacteristic
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C4×C107—21no—
155C4×C107—21no—
156C4×C107—41no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C107H(5,𝔽2)11yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—21no—
164Dic107—41no—
165Dic107—41no—
166Dic107—41no—
167Dic107#13712yes—
168C2×C107—81no—
169C2×C107#55011yescharacteristic
170C2×C107C2×#13711yescharacteristic
171C2×C107#57711yescharacteristic
172D107—81no—
173#3400—1071nomaximal, Sylow
174C107#340011yescharacteristic, complemented, Sylow
175#559—1071no—
176#557—1071no—
177#497—1071no—
178#497—1071no—
179#492—1071no—
180#464—1071no—
181#464D10711yescharacteristic, complemented
182#462—1071no—
183#461—1071no—
184#461—1071no—
185#460—1071no—
186#446—1071no—
187#440—1071no—
188#440—1071no—
189C4×Q16—1071no—
190C2×Q16—1071no—
191C2×Q16—1071no—
192C2×Q16C2×D10711yescharacteristic
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16—1071no—
196C2×SD16—1071no—
197C2×SD16—1071no—
198C2×SD16C2×D10711yescharacteristic
199C4⋊Q8—1071no—
200C4⋊Q8—1071no—
201C4⋊Q8—1071no—
202#127—1071no—
203#125—2141no—
204#125—2141no—
205#125—2141no—
206#124—1071no—
207#124—2141no—
208#124C2×D10711yescharacteristic
209#123—2141no—
210#123—2141no—
211C4×Q8—1071no—
212C4×Q8—1071no—
213C4×Q8—1071no—
214C4×Q8—1071no—
215C4×Q8—2141no—
216C4×Q8C2×D10711yescharacteristic
217C4×D4—1071no—
218C4×D4—1071no—
219C4×D4—2141no—
220#107—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#102C2×D10711yescharacteristic
230#97—1071no—
231#97—1071no—
232C4×C8—1071no—
233C2×Q8—1071no—
234C2×Q8—1071no—
235C2×Q8—1071no—
236C2×Q8C22×D10711yescharacteristic
237C2×Q8C22×D10711yescharacteristic
238C2×D4—1071no—
239C2×D4—2141no—
240C2×D4—2141no—
241C2×D4C22×D10711yescharacteristic
242C22×C4—2141no—
243C22×C4—2141no—
244Q16—41no—
245Q16—4281no—
246Q16—2142no—
247SD16—41no—
248SD16—4281no—
249SD16—4281no—
250SD16—4281no—
251SD16—2142no—
252SD16—2142no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8C22×D10711yescharacteristic
256C2×C8C22×D10711yescharacteristic
257C4⋊C4—21no—
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⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4C22×D10711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C42—21no—
279C42—1071no—
280C42—1071no—
281C42—2141no—
282C42—2141no—
283C42—2141no—
284C42C22×D10711yescharacteristic
285C23—21no—
286C23—2141no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—2142no—
292Q8—2141no—
293Q8—2141no—
294Q8—2141no—
295Q8—2141no—
296D4—21no—
297D4—21no—
298D4—8561no—
299D4—4282no—
300D4—2141no—
301D4—2141no—
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×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1072no—
320C8—2141no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—4281no—
325C22—4281no—
326C22—4281no—
327C22C2×D4×D10711yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—1072no—
334C4—4281no—
335C4—4281no—
336C4—4281no—
337C4—2141no—
338C4—2141no—
339C4—2141no—
340C4#41×D10711yescharacteristic
341C4#1681215411yescharacteristic
342C2—81no—
343C2—8561no—
344C2#2029329511yescharacteristic
345C2#137×D10711yescharacteristic
346C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radical#16812105C22
hypercenter#464D107
hypocenterC107#3400
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
2. #20293355C2semidirect
3. #20293022C2semidirect
4. #20293020C2semidirect
6. #20292852C2semidirect
7. #20292835C2semidirect
8. #20292833C2semidirect
9. #20292830C2semidirect
11. #20292756C2semidirect
12. #20292725C2semidirect
13. #20292724C2semidirect
14. #20292719C2semidirect
15. #464×C107C2semidirect
174. C107#3400semidirect
181. #464D107semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202933552
#202930222
#202930202
#202929982
#202928522
#202928352
#202928332
#202928302
#202928262
#202927562
#202927252
#202927242
#202927192
#464×C1072
#3400107