Finite Group Databasebeta

← #28348304

Subgroups — #28348304

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293324C211yesmaximal, characteristic, complemented
2#20293265C211yesmaximal, characteristic, complemented
3#20292983C211yesmaximal, characteristic, complemented
4#20292961C211yesmaximal, characteristic, complemented
5#20292952C211yesmaximal, characteristic, complemented
6#20292929C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292853C211yesmaximal, characteristic, complemented
9#20292844C211yesmaximal, characteristic, complemented
10#20292759C211yesmaximal, characteristic, complemented
11D8×Dic107C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic, complemented
19C2×C8×D107C2211yescharacteristic
20#16812102—21no—
21#16812100C2211yescharacteristic, complemented
22#16812098C2211yescharacteristic
23#16812092—21no—
24#16812090—21no—
25#16812085C2211yescharacteristic
26#16812083C2211yescharacteristic
27#16812081—21no—
28#16812080—21no—
29#16812080C2211yescharacteristic
30#16812077—21no—
31D4×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic, complemented
33C4×C22⋊D107C2211yescharacteristic
34(C22⋊D107)⋊C4—21no—
35#16812068C2211yescharacteristic, complemented
36C4⋊C4×D107C2211yescharacteristic
37C2×(C2×D107)⋊C4—21no—
38#16812043C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic, complemented
52C2×#41×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107D411yescharacteristic
65D4×D107—22no—
66D4×D107—41no—
67C22×Dic107—21no—
68C22×Dic107—21no—
69C2×C4×D107—21no—
70C2×C4×D107C2311yescharacteristic
71#13133803—41no—
72C2×C107⋊C8—41no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C8×D107—41no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107D411yescharacteristic
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96#41×C107—22no—
97#41×C107—41no—
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—21no—
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107C2311yescharacteristic
105SD16×C107—41no—
106D8×C107—22no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107—81no—
118C22×D107C2×D411yescharacteristic
119Q8×C107—21no—
120Q8×C107—21no—
121D4×C107—21no—
122D4×C107—21no—
123D4×C107—21no—
124D4×C107—42no—
125D4×C107—81no—
126D4×C107#4112yes—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—42no—
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2411yescharacteristic
134C22⋊D107—22no—
135C22⋊D107—42no—
136C22⋊D107—42no—
137C22⋊D107—81no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—81no—
142C2×Dic107—81no—
143C2×Dic107C2×D411yescharacteristic
144C2×Dic107C2×D411yescharacteristic
145C2×Dic107C2×D411yescharacteristic
146D428—21no—
147D428—21no—
148C4×D107—21no—
149C4×D107—21no—
150C4×D107—81no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—22no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—21no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107—41no—
163C22×C107—41no—
164C22×C107C22×D411yescharacteristic
165C2×D107—21no—
166C2×D107—21no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—84no—
171C2×D107#13512yes—
172C4×C107—21no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107C2×#4111yescharacteristic
176C4×C107H(5,𝔽2)11yescharacteristic
177Dic107—21no—
178Dic107—21no—
179Dic107—21no—
180Dic107—41no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—42no—
184C2×C107—41no—
185C2×C107—81no—
186C2×C107#54611yescharacteristic
187C2×C107C2×#13511yescharacteristic
188C2×C107#57611yescharacteristic
189D107—22no—
190D107—81no—
191#3297—1071nomaximal, Sylow
192C107#329711yescharacteristic, complemented, Sylow
193#553—1071no—
194#546—1071no—
195#488—1071no—
196#484—1071no—
197#482—1071no—
198#478—1071no—
199#466—1071no—
200#465—1071no—
201#463—1071no—
202#449D10711yescharacteristic, complemented
203#447—1071no—
204C4×D8—1071no—
205C8×D4—1071no—
206#416—1071no—
207C2×#102—1071no—
208C2×#41C2×D10711yescharacteristic
209C22×D4—1071no—
210C2×SD16C2×D10711yescharacteristic, complemented
211C2×D8—1071no—
212C2×D8C2×D10711yescharacteristic
213C22×C8—1071no—
214C22×C8—1071no—
215#126—2141no—
216#125—1071no—
217#124—1071no—
218#123—2141no—
219#123—2141no—
220C22⋊Q8—1071no—
221#121—1071no—
222#121—1071no—
223#121—2141no—
224#121—2141no—
225#121C2×D10711yescharacteristic, complemented
226#120—2141no—
227C4×D4—1071no—
228C4×D4—1071no—
229C4×D4—1071no—
230C4×D4—2141no—
231#117—1071no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234#107—1071no—
235#107—1071no—
236#106—1071no—
237C4⋊C8—1071no—
238#103—1071no—
239#103—1071no—
240#103C2×D10711yescharacteristic, complemented
241#102—1071no—
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102C2×D10711yescharacteristic
248C22⋊C8—1071no—
249C22⋊C8C2×D10711yescharacteristic, complemented
250C4×C8—1071no—
251C24—2141no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—4281no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2142no—
264C2×D4C22×D10711yescharacteristic
265C2×D4C22×D10711yescharacteristic
266C22×C4—21no—
267C22×C4—1071no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4—2141no—
271C22×C4C22×D10711yescharacteristic
272SD16—41no—
273D8—22no—
274D8—4281no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8—4281no—
278C2×C8—4281no—
279C2×C8C22×D10711yescharacteristic
280C2×C8C22×D10711yescharacteristic
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—1071no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4—2141no—
289C4⋊C4C22×D10711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1071no—
292C22⋊C4—1071no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C42—1071no—
302C42—2141no—
303C23—21no—
304C23—21no—
305C23—8561no—
306C23—1071no—
307C23—4282no—
308C23—4282no—
309C23—2141no—
310C23D4×D10711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—8561no—
319D4—4282no—
320D4—4282no—
321D4—2142no—
322D4—2141no—
323D4—2141no—
324D4#1313381712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—8561no—
331C2×C4—8561no—
332C2×C4—8561no—
333C2×C4—1071no—
334C2×C4—1071no—
335C2×C4—1071no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4D4×D10711yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2141no—
346C8—2141no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—8564no—
357C22—1072no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—2141no—
362C22—2141no—
363C22C2×D4×D10711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—4281no—
368C4—4281no—
369C4—4281no—
370C4—2141no—
371C4—2141no—
372C4—2141no—
373C4C2×#1313381711yescharacteristic
374C4#1681215411yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—8561no—
379C2—2142no—
380C2#2029326411yescharacteristic
381C2#135×D10711yescharacteristic
382C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20293324C2semidirect
2. #20293265C2semidirect
3. #20292983C2semidirect
4. #20292961C2semidirect
5. #20292952C2semidirect
6. #20292929C2semidirect
7. #20292861C2semidirect
8. #20292853C2semidirect
9. #20292844C2semidirect
10. #20292759C2semidirect
11. D8×Dic107C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. #20292623C2semidirect
14. #102×D107C2semidirect
15. #449×C107C2semidirect
18. C22×C107⋊C8C22semidirect
21. #16812100C22semidirect
32. D4×Dic107C22semidirect
35. #16812068C22semidirect
38. #16812043C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. Dic107⋊C8C22semidirect
42. #16812033C22semidirect
43. #16812033C22semidirect
48. #16812030C22semidirect
51. C8×Dic107C22semidirect
53. C2×SD16×C107C22semidirect
55. #121×C107C22semidirect
56. #103×C107C22semidirect
58. C22⋊C8×C107C22semidirect
192. C107#3297semidirect
202. #449D107semidirect
210. C2×SD16C2×D107semidirect
225. #121C2×D107semidirect
240. #103C2×D107semidirect
249. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933242
#202932652
#202929832
#202929612
#202929522
#202929292
#202928612
#202928532
#202928442
#202927592
D8×Dic1072
(C22⋊D107)⋊C82
#202926232
#102×D1072
#449×C1072
#3297107