Finite Group Databasebeta

← #28374875

Subgroups — #28374875

Order 13952.

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#20295723C211yesmaximal, characteristic, complemented
2#20295664C211yesmaximal, characteristic, complemented
3#20295382C211yesmaximal, characteristic, complemented
4#20295360C211yesmaximal, characteristic, complemented
5#20295351C211yesmaximal, characteristic, complemented
6#20295328C211yesmaximal, characteristic, complemented
7#20295260C211yesmaximal, characteristic, complemented
8#20295252C211yesmaximal, characteristic, complemented
9#20295243C211yesmaximal, characteristic, complemented
10#20295158C211yesmaximal, characteristic, complemented
11D8×Dic109C211yesmaximal, characteristic, complemented
12#20295089C211yesmaximal, characteristic, complemented
13#20295022C211yesmaximal, characteristic, complemented
14#102×D109C211yesmaximal, characteristic, complemented
15#449×C109C211yesmaximal, characteristic, complemented
16C2×D4×D109C2211yescharacteristic
17C2×#13182014C2211yescharacteristic
18C22×#11724784C2211yescharacteristic, complemented
19C2×C8×D109C2211yescharacteristic
20#16812692—21no—
21#16812690C2211yescharacteristic, complemented
22#16812688C2211yescharacteristic
23#16812682—21no—
24#16812680—21no—
25#16812675C2211yescharacteristic
26#16812673C2211yescharacteristic
27#16812671—21no—
28#16812670—21no—
29#16812670C2211yescharacteristic
30#16812667—21no—
31D4×Dic109C2211yescharacteristic
32D4×Dic109C2211yescharacteristic, complemented
33C4×C22⋊D109C2211yescharacteristic
34#16812659—21no—
35#16812658C2211yescharacteristic, complemented
36C4⋊C4×D109C2211yescharacteristic
37C2×#13182006—21no—
38#16812633C2211yescharacteristic, complemented
39#16812632C2211yescharacteristic, complemented
40#16812631C2211yescharacteristic, complemented
41#16812628C2211yescharacteristic, complemented
42#16812623C2211yescharacteristic, complemented
43#16812623C2211yescharacteristic, complemented
44#16812621C2211yescharacteristic
45#16812621C2211yescharacteristic
46#16812620C2211yescharacteristic
47#16812620C2211yescharacteristic
48#16812620C2211yescharacteristic, complemented
49#16812619C2211yescharacteristic
50#16812611C2211yescharacteristic
51C8×Dic109C2211yescharacteristic, complemented
52C2×#41×C109C2211yescharacteristic
53C2×SD16×C109C2211yescharacteristic, complemented
54C2×D8×C109C2211yescharacteristic
55#121×C109C2211yescharacteristic, complemented
56#103×C109C2211yescharacteristic, complemented
57#102×C109C2211yescharacteristic
58C22⋊C8×C109C2211yescharacteristic, complemented
59C23×D109—21no—
60C2×D436C2311yescharacteristic
61C2×C22⋊D109—21no—
62C2×C22⋊D109—21no—
63C2×C22⋊D109—21no—
64C2×C22⋊D109D411yescharacteristic
65D4×D109—22no—
66D4×D109—41no—
67C22×Dic109—21no—
68C22×Dic109—21no—
69C2×C4×D109—21no—
70C2×C4×D109C2311yescharacteristic
71#13182014—41no—
72C2×#11724784—41no—
73C2×#11724784C2311yescharacteristic
74C2×#11724784C2311yescharacteristic
75C8×D109—41no—
76C4⋊Dic109—21no—
77C4⋊Dic109C2311yescharacteristic
78C4⋊Dic109C2311yescharacteristic
79C4⋊Dic109C2311yescharacteristic
80#13182008—21no—
81#13182008—21no—
82#13182008—21no—
83#13182008D411yescharacteristic
84C22⋊Dic109—21no—
85C22⋊Dic109—21no—
86C22⋊Dic109—21no—
87C22⋊Dic109—21no—
88C22⋊Dic109D411yescharacteristic
89#13182006—21no—
90#13182006—21no—
91#13182006—21no—
92#13182006—21no—
93#13182006D411yescharacteristic
94C4×Dic109—21no—
95C4×Dic109C2311yescharacteristic
96#41×C109—22no—
97#41×C109—41no—
98C2×Q8×C109C2311yescharacteristic
99C2×D4×C109—21no—
100C2×D4×C109—21no—
101C2×D4×C109C2311yescharacteristic
102C2×D4×C109C2311yescharacteristic
103C22×C4×C109—21no—
104C22×C4×C109C2311yescharacteristic
105SD16×C109—41no—
106D8×C109—22no—
107C2×C8×C109C2311yescharacteristic
108C2×C8×C109C2311yescharacteristic
109C4⋊C4×C109C2311yescharacteristic
110C22⋊C4×C109—21no—
111C23×C109—21no—
112C23×C109—21no—
113C23×C109C2×D411yescharacteristic
114C22×D109—21no—
115C22×D109—42no—
116C22×D109—42no—
117C22×D109—81no—
118C22×D109C2×D411yescharacteristic
119Q8×C109—21no—
120Q8×C109—21no—
121D4×C109—21no—
122D4×C109—21no—
123D4×C109—21no—
124D4×C109—42no—
125D4×C109—81no—
126D4×C109#4112yes—
127C2×C4×C109—21no—
128C2×C4×C109—21no—
129C2×C4×C109—21no—
130C2×C4×C109—21no—
131C2×C4×C109—42no—
132C2×C4×C109C2×D411yescharacteristic
133C2×C4×C109C2411yescharacteristic
134C22⋊D109—22no—
135C22⋊D109—42no—
136C22⋊D109—42no—
137C22⋊D109—81no—
138C2×Dic109—21no—
139C2×Dic109—21no—
140C2×Dic109—21no—
141C2×Dic109—81no—
142C2×Dic109—81no—
143C2×Dic109C2×D411yescharacteristic
144C2×Dic109C2×D411yescharacteristic
145C2×Dic109C2×D411yescharacteristic
146D436—21no—
147D436—21no—
148C4×D109—21no—
149C4×D109—21no—
150C4×D109—81no—
151C8×C109—21no—
152C8×C109—21no—
153#11724784—21no—
154#11724784—21no—
155C22×C109—22no—
156C22×C109—21no—
157C22×C109—21no—
158C22×C109—21no—
159C22×C109—41no—
160C22×C109—41no—
161C22×C109—41no—
162C22×C109—41no—
163C22×C109—41no—
164C22×C109C22×D411yescharacteristic
165C2×D109—21no—
166C2×D109—21no—
167C2×D109—41no—
168C2×D109—41no—
169C2×D109—41no—
170C2×D109—84no—
171C2×D109#13512yes—
172C4×C109—21no—
173C4×C109—41no—
174C4×C109—41no—
175C4×C109C2×#4111yescharacteristic
176C4×C109H(5,𝔽2)11yescharacteristic
177Dic109—21no—
178Dic109—21no—
179Dic109—21no—
180Dic109—41no—
181Dic109—41no—
182Dic109—41no—
183C2×C109—42no—
184C2×C109—41no—
185C2×C109—81no—
186C2×C109#54611yescharacteristic
187C2×C109C2×#13511yescharacteristic
188C2×C109#57611yescharacteristic
189D109—22no—
190D109—81no—
191#3297—1091nomaximal, Sylow
192C109#329711yescharacteristic, complemented, Sylow
193#553—1091no—
194#546—1091no—
195#488—1091no—
196#484—1091no—
197#482—1091no—
198#478—1091no—
199#466—1091no—
200#465—1091no—
201#463—1091no—
202#449D10911yescharacteristic, complemented
203#447—1091no—
204C4×D8—1091no—
205C8×D4—1091no—
206#416—1091no—
207C2×#102—1091no—
208C2×#41C2×D10911yescharacteristic
209C22×D4—1091no—
210C2×SD16C2×D10911yescharacteristic, complemented
211C2×D8—1091no—
212C2×D8C2×D10911yescharacteristic
213C22×C8—1091no—
214C22×C8—1091no—
215#126—2181no—
216#125—1091no—
217#124—1091no—
218#123—2181no—
219#123—2181no—
220C22⋊Q8—1091no—
221#121—1091no—
222#121—1091no—
223#121—2181no—
224#121—2181no—
225#121C2×D10911yescharacteristic, complemented
226#120—2181no—
227C4×D4—1091no—
228C4×D4—1091no—
229C4×D4—1091no—
230C4×D4—2181no—
231#117—1091no—
232C2×C4⋊C4—1091no—
233C2×C22⋊C4—2181no—
234#107—1091no—
235#107—1091no—
236#106—1091no—
237C4⋊C8—1091no—
238#103—1091no—
239#103—1091no—
240#103C2×D10911yescharacteristic, complemented
241#102—1091no—
242#102—1091no—
243#102—1091no—
244#102—1091no—
245#102—1091no—
246#102—1091no—
247#102C2×D10911yescharacteristic
248C22⋊C8—1091no—
249C22⋊C8C2×D10911yescharacteristic, complemented
250C4×C8—1091no—
251C24—2181no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10911yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1091no—
258C2×D4—1091no—
259C2×D4—4361no—
260C2×D4—2181no—
261C2×D4—2181no—
262C2×D4—2181no—
263C2×D4—2182no—
264C2×D4C22×D10911yescharacteristic
265C2×D4C22×D10911yescharacteristic
266C22×C4—21no—
267C22×C4—1091no—
268C22×C4—2181no—
269C22×C4—2181no—
270C22×C4—2181no—
271C22×C4C22×D10911yescharacteristic
272SD16—41no—
273D8—22no—
274D8—4361no—
275C2×C8—1091no—
276C2×C8—1091no—
277C2×C8—4361no—
278C2×C8—4361no—
279C2×C8C22×D10911yescharacteristic
280C2×C8C22×D10911yescharacteristic
281C4⋊C4—1091no—
282C4⋊C4—1091no—
283C4⋊C4—1091no—
284C4⋊C4—1091no—
285C4⋊C4—2181no—
286C4⋊C4—2181no—
287C4⋊C4—2181no—
288C4⋊C4—2181no—
289C4⋊C4C22×D10911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1091no—
292C22⋊C4—1091no—
293C22⋊C4—2181no—
294C22⋊C4—2181no—
295C22⋊C4—2181no—
296C22⋊C4—2181no—
297C22⋊C4—2181no—
298C22⋊C4—2181no—
299C22⋊C4—2181no—
300C22⋊C4—2181no—
301C42—1091no—
302C42—2181no—
303C23—21no—
304C23—21no—
305C23—8721no—
306C23—1091no—
307C23—4362no—
308C23—4362no—
309C23—2181no—
310C23D4×D10911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—8721no—
319D4—4362no—
320D4—4362no—
321D4—2182no—
322D4—2181no—
323D4—2181no—
324D4#1318202812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—8721no—
331C2×C4—8721no—
332C2×C4—8721no—
333C2×C4—1091no—
334C2×C4—1091no—
335C2×C4—1091no—
336C2×C4—2181no—
337C2×C4—2181no—
338C2×C4—2181no—
339C2×C4—2181no—
340C2×C4—2181no—
341C2×C4D4×D10911yescharacteristic
342C2×C4C23×D10911yescharacteristic
343C8—21no—
344C8—21no—
345C8—2181no—
346C8—2181no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—8724no—
357C22—1092no—
358C22—4361no—
359C22—4361no—
360C22—4361no—
361C22—2181no—
362C22—2181no—
363C22C2×D4×D10911yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—4361no—
368C4—4361no—
369C4—4361no—
370C4—2181no—
371C4—2181no—
372C4—2181no—
373C4C2×#1318202811yescharacteristic
374C4#1681274411yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—8721no—
379C2—2182no—
380C2#2029566311yescharacteristic
381C2#135×D10911yescharacteristic
382C2#2029583011yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#449×C109C2
socleC22×C109C22×D4
hypercenter#449D109
hypocenterC109#3297
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20295723C2semidirect
2. #20295664C2semidirect
3. #20295382C2semidirect
4. #20295360C2semidirect
5. #20295351C2semidirect
6. #20295328C2semidirect
7. #20295260C2semidirect
8. #20295252C2semidirect
9. #20295243C2semidirect
10. #20295158C2semidirect
11. D8×Dic109C2semidirect
12. #20295089C2semidirect
13. #20295022C2semidirect
14. #102×D109C2semidirect
15. #449×C109C2semidirect
18. C22×#11724784C22semidirect
21. #16812690C22semidirect
32. D4×Dic109C22semidirect
35. #16812658C22semidirect
38. #16812633C22semidirect
39. #16812632C22semidirect
40. #16812631C22semidirect
41. #16812628C22semidirect
42. #16812623C22semidirect
43. #16812623C22semidirect
48. #16812620C22semidirect
51. C8×Dic109C22semidirect
53. C2×SD16×C109C22semidirect
55. #121×C109C22semidirect
56. #103×C109C22semidirect
58. C22⋊C8×C109C22semidirect
192. C109#3297semidirect
202. #449D109semidirect
210. C2×SD16C2×D109semidirect
225. #121C2×D109semidirect
240. #103C2×D109semidirect
249. C22⋊C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957232
#202956642
#202953822
#202953602
#202953512
#202953282
#202952602
#202952522
#202952432
#202951582
D8×Dic1092
#202950892
#202950222
#102×D1092
#449×C1092
#3297109