Finite Group Databasebeta

← #28374869

Subgroups — #28374869

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295716C211yesmaximal, characteristic, complemented
2#20295672C211yesmaximal, characteristic
3#20295381C211yesmaximal, characteristic, complemented
4#20295357C211yesmaximal, characteristic, complemented
5#20295351C211yesmaximal, characteristic, complemented
6#20295327C211yesmaximal, characteristic, complemented
7#20295267C211yesmaximal, characteristic, complemented
8#20295255C211yesmaximal, characteristic, complemented
9#20295249C211yesmaximal, characteristic, complemented
10#20295190C211yesmaximal, characteristic, complemented
11#20295179C211yesmaximal, characteristic, complemented
12C4×#13182018C211yesmaximal, characteristic, complemented
13#20295023C211yesmaximal, characteristic, complemented
14C2×#16812621C211yesmaximal, characteristic, complemented
15C8×D4×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×D436C2211yescharacteristic
18C2×Dic436C2211yescharacteristic, complemented
19C2×#13182018C2211yescharacteristic
20C2×#13182018C2211yescharacteristic
21#16812692—21no—
22#16812691C2211yescharacteristic, complemented
23#16812689—21no—
24#16812687C2211yescharacteristic
25#16812684—21no—
26#16812680—21no—
27#16812678C2211yescharacteristic
28#16812678C2211yescharacteristic
29#16812678C2211yescharacteristic, complemented
30#16812673C2211yescharacteristic
31#16812672—21no—
32#16812667—21no—
33C4×Dic218C2211yescharacteristic, complemented
34C4×D436C2211yescharacteristic
35#16812658C2211yescharacteristic, complemented
36#16812655—21no—
37C2×C4⋊Dic109C2211yescharacteristic
38C22⋊C4×D109—21no—
39#16812633C2211yescharacteristic, complemented
40#16812631C2211yescharacteristic, complemented
41#16812631C2211yescharacteristic, complemented
42#16812624C2211yescharacteristic, complemented
43#16812624C2211yescharacteristic, complemented
44#16812624C2211yescharacteristic, complemented
45#16812624C2211yescharacteristic, complemented
46#16812621C2211yescharacteristic
47#16812621C2211yescharacteristic
48#16812621C2211yescharacteristic
49#16812621C2211yescharacteristic
50#16812621C2211yescharacteristic
51#16812621C2211yescharacteristic
52C22×C8×C109C2211yescharacteristic
53C22×C8×C109C2211yescharacteristic, complemented
54C4×D4×C109C2211yescharacteristic
55C4⋊C8×C109C2211yescharacteristic, complemented
56C22⋊C8×C109C2211yescharacteristic
57C22⋊C8×C109C2211yescharacteristic, complemented
58C4×C8×C109C2211yescharacteristic, complemented
59C23×D109—21no—
60#13182029—22no—
61#13182029—41no—
62C2×Dic218C2311yescharacteristic
63C2×Dic218C2311yescharacteristic
64C2×D436—22no—
65C2×D436—21no—
66C2×D436—41no—
67C2×D436C2311yescharacteristic
68C2×D436C2311yescharacteristic
69C2×C22⋊D109—21no—
70C2×C22⋊D109—21no—
71C22×Dic109—21no—
72C2×C4×D109—21no—
73C2×C4×D109—21no—
74Dic436—41no—
75#13182018—22no—
76#13182018—41no—
77C4⋊Dic109—21no—
78C4⋊Dic109C2311yescharacteristic
79C4⋊Dic109C2311yescharacteristic
80C4⋊Dic109C2311yescharacteristic
81C4⋊Dic109C2311yescharacteristic
82#13182008—21no—
83#13182008—21no—
84#13182008—21no—
85#13182008—21no—
86#13182008—21no—
87#13182008—21no—
88C22⋊Dic109—21no—
89C22⋊Dic109—21no—
90C22⋊Dic109—21no—
91C22⋊Dic109—21no—
92#13182006—21no—
93#13182006—21no—
94#13182006—21no—
95#13182006—21no—
96C4×Dic109—21no—
97C4×Dic109—21no—
98C2×D4×C109D411yescharacteristic
99C22×C4×C109C2311yescharacteristic
100C22×C4×C109C2311yescharacteristic
101C2×C8×C109—41no—
102C2×C8×C109—41no—
103C2×C8×C109C2311yescharacteristic
104C2×C8×C109C2311yescharacteristic
105C2×C8×C109C2311yescharacteristic
106C2×C8×C109C2311yescharacteristic
107C4⋊C4×C109D411yescharacteristic
108C22⋊C4×C109D411yescharacteristic
109C22⋊C4×C109D411yescharacteristic
110C42×C109C2311yescharacteristic
111C23×C109C2×D411yescharacteristic
112C23×C109C2×D411yescharacteristic
113C22×D109—21no—
114C22×D109—21no—
115C22×D109—42no—
116C22×D109—42no—
117C22×D109—81no—
118D4×C109—22no—
119C2×C4×C109—21no—
120C2×C4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109C2×D411yescharacteristic
124C2×C4×C109C2×D411yescharacteristic
125C2×C4×C109C2×D411yescharacteristic
126C2×C4×C109C2×D411yescharacteristic
127C2×C4×C109C2411yescharacteristic
128C22⋊D109—42no—
129C22⋊D109—81no—
130C2×Dic109—21no—
131C2×Dic109—21no—
132C2×Dic109—21no—
133C2×Dic109—21no—
134C2×Dic109—21no—
135C2×Dic109—21no—
136C2×Dic109—42no—
137D436—21no—
138D436—21no—
139D436—21no—
140D436—42no—
141D436#4112yes—
142C4×D109—42no—
143C4×D109—81no—
144Dic218—21no—
145Dic218—21no—
146Dic218—21no—
147Dic218—21no—
148C8×C109—21no—
149C8×C109—21no—
150C8×C109—21no—
151C8×C109—21no—
152C22×C109—21no—
153C22×C109—21no—
154C22×C109—21no—
155C22×C109—21no—
156C22×C109—21no—
157C22×C109#13512yes—
158C22×C109C22×D411yescharacteristic
159C2×D109—22no—
160C2×D109—41no—
161C2×D109—41no—
162C2×D109—41no—
163C2×D109—41no—
164C2×D109—41no—
165C2×D109—84no—
166C4×C109—21no—
167C4×C109—21no—
168C4×C109—21no—
169C4×C109—21no—
170C4×C109C2×#4111yescharacteristic
171C4×C109H(5,𝔽2)11yescharacteristic
172Dic109—41no—
173Dic109—41no—
174Dic109—41no—
175Dic109—41no—
176Dic109—41no—
177Dic109—41no—
178C2×C109—22no—
179C2×C109—41no—
180C2×C109#54611yescharacteristic
181C2×C109C2×#13511yescharacteristic
182C2×C109#57711yescharacteristic
183D109—42no—
184D109—81no—
185#3296—1091nomaximal, Sylow
186C109#329611yescharacteristic, complemented, Sylow
187#552—1091no—
188#546—1091no—
189#488—1091no—
190#483—1091no—
191#482—1091no—
192#478—1091no—
193#467—1091no—
194#465—1091no—
195#464—1091no—
196#452—1091no—
197#450—1091no—
198C4×SD16—1091no—
199C8×D4D10911yescharacteristic, complemented
200#416—1091no—
201C2×#102—1091no—
202C2×#41—1091no—
203C22×D4—1091no—
204C2×Q16—1091no—
205C2×SD16—1091no—
206C2×SD16—1091no—
207C22×C8C2×D10911yescharacteristic
208C22×C8C2×D10911yescharacteristic, complemented
209#126—2181no—
210#125—1091no—
211#125—2181no—
212#124—1091no—
213#123—2181no—
214#123—2181no—
215C22⋊Q8—1091no—
216C22⋊Q8—1091no—
217C22⋊Q8—1091no—
218#121—1091no—
219#121—2181no—
220#120—2181no—
221C4×Q8—1091no—
222C4×D4—1091no—
223C4×D4C2×D10911yescharacteristic
224#117—1091no—
225#117—2181no—
226C2×C4⋊C4—1091no—
227C2×C22⋊C4—2181no—
228#107—1091no—
229#106—1091no—
230#106—1091no—
231C4⋊C8C2×D10911yescharacteristic, complemented
232#103—1091no—
233#103—1091no—
234#103—1091no—
235#103—1091no—
236#102—1091no—
237#102—1091no—
238#102—1091no—
239#102—1091no—
240#102—1091no—
241#102—1091no—
242C22⋊C8C2×D10911yescharacteristic
243C22⋊C8C2×D10911yescharacteristic, complemented
244C4×C8C2×D10911yescharacteristic, complemented
245C24—2181no—
246#41—4361no—
247#41—2182no—
248C2×Q8—1091no—
249C2×Q8—1091no—
250C2×D4—1091no—
251C2×D4—1091no—
252C2×D4—4361no—
253C2×D4—2182no—
254C2×D4—2181no—
255C2×D4—2181no—
256C2×D4—2181no—
257C2×D4D43611yescharacteristic
258C22×C4—2181no—
259C22×C4—2181no—
260C22×C4—2181no—
261C22×C4C22×D10911yescharacteristic
262C22×C4C22×D10911yescharacteristic
263Q16—4361no—
264SD16—4361no—
265SD16—2182no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D10911yescharacteristic
269C2×C8C22×D10911yescharacteristic
270C2×C8C22×D10911yescharacteristic
271C2×C8C22×D10911yescharacteristic
272C4⋊C4—1091no—
273C4⋊C4—1091no—
274C4⋊C4—1091no—
275C4⋊C4—1091no—
276C4⋊C4—2181no—
277C4⋊C4—2181no—
278C4⋊C4—2181no—
279C4⋊C4—2181no—
280C4⋊C4—2181no—
281C4⋊C4—2181no—
282C4⋊C4—2181no—
283C4⋊C4D43611yescharacteristic
284C22⋊C4—2181no—
285C22⋊C4—2181no—
286C22⋊C4—2181no—
287C22⋊C4—2181no—
288C22⋊C4—2181no—
289C22⋊C4—2181no—
290C22⋊C4—2181no—
291C22⋊C4—2181no—
292C22⋊C4D43611yescharacteristic
293C22⋊C4D43611yescharacteristic
294C42—2181no—
295C42—2181no—
296C42C22×D10911yescharacteristic
297C23—8721no—
298C23—4362no—
299C23—4362no—
300C23—2181no—
301C23—2181no—
302C23C2×D43611yescharacteristic
303C23C2×D43611yescharacteristic
304Q8—2181no—
305Q8—2181no—
306Q8—2181no—
307Q8—2181no—
308D4—22no—
309D4—8721no—
310D4—1092no—
311D4—4362no—
312D4—4362no—
313D4—2181no—
314D4—2181no—
315D4—2181no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—8721no—
321C2×C4—4362no—
322C2×C4—4362no—
323C2×C4—2181no—
324C2×C4—2181no—
325C2×C4—2181no—
326C2×C4—2181no—
327C2×C4—2181no—
328C2×C4—2181no—
329C2×C4C2×D43611yescharacteristic
330C2×C4C2×D43611yescharacteristic
331C2×C4C2×D43611yescharacteristic
332C2×C4C2×D43611yescharacteristic
333C2×C4C23×D10911yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—8724no—
344C22—4361no—
345C22—4361no—
346C22—4361no—
347C22—4361no—
348C22—4361no—
349C22—2182no—
350C22#1681272112yes—
351C22C22×D43611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—4361no—
357C4—4361no—
358C4—4361no—
359C4—4361no—
360C4—4361no—
361C4—4361no—
362C4#41×D10911yescharacteristic
363C4#1681274411yescharacteristic
364C2—22no—
365C2—41no—
366C2—8721no—
367C2—4362no—
368C2#2029567111yescharacteristic
369C2C2×#1681272111yescharacteristic
370C2#2029583811yescharacteristic
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC8×D4×C109C2
socleC22×C109C22×D4
prime radical#20295672C2
hypercenterC8×D4D109
hypocenterC109#3296
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20295716C2semidirect
3. #20295381C2semidirect
4. #20295357C2semidirect
5. #20295351C2semidirect
6. #20295327C2semidirect
7. #20295267C2semidirect
8. #20295255C2semidirect
9. #20295249C2semidirect
10. #20295190C2semidirect
11. #20295179C2semidirect
12. C4×#13182018C2semidirect
13. #20295023C2semidirect
14. C2×#16812621C2semidirect
15. C8×D4×C109C2semidirect
18. C2×Dic436C22semidirect
22. #16812691C22semidirect
29. #16812678C22semidirect
33. C4×Dic218C22semidirect
35. #16812658C22semidirect
39. #16812633C22semidirect
40. #16812631C22semidirect
41. #16812631C22semidirect
42. #16812624C22semidirect
43. #16812624C22semidirect
44. #16812624C22semidirect
45. #16812624C22semidirect
53. C22×C8×C109C22semidirect
55. C4⋊C8×C109C22semidirect
57. C22⋊C8×C109C22semidirect
58. C4×C8×C109C22semidirect
186. C109#3296semidirect
199. C8×D4D109semidirect
208. C22×C8C2×D109semidirect
231. C4⋊C8C2×D109semidirect
243. C22⋊C8C2×D109semidirect
244. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957162
#202956722
#202953812
#202953572
#202953512
#202953272
#202952672
#202952552
#202952492
#202951902
#202951792
C4×#131820182
#202950232
C2×#168126212
C8×D4×C1092
#3296109