Finite Group Databasebeta

← #28373216

Subgroups — #28373216

Order 13952.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295647C211yesmaximal, characteristic, complemented
2#20295367C211yesmaximal, characteristic, complemented
3#20295350C211yesmaximal, characteristic, complemented
4#20295316C211yesmaximal, characteristic, complemented
5#20295306C211yesmaximal, characteristic, complemented
6#20295241C211yesmaximal, characteristic, complemented
7#20295241C211yesmaximal, characteristic, complemented
8#20295177C211yesmaximal, characteristic, complemented
9#20295160C211yesmaximal, characteristic
10#20295136C211yesmaximal, characteristic, complemented
11#20295126C211yesmaximal, characteristic, complemented
12C4×#13182017C211yesmaximal, characteristic, complemented
13C4×#13182014C211yesmaximal, characteristic, complemented
14#20295087C211yesmaximal, characteristic, complemented
15#518×C109C211yesmaximal, characteristic, complemented
16C2×#13182017C2211yescharacteristic
17C2×#13182017C2211yescharacteristic
18C2×#13182014C2211yescharacteristic
19C2×#13182014C2211yescharacteristic
20#16812699C2211yescharacteristic, complemented
21#16812694—21no—
22#16812687C2211yescharacteristic, complemented
23#16812685—21no—
24#16812681—21no—
25#16812678C2211yescharacteristic
26#16812677—21no—
27#16812673C2211yescharacteristic
28C4×Dic218C2211yescharacteristic, complemented
29C4×D436C2211yescharacteristic, complemented
30#16812632C2211yescharacteristic, complemented
31#16812632C2211yescharacteristic, complemented
32#16812630C2211yescharacteristic, complemented
33#16812630C2211yescharacteristic, complemented
34C4⋊#11724784C2211yescharacteristic, complemented
35C4⋊#11724784C2211yescharacteristic, complemented
36#16812622C2211yescharacteristic, complemented
37#16812622C2211yescharacteristic, complemented
38#16812620C2211yescharacteristic
39#16812620C2211yescharacteristic
40#16812620C2211yescharacteristic
41#16812620C2211yescharacteristic
42#16812619C2211yescharacteristic, complemented
43#16812619C2211yescharacteristic, complemented
44C22⋊#11724784C2211yescharacteristic
45C22⋊#11724784C2211yescharacteristic
46#16812610C2211yescharacteristic, complemented
47C4×#11724784C2211yescharacteristic, complemented
48C22×D4×C109C2211yescharacteristic
49C4×D4×C109—21no—
50C4×D4×C109—21no—
51C4×D4×C109C2211yescharacteristic
52C4×D4×C109C2211yescharacteristic
53C4×D4×C109C2211yescharacteristic
54C4×D4×C109C2211yescharacteristic
55#117×C109C2211yescharacteristic
56#117×C109C2211yescharacteristic
57C2×C22⋊C4×C109—21no—
58C2×C22⋊C4×C109—21no—
59C2×Dic218—21no—
60C2×Dic218C2311yescharacteristic
61C2×D436C2311yescharacteristic
62C2×C22⋊D109—21no—
63C2×C4×D109—21no—
64#13182017—22no—
65#13182017—22no—
66#13182014—22no—
67#13182014—22no—
68C2×#11724784C2311yescharacteristic
69C2×#11724784C2311yescharacteristic
70C2×#11724784C2311yescharacteristic
71C2×#11724784C2311yescharacteristic
72C4⋊Dic109—21no—
73C4⋊Dic109C2311yescharacteristic
74C4⋊Dic109C2311yescharacteristic
75#13182008—21no—
76#13182008—21no—
77C22⋊Dic109—21no—
78#13182006—21no—
79#13182006—21no—
80#13182006—21no—
81#13182006—21no—
82C4×Dic109—21no—
83C24×C109—21no—
84C2×D4×C109—22no—
85C2×D4×C109—22no—
86C2×D4×C109—21no—
87C2×D4×C109C2311yescharacteristic
88C2×D4×C109C2311yescharacteristic
89C22×C4×C109—21no—
90C22×C4×C109—21no—
91C22×C4×C109—21no—
92C22×C4×C109—21no—
93C22×C4×C109C2311yescharacteristic
94C4⋊C4×C109D411yescharacteristic
95C4⋊C4×C109D411yescharacteristic
96C4⋊C4×C109C2311yescharacteristic
97C4⋊C4×C109C2311yescharacteristic
98C22⋊C4×C109—21no—
99C22⋊C4×C109—21no—
100C22⋊C4×C109—21no—
101C22⋊C4×C109—21no—
102C22⋊C4×C109—21no—
103C22⋊C4×C109D411yescharacteristic
104C22⋊C4×C109D411yescharacteristic
105C42×C109—21no—
106C42×C109C2311yescharacteristic
107C42×C109C2311yescharacteristic
108C23×C109—24no—
109C23×C109—21no—
110C23×C109—21no—
111C23×C109—81no—
112C23×C109—81no—
113C23×C109C2×D411yescharacteristic
114C22×D109—21no—
115D4×C109—24no—
116D4×C109—21no—
117D4×C109—21no—
118D4×C109#4112yes—
119D4×C109#4112yes—
120C2×C4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—81no—
125C2×C4×C109—81no—
126C2×C4×C109—81no—
127C2×C4×C109—81no—
128C2×C4×C109C2×D411yescharacteristic
129C2×C4×C109C2×D411yescharacteristic
130C2×C4×C109C2×D411yescharacteristic
131C2×C4×C109C2×D411yescharacteristic
132C2×C4×C109C2×D411yescharacteristic
133C2×C4×C109C2411yescharacteristic
134C22⋊D109—81no—
135C2×Dic109—21no—
136C2×Dic109—21no—
137C2×Dic109—21no—
138D436—21no—
139D436—21no—
140C4×D109—81no—
141Dic218—21no—
142Dic218—21no—
143Dic218—81no—
144#11724784—21no—
145#11724784—21no—
146#11724784—21no—
147#11724784—21no—
148C22×C109—21no—
149C22×C109—22no—
150C22×C109—22no—
151C22×C109—21no—
152C22×C109—21no—
153C22×C109—41no—
154C22×C109—41no—
155C22×C109—41no—
156C22×C109—41no—
157C22×C109—84no—
158C22×C109C22×D411yescharacteristic
159C2×D109—41no—
160C2×D109—41no—
161C2×D109—41no—
162C4×C109—21no—
163C4×C109—21no—
164C4×C109—21no—
165C4×C109—21no—
166C4×C109—21no—
167C4×C109—41no—
168C4×C109—41no—
169C4×C109C2×#4111yescharacteristic
170C4×C109C2×#4111yescharacteristic
171Dic109—41no—
172Dic109—41no—
173Dic109—41no—
174C2×C109—41no—
175C2×C109—42no—
176C2×C109—42no—
177C2×C109#52511yescharacteristic
178C2×C109#57611yescharacteristic
179C2×C109#57711yescharacteristic
180D109—81no—
181#3118—1091nomaximal, Sylow
182C109#311811yescharacteristic, complemented, Sylow
183#542—1091no—
184#518D10911yescharacteristic, complemented
185#485—1091no—
186#482—1091no—
187#476—1091no—
188#474—1091no—
189#463—1091no—
190#463—1091no—
191#450—1091no—
192#447—1091no—
193#442—1091no—
194#440—1091no—
195C4×SD16—1091no—
196C4×D8—1091no—
197#433—1091no—
198C22×D4C2×D10911yescharacteristic
199C2×SD16—1091no—
200C2×SD16—1091no—
201C2×D8—1091no—
202C2×D8—1091no—
203C4⋊Q8—1091no—
204#126—2181no—
205#124—1091no—
206#124—2181no—
207#123—2181no—
208C22⋊Q8—1091no—
209C22⋊Q8—2181no—
210#121—1091no—
211C4×Q8—1091no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1091no—
215C4×D4C2×D10911yescharacteristic
216C4×D4C2×D10911yescharacteristic
217C4×D4C2×D10911yescharacteristic
218C4×D4C2×D10911yescharacteristic
219#117C2×D10911yescharacteristic
220#117C2×D10911yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1091no—
224#107—1091no—
225#106—1091no—
226#106—1091no—
227C4⋊C8—1091no—
228C4⋊C8—1091no—
229#103—1091no—
230#103—1091no—
231#102—1091no—
232#102—1091no—
233#102—1091no—
234#102—1091no—
235#102—1091no—
236#102—1091no—
237C22⋊C8—1091no—
238C22⋊C8—1091no—
239#97—1091no—
240C4×C8—1091no—
241C24—21no—
242C2×Q8—1091no—
243C2×Q8—2181no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—1091no—
248C2×D4—2181no—
249C2×D4C22×D10911yescharacteristic
250C2×D4C22×D10911yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2181no—
256C22×C4C22×D10911yescharacteristic
257SD16—2182no—
258SD16—2182no—
259D8—2182no—
260D8—2182no—
261C2×C8—1091no—
262C2×C8—1091no—
263C2×C8—1091no—
264C2×C8—1091no—
265C4⋊C4—1091no—
266C4⋊C4—1091no—
267C4⋊C4—2181no—
268C4⋊C4—2181no—
269C4⋊C4—2181no—
270C4⋊C4C22⋊D10911yescharacteristic
271C4⋊C4C22⋊D10911yescharacteristic
272C4⋊C4C22×D10911yescharacteristic
273C4⋊C4C22×D10911yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2181no—
280C22⋊C4—2181no—
281C22⋊C4—2181no—
282C22⋊C4—2181no—
283C22⋊C4—2181no—
284C22⋊C4C22⋊D10911yescharacteristic
285C22⋊C4C22⋊D10911yescharacteristic
286C42—21no—
287C42—2181no—
288C42C22×D10911yescharacteristic
289C42C22×D10911yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—2181no—
296C23C2×C22⋊D10911yescharacteristic
297Q8—8721no—
298Q8—2181no—
299Q8—2181no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—8721no—
304D4—2181no—
305D4—2181no—
306D4#1318202912yes—
307D4#1318202912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—8721no—
317C2×C4—2181no—
318C2×C4—2181no—
319C2×C4—2181no—
320C2×C4C2×C22⋊D10911yescharacteristic
321C2×C4C2×C22⋊D10911yescharacteristic
322C2×C4C2×C22⋊D10911yescharacteristic
323C2×C4C2×C22⋊D10911yescharacteristic
324C2×C4C2×C22⋊D10911yescharacteristic
325C2×C4C23×D10911yescharacteristic
326C8—2181no—
327C8—2181no—
328C8—2181no—
329C8—2181no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—4361no—
341C22—4361no—
342C22—4361no—
343C22C22×C22⋊D10911yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—4361no—
352C4—4361no—
353C4—4361no—
354C4C2×#1318202911yescharacteristic
355C4C2×#1318202911yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—8721no—
360C2#2029553711yescharacteristic
361C2#2029582911yescharacteristic
362C2#2029583711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#518×C109C2
socleC22×C109C22×D4
prime radical#20295160C2
hypercenter#518D109
hypocenterC109#3118
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20295647C2semidirect
2. #20295367C2semidirect
3. #20295350C2semidirect
4. #20295316C2semidirect
5. #20295306C2semidirect
6. #20295241C2semidirect
7. #20295241C2semidirect
8. #20295177C2semidirect
10. #20295136C2semidirect
11. #20295126C2semidirect
12. C4×#13182017C2semidirect
13. C4×#13182014C2semidirect
14. #20295087C2semidirect
15. #518×C109C2semidirect
20. #16812699C22semidirect
22. #16812687C22semidirect
28. C4×Dic218C22semidirect
29. C4×D436C22semidirect
30. #16812632C22semidirect
31. #16812632C22semidirect
32. #16812630C22semidirect
33. #16812630C22semidirect
34. C4⋊#11724784C22semidirect
35. C4⋊#11724784C22semidirect
36. #16812622C22semidirect
37. #16812622C22semidirect
42. #16812619C22semidirect
43. #16812619C22semidirect
46. #16812610C22semidirect
47. C4×#11724784C22semidirect
182. C109#3118semidirect
184. #518D109semidirect

Maximal subgroups

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

subgroupindex
#202956472
#202953672
#202953502
#202953162
#202953062
#202952412
#202951772
#202951602
#202951362
#202951262
C4×#131820172
C4×#131820142
#202950872
#518×C1092
#3118109