Finite Group Databasebeta

← #28374861

Subgroups — #28374861

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#20295672C211yesmaximal, characteristic
2#20295383C211yesmaximal, characteristic, complemented
3#20295356C211yesmaximal, characteristic, complemented
4#20295350C211yesmaximal, characteristic, complemented
5#20295323C211yesmaximal, characteristic, complemented
6#20295265C211yesmaximal, characteristic, complemented
7#20295257C211yesmaximal, characteristic, complemented
8#20295248C211yesmaximal, characteristic, complemented
9#20295186C211yesmaximal, characteristic, complemented
10#20295176C211yesmaximal, characteristic, complemented
11C4×#13182016C211yesmaximal, characteristic, complemented
12D4×#11724784C211yesmaximal, characteristic, complemented
13#20295020C211yesmaximal, characteristic, complemented
14C2×#16812619C211yesmaximal, characteristic, complemented
15#552×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×D436C2211yescharacteristic
18C2×#13182019C2211yescharacteristic, complemented
19C2×#13182016C2211yescharacteristic
20C2×#13182016C2211yescharacteristic
21C22×#11724784C2211yescharacteristic
22C22×#11724784C2211yescharacteristic, complemented
23#16812687C2211yescharacteristic
24#16812684—21no—
25#16812678C2211yescharacteristic
26#16812673C2211yescharacteristic
27#16812672—21no—
28#16812667—21no—
29C4×D436C2211yescharacteristic
30C22⋊C4×D109—21no—
31#16812632C2211yescharacteristic, complemented
32#16812630C2211yescharacteristic, complemented
33#16812630C2211yescharacteristic, complemented
34C4⋊#11724784C2211yescharacteristic, complemented
35#16812623C2211yescharacteristic, complemented
36#16812622C2211yescharacteristic, complemented
37#16812622C2211yescharacteristic, complemented
38#16812622C2211yescharacteristic, complemented
39#16812619C2211yescharacteristic
40#16812619C2211yescharacteristic
41#16812619C2211yescharacteristic
42#16812619C2211yescharacteristic
43#16812619C2211yescharacteristic
44#16812619C2211yescharacteristic
45C22⋊#11724784C2211yescharacteristic
46C22⋊#11724784C2211yescharacteristic, complemented
47C4×#11724784C2211yescharacteristic, complemented
48#126×C109—21no—
49#125×C109—21no—
50#125×C109C2211yescharacteristic, complemented
51#123×C109—21no—
52C22⋊Q8×C109C2211yescharacteristic
53C22⋊Q8×C109C2211yescharacteristic, complemented
54C4×Q8×C109C2211yescharacteristic, complemented
55C4×D4×C109C2211yescharacteristic
56#117×C109—21no—
57#117×C109C2211yescharacteristic, complemented
58C2×C4⋊C4×C109C2211yescharacteristic
59C23×D109—21no—
60#13182029—22no—
61#13182029—41no—
62C2×Dic218C2311yescharacteristic
63C2×D436—22no—
64C2×D436—21no—
65C2×D436—41no—
66C2×D436C2311yescharacteristic
67C2×D436C2311yescharacteristic
68C2×C22⋊D109—21no—
69C2×C22⋊D109—21no—
70C2×C4×D109—21no—
71C2×C4×D109—21no—
72#13182019—41no—
73#13182016—22no—
74#13182016—41no—
75C2×#11724784—41no—
76C2×#11724784—41no—
77C2×#11724784C2311yescharacteristic
78C2×#11724784C2311yescharacteristic
79C2×#11724784C2311yescharacteristic
80C2×#11724784C2311yescharacteristic
81C4⋊Dic109C2311yescharacteristic
82#13182008—21no—
83C22⋊Dic109—21no—
84#13182006—21no—
85#13182006—21no—
86#13182006—21no—
87#13182006—21no—
88C2×Q8×C109C2311yescharacteristic
89C2×D4×C109D411yescharacteristic
90C22×C4×C109—21no—
91C22×C4×C109C2311yescharacteristic
92C22×C4×C109C2311yescharacteristic
93C4⋊C4×C109—21no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—21no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109D411yescharacteristic
100C4⋊C4×C109C2311yescharacteristic
101C4⋊C4×C109C2311yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C22⋊C4×C109—21no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C22⋊C4×C109D411yescharacteristic
107C22⋊C4×C109D411yescharacteristic
108C42×C109—21no—
109C42×C109—21no—
110C42×C109C2311yescharacteristic
111C23×C109C2×D411yescharacteristic
112C23×C109C2×D411yescharacteristic
113C22×D109—21no—
114C22×D109—21no—
115C22×D109—42no—
116C22×D109—42no—
117C22×D109—81no—
118Q8×C109—21no—
119Q8×C109—21no—
120D4×C109—22no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—21no—
126C2×C4×C109—21no—
127C2×C4×C109—21no—
128C2×C4×C109—21no—
129C2×C4×C109—42no—
130C2×C4×C109C2×D411yescharacteristic
131C2×C4×C109C2×D411yescharacteristic
132C2×C4×C109C2×D411yescharacteristic
133C2×C4×C109C2×D411yescharacteristic
134C2×C4×C109C2411yescharacteristic
135C22⋊D109—42no—
136C22⋊D109—81no—
137C2×Dic109—21no—
138C2×Dic109—21no—
139D436—21no—
140D436—21no—
141D436—21no—
142D436—42no—
143D436#4112yes—
144C4×D109—42no—
145C4×D109—81no—
146Dic218—21no—
147Dic218—21no—
148#11724784—21no—
149#11724784—21no—
150#11724784—21no—
151#11724784—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×C109—41no—
171C4×C109—41no—
172C4×C109—41no—
173C4×C109—41no—
174C4×C109C2×#4111yescharacteristic
175C4×C109H(5,𝔽2)11yescharacteristic
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#552D10911yescharacteristic, complemented
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×D4—1091no—
200#416—1091no—
201C2×#102—1091no—
202C2×#41—1091no—
203C22×D4—1091no—
204C2×Q16—1091no—
205C2×SD16—1091no—
206C2×SD16—1091no—
207C22×C8—1091no—
208C22×C8—1091no—
209#126—21no—
210#125—21no—
211#125C2×D10911yescharacteristic, complemented
212#124—1091no—
213#123—21no—
214#123—2181no—
215C22⋊Q8—1091no—
216C22⋊Q8C2×D10911yescharacteristic
217C22⋊Q8C2×D10911yescharacteristic, complemented
218#121—1091no—
219#121—2181no—
220#120—2181no—
221C4×Q8C2×D10911yescharacteristic, complemented
222C4×D4—1091no—
223C4×D4C2×D10911yescharacteristic
224#117—21no—
225#117C2×D10911yescharacteristic, complemented
226C2×C4⋊C4C2×D10911yescharacteristic
227C2×C22⋊C4—2181no—
228#107—1091no—
229#106—1091no—
230#106—1091no—
231C4⋊C8—1091no—
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⋊C8—1091no—
243C22⋊C8—1091no—
244C4×C8—1091no—
245C24—2181no—
246#41—4361no—
247#41—2182no—
248C2×Q8—1091no—
249C2×Q8C22×D10911yescharacteristic
250C2×D4—1091no—
251C2×D4—1091no—
252C2×D4—4361no—
253C2×D4—2182no—
254C2×D4—2181no—
255C2×D4—2181no—
256C2×D4—2181no—
257C2×D4C22⋊D10911yescharacteristic
258C22×C4—21no—
259C22×C4—2181no—
260C22×C4—2181no—
261C22×C4C22×D10911yescharacteristic
262C22×C4C22×D10911yescharacteristic
263Q16—4361no—
264SD16—4361no—
265SD16—2182no—
266C2×C8—1091no—
267C2×C8—1091no—
268C2×C8—1091no—
269C2×C8—1091no—
270C2×C8—4361no—
271C2×C8—4361no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—1091no—
279C4⋊C4—2181no—
280C4⋊C4C22⋊D10911yescharacteristic
281C4⋊C4C22×D10911yescharacteristic
282C4⋊C4C22×D10911yescharacteristic
283C4⋊C4C22×D10911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2181no—
288C22⋊C4—2181no—
289C22⋊C4—2181no—
290C22⋊C4—2181no—
291C22⋊C4—2181no—
292C22⋊C4C22⋊D10911yescharacteristic
293C22⋊C4C22⋊D10911yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D10911yescharacteristic
297C23—8721no—
298C23—4362no—
299C23—4362no—
300C23—2181no—
301C23—2181no—
302C23C2×C22⋊D10911yescharacteristic
303C23C2×C22⋊D10911yescharacteristic
304Q8—21no—
305Q8—21no—
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—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—8721no—
326C2×C4—4362no—
327C2×C4—2181no—
328C2×C4—2181no—
329C2×C4C2×C22⋊D10911yescharacteristic
330C2×C4C2×C22⋊D10911yescharacteristic
331C2×C4C2×C22⋊D10911yescharacteristic
332C2×C4C2×C22⋊D10911yescharacteristic
333C2×C4C23×D10911yescharacteristic
334C8—2181no—
335C8—2181no—
336C8—2181no—
337C8—2181no—
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#1681271912yes—
351C22C22×C22⋊D10911yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4361no—
361C4—4361no—
362C4#41×D10911yescharacteristic
363C4#1681274511yescharacteristic
364C2—22no—
365C2—41no—
366C2—8721no—
367C2—4362no—
368C2#2029566411yescharacteristic
369C2C2×#1681271911yescharacteristic
370C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#552×C109C2
socleC22×C109C22×D4
prime radical#20295672C2
hypercenter#552D109
hypocenterC109#3296
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20295383C2semidirect
3. #20295356C2semidirect
4. #20295350C2semidirect
5. #20295323C2semidirect
6. #20295265C2semidirect
7. #20295257C2semidirect
8. #20295248C2semidirect
9. #20295186C2semidirect
10. #20295176C2semidirect
11. C4×#13182016C2semidirect
12. D4×#11724784C2semidirect
13. #20295020C2semidirect
14. C2×#16812619C2semidirect
15. #552×C109C2semidirect
18. C2×#13182019C22semidirect
22. C22×#11724784C22semidirect
31. #16812632C22semidirect
32. #16812630C22semidirect
33. #16812630C22semidirect
34. C4⋊#11724784C22semidirect
35. #16812623C22semidirect
36. #16812622C22semidirect
37. #16812622C22semidirect
38. #16812622C22semidirect
46. C22⋊#11724784C22semidirect
47. C4×#11724784C22semidirect
50. #125×C109C22semidirect
53. C22⋊Q8×C109C22semidirect
54. C4×Q8×C109C22semidirect
57. #117×C109C22semidirect
186. C109#3296semidirect
187. #552D109semidirect
211. #125C2×D109semidirect
217. C22⋊Q8C2×D109semidirect
221. C4×Q8C2×D109semidirect
225. #117C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202956722
#202953832
#202953562
#202953502
#202953232
#202952652
#202952572
#202952482
#202951862
#202951762
C4×#131820162
D4×#117247842
#202950202
C2×#168126192
#552×C1092
#3296109