Finite Group Databasebeta

← #28375762

Subgroups — #28375762

Order 13952.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295752C211yesmaximal, characteristic, complemented
2#20295697C211yesmaximal, characteristic
3#20295391C211yesmaximal, characteristic, complemented
4#20295389C211yesmaximal, characteristic, complemented
5#20295371C211yesmaximal, characteristic, complemented
6#20295329C211yesmaximal, characteristic, complemented
7#20295326C211yesmaximal, characteristic, complemented
8#20295305C211yesmaximal, characteristic, complemented
9#20295226C211yesmaximal, characteristic, complemented
10#20295224C211yesmaximal, characteristic, complemented
11#20295224C211yesmaximal, characteristic, complemented
12#20295154C211yesmaximal, characteristic, complemented
13#20295129C211yesmaximal, characteristic, complemented
14#20295111C211yesmaximal, characteristic, complemented
15#440×C109C211yesmaximal, characteristic, complemented
16C2×#13182029—21no—
17C2×#13182016C2211yescharacteristic
18C2×#13182016C2211yescharacteristic
19#16812698—21no—
20#16812698C2211yescharacteristic
21#16812696C2211yescharacteristic
22#16812695C2211yescharacteristic
23#16812695C2211yescharacteristic
24#16812691—21no—
25#16812691C2211yescharacteristic
26#16812690—21no—
27#16812690C2211yescharacteristic
28#16812673—21no—
29#16812672—21no—
30#16812672—21no—
31Q8×Dic109C2211yescharacteristic
32Q8×Dic109C2211yescharacteristic
33C4×Dic218—21no—
34C4×Dic218C2211yescharacteristic
35#16812661C2211yescharacteristic
36#16812661C2211yescharacteristic
37#16812632C2211yescharacteristic
38#16812630C2211yescharacteristic
39C4⋊#11724784C2211yescharacteristic
40#16812628C2211yescharacteristic
41#16812628C2211yescharacteristic
42#16812621C2211yescharacteristic
43#16812621C2211yescharacteristic
44#16812621C2211yescharacteristic
45#16812621C2211yescharacteristic
46#16812620C2211yescharacteristic
47#16812620C2211yescharacteristic
48#16812619C2211yescharacteristic
49#16812619C2211yescharacteristic
50#16812609C2211yescharacteristic
51C8×Dic109C2211yescharacteristic
52C2×SD16×C109C2211yescharacteristic
53C4×Q8×C109C2211yescharacteristic
54C4×D4×C109C2211yescharacteristic
55#107×C109C2211yescharacteristic
56#103×C109C2211yescharacteristic
57#102×C109C2211yescharacteristic
58#97×C109C2211yescharacteristic
59#13182029—44no—
60C2×Dic218—21no—
61C2×Dic218D411yescharacteristic
62C2×D436—21no—
63C2×D436—21no—
64C2×D436C2311yescharacteristic
65C2×D436C2311yescharacteristic
66C2×C22⋊D109—21no—
67C2×C22⋊D109—21no—
68C2×C22⋊D109—21no—
69C2×C22⋊D109—21no—
70C2×C4×D109—21no—
71C2×C4×D109—21no—
72#13182016—22no—
73#13182016—22no—
74C2×#11724784C2311yescharacteristic
75C2×#11724784C2311yescharacteristic
76C4⋊Dic109—21no—
77C4⋊Dic109—21no—
78C4⋊Dic109—21no—
79C4⋊Dic109D411yescharacteristic
80C4⋊Dic109C2311yescharacteristic
81C4⋊Dic109C2311yescharacteristic
82#13182008—21no—
83#13182008—21no—
84#13182008—21no—
85#13182008—21no—
86#13182008D411yescharacteristic
87#13182008D411yescharacteristic
88#13182006—21no—
89#13182006—21no—
90C4×Dic109—21no—
91C4×Dic109—21no—
92C4×Dic109C2311yescharacteristic
93C4×Dic109C2311yescharacteristic
94C2×Q8×C109C2311yescharacteristic
95C2×D4×C109C2311yescharacteristic
96C22×C4×C109—21no—
97SD16×C109—22no—
98C2×C8×C109C2311yescharacteristic
99C2×C8×C109C2311yescharacteristic
100C4⋊C4×C109—21no—
101C4⋊C4×C109C2311yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C22⋊C4×C109—21no—
104C42×C109—21no—
105C42×C109C2311yescharacteristic
106C23×C109—21no—
107C22×D109—21no—
108C22×D109—21no—
109Q8×C109—21no—
110Q8×C109#4112yescomplemented
111D4×C109—21no—
112D4×C109—21no—
113C2×C4×C109—21no—
114C2×C4×C109—21no—
115C2×C4×C109—21no—
116C2×C4×C109—42no—
117C2×C4×C109C2×D411yescharacteristic
118C2×C4×C109C2×D411yescharacteristic
119C2×C4×C109C2411yescharacteristic
120C22⋊D109—42no—
121C22⋊D109—42no—
122C22⋊D109—81no—
123C22⋊D109—81no—
124C2×Dic109—21no—
125C2×Dic109—21no—
126C2×Dic109C2×D411yescharacteristic
127C2×Dic109C2×D411yescharacteristic
128C2×Dic109C2×D411yescharacteristic
129C2×Dic109C2×D411yescharacteristic
130D436—21no—
131D436—21no—
132D436—21no—
133D436—21no—
134D436—42no—
135D436—81no—
136C4×D109—42no—
137C4×D109—42no—
138Dic218—22no—
139Dic218—42no—
140C8×C109—21no—
141C8×C109—21no—
142#11724784—21no—
143#11724784—21no—
144C22×C109—41no—
145C22×C109—41no—
146C22×C109—41no—
147C22×C109C22×D411yescharacteristic
148C2×D109—41no—
149C2×D109—41no—
150C2×D109—41no—
151C2×D109—41no—
152C2×D109—41no—
153C2×D109—41no—
154C4×C109—22no—
155C4×C109—21no—
156C4×C109—21no—
157C4×C109—41no—
158C4×C109—41no—
159C4×C109C2×#4111yescharacteristic
160C4×C1092−1+411yescharacteristic
161Dic109—21no—
162Dic109—21no—
163Dic109—21no—
164Dic109—41no—
165Dic109—41no—
166Dic109AGL(1,ℤ/8)12yes—
167C2×C109—81no—
168C2×C109#54811yescharacteristic
169C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C109#57711yescharacteristic
171D109—81no—
172D109—81no—
173#3368—1091nomaximal, Sylow
174C109#336811yescharacteristic, complemented, Sylow
175#557—1091no—
176#550—1091no—
177#490—1091no—
178#490—1091no—
179#486—1091no—
180#478—1091no—
181#478—1091no—
182#474—1091no—
183#460—1091no—
184#460—1091no—
185#460—1091no—
186#446—1091no—
187#440—1091no—
188#440D10911yescharacteristic, complemented
189C4×SD16—1091no—
190C2×#41—2181no—
191C2×SD16—1091no—
192C2×SD16—1091no—
193C2×SD16C2×D10911yescharacteristic
194C4⋊Q8—1091no—
195C4⋊Q8—2181no—
196#127—1091no—
197#127—1091no—
198#127—1091no—
199#125—1091no—
200#125—1091no—
201#125—2181no—
202#125—2181no—
203#121—2181no—
204#121—2181no—
205#121—2181no—
206C4×Q8—1091no—
207C4×Q8—1091no—
208C4×Q8—1091no—
209C4×Q8—2181no—
210C4×Q8C2×D10911yescharacteristic
211C4×D4—1091no—
212C4×D4—1091no—
213C4×D4C2×D10911yescharacteristic
214#107—1091no—
215#107C2×D10911yescharacteristic
216#106—1091no—
217C4⋊C8—1091no—
218C4⋊C8—1091no—
219C4⋊C8—1091no—
220#103C2×D10911yescharacteristic
221#102—1091no—
222#102—1091no—
223#102—1091no—
224#102—1091no—
225#102—1091no—
226#102—1091no—
227#102—1091no—
228#102—1091no—
229#102C2×D10911yescharacteristic
230#97—1091no—
231#97C2×D10911yescharacteristic
232C4×C8—1091no—
233#41—4364no—
234C2×Q8—1091no—
235C2×Q8—2181no—
236C2×Q8C22×D10911yescharacteristic
237C2×D4—1091no—
238C2×D4—1091no—
239C2×D4—2181no—
240C2×D4—2181no—
241C2×D4—2181no—
242C2×D4—2181no—
243C2×D4—2181no—
244C2×D4—2181no—
245C2×D4C22×D10911yescharacteristic
246C22×C4—21no—
247C22×C4—2181no—
248C22×C4—2181no—
249SD16—22no—
250SD16—2182no—
251SD16—2182no—
252C2×C8—1091no—
253C2×C8—1091no—
254C2×C8C22×D10911yescharacteristic
255C2×C8C22×D10911yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1091no—
258C4⋊C4—1091no—
259C4⋊C4—1091no—
260C4⋊C4—1091no—
261C4⋊C4—1091no—
262C4⋊C4—2181no—
263C4⋊C4—2181no—
264C4⋊C4—2181no—
265C4⋊C4—2181no—
266C4⋊C4—2181no—
267C4⋊C4—2181no—
268C4⋊C4—2181no—
269C4⋊C4C22×D10911yescharacteristic
270C4⋊C4C22×D10911yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—2181no—
273C22⋊C4—2181no—
274C42—21no—
275C42—1091no—
276C42—1091no—
277C42—2181no—
278C42—2181no—
279C42C22×D10911yescharacteristic
280C23—21no—
281C23—2181no—
282C23—2181no—
283Q8—21no—
284Q8—4362no—
285Q8—2182no—
286Q8#1318202912yescomplemented
287D4—21no—
288D4—21no—
289D4—8721no—
290D4—8721no—
291D4—8721no—
292D4—4362no—
293D4—4362no—
294D4—4362no—
295D4—2181no—
296D4—2181no—
297D4—2181no—
298D4—2181no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—1091no—
304C2×C4—1091no—
305C2×C4—1091no—
306C2×C4—1091no—
307C2×C4—4362no—
308C2×C4—4362no—
309C2×C4—2181no—
310C2×C4—2181no—
311C2×C4D4×D10911yescharacteristic
312C2×C4D4×D10911yescharacteristic
313C2×C4C23×D10911yescharacteristic
314C8—21no—
315C8—21no—
316C8—2181no—
317C8—2181no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4361no—
322C22—4361no—
323C22—4361no—
324C22—4361no—
325C22—4361no—
326C22—4361no—
327C22C2×D4×D10911yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—1092no—
334C4—4361no—
335C4—4361no—
336C4—2181no—
337C4—2181no—
338C4—2181no—
339C4C2×#1318202911yescharacteristic
340C4#1681274611yescharacteristic
341C2—81no—
342C2—8721no—
343C2—8721no—
344C2#2029568511yescharacteristic
345C2AGL(1,ℤ/8)×D10911yescharacteristic
346C2#2029583511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#440×C109C2
socleC22×C109C22×D4
prime radical#20295697C2
hypercenter#440D109
hypocenterC109#3368
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20295752C2semidirect
3. #20295391C2semidirect
4. #20295389C2semidirect
5. #20295371C2semidirect
6. #20295329C2semidirect
7. #20295326C2semidirect
8. #20295305C2semidirect
9. #20295226C2semidirect
10. #20295224C2semidirect
11. #20295224C2semidirect
12. #20295154C2semidirect
13. #20295129C2semidirect
14. #20295111C2semidirect
15. #440×C109C2semidirect
110. Q8×C109#41semidirect
174. C109#3368semidirect
188. #440D109semidirect
286. Q8#13182029semidirect

Maximal subgroups

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

subgroupindex
#202957522
#202956972
#202953912
#202953892
#202953712
#202953292
#202953262
#202953052
#202952262
#202952242
#202951542
#202951292
#202951112
#440×C1092
#3368109