Finite Group Databasebeta

← #28372298

Subgroups — #28372298

Order 13952.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812745C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic, complemented
3C22×#13182014C211yesmaximal, characteristic, complemented
4#20295169C214yesmaximal, complemented
5#20295159C214yesmaximal, complemented
6#20295030C211yesmaximal, characteristic, complemented
7#20295020C211yesmaximal, characteristic, complemented
8#20294993C211yesmaximal, characteristic, complemented
9#535×C109C211yesmaximal, characteristic, complemented
10#16812745—24no—
11#16812745—24no—
12C2×#13182030—21no—
13C2×#13182029C2211yescharacteristic
14C2×#13182029C2212yescomplemented
15C22×D436C2211yescharacteristic
16C22×D436C2212yescomplemented
17C2×D4×D109—22no—
18C2×D4×D109—21no—
19#16812725—24no—
20C2×#13182017C2212yescomplemented
21C2×#13182014—24no—
22C2×#13182014C2214yescomplemented
23C2×#13182014C2212yescomplemented
24C2×#13182013C2211yescharacteristic, complemented
25C22×#11724784C2211yescharacteristic, complemented
26#16812622C2212yescomplemented
27#16812619C2214yescomplemented
28#16812619C2212yescomplemented
29C22⋊#11724784C2214yescomplemented
30C2×#41×C109C2211yescharacteristic
31C22×D4×C109C2211yescharacteristic, complemented
32#127×C109—21no—
33#124×C109—21no—
34#121×C109C2214yescomplemented
35#120×C109—22no—
36#117×C109C2211yescharacteristic, complemented
37C23×D109—21no—
38C23×D109—22no—
39#13182030—28no—
40#13182029—22no—
41#13182029—24no—
42#13182029—24no—
43#13182029D414yescomplemented
44C2×Dic218C2311yescharacteristic
45C2×D436—21no—
46C2×D436—22no—
47C2×D436—22no—
48C2×D436—24no—
49C2×D436—24no—
50C2×D436D414yescomplemented
51C2×D436C2312yes—
52C2×D436C2311yescharacteristic
53C2×C22⋊D109—21no—
54C2×C22⋊D109—22no—
55D4×D109—216no—
56D4×D109—28no—
57C2×C4×D109—21no—
58C2×C4×D109—22no—
59#13182017—24no—
60#13182014—28no—
61#13182014—24no—
62#13182013—22no—
63C2×#11724784—22no—
64C2×#11724784C2312yes—
65C2×#11724784C2312yes—
66C24×C109—21no—
67#41×C109—22no—
68#41×C109—22no—
69C2×Q8×C109D411yescharacteristic
70C2×D4×C109—24no—
71C2×D4×C109—22no—
72C2×D4×C109—22no—
73C2×D4×C109—21no—
74C2×D4×C109D412yes—
75C2×D4×C109D411yescharacteristic
76C2×D4×C109C2312yes—
77C22×C4×C109C2312yes—
78C22×C4×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—21no—
81C22⋊C4×C109—24no—
82C42×C109—21no—
83C23×C109—22no—
84C23×C109—42no—
85C23×C109—82no—
86C23×C109C2×D412yes—
87C23×C109C2×D411yescharacteristic
88C22×D109—21no—
89C22×D109—22no—
90C22×D109—28no—
91C22×D109—24no—
92C22×D109—82no—
93C22×D109—84no—
94Q8×C109—22no—
95D4×C109—24no—
96D4×C109—22no—
97D4×C109—22no—
98D4×C109—22no—
99D4×C109—42no—
100D4×C109—84no—
101C2×C4×C109—22no—
102C2×C4×C109—22no—
103C2×C4×C109—22no—
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2×D412yes—
107C2×C4×C109C2×D411yescharacteristic
108C2×C4×C109C2411yescharacteristic
109C22⋊D109—28no—
110C22⋊D109—24no—
111C2×Dic109—21no—
112D436—28no—
113D436—24no—
114D436—21no—
115D436—22no—
116D436C2×D412yes—
117D436C2×D414yes—
118C4×D109—28no—
119C4×D109—24no—
120Dic218—21no—
121Dic218C2×D412yes—
122#11724784—22no—
123#11724784—22no—
124C22×C109—22no—
125C22×C109—22no—
126C22×C109—24no—
127C22×C109—42no—
128C22×C109—44no—
129C22×C109—84no—
130C22×C109#12012yes—
131C22×C109C22×D411yescharacteristic
132C2×D109—24no—
133C2×D109—22no—
134C2×D109—44no—
135C2×D109—42no—
136C2×D109—84no—
137C2×D109—88no—
138C4×C109—22no—
139C4×C109—42no—
140C4×C109#12012yes—
141C4×C109C22×D411yescharacteristic
142C4×C109C22×D411yescharacteristic
143Dic109—22no—
144C2×C109—22no—
145C2×C109—42no—
146C2×C109—82no—
147C2×C109C2×#12011yescharacteristic
148C2×C109#57612yes—
149D109—44no—
150D109—42no—
151#3008—1091nomaximal, Sylow
152C109#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1091no—
154C2×AGL(1,ℤ/8)—1091no—
155C22×D8—1091no—
156#535D10911yescharacteristic, complemented
157#449—1094no—
158#447—1094no—
159#418—1091no—
160#416—1091no—
161#408—1091no—
162H(5,𝔽2)—2184no—
163H(5,𝔽2)—2184no—
164C2×#41—1091no—
165C2×#41—1092no—
166C2×#41—2181no—
167C2×#41C2×D10911yescharacteristic
168C22×D4—1091no—
169C22×D4—1092no—
170C22×D4—2182no—
171C22×D4—2181no—
172C22×D4C2×D10911yescharacteristic, complemented
173AGL(1,ℤ/8)—2184no—
174C2×SD16—1092no—
175C2×D8—1094no—
176C2×D8—1092no—
177C2×D8—2184no—
178C2×M16—1091no—
179C22×C8—1091no—
180#127—21no—
181#124—21no—
182#121C2×D10914yescomplemented
183#120—22no—
184#117C2×D10911yescharacteristic, complemented
185#103—1092no—
186#102—1092no—
187#102—1094no—
188C22⋊C8—1094no—
189C24—21no—
190C24—2182no—
191C24—2181no—
192#41—22no—
193#41—22no—
194#41—1094no—
195#41—2184no—
196#41—2184no—
197#41—2188no—
198#41—2182no—
199C2×Q8—1091no—
200C2×Q8C22⋊D10911yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—1094no—
206C2×D4—1092no—
207C2×D4—1091no—
208C2×D4—2184no—
209C2×D4—2184no—
210C2×D4—21816no—
211C2×D4—2188no—
212C2×D4—2182no—
213C2×D4—2181no—
214C2×D4—2182no—
215C2×D4—2182no—
216C2×D4—2181no—
217C2×D4C22⋊D10912yes—
218C2×D4C22⋊D10911yescharacteristic
219C2×D4C22×D10912yes—
220C22×C4—2182no—
221C22×C4—2181no—
222C22×C4C22×D10912yes—
223C22×C4C22×D10911yescharacteristic
224SD16—2184no—
225D8—2188no—
226D8—2184no—
227M16—2182no—
228C2×C8—1092no—
229C2×C8—1092no—
230C2×C8—2182no—
231C4⋊C4C22×D10912yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—8722no—
239C23—8724no—
240C23—2188no—
241C23—2184no—
242C23—2182no—
243C23—2181no—
244C23C2×C22⋊D10912yes—
245C23C2×C22⋊D10911yescharacteristic
246Q8—22no—
247Q8—1092no—
248Q8—2181no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—1094no—
256D4—1092no—
257D4—2188no—
258D4—2188no—
259D4—2184no—
260D4—2184no—
261D4—2181no—
262D4—2182no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2188no—
267C2×C4—2181no—
268C2×C4—2184no—
269C2×C4D4×D10912yes—
270C2×C4D4×D10912yes—
271C2×C4C2×C22⋊D10912yes—
272C2×C4C2×C22⋊D10911yescharacteristic
273C2×C4C23×D10911yescharacteristic
274C8—2182no—
275C8—2182no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—8724no—
283C22—8728no—
284C22—4364no—
285C22—4362no—
286C22—2184no—
287C22—2182no—
288C22#1681266712yes—
289C22C22×C22⋊D10911yescharacteristic
290C4—22no—
291C4—42no—
292C4—2182no—
293C4#1681266712yes—
294C4C2×D4×D10911yescharacteristic
295C4C2×D4×D10911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—4364no—
300C2—4362no—
301C2C2×#1681266711yescharacteristic
302C2#2029582912yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#535×C109C2
socleC22×C109C22×D4
hypercenter#535D109
hypocenterC109#3008
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812745C2semidirect
2. C2×#16812725C2semidirect
3. C22×#13182014C2semidirect
4. #20295169C2semidirect
5. #20295159C2semidirect
6. #20295030C2semidirect
7. #20295020C2semidirect
8. #20294993C2semidirect
9. #535×C109C2semidirect
14. C2×#13182029C22semidirect
16. C22×D436C22semidirect
20. C2×#13182017C22semidirect
22. C2×#13182014C22semidirect
23. C2×#13182014C22semidirect
24. C2×#13182013C22semidirect
25. C22×#11724784C22semidirect
26. #16812622C22semidirect
27. #16812619C22semidirect
28. #16812619C22semidirect
29. C22⋊#11724784C22semidirect
31. C22×D4×C109C22semidirect
34. #121×C109C22semidirect
36. #117×C109C22semidirect
43. #13182029D4semidirect
50. C2×D436D4semidirect
152. C109#3008semidirect
156. #535D109semidirect
172. C22×D4C2×D109semidirect
182. #121C2×D109semidirect
184. #117C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127452
C2×#168127252
C22×#131820142
#202951692
#202951592
#202950302
#202950202
#202949932
#535×C1092
#3008109