Finite Group Databasebeta

← #28372959

Subgroups — #28372959

Order 13952.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#20295539C211yesmaximal, characteristic
2C2×C4⋊C4×D109C211yesmaximal, characteristic, complemented
3#20295359C211yesmaximal, characteristic, complemented
4#20295355C212yesmaximal, complemented
5#20295345C211yesmaximal, characteristic, complemented
6#20295344C211yesmaximal, characteristic, complemented
7#20295341C212yesmaximal, complemented
8#20295067C212yesmaximal, complemented
9#20295026C212yesmaximal, complemented
10#20294988C211yesmaximal, characteristic, complemented
11#484×C109C211yesmaximal, characteristic, complemented
12C2×#13182030C2211yescharacteristic
13C22×C4×D109—21no—
14C22×C4×D109C2211yescharacteristic
15C2×#13182012C2212yes—
16#16812684—21no—
17#16812675C2211yescharacteristic
18#16812674C2211yescharacteristic
19#16812673C2211yescharacteristic
20#16812669—21no—
21#16812661—21no—
22#16812659—21no—
23#16812657C2211yescharacteristic
24C2×C4⋊Dic109C2211yescharacteristic
25C2×#13182008—21no—
26C4⋊C4×D109—21no—
27C4⋊C4×D109—21no—
28C4⋊C4×D109C2212yes—
29C4⋊C4×D109C2212yes—
30#16812632C2212yescomplemented
31#16812631C2212yescomplemented
32#16812630C2212yescomplemented
33#16812623C2212yescomplemented
34#16812621C2212yescomplemented
35#16812619C2212yescomplemented
36C22⋊#11724784C2211yescharacteristic
37#16812611C2212yes—
38C22⋊Q8×C109C2211yescharacteristic
39C2×C4⋊C4×C109C2211yescharacteristic
40#107×C109C2212yescomplemented
41#103×C109C2212yescomplemented
42C22⋊C8×C109C2211yescharacteristic
43C23×D109D411yescharacteristic
44#13182030—24no—
45C2×D436—21no—
46C2×D436—21no—
47C2×D436C2311yescharacteristic
48C2×C22⋊D109—21no—
49C22×Dic109—21no—
50C22×Dic109D411yescharacteristic
51C2×C4×D109—24no—
52C2×C4×D109—22no—
53C2×C4×D109—22no—
54C2×C4×D109—21no—
55C2×C4×D109—44no—
56C2×C4×D109D411yescharacteristic
57C2×C4×D109D411yescharacteristic
58C2×C4×D109C2311yescharacteristic
59C2×C4×D109C2311yescharacteristic
60#13182012—24no—
61C2×#11724784C2312yes—
62C4⋊Dic109—21no—
63C4⋊Dic109C2312yes—
64C4⋊Dic109C2311yescharacteristic
65#13182008—22no—
66#13182008—22no—
67#13182008—21no—
68#13182006—21no—
69#13182006—21no—
70#13182006—21no—
71#13182006—21no—
72C4×Dic109—21no—
73C2×Q8×C109C2311yescharacteristic
74C22×C4×C109—21no—
75C22×C4×C109C2311yescharacteristic
76C2×C8×C109C2312yes—
77C4⋊C4×C109—21no—
78C4⋊C4×C109—21no—
79C4⋊C4×C109C2312yes—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C23×C109C2×D411yescharacteristic
83C22×D109—22no—
84C22×D109—22no—
85C22×D109—22no—
86C22×D109—21no—
87C22×D109C2×D411yescharacteristic
88C22×D109C2×D411yescharacteristic
89Q8×C109—22no—
90C2×C4×C109—22no—
91C2×C4×C109—21no—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109—42no—
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2411yescharacteristic
97C22⋊D109—42no—
98C2×Dic109—22no—
99C2×Dic109—21no—
100C2×Dic109—22no—
101C2×Dic109—42no—
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104D436—22no—
105D436—42no—
106D436—42no—
107C4×D109—22no—
108C4×D109—22no—
109C4×D109—22no—
110C4×D109—44no—
111C4×D109—44no—
112C4×D109—42no—
113C4×D109#4114yes—
114C8×C109—22no—
115#11724784—22no—
116C22×C109—21no—
117C22×C109—21no—
118C22×C109#13712yes—
119C22×C109C22×D411yescharacteristic
120C2×D109—28no—
121C2×D109—21no—
122C2×D109—21no—
123C2×D109—21no—
124C2×D109—21no—
125C2×D109—41no—
126C2×D109—41no—
127C2×D109—41no—
128C2×D109#12312yes—
129C2×D109AGL(1,ℤ/8)12yes—
130C4×C109—21no—
131C4×C109—41no—
132C4×C109—42no—
133C4×C109—41no—
134C4×C109C2×#4112yes—
135Dic109—21no—
136Dic109—42no—
137Dic109—41no—
138Dic109#12312yes—
139C2×C109—22no—
140C2×C109C2×#12311yescharacteristic
141C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C109C2×#13711yescharacteristic
143D109—22no—
144D109—22no—
145D109—81no—
146#3092—1091nomaximal, Sylow
147C109#309211yescharacteristic, complemented, Sylow
148#525—1091no—
149C22×C4⋊C4—1091no—
150#484—1091no—
151#484D10911yescharacteristic, complemented
152#483—1092no—
153#481—1091no—
154#481—1091no—
155#480—1092no—
156#428—1092no—
157#417—1092no—
158#407—1091no—
159C2×#41—1091no—
160C23×C4—1091no—
161C23×C4—2181no—
162C2×M16—1092no—
163#123—2181no—
164C22⋊Q8—1091no—
165C22⋊Q8C2×D10911yescharacteristic
166#121—1091no—
167#121—1091no—
168#120—2181no—
169C4×D4—2181no—
170C4×D4—2181no—
171#117—1091no—
172C2×C4⋊C4—1091no—
173C2×C4⋊C4—1092no—
174C2×C4⋊C4—1092no—
175C2×C4⋊C4—2181no—
176C2×C4⋊C4—2181no—
177C2×C4⋊C4—2181no—
178C2×C4⋊C4C2×D10911yescharacteristic
179#107—1092no—
180#107C2×D10912yescomplemented
181#106—1092no—
182#106—1092no—
183#103—1092no—
184#103C2×D10912yescomplemented
185#102—1092no—
186#102—1092no—
187C22⋊C8—1091no—
188C22⋊C8—1092no—
189C22⋊C8C2×D10911yescharacteristic
190C24—1091no—
191#41—2184no—
192C2×Q8C22×D10911yescharacteristic
193C2×D4—1091no—
194C2×D4—2181no—
195C2×D4—2181no—
196C2×D4—2181no—
197C22×C4—21no—
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—1091no—
201C22×C4—1091no—
202C22×C4—1091no—
203C22×C4—4364no—
204C22×C4—2181no—
205C22×C4—2184no—
206C22×C4—2182no—
207C22×C4—2182no—
208C22×C4—2181no—
209C22×C4C22×D10911yescharacteristic
210M16—2184no—
211C2×C8—1092no—
212C2×C8C22×D10912yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1092no—
216C4⋊C4—1091no—
217C4⋊C4—2182no—
218C4⋊C4—2182no—
219C4⋊C4—2181no—
220C4⋊C4—2181no—
221C4⋊C4C22×D10912yes—
222C4⋊C4C22×D10911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2181no—
225C22⋊C4—2181no—
226C22⋊C4—2181no—
227C22⋊C4—2181no—
228C42—2181no—
229C23—1091no—
230C23—1091no—
231C23—2182no—
232C23—2182no—
233C23—2181no—
234C23—2182no—
235C23D4×D10911yescharacteristic
236Q8—22no—
237D4—4362no—
238D4—4362no—
239D4—4362no—
240D4—2182no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—1091no—
247C2×C4—1094no—
248C2×C4—1091no—
249C2×C4—4362no—
250C2×C4—4364no—
251C2×C4—4364no—
252C2×C4—4362no—
253C2×C4—2182no—
254C2×C4—2182no—
255C2×C4—2182no—
256C2×C4—2181no—
257C2×C4—2182no—
258C2×C4—2182no—
259C2×C4D4×D10911yescharacteristic
260C2×C4C23×D10911yescharacteristic
261C8—22no—
262C8—2182no—
263C22—21no—
264C22—21no—
265C22—1092no—
266C22—1092no—
267C22—4361no—
268C22—4361no—
269C22—4361no—
270C22—2188no—
271C22—2181no—
272C22—2181no—
273C22—2181no—
274C22—2181no—
275C22#1681273112yes—
276C22C2×D4×D10911yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—1092no—
282C4—4362no—
283C4—4361no—
284C4—2181no—
285C4#41×D10912yes—
286C2—22no—
287C2—8721no—
288C2—2182no—
289C2—2182no—
290C2#123×D10911yescharacteristic
291C2AGL(1,ℤ/8)×D10911yescharacteristic
292C2C2×#1681273111yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#484×C109C2
socleC22×C109C22×D4
prime radical#20295539C2
hypercenter#484D109
hypocenterC109#3092
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D109C2semidirect
3. #20295359C2semidirect
4. #20295355C2semidirect
5. #20295345C2semidirect
6. #20295344C2semidirect
7. #20295341C2semidirect
8. #20295067C2semidirect
9. #20295026C2semidirect
10. #20294988C2semidirect
11. #484×C109C2semidirect
30. #16812632C22semidirect
31. #16812631C22semidirect
32. #16812630C22semidirect
33. #16812623C22semidirect
34. #16812621C22semidirect
35. #16812619C22semidirect
40. #107×C109C22semidirect
41. #103×C109C22semidirect
147. C109#3092semidirect
151. #484D109semidirect
180. #107C2×D109semidirect
184. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955392
C2×C4⋊C4×D1092
#202953592
#202953552
#202953452
#202953442
#202953412
#202950672
#202950262
#202949882
#484×C1092
#3092109