Finite Group Databasebeta

← #28372954

Subgroups — #28372954

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#20295536C211yesmaximal, characteristic
2C2×C4⋊C4×D109C211yesmaximal, characteristic, complemented
3#20295362C211yesmaximal, characteristic, complemented
4#20295361C211yesmaximal, characteristic, complemented
5#20295358C212yesmaximal, complemented
6#20295342C211yesmaximal, characteristic, complemented
7#20295338C212yesmaximal, complemented
8#20295065C212yesmaximal, complemented
9#20295024C212yesmaximal, complemented
10#20294988C211yesmaximal, characteristic, complemented
11#481×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2211yescharacteristic
13C22×C4×D109—21no—
14C22×C4×D109C2211yescharacteristic
15C2×#13182012C2212yes—
16#16812681—21no—
17#16812679C2211yescharacteristic
18#16812678C2211yescharacteristic
19#16812670C2211yescharacteristic
20#16812667—21no—
21D4×Dic109—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#16812633C2212yescomplemented
31#16812632C2212yescomplemented
32#16812630C2212yescomplemented
33#16812624C2212yescomplemented
34#16812622C2212yescomplemented
35#16812620C2212yescomplemented
36C22⋊#11724784C2211yescharacteristic
37#16812611C2212yes—
38#121×C109C2211yescharacteristic
39C2×C4⋊C4×C109C2211yescharacteristic
40#106×C109C2212yescomplemented
41#102×C109C2212yescomplemented
42C22⋊C8×C109C2211yescharacteristic
43C23×D109D411yescharacteristic
44#13182028—24no—
45C2×Dic218C2311yescharacteristic
46C2×C22⋊D109—21no—
47C2×C22⋊D109—21no—
48C22×Dic109—21no—
49C22×Dic109—21no—
50C22×Dic109D411yescharacteristic
51C2×C4×D109—24no—
52C2×C4×D109—22no—
53C2×C4×D109—22no—
54C2×C4×D109—44no—
55C2×C4×D109D411yescharacteristic
56C2×C4×D109D411yescharacteristic
57C2×C4×D109C2311yescharacteristic
58C2×C4×D109C2311yescharacteristic
59#13182012—24no—
60C2×#11724784C2312yes—
61C4⋊Dic109—21no—
62C4⋊Dic109—21no—
63C4⋊Dic109C2312yes—
64C4⋊Dic109C2311yescharacteristic
65#13182008—21no—
66#13182008—22no—
67#13182008—22no—
68C22⋊Dic109—21no—
69C22⋊Dic109—21no—
70#13182006—21no—
71#13182006—21no—
72C4×Dic109—21no—
73C2×D4×C109—21no—
74C2×D4×C109C2311yescharacteristic
75C22×C4×C109—21no—
76C22×C4×C109C2311yescharacteristic
77C2×C8×C109C2312yes—
78C4⋊C4×C109—21no—
79C4⋊C4×C109C2312yes—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C23×C109—21no—
83C23×C109C2×D411yescharacteristic
84C22×D109—22no—
85C22×D109—22no—
86C22×D109—22no—
87C22×D109C2×D411yescharacteristic
88C22×D109C2×D411yescharacteristic
89D4×C109—22no—
90D4×C109—42no—
91C2×C4×C109—22no—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109—42no—
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2411yescharacteristic
97C22⋊D109—42no—
98C22⋊D109—42no—
99C2×Dic109—22no—
100C2×Dic109—21no—
101C2×Dic109—21no—
102C2×Dic109—22no—
103C2×Dic109—42no—
104C2×Dic109—42no—
105C2×Dic109C2×D411yescharacteristic
106C2×Dic109C2×D411yescharacteristic
107C4×D109—22no—
108C4×D109—22no—
109C4×D109—22no—
110C4×D109—44no—
111C4×D109—44no—
112C4×D109#4114yes—
113Dic218—22no—
114C8×C109—22no—
115#11724784—22no—
116C22×C109—21no—
117C22×C109—21no—
118C22×C109—41no—
119C22×C109—41no—
120C22×C109—41no—
121C22×C109#13712yes—
122C22×C109C22×D411yescharacteristic
123C2×D109—28no—
124C2×D109—21no—
125C2×D109—21no—
126C2×D109—21no—
127C2×D109—21no—
128C2×D109#12312yes—
129C2×D109AGL(1,ℤ/8)12yes—
130C4×C109—21no—
131C4×C109—42no—
132C4×C109—41no—
133C4×C109C2×#4112yes—
134Dic109—21no—
135Dic109—41no—
136Dic109—42no—
137Dic109—41no—
138Dic109#12312yes—
139C2×C109—22no—
140C2×C109—81no—
141C2×C109C2×#12311yescharacteristic
142C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C109C2×#13711yescharacteristic
144D109—22no—
145D109—22no—
146#3092—1091nomaximal, Sylow
147C109#309211yescharacteristic, complemented, Sylow
148#525—1091no—
149C22×C4⋊C4—1091no—
150#484—1091no—
151#484—1091no—
152#483—1092no—
153#481—1091no—
154#481D10911yescharacteristic, complemented
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⋊Q8—1091no—
166#121—1091no—
167#121C2×D10911yescharacteristic
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#107—1092no—
181#106—1092no—
182#106C2×D10912yescomplemented
183#103—1092no—
184#103—1092no—
185#102—1092no—
186#102C2×D10912yescomplemented
187C22⋊C8—1091no—
188C22⋊C8—1092no—
189C22⋊C8C2×D10911yescharacteristic
190C24—1091no—
191#41—2184no—
192C2×Q8—1091no—
193C2×D4—21no—
194C2×D4—2181no—
195C2×D4—2181no—
196C2×D4C22×D10911yescharacteristic
197C22×C4—21no—
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—1091no—
201C22×C4—1091no—
202C22×C4—1091no—
203C22×C4—4364no—
204C22×C4—2184no—
205C22×C4—2182no—
206C22×C4—2182no—
207C22×C4—2181no—
208C22×C4—2181no—
209C22×C4C22×D10911yescharacteristic
210M16—2184no—
211C2×C8—1092no—
212C2×C8C22×D10912yes—
213C4⋊C4—21no—
214C4⋊C4—1092no—
215C4⋊C4—1091no—
216C4⋊C4—2182no—
217C4⋊C4—2182no—
218C4⋊C4—2181no—
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—21no—
230C23—1091no—
231C23—1091no—
232C23—2182no—
233C23—2182no—
234C23—2182no—
235C23D4×D10911yescharacteristic
236Q8—2182no—
237D4—22no—
238D4—42no—
239D4—4362no—
240D4—4362no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—1091no—
246C2×C4—1094no—
247C2×C4—1091no—
248C2×C4—4364no—
249C2×C4—4364no—
250C2×C4—4362no—
251C2×C4—4362no—
252C2×C4—2182no—
253C2×C4—2182no—
254C2×C4—2182no—
255C2×C4—2181no—
256C2×C4—2181no—
257C2×C4—2182no—
258C2×C4—2182no—
259C2×C4D4×D10911yescharacteristic
260C2×C4C23×D10911yescharacteristic
261C8—22no—
262C8—2182no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—1092no—
269C22—1092no—
270C22—2188no—
271C22—2181no—
272C22—2181no—
273C22—2181no—
274C22—2181no—
275C22#1681272812yes—
276C22C2×D4×D10911yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—1092no—
281C4—4361no—
282C4—4362no—
283C4—4361no—
284C4—2181no—
285C4#41×D10912yes—
286C2—22no—
287C2—81no—
288C2—2182no—
289C2—2182no—
290C2#123×D10911yescharacteristic
291C2AGL(1,ℤ/8)×D10911yescharacteristic
292C2C2×#1681272811yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#481×C109C2
socleC22×C109C22×D4
prime radical#20295536C2
hypercenter#481D109
hypocenterC109#3092
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D109C2semidirect
3. #20295362C2semidirect
4. #20295361C2semidirect
5. #20295358C2semidirect
6. #20295342C2semidirect
7. #20295338C2semidirect
8. #20295065C2semidirect
9. #20295024C2semidirect
10. #20294988C2semidirect
11. #481×C109C2semidirect
30. #16812633C22semidirect
31. #16812632C22semidirect
32. #16812630C22semidirect
33. #16812624C22semidirect
34. #16812622C22semidirect
35. #16812620C22semidirect
40. #106×C109C22semidirect
41. #102×C109C22semidirect
147. C109#3092semidirect
154. #481D109semidirect
182. #106C2×D109semidirect
186. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955362
C2×C4⋊C4×D1092
#202953622
#202953612
#202953582
#202953422
#202953382
#202950652
#202950242
#202949882
#481×C1092
#3092109