Finite Group Databasebeta

← #30675922

Subgroups — #30675922

Order 16768.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#22553881C211yesmaximal, characteristic
2#22553792C211yesmaximal, characteristic, complemented
3#22553639C212yesmaximal, complemented
4#22553638C212yesmaximal, complemented
5#22553632C212yesmaximal, complemented
6#22553627C212yesmaximal, complemented
7#22553355C211yesmaximal, characteristic, complemented
8#22553316C211yesmaximal, characteristic, complemented
9C2×#16862168C211yesmaximal, characteristic, complemented
10#22553278C211yesmaximal, characteristic, complemented
11#427×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C2×#14523497C2211yescharacteristic
15#16862243—21no—
16#16862234—21no—
17#16862221C2212yes—
18#16862221C2212yes—
19#16862216—22no—
20C4×Dic262—21no—
21C4×D524—21no—
22(C22⋊D131)⋊C4—22no—
23#16862204C2212yescomplemented
24C2×C4⋊Dic131C2211yescharacteristic
25C4⋊C4×D131C2212yescomplemented
26#16862179C2212yescomplemented
27#16862177C2212yescomplemented
28#16862168C2212yes—
29#16862168C2212yes—
30#16862166C2212yes—
31#16862166C2212yes—
32(C2×D131)⋊C8C2212yescomplemented
33(C2×D131)⋊C8C2212yescomplemented
34C22×C8×C131C2211yescharacteristic
35#117×C131C2211yescharacteristic
36#117×C131C2211yescharacteristic
37#107×C131C2211yescharacteristic, complemented
38#107×C131C2211yescharacteristic, complemented
39#106×C131C2211yescharacteristic, complemented
40#106×C131C2211yescharacteristic, complemented
41C23×D131—21no—
42#14523513—22no—
43#14523513—22no—
44C2×Dic262D411yescharacteristic
45C2×D524—22no—
46C2×D524—22no—
47C2×D524—21no—
48C2×D524—22no—
49C2×D524—22no—
50C2×D524D411yescharacteristic
51C2×D524C2312yes—
52C2×C22⋊D131D412yes—
53C22×Dic131—21no—
54C2×C4×D131—22no—
55C2×C4×D131C2312yes—
56#14523497—22no—
57C2×C131⋊C8C2312yes—
58C4⋊Dic131—21no—
59C4⋊Dic131C2312yes—
60Dic131⋊C4—22no—
61(C2×D131)⋊C4—22no—
62(C2×D131)⋊C4—22no—
63(C2×D131)⋊C4—22no—
64C4×Dic131—22no—
65C22×C4×C131C2311yescharacteristic
66C2×C8×C131—22no—
67C2×C8×C131C2311yescharacteristic
68C2×C8×C131C2311yescharacteristic
69C4⋊C4×C131C2311yescharacteristic
70C4⋊C4×C131C2311yescharacteristic
71C4⋊C4×C131C2311yescharacteristic
72C4⋊C4×C131C2311yescharacteristic
73C22⋊C4×C131—21no—
74C22⋊C4×C131—21no—
75C42×C131—21no—
76C42×C131—21no—
77C22×D131—22no—
78C22×D131—42no—
79C22×D131—81no—
80C22×D131—81no—
81C22×D131C2×D412yes—
82Dic262—22no—
83D524—22no—
84D524—22no—
85D524—22no—
86D524—42no—
87D524—82no—
88D524—82no—
89C22⋊D131—24no—
90C2×Dic131—22no—
91C2×Dic131—81no—
92C2×Dic131C2×D412yes—
93C4×D131—22no—
94C4×D131—22no—
95C4×D131—82no—
96C131⋊C8—22no—
97C23×C131C2×D411yescharacteristic
98C2×C4×C131—21no—
99C2×C4×C131—21no—
100C2×C4×C131—21no—
101C2×C4×C131—21no—
102C2×C4×C131C2×D411yescharacteristic
103C2×C4×C131#4112yes—
104C2×C4×C131#4112yes—
105C2×C4×C131C2411yescharacteristic
106C8×C131—21no—
107C8×C131—21no—
108C22×C131—21no—
109C22×C131—21no—
110C22×C131#12312yes—
111C22×C131C22×D411yescharacteristic
112C2×D131—22no—
113C2×D131—22no—
114C2×D131—22no—
115C2×D131—42no—
116C2×D131—42no—
117C2×D131—42no—
118C2×D131—84no—
119C4×C131—41no—
120C4×C131—41no—
121C4×C131—41no—
122C4×C131—41no—
123C4×C131#12312yes—
124C4×C131C2×#4111yescharacteristic
125C4×C131C2×#4111yescharacteristic
126Dic131—22no—
127Dic131—42no—
128C2×C131—22no—
129C2×C131C2×#12311yescharacteristic
130C2×C131#57611yescharacteristic
131C2×C131#57711yescharacteristic
132D131—42no—
133D131—82no—
134C131#309611yescharacteristic, complemented, Sylow
135#3096—1311nomaximal, Sylow
136#535—1311no—
137#520—1311no—
138#482—1312no—
139#482—1312no—
140#481—1312no—
141#480—1312no—
142#428—1311no—
143#427D13111yescharacteristic, complemented
144#418—1311no—
145C2×#102—1311no—
146#408—1311no—
147C2×#41—1311no—
148C22×D4—1311no—
149C2×M16—1311no—
150C22×C8C2×D13111yescharacteristic
151#127—2621no—
152#124—2621no—
153#121—1312no—
154#121—1312no—
155#120—2622no—
156C4×Q8—2621no—
157C4×D4—2621no—
158C4×D4—2622no—
159#117—1312no—
160#117C2×D13111yescharacteristic
161#117C2×D13111yescharacteristic
162C2×C4⋊C4—1311no—
163C2×C4⋊C4—1312no—
164#107—1312no—
165#107C2×D13111yescharacteristic, complemented
166#107C2×D13111yescharacteristic, complemented
167#106—1312no—
168#106C2×D13111yescharacteristic, complemented
169#106C2×D13111yescharacteristic, complemented
170#102—1312no—
171#102—1312no—
172#102—1312no—
173#102—1312no—
174C22⋊C8—1312no—
175C22⋊C8—1312no—
176C24—2621no—
177#41—2622no—
178#41—2622no—
179C2×Q8—1311no—
180C2×D4—2622no—
181C2×D4—2622no—
182C2×D4—2621no—
183C2×D4—2622no—
184C2×D4—2622no—
185C2×D4—1312no—
186C2×D4—1311no—
187C2×D4—1312no—
188C22×C4—2622no—
189C22×C4—2621no—
190C22×C4—1312no—
191C22×C4C22×D13111yescharacteristic
192M16—2622no—
193C2×C8—22no—
194C2×C8—1312no—
195C2×C8C22×D13111yescharacteristic
196C2×C8C22×D13111yescharacteristic
197C4⋊C4—2622no—
198C4⋊C4—2621no—
199C4⋊C4—1312no—
200C4⋊C4C22×D13111yescharacteristic
201C4⋊C4C22×D13111yescharacteristic
202C4⋊C4C22×D13111yescharacteristic
203C4⋊C4C22×D13111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—2622no—
207C22⋊C4—2622no—
208C22⋊C4—2622no—
209C42—21no—
210C42—21no—
211C42—2622no—
212C23—2622no—
213C23—5242no—
214C23—10481no—
215C23—10481no—
216C23—1312no—
217C23D4×D13111yescharacteristic
218Q8—2622no—
219D4—2624no—
220D4—2622no—
221D4—2622no—
222D4—2622no—
223D4—5242no—
224D4—10482no—
225D4—10482no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—2622no—
233C2×C4—10482no—
234C2×C4—10481no—
235C2×C4—1312no—
236C2×C4D4×D13111yescharacteristic
237C2×C4#1452351312yes—
238C2×C4#1452351412yes—
239C2×C4C23×D13111yescharacteristic
240C8—21no—
241C8—21no—
242C8—2622no—
243C22—21no—
244C22—21no—
245C22—2622no—
246C22—2622no—
247C22—2622no—
248C22—5242no—
249C22—5242no—
250C22—5242no—
251C22—10484no—
252C22#1686223112yes—
253C22C2×D4×D13111yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—2622no—
259C4—5242no—
260C4#1686223112yes—
261C4C2×#1452351311yescharacteristic
262C4C2×#1452351411yescharacteristic
263C2—22no—
264C2—5242no—
265C2—10482no—
266C2C2×#1686223111yescharacteristic
267C2#2255411411yescharacteristic
268C2#2255411911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#427×C131C2
socleC22×C131C22×D4
prime radical#22553881C2
hypercenter#427D131
hypocenterC131#3096
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #22553792C2semidirect
3. #22553639C2semidirect
4. #22553638C2semidirect
5. #22553632C2semidirect
6. #22553627C2semidirect
7. #22553355C2semidirect
8. #22553316C2semidirect
9. C2×#16862168C2semidirect
10. #22553278C2semidirect
11. #427×C131C2semidirect
23. #16862204C22semidirect
25. C4⋊C4×D131C22semidirect
26. #16862179C22semidirect
27. #16862177C22semidirect
32. (C2×D131)⋊C8C22semidirect
33. (C2×D131)⋊C8C22semidirect
37. #107×C131C22semidirect
38. #107×C131C22semidirect
39. #106×C131C22semidirect
40. #106×C131C22semidirect
134. C131#3096semidirect
143. #427D131semidirect
165. #107C2×D131semidirect
166. #107C2×D131semidirect
168. #106C2×D131semidirect
169. #106C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538812
#225537922
#225536392
#225536382
#225536322
#225536272
#225533552
#225533162
C2×#168621682
#225532782
#427×C1312
#3096131