Finite Group Databasebeta

← #30675340

Subgroups — #30675340

Order 16768.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16862276C212yesmaximal, complemented
2#22553883C211yesmaximal, characteristic, complemented
3#22553495C214yesmaximal, complemented
4#22553490C214yesmaximal, complemented
5#22553329C212yesmaximal, complemented
6#22553296C211yesmaximal, characteristic, complemented
7#585×C131C211yesmaximal, characteristic
8C22×Dic262C2211yescharacteristic, complemented
9#16862276—24no—
10#16862276C2218yescomplemented
11C2×#14523503—22no—
12C2×#14523501—22no—
13C2×#14523497C2212yescomplemented
14#16862244C2212yescomplemented
15#16862233C2212yescomplemented
16#16862227—22no—
17#16862225—22no—
18#16862205C2211yescharacteristic, complemented
19#16862173C2218yescomplemented
20#16862165C2212yescomplemented
21#16862163C2212yescomplemented
222−1+4×C131—22no—
232−1+4×C131C2212yes—
24H(5,𝔽2)×C131—22no—
25H(5,𝔽2)×C131—22no—
26H(5,𝔽2)×C131C2212yes—
27C2×#41×C131—24no—
28C2×#41×C131—22no—
29C2×#41×C131C2212yes—
30C2×#41×C131C2211yescharacteristic
31C2×Dic262—21no—
32C2×Dic262—22no—
33C2×Dic262—22no—
34C2×Dic262C2312yes—
35C22×Dic131—21no—
36#14523503—28no—
37#14523501—28no—
38#14523497—22no—
39#14523497C2314yes—
40C2×C131⋊C8—22no—
41C4⋊Dic131—21no—
42Dic131⋊C4—24no—
43C22⋊Dic131—24no—
44C22⋊Dic131—21no—
45C4×Dic131C2312yes—
46#41×C131—24no—
47#41×C131—28no—
48#41×C131—24no—
49#41×C131—24no—
50#41×C131—24no—
51#41×C131—22no—
52#41×C131—22no—
53#41×C131—24no—
54#41×C131—22no—
55#41×C131—22no—
56#41×C131D414yes—
57#41×C131C2314yes—
58C2×Q8×C131—22no—
59C2×Q8×C131—24no—
60C2×Q8×C131D412yes—
61C2×Q8×C131C2311yescharacteristic
62C2×D4×C131—28no—
63C2×D4×C131—22no—
64C2×D4×C131—24no—
65C2×D4×C131—22no—
66C2×D4×C131—24no—
67C2×D4×C131D412yes—
68C2×D4×C131D412yes—
69C2×D4×C131C2311yescharacteristic
70C22×C4×C131—24no—
71C22×C4×C131—22no—
72C22×C4×C131D412yes—
73C22×C4×C131C2311yescharacteristic
74Dic262—24no—
75Dic262—41no—
76Dic262—41no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—42no—
80C131⋊C8—24no—
81C23×C131—24no—
82C23×C131—22no—
83C23×C131C2×D412yes—
84C23×C131C2×D411yescharacteristic
85Q8×C131—24no—
86Q8×C131—22no—
87Q8×C131—22no—
88Q8×C131C2×D414yes—
89D4×C131—28no—
90D4×C131—24no—
91D4×C131—24no—
92D4×C131—24no—
93D4×C131—24no—
94D4×C131—22no—
95D4×C131—22no—
96D4×C131C2×D414yes—
97C2×C4×C131—24no—
98C2×C4×C131—28no—
99C2×C4×C131—22no—
100C2×C4×C131—24no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—24no—
104C2×C4×C131C2×D412yes—
105C2×C4×C131C2×D411yescharacteristic
106C2×C4×C131C2×D412yes—
107C2×C4×C131C2×D412yes—
108C2×C4×C131C2411yescharacteristic
109C22×C131—22no—
110C22×C131—24no—
111C22×C131—41no—
112C22×C131—42no—
113C22×C131—82no—
114C22×C131—84no—
115C22×C131#12012yes—
116C22×C131C22×D411yescharacteristic
117C4×C131—24no—
118C4×C131—22no—
119C4×C131#12011yescharacteristic
120C4×C131#12011yescharacteristic
121C4×C131C22×D412yes—
122Dic131—42no—
123Dic131—42no—
124C2×C131—21no—
125C2×C131—22no—
126C2×C131—42no—
127C2×C131—44no—
128C2×C131C2×#12011yescharacteristic
129C131#302211yescharacteristic, complemented, Sylow
130#3022—1311nomaximal, Sylow
131#585D13111yescharacteristic
132C2×#137—1312no—
133#536—1311no—
134#456—1314no—
135#455—1314no—
136#421—1312no—
137#413—1311no—
1382−1+4—22no—
1392−1+4C2×D13112yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D13112yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D13112yes—
146C2×#41C2×D13111yescharacteristic
147C22×Q8—1311no—
148#137—2624no—
149#137—1318no—
150C2×Q16—2622no—
151C2×SD16—2622no—
152C2×M16—1312no—
153C4⋊Q8—1312no—
154#124—1312no—
155#123—2622no—
156C22⋊Q8—2622no—
157#117—1311no—
158#104—1318no—
159#101—1312no—
160#100—1312no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D13114yes—
172#41C22×D13114yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—2621no—
176C2×Q8—2622no—
177C2×Q8—2622no—
178C2×Q8—1312no—
179C2×Q8C22⋊D13112yes—
180C2×Q8C22×D13111yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D13112yes—
187C2×D4C22⋊D13112yes—
188C2×D4C22×D13111yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—2621no—
192C22×C4C22⋊D13112yes—
193C22×C4C22×D13111yescharacteristic
194Q16—2628no—
195SD16—2628no—
196M16—2622no—
197M16—1314no—
198C2×C8—2622no—
199C4⋊C4—2624no—
200C4⋊C4—2621no—
201C22⋊C4—2621no—
202C22⋊C4—2624no—
203C42—1312no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D13112yes—
207C23C2×C22⋊D13111yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—2624no—
212Q8—5241no—
213Q8—5241no—
214Q8C2×C22⋊D13114yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D13114yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—5242no—
233C2×C4C2×C22⋊D13112yes—
234C2×C4C2×C22⋊D13111yescharacteristic
235C2×C4C2×C22⋊D13112yes—
236C2×C4C2×C22⋊D13112yes—
237C2×C4C23×D13111yescharacteristic
238C8—2624no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1686221512yes—
246C22C22×C22⋊D13111yescharacteristic
247C4—24no—
248C4—22no—
249C4—5242no—
250C4—5242no—
251C4#1686221511yescharacteristic
252C4#1686221511yescharacteristic
253C4C22×C22⋊D13112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1686221511yescharacteristic
subgroupG/H
centerC2C2×#16862215
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting, prime radical#585×C131C2
socleC2×C131C2×#120
hypercenter#585D131
hypocenterC131#3022
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16862276C2semidirect
2. #22553883C2semidirect
3. #22553495C2semidirect
4. #22553490C2semidirect
5. #22553329C2semidirect
6. #22553296C2semidirect
8. C22×Dic262C22semidirect
10. #16862276C22semidirect
13. C2×#14523497C22semidirect
14. #16862244C22semidirect
15. #16862233C22semidirect
18. #16862205C22semidirect
19. #16862173C22semidirect
20. #16862165C22semidirect
21. #16862163C22semidirect
129. C131#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#168622762
#225538832
#225534952
#225534902
#225533292
#225532962
#585×C1312
#3022131