Finite Group Databasebeta

← #30675201

Subgroups — #30675201

Order 16768.

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

normal subgroups only

This group has 758 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
1C22×#14523513C211yesmaximal, characteristic, complemented
2C2×#16862276C211yesmaximal, characteristic, complemented
3C2×#16862271C211yesmaximal, characteristic, complemented
4#22553472C214yesmaximal, complemented
5#22553455C214yesmaximal, complemented
6#22553320C211yesmaximal, characteristic, complemented
7#22553316C211yesmaximal, characteristic, complemented
8#22553276C211yesmaximal, characteristic, complemented
9#525×C131C211yesmaximal, characteristic, complemented
10C2×#14523513—24no—
11C2×#14523513—24no—
12C2×#14523513—22no—
13C2×#14523513C2214yes—
14C22×Dic262C2211yescharacteristic
15C22×D524C2211yescharacteristic
16C22×C22⋊D131—21no—
17C22×C4×D131—21no—
18#16862276—24no—
19#16862271—24no—
20C2×#14523503C2212yescomplemented
21C2×#14523501C2212yescomplemented
22C2×#14523500C2212yescomplemented
23C2×#14523498C2212yescomplemented
24C2×#14523497C2211yescharacteristic
25C2×#14523497C2211yescharacteristic
26#16862169C2214yescomplemented
27#16862166C2214yescomplemented
28C22⋊(C131⋊C8)C2214yescomplemented
29C2×#41×C131C2211yescharacteristic
30C23×C4×C131C2211yescharacteristic
31#123×C131—21no—
32C22⋊Q8×C131C2212yescomplemented
33#121×C131C2212yescomplemented
34#120×C131—21no—
35C4×D4×C131—22no—
36#117×C131C2211yescharacteristic
37C23×D131—21no—
38#14523513—216no—
39#14523513—28no—
40#14523513—28no—
41C2×Dic262—22no—
42C2×Dic262—21no—
43C2×Dic262D414yescomplemented
44C2×Dic262C2312yes—
45C2×D524—22no—
46C2×D524—21no—
47C2×D524D414yescomplemented
48C2×D524C2312yes—
49C2×C22⋊D131—28no—
50C2×C22⋊D131—24no—
51C22×Dic131—21no—
52C2×C4×D131—28no—
53C2×C4×D131—24no—
54#14523503—24no—
55#14523501—24no—
56#14523500—24no—
57#14523498—24no—
58#14523497—22no—
59#14523497—22no—
60C2×C131⋊C8C2312yes—
61C2×C131⋊C8C2312yes—
62C24×C131D411yescharacteristic
63#41×C131—22no—
64#41×C131—22no—
65C2×Q8×C131C2311yescharacteristic
66C2×D4×C131—22no—
67C2×D4×C131—21no—
68C2×D4×C131C2311yescharacteristic
69C22×C4×C131—22no—
70C22×C4×C131—22no—
71C22×C4×C131—21no—
72C22×C4×C131D411yescharacteristic
73C22×C4×C131D412yes—
74C22×C4×C131C2312yes—
75C22×C4×C131C2311yescharacteristic
76C4⋊C4×C131—22no—
77C4⋊C4×C131C2312yes—
78C22⋊C4×C131—21no—
79C22⋊C4×C131—22no—
80C22⋊C4×C131—22no—
81C42×C131—21no—
82C22×D131—24no—
83C22×D131—22no—
84C22×D131—44no—
85Dic262—24no—
86Dic262—22no—
87Dic262C2×D414yes—
88D524—24no—
89D524—22no—
90D524C2×D414yes—
91C22⋊D131—216no—
92C2×Dic131—24no—
93C2×Dic131—22no—
94C4×D131—216no—
95C131⋊C8—22no—
96C131⋊C8—22no—
97C23×C131—22no—
98C23×C131—21no—
99C23×C131—42no—
100C23×C131C2×D412yes—
101C23×C131C2×D411yescharacteristic
102Q8×C131—21no—
103Q8×C131—21no—
104D4×C131—21no—
105D4×C131—21no—
106D4×C131—44no—
107D4×C131—42no—
108C2×C4×C131—22no—
109C2×C4×C131—24no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—21no—
114C2×C4×C131—42no—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131C2411yescharacteristic
120C22×C131—22no—
121C22×C131—22no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—44no—
125C22×C131—41no—
126C22×C131—42no—
127C22×C131#12012yes—
128C22×C131C22×D411yescharacteristic
129C2×D131—24no—
130C2×D131—48no—
131C2×D131—44no—
132C4×C131—22no—
133C4×C131—41no—
134C4×C131—42no—
135C4×C131#12012yes—
136C4×C131C22×D411yescharacteristic
137C4×C131C22×D411yescharacteristic
138Dic131—24no—
139C2×C131—22no—
140C2×C131—42no—
141C2×C131—81no—
142C2×C131C2×#12011yescharacteristic
143C2×C131#57512yes—
144D131—44no—
145C131#300511yescharacteristic, complemented, Sylow
146#3005—1311nomaximal, Sylow
147C22×#41—1311no—
148C2×#137—1311no—
149C2×AGL(1,ℤ/8)—1311no—
150#525D13111yescharacteristic, complemented
151#452—1314no—
152#449—1314no—
153#419—1311no—
154#418—1311no—
155#407—1311no—
156C2×#41—2624no—
157C2×#41—2624no—
158C2×#41—2622no—
159C2×#41—1314no—
160C2×#41C2×D13111yescharacteristic
161C22×Q8—1311no—
162C22×D4—2621no—
163C22×D4—1311no—
164C23×C4—2621no—
165C23×C4C2×D13111yescharacteristic
166#137—2624no—
167AGL(1,ℤ/8)—2624no—
168C2×Q16—1312no—
169C2×SD16—1312no—
170C2×SD16—1312no—
171C2×D8—1312no—
172C2×M16—1311no—
173C2×M16—1311no—
174#123—21no—
175C22⋊Q8C2×D13112yescomplemented
176#121C2×D13112yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D13111yescharacteristic
180#103—1314no—
181#102—1314no—
182C22⋊C8—1314no—
183C24—2621no—
184C24C22⋊D13111yescharacteristic
185#41—22no—
186#41—22no—
187#41—26216no—
188#41—2628no—
189#41—2628no—
190C2×Q8—2622no—
191C2×Q8—2621no—
192C2×Q8—1314no—
193C2×Q8—1312no—
194C2×Q8C22×D13111yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—2628no—
198C2×D4—2624no—
199C2×D4—2622no—
200C2×D4—2621no—
201C2×D4—1314no—
202C2×D4—1312no—
203C2×D4C22×D13111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—2628no—
208C22×C4—2621no—
209C22×C4—2624no—
210C22×C4C22⋊D13112yes—
211C22×C4C22⋊D13111yescharacteristic
212C22×C4C22×D13112yes—
213C22×C4C22×D13111yescharacteristic
214Q16—2624no—
215SD16—2624no—
216SD16—2624no—
217D8—2624no—
218M16—2622no—
219M16—2622no—
220C2×C8—1312no—
221C2×C8—1312no—
222C4⋊C4—22no—
223C4⋊C4C22×D13112yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—2624no—
232C23—2622no—
233C23—5244no—
234C23C2×C22⋊D13112yes—
235C23C2×C22⋊D13111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—2624no—
239Q8—2622no—
240Q8—1314no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—26216no—
246D4—2624no—
247D4—2622no—
248D4—1314no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—26216no—
257C2×C4—2624no—
258C2×C4—2622no—
259C2×C4D4×D13112yes—
260C2×C4D4×D13112yes—
261C2×C4C2×C22⋊D13112yes—
262C2×C4C2×C22⋊D13111yescharacteristic
263C2×C4C23×D13111yescharacteristic
264C8—2622no—
265C8—2622no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—2624no—
274C22—5248no—
275C22—5244no—
276C22#1686221412yes—
277C22C22×C22⋊D13111yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—2624no—
282C4#1686221412yes—
283C4C2×D4×D13111yescharacteristic
284C4C2×D4×D13111yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—5244no—
289C2C2×#1686221411yescharacteristic
290C2#2255411012yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#525×C131C2
socleC22×C131C22×D4
hypercenter#525D131
hypocenterC131#3005
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C2×#16862276C2semidirect
3. C2×#16862271C2semidirect
4. #22553472C2semidirect
5. #22553455C2semidirect
6. #22553320C2semidirect
7. #22553316C2semidirect
8. #22553276C2semidirect
9. #525×C131C2semidirect
20. C2×#14523503C22semidirect
21. C2×#14523501C22semidirect
22. C2×#14523500C22semidirect
23. C2×#14523498C22semidirect
26. #16862169C22semidirect
27. #16862166C22semidirect
28. C22⋊(C131⋊C8)C22semidirect
32. C22⋊Q8×C131C22semidirect
33. #121×C131C22semidirect
43. C2×Dic262D4semidirect
47. C2×D524D4semidirect
145. C131#3005semidirect
150. #525D131semidirect
175. C22⋊Q8C2×D131semidirect
176. #121C2×D131semidirect

Maximal subgroups

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

subgroupindex
C22×#145235132
C2×#168622762
C2×#168622712
#225534722
#225534552
#225533202
#225533162
#225532762
#525×C1312
#3005131