Finite Group Databasebeta

← #30674286

Subgroups — #30674286

Order 16768.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16862230C211yesmaximal, characteristic
2C22×C4×Dic131C211yesmaximal, characteristic, complemented
3#22553202C212yesmaximal, complemented
4#22553201C212yesmaximal, complemented
5#22553199C212yesmaximal, complemented
6#22553198C212yesmaximal, complemented
7#22553191C211yesmaximal, characteristic
8#22553189C211yesmaximal, characteristic
9#22553184C211yesmaximal, characteristic, complemented
10#22553173C211yesmaximal, characteristic, complemented
11#385×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15#16862230—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×C4⋊Dic131C2211yescharacteristic
18C2×Dic131⋊C4C2212yes—
19C2×C22⋊Dic131C2211yescharacteristic
20C2×(C2×D131)⋊C4C2212yes—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2211yescharacteristic
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×C4×Dic131—24no—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C23×C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2212yes—
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C2×C22⋊D131—22no—
39C2×C22⋊D131—22no—
40C2×C22⋊D131—22no—
41C22×Dic131—24no—
42C22×Dic131—22no—
43C22×Dic131—22no—
44C22×Dic131—22no—
45C22×Dic131D414yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C22×Dic131C2312yes—
51C4⋊Dic131—22no—
52C4⋊Dic131—22no—
53Dic131⋊C4—24no—
54C22⋊Dic131—22no—
55(C2×D131)⋊C4—22no—
56(C2×D131)⋊C4—22no—
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4—24no—
59C4×Dic131—24no—
60C4×Dic131—24no—
61C24×C131C2311yescharacteristic
62C22×C4×C131—22no—
63C22×C4×C131—24no—
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2312yes—
66C22×C4×C131C2311yescharacteristic
67C22×C4×C131C2312yes—
68C4⋊C4×C131—24no—
69C22⋊C4×C131—22no—
70C22⋊C4×C131—22no—
71C22×D131—21no—
72C22×D131—22no—
73C22×D131—22no—
74C22×D131—21no—
75C22×D131—21no—
76C22⋊D131—44no—
77C2×Dic131—28no—
78C2×Dic131—28no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—21no—
82C2×Dic131—21no—
83C2×Dic131—21no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131—22no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—21no—
90C2×Dic131—21no—
91C2×Dic131C2×D412yes—
92C2×Dic131C2×D412yes—
93C2×Dic131#4114yes—
94C23×C131—22no—
95C23×C131—22no—
96C23×C131C2×D412yes—
97C23×C131#4112yes—
98C23×C131#4112yes—
99C23×C131C2411yescharacteristic
100C2×C4×C131—28no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—21no—
104C2×C4×C131—21no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131#4114yes—
111C22×C131—22no—
112C22×C131—22no—
113C22×C131—24no—
114C22×C131—24no—
115C22×C131#12314yes—
116C22×C131C22×D411yescharacteristic
117C22×C131C2×#4112yes—
118C22×C131C2×#4112yes—
119C22×C131C2×#4111yescharacteristic
120C22×C131C2×#4111yescharacteristic
121C2×D131—41no—
122C2×D131—41no—
123C2×D131—41no—
124C2×D131—42no—
125C2×D131—42no—
126C4×C131—24no—
127C4×C131—41no—
128C4×C131—41no—
129C4×C131—42no—
130Dic131—22no—
131Dic131—22no—
132Dic131—24no—
133Dic131—41no—
134Dic131—42no—
135C2×C131—24no—
136C2×C131C2×#12311yescharacteristic
137C2×C131#52511yescharacteristic
138C2×C131#52511yescharacteristic
139C2×C131#52912yes—
140C2×C131#52912yes—
141D131—81no—
142C131#285311yescharacteristic, complemented, Sylow
143#2853—1311nomaximal, Sylow
144C2×#123—1311no—
145C22×C42—1311no—
146#388—1312no—
147#388—1312no—
148#387—1312no—
149#387—1312no—
150#386—1311no—
151#386—1311no—
152#385—1311no—
153#385D13111yescharacteristic, complemented
154#381—1311no—
155C22×D4—1311no—
156C23×C4—1311no—
157C23×C4—1311no—
158C23×C4C2×D13111yescharacteristic
159#123—2624no—
160C2×C4⋊C4—1311no—
161C2×C4⋊C4—1311no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4C2×D13112yes—
164C2×C22⋊C4—1311no—
165C2×C22⋊C4—1311no—
166C2×C22⋊C4—1311no—
167C2×C22⋊C4—1312no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4C2×D13111yescharacteristic
170C2×C22⋊C4C2×D13111yescharacteristic
171C2×C42—2624no—
172C2×C42—1312no—
173C2×C42—1312no—
174#95—1312no—
175#95—1312no—
176#95—1312no—
177#95—1312no—
178#95—1312no—
179#95C2×D13112yes—
180C24—1311no—
181C24C22×D13111yescharacteristic
182C2×D4—2622no—
183C2×D4—2622no—
184C2×D4—2622no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—2622no—
188C22×C4—2622no—
189C22×C4—2622no—
190C22×C4—2624no—
191C22×C4—1312no—
192C22×C4—1311no—
193C22×C4—1314no—
194C22×C4—1312no—
195C22×C4—1311no—
196C22×C4—1311no—
197C22×C4C22×D13112yes—
198C22×C4C22×D13111yescharacteristic
199C22×C4C22×D13111yescharacteristic
200C22×C4C22×D13112yes—
201C4⋊C4—24no—
202C4⋊C4—2624no—
203C4⋊C4—2622no—
204C4⋊C4—2622no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2624no—
208C22⋊C4—2622no—
209C22⋊C4—2622no—
210C22⋊C4—2622no—
211C22⋊C4—2624no—
212C42—2624no—
213C42—2624no—
214C23—22no—
215C23—22no—
216C23—2621no—
217C23—2622no—
218C23—2622no—
219C23—2621no—
220C23—2621no—
221C23D4×D13112yes—
222C23#1452351312yes—
223C23#1452351412yes—
224C23C23×D13111yescharacteristic
225D4—5244no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—2628no—
237C2×C4—2628no—
238C2×C4—2622no—
239C2×C4—2622no—
240C2×C4—2621no—
241C2×C4—2621no—
242C2×C4—2621no—
243C2×C4—2621no—
244C2×C4—2621no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2621no—
248C2×C4—2622no—
249C2×C4—2621no—
250C2×C4—1312no—
251C2×C4—1312no—
252C2×C4—1314no—
253C2×C4#1452351214yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—5241no—
259C22—5241no—
260C22—5241no—
261C22—5242no—
262C22—5242no—
263C22#1686223114yes—
264C22C2×D4×D13111yescharacteristic
265C22#41×D13112yes—
266C22C2×#1452351212yes—
267C22C2×#1452351311yescharacteristic
268C22C2×#1452351411yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—2622no—
274C4—2622no—
275C4—2624no—
276C4—5241no—
277C4—5242no—
278C2—24no—
279C2—10481no—
280C2C2×#1686223111yescharacteristic
281C2#2255381811yescharacteristic
282C2#2255382011yescharacteristic
283C2#2255384212yes—
284C2#2255384812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#385×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#385D131
hypocenterC131#2853
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic131C2semidirect
3. #22553202C2semidirect
4. #22553201C2semidirect
5. #22553199C2semidirect
6. #22553198C2semidirect
9. #22553184C2semidirect
10. #22553173C2semidirect
11. #385×C131C2semidirect
142. C131#2853semidirect
153. #385D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622302
C22×C4×Dic1312
#225532022
#225532012
#225531992
#225531982
#225531912
#225531892
#225531842
#225531732
#385×C1312
#2853131