Finite Group Databasebeta

← #30674282

Subgroups — #30674282

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×#16862227C211yesmaximal, characteristic
2C22×C4×Dic131C211yesmaximal, characteristic, complemented
3#22553206C212yesmaximal, complemented
4#22553205C212yesmaximal, complemented
5#22553195C212yesmaximal, complemented
6#22553194C212yesmaximal, complemented
7#22553192C211yesmaximal, characteristic
8#22553187C211yesmaximal, characteristic, complemented
9#22553185C211yesmaximal, characteristic, complemented
10#22553173C211yesmaximal, characteristic, complemented
11#386×C131C211yesmaximal, characteristic
12C23×Dic131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14#16862227—24no—
15C2×C4⋊Dic131C2212yes—
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2211yescharacteristic
18C2×Dic131⋊C4C2211yescharacteristic
19C2×C22⋊Dic131C2211yescharacteristic
20C2×C22⋊Dic131C2211yescharacteristic
21C2×C22⋊Dic131C2212yes—
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2211yescharacteristic
24C2×C4×Dic131—24no—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C22×D4×C131C2211yescharacteristic
33C23×C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36#95×C131C2212yes—
37C22×Dic131—24no—
38C22×Dic131—22no—
39C22×Dic131—22no—
40C22×Dic131—22no—
41C22×Dic131D414yes—
42C22×Dic131C2312yes—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C4⋊Dic131—24no—
49Dic131⋊C4—22no—
50Dic131⋊C4—22no—
51Dic131⋊C4—24no—
52C22⋊Dic131—22no—
53C22⋊Dic131—22no—
54C22⋊Dic131—24no—
55C22⋊Dic131—22no—
56C22⋊Dic131—22no—
57C4×Dic131—24no—
58C4×Dic131—24no—
59C24×C131C2311yescharacteristic
60C24×C131C2311yescharacteristic
61C2×D4×C131—22no—
62C2×D4×C131—22no—
63C2×D4×C131—22no—
64C22×C4×C131—22no—
65C22×C4×C131—24no—
66C22×C4×C131C2311yescharacteristic
67C22×C4×C131C2312yes—
68C22×C4×C131C2312yes—
69C22⋊C4×C131—22no—
70C22⋊C4×C131—24no—
71C2×Dic131—28no—
72C2×Dic131—28no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—21no—
76C2×Dic131—21no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—21no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131C2×D412yes—
87C2×Dic131C2×D412yes—
88C2×Dic131#4114yes—
89C23×C131—22no—
90C23×C131—22no—
91C23×C131—21no—
92C23×C131—22no—
93C23×C131—21no—
94C23×C131—21no—
95C23×C131—22no—
96C23×C131C2×D412yes—
97C23×C131#4112yes—
98C23×C131#4112yes—
99C23×C131C2411yescharacteristic
100D4×C131—44no—
101C2×C4×C131—28no—
102C2×C4×C131—22no—
103C2×C4×C131—22no—
104C2×C4×C131—21no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—21no—
108C2×C4×C131—21no—
109C2×C4×C131—22no—
110C2×C4×C131#4114yes—
111C22×C131—22no—
112C22×C131—22no—
113C22×C131—24no—
114C22×C131—24no—
115C22×C131—41no—
116C22×C131—41no—
117C22×C131—41no—
118C22×C131—42no—
119C22×C131—42no—
120C22×C131#12314yes—
121C22×C131C22×D411yescharacteristic
122C22×C131C2×#4112yes—
123C22×C131C2×#4111yescharacteristic
124C22×C131C2×#4112yes—
125C22×C131C2×#4111yescharacteristic
126C4×C131—24no—
127C4×C131—41no—
128C4×C131—42no—
129Dic131—22no—
130Dic131—22no—
131Dic131—24no—
132Dic131—41no—
133Dic131—41no—
134Dic131—42no—
135C2×C131—24no—
136C2×C131—81no—
137C2×C131C2×#12311yescharacteristic
138C2×C131#52511yescharacteristic
139C2×C131#52511yescharacteristic
140C2×C131#52912yes—
141C2×C131#52912yes—
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#386D13111yescharacteristic
152#385—1311no—
153#385—1311no—
154#381—1311no—
155C22×D4C2×D13111yescharacteristic
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⋊C4—1312no—
164C2×C22⋊C4—1311no—
165C2×C22⋊C4—1311no—
166C2×C22⋊C4—1311no—
167C2×C22⋊C4—1311no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4C2×D13111yescharacteristic
170C2×C22⋊C4C2×D13112yes—
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—
180C24C22×D13111yescharacteristic
181C24C22×D13111yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
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—1311no—
194C22×C4—1314no—
195C22×C4—1312no—
196C22×C4—1311no—
197C22×C4—1311no—
198C22×C4C22×D13112yes—
199C22×C4C22×D13111yescharacteristic
200C22×C4C22×D13112yes—
201C4⋊C4—2622no—
202C4⋊C4—2622no—
203C4⋊C4—2624no—
204C4⋊C4—2624no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—2622no—
208C22⋊C4—2622no—
209C22⋊C4—2624no—
210C22⋊C4—2622no—
211C22⋊C4—2622no—
212C42—2624no—
213C42—2624no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D13112yes—
222C23#1452351212yes—
223C23#1452351312yes—
224C23C23×D13111yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—2628no—
236C2×C4—2628no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—2621no—
240C2×C4—2621no—
241C2×C4—2621no—
242C2×C4—2621no—
243C2×C4—2621no—
244C2×C4—2621no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2622no—
248C2×C4—2622no—
249C2×C4—2622no—
250C2×C4—1312no—
251C2×C4—1312no—
252C2×C4—1314no—
253C2×C4#1452351214yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1686222814yes—
264C22C2×D4×D13111yescharacteristic
265C22#41×D13112yes—
266C22C2×#1452351211yescharacteristic
267C22C2×#1452351212yes—
268C22C2×#1452351311yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—2622no—
273C4—2622no—
274C4—2624no—
275C4—5241no—
276C4—5241no—
277C4—5242no—
278C2—24no—
279C2—81no—
280C2C2×#1686222811yescharacteristic
281C2#2255381811yescharacteristic
282C2#2255382011yescharacteristic
283C2#2255384212yes—
284C2#2255384312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#386×C131C2
prime radicalC22×D4×C131C22
hypercenter#386D131
hypocenterC131#2853
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic131C2semidirect
3. #22553206C2semidirect
4. #22553205C2semidirect
5. #22553195C2semidirect
6. #22553194C2semidirect
8. #22553187C2semidirect
9. #22553185C2semidirect
10. #22553173C2semidirect
142. C131#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#168622272
C22×C4×Dic1312
#225532062
#225532052
#225531952
#225531942
#225531922
#225531872
#225531852
#225531732
#386×C1312
#2853131