Finite Group Databasebeta

← #30670422

Subgroups — #30670422

Order 16768.

568 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16862217C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic, complemented
3#22553227C214yesmaximal, complemented
4#22553215C211yesmaximal, characteristic, complemented
5#22553206C214yesmaximal, complemented
6#22553181C211yesmaximal, characteristic, complemented
7#22553178C211yesmaximal, characteristic, complemented
8C4×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
9#390×C131C211yesmaximal, characteristic, complemented
10C22×C22⋊D131C2212yes—
11C22×C4×D131C2211yescharacteristic
12#16862217—24no—
13#16862204—24no—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2214yes—
17C2×C22⋊Dic131C2212yes—
18C2×C22⋊Dic131C2214yes—
19C2×(C2×D131)⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×C4×Dic131C2211yescharacteristic
22C2×C4×Dic131C2212yes—
23C131⋊#95C2212yes—
24C131⋊#95C2214yes—
25C22×D4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C22⋊C4×C131C2214yes—
28C2×C42×C131C2211yescharacteristic
29C23×D131C2311yescharacteristic
30C2×C22⋊D131—24no—
31C2×C22⋊D131—44no—
32C22×Dic131C2312yes—
33C22×Dic131C2312yes—
34C22×Dic131C2311yescharacteristic
35C22×Dic131C2312yes—
36C2×C4×D131—22no—
37C2×C4×D131—22no—
38C2×C4×D131D414yes—
39C4⋊Dic131—22no—
40C4⋊Dic131—22no—
41Dic131⋊C4—44no—
42C22⋊Dic131—24no—
43C22⋊Dic131—28no—
44(C2×D131)⋊C4—24no—
45(C2×D131)⋊C4—24no—
46C4×Dic131—24no—
47C4×Dic131—22no—
48C24×C131C2312yes—
49C2×D4×C131—22no—
50C2×D4×C131—22no—
51C2×D4×C131—22no—
52C22×C4×C131C2312yes—
53C22×C4×C131C2312yes—
54C22×C4×C131C2311yescharacteristic
55C4⋊C4×C131—22no—
56C22⋊C4×C131—44no—
57C42×C131—22no—
58C22×D131—22no—
59C22×D131—21no—
60C22×D131—21no—
61C22×D131—22no—
62C22×D131C2×D412yes—
63C22⋊D131—48no—
64C2×Dic131—22no—
65C2×Dic131—21no—
66C2×Dic131—21no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—24no—
71C2×Dic131—24no—
72C2×Dic131C2×D412yes—
73C2×Dic131#4114yes—
74C4×D131—28no—
75C23×C131—22no—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131—24no—
79C23×C131—24no—
80C23×C131C2411yescharacteristic
81D4×C131—44no—
82C2×C4×C131—22no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—24no—
87C2×C4×C131C2×D412yes—
88C2×C4×C131#4112yes—
89C2×C4×C131#4112yes—
90C22×C131—42no—
91C22×C131—42no—
92C22×C131—42no—
93C22×C131—44no—
94C22×C131—44no—
95C22×C131C22×D411yescharacteristic
96C22×C131C2×#4112yes—
97C22×C131C2×#4111yescharacteristic
98C22×C131C2×#4111yescharacteristic
99C22×C131H(5,𝔽2)12yes—
100C2×D131—22no—
101C2×D131—22no—
102C2×D131—22no—
103C2×D131—42no—
104C2×D131—42no—
105C4×C131—42no—
106C4×C131—42no—
107C4×C131#12314yes—
108Dic131—22no—
109Dic131—24no—
110Dic131—42no—
111Dic131—42no—
112C2×C131—82no—
113C2×C131C2×#12311yescharacteristic
114C2×C131#53211yescharacteristic
115C2×C131#53511yescharacteristic
116C2×C131#55312yes—
117C2×C131#55512yes—
118D131—42no—
119C131#250011yescharacteristic, complemented, Sylow
120#2500—1311nomaximal, Sylow
121C2×#121—1311no—
122C2×#117—1311no—
123#394—1314no—
124#390—1311no—
125#390D13111yescharacteristic, complemented
126#388—1314no—
127#384—1311no—
128#382—1311no—
129C4×C22⋊C4—1311no—
130C22×D4—1312no—
131C22×D4C2×D13111yescharacteristic
132C23×C4—1311no—
133#121—2624no—
134#117—2624no—
135C2×C4⋊C4—1311no—
136C2×C4⋊C4—1311no—
137C2×C4⋊C4—1314no—
138C2×C4⋊C4C2×D13111yescharacteristic
139C2×C22⋊C4—1312no—
140C2×C22⋊C4—1314no—
141C2×C22⋊C4—1312no—
142C2×C22⋊C4—1312no—
143C2×C22⋊C4C2×D13114yes—
144C2×C42—1312no—
145C2×C42—1311no—
146C2×C42C2×D13111yescharacteristic
147#95—1312no—
148#95—1314no—
149C24—1311no—
150C24C22×D13112yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—2624no—
155C2×D4—5244no—
156C22×C4—2622no—
157C22×C4—2622no—
158C22×C4—1312no—
159C22×C4—1312no—
160C22×C4—1311no—
161C22×C4—1312no—
162C22×C4—1314no—
163C22×C4C22×D13112yes—
164C22×C4C22×D13112yes—
165C22×C4C22×D13111yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—2622no—
168C4⋊C4—2622no—
169C4⋊C4—5244no—
170C22⋊C4—44no—
171C22⋊C4—2624no—
172C22⋊C4—2624no—
173C22⋊C4—2628no—
174C22⋊C4—2624no—
175C42—22no—
176C42—2624no—
177C42—2622no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—22no—
182C23—24no—
183C23—2622no—
184C23—2621no—
185C23—2621no—
186C23—2622no—
187C23—1312no—
188C23C23×D13111yescharacteristic
189D4—44no—
190D4—5248no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—2621no—
197C2×C4—2622no—
198C2×C4—2621no—
199C2×C4—2622no—
200C2×C4—2628no—
201C2×C4—2622no—
202C2×C4—2622no—
203C2×C4—2624no—
204C2×C4—2624no—
205C2×C4—1312no—
206C2×C4—1314no—
207C2×C4D4×D13112yes—
208C2×C4#1452351212yes—
209C2×C4#1452351312yes—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—2622no—
216C22—2622no—
217C22—2622no—
218C22—5242no—
219C22—5242no—
220C22C2×D4×D13111yescharacteristic
221C22#41×D13112yes—
222C22C2×#1452351211yescharacteristic
223C22C2×#1452351311yescharacteristic
224C22#1686229112yes—
225C4—42no—
226C4—42no—
227C4—2624no—
228C4—2622no—
229C4—5242no—
230C4—5242no—
231C4#1686222814yes—
232C2—82no—
233C2—5242no—
234C2C2×#1686222811yescharacteristic
235C2#2255386111yescharacteristic
236C2#2255387511yescharacteristic
237C2#2255401212yes—
238C2#2255402612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#390×C131C2
prime radicalC2×#16862217C2
hypercenter#390D131
hypocenterC131#2500
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16862204C2semidirect
3. #22553227C2semidirect
4. #22553215C2semidirect
5. #22553206C2semidirect
6. #22553181C2semidirect
7. #22553178C2semidirect
8. C4×(C2×D131)⋊C4C2semidirect
9. #390×C131C2semidirect
119. C131#2500semidirect
125. #390D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622172
C2×#168622042
#225532272
#225532152
#225532062
#225531812
#225531782
C4×(C2×D131)⋊C42
#390×C1312
#2500131