Finite Group Databasebeta

← #30670398

Subgroups — #30670398

Order 16768.

546 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16862228C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic, complemented
3#22553250C211yesmaximal, characteristic, complemented
4#22553244C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553205C212yesmaximal, complemented
7#22553198C212yesmaximal, complemented
8#22553188C212yesmaximal
9#22553169C212yesmaximal, complemented
10#95×D131C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C22×C4×D131C2212yes—
14C22×C4×D131C2211yescharacteristic
15#16862228—24no—
16#16862204—24no—
17C2×C4⋊Dic131C2211yescharacteristic
18C2×C4⋊Dic131C2211yescharacteristic
19C2×C4⋊Dic131C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2212yes—
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×(C2×D131)⋊C4C2212yes—
25C2×C4×Dic131C2212yes—
26C131⋊#95C2212yes—
27C131⋊#95C2211yescharacteristic
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2212yes—
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2211yescharacteristic
37C23×D131C2311yescharacteristic
38C2×C22⋊D131—22no—
39C2×C22⋊D131—24no—
40C22×Dic131C2312yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2312yes—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C2×C4×D131—24no—
46C2×C4×D131—24no—
47C2×C4×D131—24no—
48C2×C4×D131—42no—
49C2×C4×D131D414yes—
50C4⋊Dic131—22no—
51C4⋊Dic131—22no—
52C4⋊Dic131—42no—
53Dic131⋊C4—22no—
54C22⋊Dic131—22no—
55C22⋊Dic131—42no—
56(C2×D131)⋊C4—22no—
57(C2×D131)⋊C4—24no—
58C4×Dic131—24no—
59C24×C131C2311yescharacteristic
60C22×C4×C131C2312yes—
61C22×C4×C131C2311yescharacteristic
62C22×C4×C131C2311yescharacteristic
63C22×C4×C131C2312yes—
64C4⋊C4×C131—22no—
65C4⋊C4×C131—24no—
66C22⋊C4×C131—22no—
67C22⋊C4×C131—24no—
68C22×D131—22no—
69C22×D131—21no—
70C22×D131—21no—
71C22×D131C2×D412yes—
72C22×D131#4114yes—
73C22⋊D131—44no—
74C2×Dic131—22no—
75C2×Dic131—22no—
76C2×Dic131—21no—
77C2×Dic131—21no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—21no—
84C2×Dic131—22no—
85C2×Dic131—21no—
86C2×Dic131C2×D412yes—
87C4×D131—28no—
88C4×D131—48no—
89C23×C131—21no—
90C23×C131—22no—
91C23×C131—22no—
92C23×C131—21no—
93C23×C131—21no—
94C23×C131C2411yescharacteristic
95C2×C4×C131—22no—
96C2×C4×C131—22no—
97C2×C4×C131—22no—
98C2×C4×C131—21no—
99C2×C4×C131—22no—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131C2×D412yes—
104C2×C4×C131#4114yes—
105C22×C131—41no—
106C22×C131—41no—
107C22×C131—41no—
108C22×C131—42no—
109C22×C131—42no—
110C22×C131C22×D411yescharacteristic
111C22×C131C2×#4112yes—
112C22×C131C2×#4112yes—
113C22×C131H(5,𝔽2)11yescharacteristic
114C22×C1312−1+411yescharacteristic
115C2×D131—22no—
116C2×D131—22no—
117C2×D131—24no—
118C2×D131—24no—
119C2×D131#12314yes—
120C4×C131—22no—
121C4×C131—24no—
122C4×C131—41no—
123C4×C131—42no—
124Dic131—22no—
125Dic131—41no—
126Dic131—42no—
127Dic131—41no—
128Dic131—42no—
129C2×C131—81no—
130C2×C131C2×#12311yescharacteristic
131C2×C131#53211yescharacteristic
132C2×C131#53711yescharacteristic
133C2×C131#55112yes—
134C2×C131#55212yes—
135D131—24no—
136C131#249811yescharacteristic, complemented, Sylow
137#2498—1311nomaximal, Sylow
138C2×#123—1311no—
139C2×#117—1311no—
140#400—1311no—
141#399D13111yescharacteristic, complemented
142#398—1311no—
143#397—1311no—
144#388—1312no—
145#387—1312no—
146#386—1312no—
147#380—1312no—
148C2×#95—1311no—
149C22×D4—1311no—
150C23×C4—1312no—
151C23×C4—1311no—
152#123—2624no—
153#117—2624no—
154C2×C4⋊C4—1311no—
155C2×C4⋊C4—1311no—
156C2×C4⋊C4—1311no—
157C2×C4⋊C4—1312no—
158C2×C4⋊C4C2×D13111yescharacteristic
159C2×C4⋊C4C2×D13112yes—
160C2×C22⋊C4—1311no—
161C2×C22⋊C4—1312no—
162C2×C22⋊C4—1311no—
163C2×C22⋊C4—1312no—
164C2×C22⋊C4C2×D13111yescharacteristic
165C2×C22⋊C4C2×D13112yes—
166C2×C42—1312no—
167#95—1312no—
168#95—1311no—
169#95—1312no—
170#95—1312no—
171#95—1312no—
172#95—1312no—
173#95C2×D13111yescharacteristic
174C24—1311no—
175C24C22×D13111yescharacteristic
176C2×D4—2622no—
177C2×D4—2624no—
178C22×C4—2624no—
179C22×C4—2624no—
180C22×C4—2624no—
181C22×C4—5242no—
182C22×C4—1312no—
183C22×C4—1311no—
184C22×C4—1312no—
185C22×C4—1311no—
186C22×C4—1311no—
187C22×C4—1314no—
188C22×C4C22×D13112yes—
189C22×C4C22×D13111yescharacteristic
190C22×C4C22×D13111yescharacteristic
191C22×C4C22×D13112yes—
192C4⋊C4—22no—
193C4⋊C4—24no—
194C4⋊C4—2622no—
195C4⋊C4—2622no—
196C4⋊C4—2622no—
197C4⋊C4—5242no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—2624no—
201C22⋊C4—2622no—
202C22⋊C4—2622no—
203C22⋊C4—5242no—
204C42—2624no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—2622no—
211C23—2621no—
212C23—2621no—
213C23—1312no—
214C23—1314no—
215C23C23×D13111yescharacteristic
216D4—5244no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—2628no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—2621no—
229C2×C4—2621no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—2622no—
233C2×C4—2622no—
234C2×C4—2622no—
235C2×C4—2621no—
236C2×C4—2622no—
237C2×C4—2621no—
238C2×C4—5248no—
239C2×C4—1312no—
240C2×C4D4×D13112yes—
241C2×C4#1452351214yes—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—2622no—
248C22—2622no—
249C22—2624no—
250C22—2624no—
251C22—1314no—
252C22C2×D4×D13111yescharacteristic
253C22#41×D13112yes—
254C22C2×#1452351212yes—
255C22#1686229211yescharacteristic
256C22#1686229411yescharacteristic
257C4—24no—
258C4—22no—
259C4—42no—
260C4—41no—
261C4—2622no—
262C4—5242no—
263C4—5241no—
264C4—5242no—
265C4—5241no—
266C2—2624no—
267C2—81no—
268C2#123×D13111yescharacteristic
269C2#2255386011yescharacteristic
270C2#2255389211yescharacteristic
271C2#2255398512yes—
272C2#2255399612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#399D131
hypocenterC131#2498
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#16862204C2semidirect
3. #22553250C2semidirect
4. #22553244C2semidirect
5. #22553237C2semidirect
6. #22553205C2semidirect
7. #22553198C2semidirect
9. #22553169C2semidirect
10. #95×D131C2semidirect
11. #399×C131C2semidirect
136. C131#2498semidirect
141. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622282
C2×#168622042
#225532502
#225532442
#225532372
#225532052
#225531982
#225531882
#225531692
#95×D1312
#399×C1312
#2498131