Finite Group Databasebeta

← #30670407

Subgroups — #30670407

Order 16768.

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

normal subgroups only

This group has 546 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×#16862205C211yesmaximal, characteristic, complemented
2#22553251C211yesmaximal, characteristic, complemented
3#22553248C211yesmaximal, characteristic, complemented
4#22553242C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553200C212yesmaximal, complemented
7#22553193C212yesmaximal, complemented
8#22553192C212yesmaximal
9#22553170C212yesmaximal, complemented
10C2×C131⋊#95C211yesmaximal, characteristic, complemented
11C2×#123×C131C211yesmaximal, characteristic
12C23×Dic131C2212yes—
13#16862205—24no—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×C22⋊Dic131C2212yes—
19C2×C22⋊Dic131C2212yes—
20C2×C22⋊Dic131C2212yes—
21C2×C4×Dic131C2212yes—
22C131⋊#95C2212yes—
23C131⋊#95C2211yescharacteristic
24C131⋊#95C2211yescharacteristic
25C131⋊#95C2212yes—
26C131⋊#95C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C22×D4×C131C2211yescharacteristic
30C23×C4×C131C2211yescharacteristic
31#123×C131—24no—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2211yescharacteristic
37C22×Dic131—24no—
38C22×Dic131—24no—
39C22×Dic131—42no—
40C22×Dic131C2312yes—
41C22×Dic131C2312yes—
42C22×Dic131C2312yes—
43C22×Dic131C2312yes—
44C4⋊Dic131—22no—
45C4⋊Dic131—22no—
46Dic131⋊C4—24no—
47Dic131⋊C4—42no—
48C22⋊Dic131—24no—
49C22⋊Dic131—24no—
50C22⋊Dic131—42no—
51C4×Dic131—24no—
52C24×C131C2311yescharacteristic
53C24×C131C2311yescharacteristic
54C2×D4×C131—22no—
55C2×D4×C131—24no—
56C22×C4×C131—24no—
57C22×C4×C131D414yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2311yescharacteristic
60C22×C4×C131C2311yescharacteristic
61C22×C4×C131C2311yescharacteristic
62C22×C4×C131C2311yescharacteristic
63C4⋊C4×C131—22no—
64C4⋊C4×C131—22no—
65C22⋊C4×C131—22no—
66C22⋊C4×C131—22no—
67C22⋊C4×C131—22no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131—22no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—48no—
80C2×Dic131#4114yes—
81C23×C131—22no—
82C23×C131—21no—
83C23×C131—21no—
84C23×C131—21no—
85C23×C131—22no—
86C23×C131—22no—
87C23×C131—21no—
88C23×C131—21no—
89C23×C131C2×D412yes—
90C23×C131#4114yes—
91C23×C131C2411yescharacteristic
92D4×C131—44no—
93C2×C4×C131—28no—
94C2×C4×C131—22no—
95C2×C4×C131—22no—
96C2×C4×C131—21no—
97C2×C4×C131—21no—
98C2×C4×C131—21no—
99C2×C4×C131—22no—
100C2×C4×C131—22no—
101C2×C4×C131—21no—
102C2×C4×C131—21no—
103C2×C4×C131C2×D412yes—
104C2×C4×C131C2×D412yes—
105C22×C131—22no—
106C22×C131—22no—
107C22×C131—24no—
108C22×C131—24no—
109C22×C131—41no—
110C22×C131—41no—
111C22×C131—41no—
112C22×C131—42no—
113C22×C131—42no—
114C22×C131#12314yes—
115C22×C131C22×D411yescharacteristic
116C22×C131C2×#4112yes—
117C22×C131C2×#4112yes—
118C22×C131H(5,𝔽2)11yescharacteristic
119C22×C1312−1+411yescharacteristic
120C4×C131—22no—
121C4×C131—22no—
122C4×C131—41no—
123C4×C131—41no—
124C4×C131—41no—
125Dic131—24no—
126Dic131—42no—
127Dic131—42no—
128Dic131—42no—
129C2×C131—24no—
130C2×C131—81no—
131C2×C131C2×#12311yescharacteristic
132C2×C131#53211yescharacteristic
133C2×C131#53711yescharacteristic
134C2×C131#55112yes—
135C2×C131#55212yes—
136C131#249811yescharacteristic, complemented, Sylow
137#2498—1311nomaximal, Sylow
138C2×#123D13111yescharacteristic
139C2×#117—1311no—
140#400—1311no—
141#399—1311no—
142#398—1311no—
143#397—1311no—
144#388—1312no—
145#387—1312no—
146#386—1312no—
147#380—1312no—
148C2×#95—1311no—
149C22×D4C2×D13111yescharacteristic
150C23×C4—1312no—
151C23×C4C2×D13111yescharacteristic
152#123—24no—
153#117—2624no—
154C2×C4⋊C4—1311no—
155C2×C4⋊C4—1311no—
156C2×C4⋊C4—1312no—
157C2×C4⋊C4—1312no—
158C2×C4⋊C4C2×D13111yescharacteristic
159C2×C4⋊C4C2×D13111yescharacteristic
160C2×C22⋊C4—1312no—
161C2×C22⋊C4—1312no—
162C2×C22⋊C4—1312no—
163C2×C22⋊C4C2×D13111yescharacteristic
164C2×C22⋊C4C2×D13111yescharacteristic
165C2×C22⋊C4C2×D13111yescharacteristic
166C2×C42—1312no—
167#95—1312no—
168#95—1311no—
169#95—1311no—
170#95—1312no—
171#95—1312no—
172#95—1312no—
173#95—1312no—
174C24C22×D13111yescharacteristic
175C24C22×D13111yescharacteristic
176C2×D4—22no—
177C2×D4—24no—
178C22×C4—24no—
179C22×C4—2624no—
180C22×C4—2624no—
181C22×C4—5242no—
182C22×C4—1312no—
183C22×C4—1312no—
184C22×C4—1312no—
185C22×C4—1312no—
186C22×C4C22⋊D13114yes—
187C22×C4C22×D13111yescharacteristic
188C22×C4C22×D13111yescharacteristic
189C22×C4C22×D13111yescharacteristic
190C22×C4C22×D13111yescharacteristic
191C22×C4C22×D13111yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—2624no—
195C4⋊C4—2622no—
196C4⋊C4—2622no—
197C4⋊C4—5242no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—2624no—
202C22⋊C4—2624no—
203C22⋊C4—5242no—
204C42—2624no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23C2×C22⋊D13112yes—
214C23#1452351214yes—
215C23C23×D13111yescharacteristic
216D4—44no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—2622no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—2622no—
233C2×C4—2622no—
234C2×C4—2622no—
235C2×C4—2622no—
236C2×C4—2622no—
237C2×C4—2622no—
238C2×C4—5248no—
239C2×C4—1314no—
240C2×C4C2×C22⋊D13112yes—
241C2×C4C2×C22⋊D13112yes—
242C22—22no—
243C22—22no—
244C22—24no—
245C22—24no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—42no—
250C22—41no—
251C22#1686222714yes—
252C22C22×C22⋊D13111yescharacteristic
253C22#41×D13112yes—
254C22C2×#1452351212yes—
255C22#1686229211yescharacteristic
256C22#1686229311yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—2624no—
263C4—5242no—
264C4—5242no—
265C4—5242no—
266C2—24no—
267C2—81no—
268C2C2×#1686222711yescharacteristic
269C2#2255385811yescharacteristic
270C2#2255389011yescharacteristic
271C2#2255398512yes—
272C2#2255399912yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
prime radicalC22×D4×C131C22
hypercenterC2×#123D131
hypocenterC131#2498
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16862205C2semidirect
2. #22553251C2semidirect
3. #22553248C2semidirect
4. #22553242C2semidirect
5. #22553237C2semidirect
6. #22553200C2semidirect
7. #22553193C2semidirect
9. #22553170C2semidirect
10. C2×C131⋊#95C2semidirect
136. C131#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#168622052
#225532512
#225532482
#225532422
#225532372
#225532002
#225531932
#225531922
#225531702
C2×C131⋊#952
C2×#123×C1312
#2498131