Finite Group Databasebeta

← #30671035

Subgroups — #30671035

Order 16768.

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

normal subgroups only

This group has 696 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×#16862232C212yesmaximal, complemented
2C2×#16862219C212yesmaximal, complemented
3#22553232C211yesmaximal, characteristic, complemented
4#22553229C211yesmaximal, characteristic, complemented
5#22553220C212yesmaximal, complemented
6#22553200C212yesmaximal, complemented
7#22553188C212yesmaximal, complemented
8C4×(C2×D131)⋊C4C212yesmaximal, complemented
9#390×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2211yescharacteristic
11C22×D524C2211yescharacteristic
12C22×C22⋊D131C2212yes—
13C22×C22⋊D131C2212yes—
14C22×C4×D131C2212yes—
15#16862232—28no—
16#16862219—28no—
17C2×Dic131⋊C4C2212yes—
18C2×C22⋊Dic131C2212yes—
19C2×C22⋊Dic131C2212yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×C4×Dic131C2212yes—
24C131⋊#95C2212yes—
25C131⋊#95C2212yes—
26C131⋊#95C2211yescharacteristic
27C131⋊#95C2211yescharacteristic
28C22×D4×C131C2211yescharacteristic
29C2×C4⋊C4×C131C2211yescharacteristic
30C2×C22⋊C4×C131C2212yes—
31C2×C22⋊C4×C131C2212yes—
32C2×C42×C131C2211yescharacteristic
33C23×D131C2312yes—
34C2×Dic262—24no—
35C2×Dic262—41no—
36C2×D524—24no—
37C2×D524—41no—
38C2×C22⋊D131—24no—
39C2×C22⋊D131—24no—
40C2×C22⋊D131—24no—
41C2×C22⋊D131—24no—
42C2×C22⋊D131—42no—
43C2×C22⋊D131—42no—
44C22×Dic131C2312yes—
45C22×Dic131C2312yes—
46C22×Dic131C2312yes—
47C2×C4×D131—24no—
48C2×C4×D131—24no—
49C2×C4×D131—42no—
50Dic131⋊C4—24no—
51C22⋊Dic131—24no—
52C22⋊Dic131—24no—
53(C2×D131)⋊C4—24no—
54(C2×D131)⋊C4—24no—
55(C2×D131)⋊C4D418yescomplemented
56C4×Dic131—24no—
57C24×C131C2312yes—
58C2×D4×C131—22no—
59C2×D4×C131—41no—
60C2×D4×C131—41no—
61C22×C4×C131C2311yescharacteristic
62C22×C4×C131C2311yescharacteristic
63C22×C4×C131C2312yes—
64C22×C4×C131C2311yescharacteristic
65C4⋊C4×C131—41no—
66C22⋊C4×C131—24no—
67C22⋊C4×C131—24no—
68C42×C131—41no—
69C22×D131—22no—
70C22×D131—22no—
71C22×D131—22no—
72C22×D131—22no—
73C22×D131—22no—
74C22×D131—22no—
75C22×D131C2×D414yes—
76Dic262—44no—
77D524—44no—
78C22⋊D131—48no—
79C22⋊D131—48no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131C2×D414yes—
88C2×Dic131#4114yes—
89C4×D131—48no—
90C23×C131—22no—
91C23×C131—22no—
92C23×C131—22no—
93C23×C131—22no—
94C23×C131—22no—
95C23×C131—22no—
96C23×C131—22no—
97C23×C131C2411yescharacteristic
98D4×C131—44no—
99C2×C4×C131—22no—
100C2×C4×C131—21no—
101C2×C4×C131—22no—
102C2×C4×C131—21no—
103C2×C4×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131C2×D414yes—
108C2×C4×C131#4112yes—
109C22×C131—42no—
110C22×C131—42no—
111C22×C131—42no—
112C22×C131—42no—
113C22×C131—42no—
114C22×C131—42no—
115C22×C131—42no—
116C22×C131C22×D412yes—
117C22×C131C2×#4112yes—
118C22×C131C2×#4111yescharacteristic
119C22×C131H(5,𝔽2)11yescharacteristic
120C22×C131H(5,𝔽2)11yescharacteristic
121C2×D131—24no—
122C2×D131—24no—
123C2×D131—24no—
124C2×D131—42no—
125C2×D131—42no—
126C2×D131—42no—
127C2×D131—42no—
128C4×C131—22no—
129C4×C131—24no—
130C4×C131—41no—
131C4×C131—41no—
132Dic131—24no—
133Dic131—24no—
134Dic131—42no—
135C2×C131—82no—
136C2×C131#53212yes—
137C2×C131#53811yescharacteristic
138C2×C131D4211yescharacteristic
139C2×C131#54612yes—
140C2×C131#55511yescharacteristic
141D131—44no—
142C131#255511yescharacteristic, complemented, Sylow
143#2555—1311nomaximal, Sylow
144C2×#124—1312no—
145C2×#121—1312no—
146#395—1311no—
147#394—1311no—
148#392—1312no—
149#390D13111yescharacteristic, complemented
150#388—1312no—
151#386—1312no—
152C4×C22⋊C4—1312no—
153C22×Q8—1311no—
154C22×D4—1312no—
155C22×D4—1312no—
156C22×D4—1311no—
157C22×D4C2×D13111yescharacteristic
158C23×C4—1312no—
159#124—2628no—
160#121—2628no—
161C2×C4⋊C4—1312no—
162C2×C4⋊C4C2×D13111yescharacteristic
163C2×C22⋊C4—1312no—
164C2×C22⋊C4—1312no—
165C2×C22⋊C4—1312no—
166C2×C22⋊C4—1312no—
167C2×C22⋊C4—1312no—
168C2×C22⋊C4C2×D13112yes—
169C2×C22⋊C4C2×D13112yes—
170C2×C42—1312no—
171C2×C42C2×D13111yescharacteristic
172#95—1312no—
173#95—1312no—
174#95—1311no—
175#95—1311no—
176C24—1312no—
177C24C22×D13112yes—
178C2×Q8—2624no—
179C2×Q8—5241no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—2624no—
184C2×D4—2624no—
185C2×D4—2624no—
186C2×D4—2624no—
187C2×D4—2624no—
188C2×D4—5241no—
189C2×D4—5242no—
190C2×D4—5242no—
191C22×C4—2624no—
192C22×C4—2624no—
193C22×C4—5242no—
194C22×C4—1312no—
195C22×C4—1312no—
196C22×C4—1312no—
197C22×C4C22×D13111yescharacteristic
198C22×C4C22×D13111yescharacteristic
199C22×C4C22×D13112yes—
200C22×C4C22×D13111yescharacteristic
201C4⋊C4—41no—
202C4⋊C4—2624no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—2624no—
206C22⋊C4—2624no—
207C22⋊C4—2624no—
208C22⋊C4—2624no—
209C22⋊C4—1318no—
210C42—41no—
211C42—2624no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—2622no—
220C23—2622no—
221C23—2622no—
222C23—2622no—
223C23—2622no—
224C23—2622no—
225C23—1314no—
226C23C23×D13111yescharacteristic
227Q8—5244no—
228D4—44no—
229D4—5244no—
230D4—5248no—
231D4—5248no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—2622no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2622no—
244C2×C4—2622no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—5248no—
248C2×C4—1314no—
249C2×C4—1314no—
250C2×C4D4×D13114yes—
251C2×C4#1452351312yes—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—2624no—
260C22—2624no—
261C22—2624no—
262C22—5242no—
263C22—5242no—
264C22—5242no—
265C22—5242no—
266C22C2×D4×D13112yes—
267C22#41×D13112yes—
268C22C2×#1452351311yescharacteristic
269C22#1686229111yescharacteristic
270C22#1686229111yescharacteristic
271C4—22no—
272C4—24no—
273C4—41no—
274C4—41no—
275C4—2624no—
276C4—2624no—
277C4—5242no—
278C2—82no—
279C2—5244no—
280C2#2255385912yes—
281C2#2255390011yescharacteristic
282C2#2255394711yescharacteristic
283C2#2255395112yes—
284C2#2255402411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#390×C131C2
hypercenter#390D131
hypocenterC131#2555
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16862232C2semidirect
2. C2×#16862219C2semidirect
3. #22553232C2semidirect
4. #22553229C2semidirect
5. #22553220C2semidirect
6. #22553200C2semidirect
7. #22553188C2semidirect
8. C4×(C2×D131)⋊C4C2semidirect
9. #390×C131C2semidirect
55. (C2×D131)⋊C4D4semidirect
142. C131#2555semidirect
149. #390D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622322
C2×#168622192
#225532322
#225532292
#225532202
#225532002
#225531882
C4×(C2×D131)⋊C42
#390×C1312
#2555131