Finite Group Databasebeta

← #19139557

Subgroups — #19139557

Order 5504.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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×#13063297C212yesmaximal, complemented
2C2×C4×C22⋊D43C211yesmaximal, characteristic, complemented
3#14582830C211yesmaximal, characteristic, complemented
4#14582825C211yesmaximal, characteristic, complemented
5#14582813C211yesmaximal, characteristic, complemented
6#14582807C211yesmaximal, characteristic, complemented
7#14582787C212yesmaximal, complemented
8#14582779C212yesmaximal, complemented
9#14582775C212yesmaximal, complemented
10C2×C43⋊#95C211yesmaximal, characteristic, complemented
11C2×#121×C43C211yesmaximal, characteristic, complemented
12C22×C22⋊D43C2212yes—
13C22×C22⋊D43C2211yescharacteristic
14C23×Dic43C2212yes—
15C22×C4×D43C2211yescharacteristic
16#13063297—28no—
17C4×C22⋊D43—24no—
18C2×Dic43⋊C4C2212yes—
19C2×Dic43⋊C4C2211yescharacteristic
20C2×C22⋊Dic43C2212yes—
21C2×C22⋊Dic43C2212yes—
22C2×C22⋊Dic43C2211yescharacteristic
23C2×C22⋊Dic43C2212yes—
24C2×(C2×D43)⋊C4C2211yescharacteristic
25C2×(C2×D43)⋊C4C2212yes—
26C2×C4×Dic43C2211yescharacteristic
27C43⋊#95C2212yes—
28C43⋊#95C2212yes—
29C43⋊#95C2211yescharacteristic
30C43⋊#95C2211yescharacteristic
31C43⋊#95C2212yes—
32C22×D4×C43C2212yes—
33C22×D4×C43C2211yescharacteristic
34C23×C4×C43C2211yescharacteristic
35#121×C43—24no—
36C2×C4⋊C4×C43C2211yescharacteristic
37C2×C22⋊C4×C43C2212yes—
38C24×C43C2312yes—
39C24×C43C2311yescharacteristic
40C23×D43C2311yescharacteristic
41C2×D4×C43—22no—
42C2×D4×C43—24no—
43C2×D4×C43—24no—
44C2×D4×C43—24no—
45C2×D4×C43—42no—
46C22×C4×C43—42no—
47C22×C4×C43D414yescomplemented
48C22×C4×C43C2312yes—
49C22×C4×C43C2311yescharacteristic
50C22×C4×C43C2311yescharacteristic
51C2×C22⋊D43—24no—
52C2×C22⋊D43—22no—
53C2×C22⋊D43—22no—
54C2×C22⋊D43—24no—
55C2×C22⋊D43—42no—
56C2×C22⋊D43D414yescomplemented
57C22×Dic43—24no—
58C22×Dic43—24no—
59C22×Dic43—42no—
60C22×Dic43C2312yes—
61C22×Dic43C2311yescharacteristic
62C22×Dic43C2311yescharacteristic
63C22×Dic43C2311yescharacteristic
64C22×Dic43C2312yes—
65C2×C4×D43—22no—
66C2×C4×D43—42no—
67C4⋊C4×C43—22no—
68C22⋊C4×C43—24no—
69C22⋊Dic43—24no—
70C22⋊Dic43—24no—
71C22⋊Dic43—22no—
72C22⋊Dic43—42no—
73(C2×D43)⋊C4—22no—
74(C2×D43)⋊C4—24no—
75Dic43⋊C4—22no—
76Dic43⋊C4—24no—
77C4×Dic43—22no—
78C23×C43—22no—
79C23×C43—22no—
80C23×C43—22no—
81C23×C43—22no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43—22no—
85C23×C43—22no—
86C23×C43—22no—
87C23×C43C2×D412yes—
88C23×C43C2×D412yes—
89C23×C43#4112yes—
90C23×C43C2411yescharacteristic
91C22×D43—21no—
92C22×D43—22no—
93C22×D43—21no—
94C22×D43—22no—
95C22×D43C2×D412yes—
96D4×C43—44no—
97D4×C43—48no—
98C2×C4×C43—22no—
99C2×C4×C43—22no—
100C2×C4×C43—22no—
101C2×C4×C43—22no—
102C2×C4×C43—22no—
103C2×C4×C43—44no—
104C2×C4×C43C2×D412yes—
105C2×C4×C43C2×D412yes—
106C22⋊D43—28no—
107C22⋊D43—48no—
108C2×Dic43—21no—
109C2×Dic43—22no—
110C2×Dic43—21no—
111C2×Dic43—22no—
112C2×Dic43—21no—
113C2×Dic43—21no—
114C2×Dic43—22no—
115C2×Dic43—22no—
116C2×Dic43—22no—
117C2×Dic43—22no—
118C2×Dic43—22no—
119C2×Dic43—48no—
120C2×Dic43C2×D412yes—
121C2×Dic43#4114yes—
122C4×D43—44no—
123C22×C43—22no—
124C22×C43—22no—
125C22×C43—24no—
126C22×C43—24no—
127C22×C43—42no—
128C22×C43—42no—
129C22×C43—42no—
130C22×C43—42no—
131C22×C43—42no—
132C22×C43—42no—
133C22×C43—42no—
134C22×C43#12114yes—
135C22×C43C22×D411yescharacteristic
136C22×C43C22×D411yescharacteristic
137C22×C43C2×#4112yes—
138C22×C43C2×#4111yescharacteristic
139C22×C43H(5,𝔽2)12yes—
140C2×D43—22no—
141C2×D43—24no—
142C2×D43—41no—
143C2×D43—41no—
144C2×D43—42no—
145C4×C43—22no—
146C4×C43—22no—
147C4×C43—42no—
148Dic43—22no—
149Dic43—24no—
150Dic43—41no—
151Dic43—41no—
152Dic43—42no—
153#2543—431nomaximal, Sylow
154C2×C43—24no—
155C2×C43—82no—
156C2×C43C2×#12111yescharacteristic
157C2×C43#53211yescharacteristic
158C2×C43#53411yescharacteristic
159C2×C43#54612yes—
160C2×C43#55312yes—
161D43—42no—
162C2×#121—432no—
163C2×#121D4311yescharacteristic, complemented
164C2×C4×D4—431no—
165#398—431no—
166#397—431no—
167#394—431no—
168#392—431no—
169#388—432no—
170#386—432no—
171#386—432no—
172C2×#95—431no—
173C43#254311yescharacteristic, complemented, Sylow
174C22×D4—431no—
175C22×D4—432no—
176C22×D4C2×D4312yes—
177C22×D4C2×D4311yescharacteristic
178C23×C4—432no—
179C23×C4—431no—
180C23×C4C2×D4311yescharacteristic
181#121—24no—
182#121—868no—
183C4×D4—864no—
184C2×C4⋊C4—431no—
185C2×C4⋊C4—432no—
186C2×C4⋊C4C2×D4311yescharacteristic
187C2×C22⋊C4—431no—
188C2×C22⋊C4—432no—
189C2×C22⋊C4—432no—
190C2×C22⋊C4—431no—
191C2×C22⋊C4—432no—
192C2×C22⋊C4—432no—
193C2×C22⋊C4C2×D4312yes—
194C2×C42—431no—
195#95—432no—
196#95—431no—
197#95—431no—
198#95—432no—
199#95—432no—
200C24—431no—
201C24C22×D4312yes—
202C24C22×D4311yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—434no—
209C2×D4—864no—
210C2×D4—862no—
211C2×D4—862no—
212C2×D4—864no—
213C2×D4—1722no—
214C22×C4—42no—
215C22×C4—432no—
216C22×C4—431no—
217C22×C4—432no—
218C22×C4—431no—
219C22×C4—431no—
220C22×C4—864no—
221C22×C4—864no—
222C22×C4—862no—
223C22×C4—1722no—
224C22×C4—1722no—
225C22×C4C22⋊D4314yescomplemented
226C22×C4C22×D4312yes—
227C22×C4C22×D4311yescharacteristic
228C22×C4C22×D4311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—864no—
231C4⋊C4—862no—
232C22⋊C4—24no—
233C22⋊C4—864no—
234C22⋊C4—864no—
235C22⋊C4—864no—
236C22⋊C4—862no—
237C22⋊C4—862no—
238C22⋊C4—1722no—
239C42—862no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—432no—
250C23—861no—
251C23—862no—
252C23—861no—
253C23—862no—
254C23D4×D4312yes—
255C23#1062190012yes—
256C23C2×C22⋊D4312yes—
257C23C23×D4311yescharacteristic
258D4—44no—
259D4—48no—
260D4—868no—
261D4—1728no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—432no—
269C2×C4—434no—
270C2×C4—862no—
271C2×C4—861no—
272C2×C4—861no—
273C2×C4—862no—
274C2×C4—862no—
275C2×C4—861no—
276C2×C4—861no—
277C2×C4—862no—
278C2×C4—862no—
279C2×C4—862no—
280C2×C4—862no—
281C2×C4—1724no—
282C2×C4—1728no—
283C2×C4C2×C22⋊D4312yes—
284C2×C4C2×C22⋊D4312yes—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—862no—
297C22—864no—
298C22—1722no—
299C22—1721no—
300C22—1721no—
301C22#1306329614yes—
302C22C2×D4×D4311yescharacteristic
303C22C22×C22⋊D4311yescharacteristic
304C22#41×D4312yes—
305C22C2×#1062190011yescharacteristic
306C22#1306337012yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—862no—
311C4—864no—
312C4—1721no—
313C4—1721no—
314C4—1722no—
315C2—24no—
316C2—82no—
317C2—1722no—
318C2C2×#1306329611yescharacteristic
319C2#1458344511yescharacteristic
320C2#1458345811yescharacteristic
321C2#1458353812yes—
322C2#1458359212yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#121×C43C2
hypercenterC2×#121D43
hypocenterC43#2543
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13063297C2semidirect
2. C2×C4×C22⋊D43C2semidirect
3. #14582830C2semidirect
4. #14582825C2semidirect
5. #14582813C2semidirect
6. #14582807C2semidirect
7. #14582787C2semidirect
8. #14582779C2semidirect
9. #14582775C2semidirect
10. C2×C43⋊#95C2semidirect
11. C2×#121×C43C2semidirect
47. C22×C4×C43D4semidirect
56. C2×C22⋊D43D4semidirect
163. C2×#121D43semidirect
173. C43#2543semidirect
225. C22×C4C22⋊D43semidirect