Finite Group Databasebeta

← #19138198

Subgroups — #19138198

Order 5504.

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

normal subgroups only

This group has 914 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
1C22×#10621900C211yesmaximal, characteristic, complemented
2C2×#13063307C211yesmaximal, characteristic, complemented
3C2×#13063296C212yesmaximal, complemented
4#14582820C211yesmaximal, characteristic, complemented
5#14582813C211yesmaximal, characteristic, complemented
6#14582811C211yesmaximal, characteristic, complemented
7#14582779C212yesmaximal, complemented
8#14582775C212yesmaximal, complemented
9#14582774C212yesmaximal, complemented
10#95×D43C211yesmaximal, characteristic, complemented
11C43×#392C211yesmaximal, characteristic, complemented
12C2×#10621900—28no—
13C2×#10621900—24no—
14C22×Dic86C2211yescharacteristic
15C22×C22⋊D43C2211yescharacteristic
16C22×C22⋊D43C2212yes—
17C23×Dic43C2212yes—
18C22×C4×D43C2212yes—
19C22×C4×D43C2211yescharacteristic
20#13063307—24no—
21#13063296—28no—
22C2×C4⋊Dic43C2212yes—
23C2×C4⋊Dic43C2211yescharacteristic
24C2×Dic43⋊C4C2211yescharacteristic
25C2×C22⋊Dic43C2212yes—
26C2×C22⋊Dic43C2212yes—
27C2×C22⋊Dic43C2211yescharacteristic
28C2×C22⋊Dic43C2212yes—
29C2×(C2×D43)⋊C4C2211yescharacteristic
30C43⋊#95C2211yescharacteristic
31C43⋊#95C2212yes—
32C43⋊#95C2212yes—
33C43⋊#95C2212yes—
34C22×D4×C43C2211yescharacteristic
35C22×D4×C43C2212yes—
36C2×C22⋊C4×C43C2211yescharacteristic
37C2×C22⋊C4×C43C2212yes—
38#95×C43C2211yescharacteristic
39C24×C43C2311yescharacteristic
40C24×C43C2312yes—
41C23×D43C2311yescharacteristic
42C2×D4×C43—22no—
43C2×D4×C43—24no—
44C2×D4×C43—24no—
45C2×D4×C43—24no—
46C2×D4×C43—42no—
47C22×C4×C43C2311yescharacteristic
48C22×C4×C43C2311yescharacteristic
49C22×C4×C43C2312yes—
50C2×C22⋊D43—24no—
51C2×C22⋊D43—24no—
52C2×C22⋊D43—24no—
53C2×C22⋊D43—22no—
54C2×C22⋊D43D418yescomplemented
55C22×Dic43—24no—
56C22×Dic43—24no—
57C22×Dic43—24no—
58C22×Dic43C2312yes—
59C22×Dic43C2311yescharacteristic
60C22×Dic43C2311yescharacteristic
61C22×Dic43C2311yescharacteristic
62C22×Dic43C2312yes—
63#10621900—232no—
64C2×C4×D43—24no—
65C2×C4×D43—24no—
66C2×C4×D43—24no—
67C2×C4×D43—42no—
68C2×C4×D43D414yescomplemented
69C2×Dic86—22no—
70C2×Dic86—24no—
71C22⋊C4×C43—22no—
72C22⋊C4×C43—24no—
73C22⋊Dic43—24no—
74C22⋊Dic43—22no—
75C22⋊Dic43—24no—
76C22⋊Dic43—42no—
77(C2×D43)⋊C4—22no—
78C4⋊Dic43—24no—
79C4⋊Dic43—22no—
80Dic43⋊C4—22no—
81C23×C43—22no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43—22no—
85C23×C43—22no—
86C23×C43—22no—
87C23×C43—21no—
88C23×C43—22no—
89C23×C43—22no—
90C23×C43—21no—
91C23×C43—21no—
92C23×C43C2×D414yes—
93C23×C43C2411yescharacteristic
94C22×D43—22no—
95C22×D43—21no—
96C22×D43—21no—
97C22×D43C2×D412yes—
98C22×D43C2×D414yes—
99D4×C43—28no—
100D4×C43—48no—
101C2×C4×C43—22no—
102C2×C4×C43—22no—
103C2×C4×C43—22no—
104C2×C4×C43—21no—
105C2×C4×C43—22no—
106C2×C4×C43C2×D412yes—
107C2×C4×C43#4114yescomplemented
108C22⋊D43—216no—
109C22⋊D43—44no—
110C2×Dic43—216no—
111C2×Dic43—22no—
112C2×Dic43—22no—
113C2×Dic43—22no—
114C2×Dic43—21no—
115C2×Dic43—21no—
116C2×Dic43—22no—
117C2×Dic43—22no—
118C2×Dic43—22no—
119C2×Dic43—21no—
120C2×Dic43—22no—
121C2×Dic43—21no—
122C2×Dic43C2×D414yes—
123C2×Dic43C2×D412yes—
124C4×D43—28no—
125C4×D43—48no—
126Dic86—28no—
127C22×C43—24no—
128C22×C43—24no—
129C22×C43—24no—
130C22×C43—42no—
131C22×C43—42no—
132C22×C43—42no—
133C22×C43—42no—
134C22×C43—41no—
135C22×C43—41no—
136C22×C43—41no—
137C22×C43—42no—
138C22×C43—42no—
139C22×C43C22×D411yescharacteristic
140C22×C43C22×D412yes—
141C22×C43C2×#4112yes—
142C22×C43H(5,𝔽2)11yescharacteristic
143C22×C432−1+411yescharacteristic
144C2×D43—22no—
145C2×D43—22no—
146C2×D43—24no—
147C2×D43—24no—
148C2×D43#12014yes—
149C4×C43—22no—
150C4×C43—24no—
151C4×C43—41no—
152Dic43—24no—
153Dic43—22no—
154Dic43—41no—
155Dic43—41no—
156Dic43—42no—
157#2430—431nomaximal, Sylow
158C2×C43—44no—
159C2×C43—81no—
160C2×C43C2×#12011yescharacteristic
161C2×C43#53211yescharacteristic
162C2×C43#53711yescharacteristic
163C2×C43#54612yes—
164C2×C43#54712yes—
165D43—24no—
166C22×#41—431no—
167C2×#123—431no—
168C2×#121—432no—
169#395—431no—
170#394—431no—
171#393—431no—
172#392D4311yescharacteristic, complemented
173#386—432no—
174#386—432no—
175#385—432no—
176C2×#95—431no—
177C43#243011yescharacteristic, complemented, Sylow
178C2×#41—864no—
179C2×#41—868no—
180C22×Q8—431no—
181C22×D4—432no—
182C22×D4—431no—
183C22×D4C2×D4311yescharacteristic
184C22×D4C2×D4312yes—
185C23×C4—431no—
186C23×C4—432no—
187C23×C4—432no—
188#123—864no—
189#121—868no—
190C2×C4⋊C4—432no—
191C2×C4⋊C4—431no—
192C2×C4⋊C4—431no—
193C2×C22⋊C4—432no—
194C2×C22⋊C4—432no—
195C2×C22⋊C4—431no—
196C2×C22⋊C4—431no—
197C2×C22⋊C4—432no—
198C2×C22⋊C4C2×D4311yescharacteristic
199C2×C22⋊C4C2×D4312yes—
200#95—432no—
201#95—431no—
202#95—432no—
203#95—432no—
204#95C2×D4311yescharacteristic
205C24—431no—
206C24C22×D4311yescharacteristic
207C24C22×D4312yes—
208#41—8632no—
209C2×Q8—864no—
210C2×Q8—862no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—42no—
216C2×D4—438no—
217C2×D4—864no—
218C2×D4—864no—
219C2×D4—864no—
220C2×D4—862no—
221C22×C4—431no—
222C22×C4—432no—
223C22×C4—434no—
224C22×C4—432no—
225C22×C4—431no—
226C22×C4—431no—
227C22×C4—864no—
228C22×C4—864no—
229C22×C4—864no—
230C22×C4—864no—
231C22×C4—864no—
232C22×C4—864no—
233C22×C4—1722no—
234C22×C4C22×D4311yescharacteristic
235C22×C4C22×D4311yescharacteristic
236C22×C4C22×D4312yes—
237C4⋊C4—864no—
238C4⋊C4—862no—
239C4⋊C4—862no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—862no—
243C22⋊C4—864no—
244C22⋊C4—862no—
245C22⋊C4—864no—
246C22⋊C4—1722no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—432no—
259C23—434no—
260C23—862no—
261C23—861no—
262C23—861no—
263C23D4×D4314yes—
264C23C23×D4311yescharacteristic
265Q8—868no—
266D4—28no—
267D4—48no—
268D4—8616no—
269D4—1724no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—432no—
276C2×C4—434no—
277C2×C4—868no—
278C2×C4—8616no—
279C2×C4—862no—
280C2×C4—862no—
281C2×C4—862no—
282C2×C4—861no—
283C2×C4—862no—
284C2×C4—862no—
285C2×C4—861no—
286C2×C4—862no—
287C2×C4—861no—
288C2×C4—862no—
289C2×C4—861no—
290C2×C4—1728no—
291C2×C4D4×D4312yes—
292C2×C4#1062190014yescomplemented
293C22—24no—
294C22—24no—
295C22—24no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—434no—
306C22—862no—
307C22—862no—
308C22—864no—
309C22—864no—
310C22C2×D4×D4311yescharacteristic
311C22C2×D4×D4312yes—
312C22C2×#1062190012yes—
313C22#1306337011yescharacteristic
314C22#1306337211yescharacteristic
315C4—22no—
316C4—24no—
317C4—41no—
318C4—862no—
319C4—864no—
320C4—1721no—
321C4—1721no—
322C4—1722no—
323C2—44no—
324C2—81no—
325C2—864no—
326C2#120×D4311yescharacteristic
327C2#1458344711yescharacteristic
328C2#1458347811yescharacteristic
329C2#1458353612yes—
330C2#1458354912yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#392C2
hypercenter#392D43
hypocenterC43#2430
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#10621900C2semidirect
2. C2×#13063307C2semidirect
3. C2×#13063296C2semidirect
4. #14582820C2semidirect
5. #14582813C2semidirect
6. #14582811C2semidirect
7. #14582779C2semidirect
8. #14582775C2semidirect
9. #14582774C2semidirect
10. #95×D43C2semidirect
11. C43×#392C2semidirect
54. C2×C22⋊D43D4semidirect
68. C2×C4×D43D4semidirect
107. C2×C4×C43#41semidirect
172. #392D43semidirect
177. C43#2430semidirect
292. C2×C4#10621900semidirect