Finite Group Databasebeta

← #19142597

Subgroups — #19142597

Order 5504.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13063299C211yesmaximal, characteristic, complemented
2#14582838C211yesmaximal, characteristic, complemented
3#14582835C211yesmaximal, characteristic, complemented
4#14582821C212yesmaximal, complemented
5#14582816C211yesmaximal, characteristic, complemented
6#14582815C212yesmaximal, complemented
7#14582814C212yesmaximal, complemented
8#14582808C211yesmaximal, characteristic, complemented
9#14582807C212yesmaximal, complemented
10#14582759C211yesmaximal, characteristic, complemented
11C2×#123×C43C211yesmaximal, characteristic, complemented
12C22×D172C2211yescharacteristic
13C22×C22⋊D43C2212yes—
14C22×C22⋊D43C2212yes—
15#13063299—24no—
16C2×C4⋊Dic43C2211yescharacteristic
17C2×C4⋊Dic43C2212yes—
18C2×Dic43⋊C4C2212yes—
19C2×C22⋊Dic43C2212yes—
20C2×C22⋊Dic43C2212yes—
21C2×(C2×D43)⋊C4C2212yes—
22C2×(C2×D43)⋊C4C2212yes—
23C2×(C2×D43)⋊C4C2212yes—
24C2×(C2×D43)⋊C4C2212yes—
25C43⋊#95C2212yes—
26C43⋊#95C2211yescharacteristic
27C43⋊#95C2212yes—
28C43⋊#95C2211yescharacteristic
29C22×D4×C43C2211yescharacteristic
30C23×C4×C43C2211yescharacteristic
31#123×C43—24no—
32C2×C4⋊C4×C43C2211yescharacteristic
33C2×C4⋊C4×C43C2211yescharacteristic
34C2×C22⋊C4×C43C2211yescharacteristic
35C2×C22⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2211yescharacteristic
37C24×C43C2311yescharacteristic
38C24×C43C2311yescharacteristic
39C23×D43C2312yes—
40C2×D4×C43—22no—
41C2×D4×C43—42no—
42C22×C4×C43—42no—
43C22×C4×C43D414yescomplemented
44C22×C4×C43C2311yescharacteristic
45C22×C4×C43C2311yescharacteristic
46C22×C4×C43C2311yescharacteristic
47C22×C4×C43C2311yescharacteristic
48C22×C4×C43C2311yescharacteristic
49C2×C22⋊D43—24no—
50C2×C22⋊D43—42no—
51C2×C22⋊D43—42no—
52C2×C22⋊D43—42no—
53C2×C22⋊D43—42no—
54C2×C22⋊D43—42no—
55C22×Dic43C2312yes—
56C22×Dic43C2312yes—
57C22×Dic43C2312yes—
58C2×D172—22no—
59C2×D172—42no—
60C4⋊C4×C43—22no—
61C4⋊C4×C43—22no—
62C22⋊C4×C43—22no—
63C22⋊C4×C43—22no—
64C22⋊C4×C43—22no—
65C22⋊Dic43—42no—
66C22⋊Dic43—42no—
67(C2×D43)⋊C4—24no—
68(C2×D43)⋊C4—42no—
69(C2×D43)⋊C4—42no—
70(C2×D43)⋊C4—42no—
71C4⋊Dic43—22no—
72C4⋊Dic43—42no—
73Dic43⋊C4—42no—
74C23×C43—21no—
75C23×C43—22no—
76C23×C43—22no—
77C23×C43—21no—
78C23×C43—21no—
79C23×C43—22no—
80C23×C43—21no—
81C23×C43—22no—
82C23×C43—21no—
83C23×C43C2×D412yes—
84C23×C43C2411yescharacteristic
85C22×D43—22no—
86C22×D43—22no—
87C22×D43—22no—
88C22×D43—22no—
89C22×D43—22no—
90C22×D43—22no—
91C22×D43—22no—
92D4×C43—44no—
93C2×C4×C43—22no—
94C2×C4×C43—22no—
95C2×C4×C43—21no—
96C2×C4×C43—21no—
97C2×C4×C43—21no—
98C2×C4×C43—22no—
99C2×C4×C43—21no—
100C2×C4×C43—22no—
101C2×C4×C43—21no—
102C2×C4×C43—44no—
103C2×C4×C43C2×D412yes—
104C2×C4×C43C2×D412yes—
105C22⋊D43—48no—
106C22⋊D43—48no—
107C2×Dic43—22no—
108C2×Dic43—22no—
109C2×Dic43—22no—
110C2×Dic43—22no—
111C2×Dic43—22no—
112C2×Dic43—22no—
113C2×Dic43—22no—
114C2×Dic43—22no—
115C2×Dic43—22no—
116D172—44no—
117C22×C43—22no—
118C22×C43—22no—
119C22×C43—22no—
120C22×C43—42no—
121C22×C43—42no—
122C22×C43—41no—
123C22×C43—41no—
124C22×C43—41no—
125C22×C43—42no—
126C22×C43—42no—
127C22×C43C22×D411yescharacteristic
128C22×C43H(5,𝔽2)12yes—
129C22×C43H(5,𝔽2)12yes—
130C22×C43H(5,𝔽2)11yescharacteristic
131C22×C432−1+411yescharacteristic
132C2×D43—42no—
133C2×D43—42no—
134C2×D43—42no—
135C2×D43—42no—
136C2×D43—42no—
137C2×D43—42no—
138C2×D43—42no—
139C4×C43—22no—
140C4×C43—22no—
141C4×C43—41no—
142C4×C43—41no—
143C4×C43—41no—
144Dic43—42no—
145Dic43—42no—
146Dic43—42no—
147#2813—431nomaximal, Sylow
148C2×C43—42no—
149C2×C43—81no—
150C2×C43#53411yescharacteristic
151C2×C43#53511yescharacteristic
152C2×C43#53711yescharacteristic
153C2×C43#56012yes—
154C2×C43#56212yes—
155D43—82no—
156C2×#123D4311yescharacteristic, complemented
157C2×#121—431no—
158#400—431no—
159#399—431no—
160#396—432no—
161#394—431no—
162#394—432no—
163#394—432no—
164#392—432no—
165#392—431no—
166#380—431no—
167C43#281311yescharacteristic, complemented, Sylow
168C22×D4—432no—
169C22×D4—432no—
170C22×D4—431no—
171C22×D4C2×D4311yescharacteristic
172C23×C4C2×D4311yescharacteristic
173#123—24no—
174#121—864no—
175C2×C4⋊C4—431no—
176C2×C4⋊C4—432no—
177C2×C4⋊C4—432no—
178C2×C4⋊C4C2×D4311yescharacteristic
179C2×C4⋊C4C2×D4311yescharacteristic
180C2×C22⋊C4—432no—
181C2×C22⋊C4—432no—
182C2×C22⋊C4—432no—
183C2×C22⋊C4—432no—
184C2×C22⋊C4—432no—
185C2×C22⋊C4—432no—
186C2×C22⋊C4C2×D4311yescharacteristic
187C2×C22⋊C4C2×D4311yescharacteristic
188C2×C22⋊C4C2×D4311yescharacteristic
189#95—431no—
190#95—431no—
191#95—432no—
192#95—432no—
193C24—432no—
194C24C22×D4311yescharacteristic
195C24C22×D4311yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—864no—
199C2×D4—862no—
200C2×D4—1722no—
201C2×D4—1722no—
202C2×D4—1722no—
203C2×D4—1722no—
204C2×D4—1722no—
205C2×D4—1722no—
206C22×C4—42no—
207C22×C4—432no—
208C22×C4—432no—
209C22×C4—432no—
210C22×C4C22⋊D4314yescomplemented
211C22×C4C22×D4311yescharacteristic
212C22×C4C22×D4311yescharacteristic
213C22×C4C22×D4311yescharacteristic
214C22×C4C22×D4311yescharacteristic
215C22×C4C22×D4311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—862no—
219C4⋊C4—1722no—
220C4⋊C4—1722no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—864no—
225C22⋊C4—1722no—
226C22⋊C4—1722no—
227C22⋊C4—1722no—
228C22⋊C4—1722no—
229C22⋊C4—1722no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—862no—
240C23—862no—
241C23—862no—
242C23—862no—
243C23—862no—
244C23—862no—
245C23—862no—
246C23C2×C22⋊D4312yes—
247C23C23×D4311yescharacteristic
248D4—44no—
249D4—1724no—
250D4—1728no—
251D4—1728no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—862no—
263C2×C4—862no—
264C2×C4—862no—
265C2×C4—862no—
266C2×C4—862no—
267C2×C4—862no—
268C2×C4—862no—
269C2×C4—862no—
270C2×C4—862no—
271C2×C4C2×C22⋊D4312yes—
272C2×C4C2×C22⋊D4312yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—1722no—
284C22—1722no—
285C22—1722no—
286C22—1722no—
287C22—1722no—
288C22—1722no—
289C22—1722no—
290C22C22×C22⋊D4311yescharacteristic
291C22#1306337012yes—
292C22#1306337111yescharacteristic
293C22#1306337112yes—
294C22#1306337211yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—1722no—
301C4—1722no—
302C4—1722no—
303C2—42no—
304C2—81no—
305C2—3442no—
306C2#1458345811yescharacteristic
307C2#1458346311yescharacteristic
308C2#1458347711yescharacteristic
309C2#1458363912yes—
310C2#1458364612yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#123×C43C2
hypercenterC2×#123D43
hypocenterC43#2813
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13063299C2semidirect
2. #14582838C2semidirect
3. #14582835C2semidirect
4. #14582821C2semidirect
5. #14582816C2semidirect
6. #14582815C2semidirect
7. #14582814C2semidirect
8. #14582808C2semidirect
9. #14582807C2semidirect
10. #14582759C2semidirect
11. C2×#123×C43C2semidirect
43. C22×C4×C43D4semidirect
156. C2×#123D43semidirect
167. C43#2813semidirect
210. C22×C4C22⋊D43semidirect