Finite Group Databasebeta

← #19140734

Subgroups — #19140734

Order 5504.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13063319C211yesmaximal, characteristic, complemented
2#14582822C212yesmaximal, complemented
3#14582818C211yesmaximal, characteristic, complemented
4#14582815C212yesmaximal, complemented
5#14582813C211yesmaximal, characteristic, complemented
6#14582810C211yesmaximal, characteristic, complemented
7#14582807C211yesmaximal, characteristic
8#14582799C212yesmaximal, complemented
9#14582793C212yesmaximal, complemented
10C4×C4⋊Dic43C211yesmaximal, characteristic, complemented
11C2×#124×C43C211yesmaximal, characteristic, complemented
12C22×C22⋊D43C2212yes—
13#13063319—24no—
14C2×C4⋊Dic43C2212yes—
15C2×Dic43⋊C4C2211yescharacteristic
16C2×Dic43⋊C4C2211yescharacteristic
17C2×Dic43⋊C4C2211yescharacteristic
18C2×Dic43⋊C4C2212yes—
19C2×C22⋊Dic43C2212yes—
20C2×C22⋊Dic43C2212yes—
21C2×C22⋊Dic43C2212yes—
22C2×(C2×D43)⋊C4C2211yescharacteristic
23C2×(C2×D43)⋊C4C2211yescharacteristic
24C2×(C2×D43)⋊C4C2211yescharacteristic
25C2×(C2×D43)⋊C4C2212yes—
26C2×C4×Dic43C2212yes—
27C43⋊#95C2212yes—
28C43⋊#95C2212yes—
29C43⋊#95C2211yescharacteristic
30C43⋊#95C2211yescharacteristic
31C22×Q8×C43C2211yescharacteristic
32C22×D4×C43C2211yescharacteristic
33#124×C43—24no—
34C2×C22⋊C4×C43C2212yes—
35C2×C22⋊C4×C43C2212yes—
36C2×C42×C43C2211yescharacteristic
37C24×C43C2312yes—
38C23×D43C2311yescharacteristic
39C2×Q8×C43—22no—
40C2×Q8×C43—42no—
41C2×D4×C43—22no—
42C2×D4×C43—42no—
43C22×C4×C43C2312yes—
44C22×C4×C43C2311yescharacteristic
45C22×C4×C43C2311yescharacteristic
46C22×C4×C43C2311yescharacteristic
47C2×C22⋊D43—24no—
48C2×C22⋊D43—42no—
49C2×C22⋊D43—42no—
50C22×Dic43C2312yes—
51C22×Dic43C2311yescharacteristic
52C22×Dic43C2311yescharacteristic
53C22×Dic43C2311yescharacteristic
54C22×Dic43C2312yes—
55C22⋊C4×C43—24no—
56C22⋊C4×C43—24no—
57C42×C43D414yescomplemented
58C22⋊Dic43—24no—
59C22⋊Dic43—42no—
60C22⋊Dic43—42no—
61(C2×D43)⋊C4—22no—
62(C2×D43)⋊C4—22no—
63(C2×D43)⋊C4—22no—
64(C2×D43)⋊C4—42no—
65C4⋊Dic43—24no—
66Dic43⋊C4—22no—
67Dic43⋊C4—22no—
68Dic43⋊C4—22no—
69Dic43⋊C4—42no—
70C4×Dic43—42no—
71C23×C43—22no—
72C23×C43—22no—
73C23×C43—22no—
74C23×C43—22no—
75C23×C43—22no—
76C23×C43—22no—
77C23×C43—22no—
78C23×C43C2411yescharacteristic
79C22×D43—21no—
80C22×D43—22no—
81C22×D43—22no—
82C22×D43—21no—
83C22×D43—21no—
84Q8×C43—44no—
85D4×C43—44no—
86C2×C4×C43—22no—
87C2×C4×C43—21no—
88C2×C4×C43—22no—
89C2×C4×C43—21no—
90C2×C4×C43—22no—
91C2×C4×C43—22no—
92C2×C4×C43—22no—
93C2×C4×C43C2×D412yes—
94C2×C4×C43C2×D412yes—
95C2×C4×C43C2×D412yes—
96C22⋊D43—48no—
97C2×Dic43—22no—
98C2×Dic43—22no—
99C2×Dic43—21no—
100C2×Dic43—21no—
101C2×Dic43—21no—
102C2×Dic43—22no—
103C2×Dic43—22no—
104C2×Dic43—22no—
105C2×Dic43—22no—
106C2×Dic43—22no—
107C2×Dic43—21no—
108C2×Dic43—21no—
109C2×Dic43#4114yes—
110C22×C43—42no—
111C22×C43—42no—
112C22×C43—42no—
113C22×C43—42no—
114C22×C43—42no—
115C22×C43—42no—
116C22×C43—42no—
117C22×C43C22×D411yescharacteristic
118C22×C43C2×#4112yes—
119C22×C43H(5,𝔽2)12yes—
120C22×C43H(5,𝔽2)11yescharacteristic
121C22×C432−1+411yescharacteristic
122C2×D43—41no—
123C2×D43—41no—
124C2×D43—41no—
125C2×D43—42no—
126C2×D43—42no—
127C4×C43—22no—
128C4×C43—22no—
129C4×C43—22no—
130C4×C43—42no—
131Dic43—24no—
132Dic43—41no—
133Dic43—41no—
134Dic43—41no—
135Dic43—42no—
136#2662—431nomaximal, Sylow
137C2×C43—82no—
138C2×C43#53211yescharacteristic
139C2×C43#53511yescharacteristic
140C2×C43#53711yescharacteristic
141C2×C43#53812yes—
142C2×C43#54212yes—
143D43—81no—
144C2×#126—431no—
145C2×#124D4311yescharacteristic, complemented
146#396—432no—
147#395—431no—
148#394—431no—
149#394—432no—
150#393—431no—
151#392—431no—
152#390—432no—
153#388—432no—
154C4×C4⋊C4—431no—
155C43#266211yescharacteristic, complemented, Sylow
156C22×Q8C2×D4311yescharacteristic
157C22×D4—432no—
158C22×D4C2×D4311yescharacteristic
159#126—864no—
160#124—24no—
161C2×C4⋊C4—431no—
162C2×C4⋊C4—431no—
163C2×C4⋊C4—431no—
164C2×C4⋊C4—432no—
165C2×C4⋊C4—432no—
166C2×C22⋊C4—432no—
167C2×C22⋊C4—431no—
168C2×C22⋊C4—431no—
169C2×C22⋊C4—431no—
170C2×C22⋊C4—432no—
171C2×C22⋊C4—432no—
172C2×C22⋊C4—432no—
173C2×C22⋊C4C2×D4312yes—
174C2×C22⋊C4C2×D4312yes—
175C2×C42—432no—
176C2×C42C2×D4311yescharacteristic
177#95—431no—
178#95—431no—
179#95—432no—
180#95—432no—
181C24—431no—
182C24C22×D4312yes—
183C2×Q8—22no—
184C2×Q8—42no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—864no—
188C2×D4—1722no—
189C2×D4—1722no—
190C22×C4—432no—
191C22×C4—432no—
192C22×C4—431no—
193C22×C4—431no—
194C22×C4—431no—
195C22×C4C22×D4312yes—
196C22×C4C22×D4311yescharacteristic
197C22×C4C22×D4311yescharacteristic
198C22×C4C22×D4311yescharacteristic
199C4⋊C4—864no—
200C4⋊C4—862no—
201C4⋊C4—862no—
202C4⋊C4—862no—
203C4⋊C4—1722no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—864no—
207C22⋊C4—862no—
208C22⋊C4—862no—
209C22⋊C4—862no—
210C22⋊C4—1722no—
211C22⋊C4—1722no—
212C22⋊C4—1722no—
213C42—1722no—
214C42C22⋊D4314yescomplemented
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—861no—
223C23—862no—
224C23—862no—
225C23—861no—
226C23—861no—
227C23C23×D4311yescharacteristic
228Q8—44no—
229D4—44no—
230D4—1728no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—434no—
239C2×C4—862no—
240C2×C4—862no—
241C2×C4—861no—
242C2×C4—861no—
243C2×C4—861no—
244C2×C4—862no—
245C2×C4—862no—
246C2×C4—862no—
247C2×C4—862no—
248C2×C4—862no—
249C2×C4—861no—
250C2×C4—861no—
251C2×C4C2×C22⋊D4312yes—
252C2×C4C2×C22⋊D4312yes—
253C2×C4C2×C22⋊D4312yes—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—1721no—
262C22—1721no—
263C22—1721no—
264C22—1722no—
265C22—1722no—
266C22C22×C22⋊D4311yescharacteristic
267C22#41×D4312yes—
268C22#1306337012yes—
269C22#1306337111yescharacteristic
270C22#1306337211yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—864no—
276C4—1721no—
277C4—1721no—
278C4—1721no—
279C4—1722no—
280C2—82no—
281C2—3441no—
282C2#1458344511yescharacteristic
283C2#1458346611yescharacteristic
284C2#1458347711yescharacteristic
285C2#1458348312yes—
286C2#1458351312yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#124×C43C2
hypercenterC2×#124D43
hypocenterC43#2662
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13063319C2semidirect
2. #14582822C2semidirect
3. #14582818C2semidirect
4. #14582815C2semidirect
5. #14582813C2semidirect
6. #14582810C2semidirect
8. #14582799C2semidirect
9. #14582793C2semidirect
10. C4×C4⋊Dic43C2semidirect
11. C2×#124×C43C2semidirect
57. C42×C43D4semidirect
145. C2×#124D43semidirect
155. C43#2662semidirect
214. C42C22⋊D43semidirect