Finite Group Databasebeta

← #72231045

Subgroups — #72231045

Order 2400.

444 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 66 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×D5⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
2C2×D3×#1044C211yesmaximal, characteristic, complemented
3#11925419C211yesmaximal, characteristic
4#11925361C211yesmaximal, characteristic, complemented
5#11925354C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#1043C211yesmaximal, characteristic, complemented
7C3×#87831C211yesmaximal, characteristic, complemented
8#11717377—31nomaximal, Hall
9C2×D3×C5⋊D5C2211yescharacteristic
10C2×C3×D52C2211yescharacteristic
11C2×C3⋊#1044C2211yescharacteristic
12C2×C3⋊#1043C2211yescharacteristic
13C2×C3×#1044C2211yescharacteristic
14C2×C3×#1043C2211yescharacteristic
15D5⋊ΓL(1,𝔽16)—22no—
16D3×#1044—22no—
17#10597126C2211yescharacteristic
18#10597123C412yescomplemented
19C5×#1201C412yescomplemented
20#67⋊AGL(1,𝔽5)—52nomaximal
21C22×#1044—31no—
22C2×D5⋊AGL(1,𝔽5)—31no—
23#87886—31no—
24C22⋊#1043—31no—
25#87832—31no—
26#87831D311yescharacteristic, complemented
27#87830—31no—
28C2×C3⋊(C5⋊D5)C2×C411yescharacteristic, complemented
29C2×C52×D3C2×C411yescharacteristic, complemented
30C2×C3×C5⋊D5C2311yescharacteristic
31C2×C3×C5×D5C2×C412yescomplemented
32D3×C5⋊D5—21no—
33D3×C5⋊D5—21no—
34C3×D52—21no—
35C3×D52—21no—
36C3⋊#1044D412yescomplemented
37C3⋊#1043—21no—
38C3×#1044—21no—
39C3×#1043—21no—
40C5×Dic15C2×C412yescomplemented
41C22×ΓL(1,𝔽16)—52no—
42C2×D3×AGL(1,𝔽5)—52no—
43D5×#67—52no—
44C3×C22⋊AGL(1,𝔽5)—52no—
45Dic3⋊AGL(1,𝔽5)—52no—
46(C2×D3)⋊AGL(1,𝔽5)—52no—
47(C2×D3)⋊AGL(1,𝔽5)—102no—
48Dic3×AGL(1,𝔽5)—52no—
49C22×C5⋊D5—31no—
50C2×D52C2×D311yescharacteristic
51C2×#1044—31no—
52C2×#1044—62no—
53C2×#1044C2×D311yescharacteristic
54C2×#1043—31no—
55C2×#1043C2×D311yescharacteristic
56D5⋊AGL(1,𝔽5)—62no—
57C5×C22⋊D5—32no—
58#6038—32no—
59#6036—31no—
60D4×AGL(1,𝔽5)—152no—
61C2×C3×C52C22×C411yescharacteristic
62C3⋊(C5⋊D5)—21no—
63C52×D3—21no—
64C3×C5⋊D5C2×D411yescharacteristic
65C3×C5⋊D5#4111yescharacteristic
66C3×C5×D5—22no—
67C22×C3×D5—52no—
68C2×D3×D5—52no—
69C2×D3×D5—102no—
70C2×ΓL(1,𝔽16)—52no—
71C2×ΓL(1,𝔽16)—52no—
72C2×ΓL(1,𝔽16)—104no—
73C2×ΓL(1,𝔽16)—102no—
74C2×C3×AGL(1,𝔽5)—52no—
75C2×C3×AGL(1,𝔽5)—52no—
76C2×C3×AGL(1,𝔽5)—102no—
77D3×AGL(1,𝔽5)—104no—
78#1220—52no—
79C5×#67—52no—
80#1202—52no—
81#1201AGL(1,𝔽5)12yescomplemented
82D5×Dic3—52no—
83C22×C52—31no—
84C2×C5⋊D5—31no—
85C2×C5⋊D5—61no—
86C2×C5⋊D5—61no—
87C2×C5⋊D5C22×D311yescharacteristic
88C2×C5×D5C4×D312yescomplemented
89D52—21no—
90D52—21no—
91#1044—21no—
92#1044—32no—
93#1043—21no—
94#1043—61no—
95C5×Dic5—32no—
96#67⋊C4—251noHall
97C22×AGL(1,𝔽5)—152no—
98C22×AGL(1,𝔽5)—152no—
99D4×D5—152no—
100C22⋊AGL(1,𝔽5)—52no—
101C22⋊AGL(1,𝔽5)—152no—
102C22⋊AGL(1,𝔽5)—302no—
103C4⋊AGL(1,𝔽5)—152no—
104C4×AGL(1,𝔽5)—152no—
105C3×C52C4×D411yescharacteristic, complemented, Hall
106C22×C3×C5—52no—
107C2×D15—52no—
108C2×D15—102no—
109C2×C5×D3—22no—
110C2×C5×D3C2×AGL(1,𝔽5)12yescomplemented
111C2×C3×D5—52no—
112C2×C3×D5—102no—
113C2×C3×D5—102no—
114C2×C3×D5—102no—
115C2×C3×D5C2×AGL(1,𝔽5)12yescomplemented
116D3×D5—102no—
117D3×D5—102no—
118D3×D5—202no—
119D3×D5—202no—
120ΓL(1,𝔽16)—54no—
121ΓL(1,𝔽16)—102no—
122ΓL(1,𝔽16)—202no—
123C3×AGL(1,𝔽5)—102no—
124C3×AGL(1,𝔽5)—102no—
125C3×AGL(1,𝔽5)—202no—
126Dic15C2×AGL(1,𝔽5)12yescomplemented
127C5×Dic3—52no—
128C2×C52—61no—
129C2×C52C2×C4×D311yescharacteristic
130C5⋊D5—61no—
131C5⋊D5D3×D411yescharacteristic
132C5⋊D5#23711yescharacteristic
133C5×D5—22no—
134C2×#67—251no—
135C22×Dic3—251no—
136C2×C4×D3—251no—
137C3×C22⋊C4—251no—
138(C2×D3)⋊C4—251no—
139Dic3⋊C4—251no—
140C4×Dic3—251no—
141C22×D5—52no—
142C22×D5—152no—
143C22×D5—302no—
144C2×AGL(1,𝔽5)—52no—
145C2×AGL(1,𝔽5)—52no—
146C2×AGL(1,𝔽5)—102no—
147C2×AGL(1,𝔽5)—152no—
148C2×AGL(1,𝔽5)—152no—
149C2×AGL(1,𝔽5)—302no—
150C2×AGL(1,𝔽5)—304no—
151C2×AGL(1,𝔽5)—304no—
152C5×D4—152no—
153C22⋊D5—32no—
154C22⋊D5—152no—
155D20—152no—
156C4×D5—152no—
157C4×D4—751noSylow
158C2×C3×C5—22no—
159C2×C3×C5—102no—
160C2×C3×C5C22×AGL(1,𝔽5)12yes—
161D15—102no—
162D15—202no—
163C3×D5—22no—
164C3×D5—52no—
165C3×D5—52no—
166C3×D5—102no—
167C3×D5—102no—
168C5×D3—22no—
169C5×D3—42no—
170C52#67⋊C411yescharacteristic, complemented, Sylow
171C23×C3—251no—
172C22×D3—251no—
173C2×C3×C4—251no—
174C2×C3×C4—251no—
175#67—52no—
176#67—252no—
177C2×Dic3—251no—
178C2×Dic3—251no—
179C2×Dic3—251no—
180C2×Dic3—502no—
181C4×D3—502no—
182C22×C5—32no—
183C22×C5—52no—
184C22×C5—62no—
185C2×D5—52no—
186C2×D5—102no—
187C2×D5—102no—
188C2×D5—102no—
189C2×D5—152no—
190C2×D5—302no—
191C2×D5—302no—
192C2×D5—302no—
193C2×D5—602no—
194C2×D5—602no—
195C2×D5D3×AGL(1,𝔽5)12yescomplemented
196AGL(1,𝔽5)—102no—
197AGL(1,𝔽5)—102no—
198AGL(1,𝔽5)—154no—
199AGL(1,𝔽5)—202no—
200AGL(1,𝔽5)—302no—
201AGL(1,𝔽5)—602no—
202C4×C5—152no—
203Dic5—32no—
204C2×D4—751no—
205C22×C4—751no—
206C22×C4—751no—
207C4⋊C4—751no—
208C22⋊C4—251no—
209C22⋊C4—751no—
210C42—751no—
211C3×C5—22no—
212C3×C5D4×AGL(1,𝔽5)12yescomplemented
213C22×C3—52no—
214C22×C3—251no—
215C22×C3—501no—
216C22×C3—501no—
217C2×D3—251no—
218C2×D3—501no—
219C2×D3—501no—
220C2×D3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
221C3×C4—501no—
222C3×C4—501no—
223Dic3—52no—
224Dic3—252no—
225Dic3—501no—
226C2×C5—22no—
227C2×C5—62no—
228C2×C5—102no—
229C2×C5—122no—
230C2×C5C2×D3×AGL(1,𝔽5)12yes—
231D5—22no—
232D5—52no—
233D5—52no—
234D5—102no—
235D5—102no—
236D5—302no—
237D5—602no—
238C23—251no—
239C23—751no—
240D4—152no—
241D4—752no—
242C2×C4—251no—
243C2×C4—251no—
244C2×C4—751no—
245C2×C4—751no—
246C2×C4—751no—
247C2×C4—1502no—
248C2×C4—1502no—
249C2×C3—102no—
250C2×C3—251no—
251C2×C3—251no—
252C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
253D3—21no—
254D3—501no—
255C5—22no—
256C5#67⋊AGL(1,𝔽5)12yescomplemented
257C22—31no—
258C22—52no—
259C22—251no—
260C22—501no—
261C22—501no—
262C22—751no—
263C22—1501no—
264C22—1501no—
265C4—152no—
266C4—501no—
267C4—501no—
268C4—752no—
269C4—1501no—
270C3#1171737711yescharacteristic, complemented, Sylow
271C2—61no—
272C2—102no—
273C2—251no—
274C2—251no—
275C2—1501no—
276C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radical#11925419C2
cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D5⋊ΓL(1,𝔽16)C2semidirect
2. C2×D3×#1044C2semidirect
4. #11925361C2semidirect
5. #11925354C2semidirect
6. (C2×D3)⋊#1043C2semidirect
7. C3×#87831C2semidirect
18. #10597123C4semidirect
19. C5×#1201C4semidirect
26. #87831D3semidirect
28. C2×C3⋊(C5⋊D5)C2×C4semidirect
29. C2×C52×D3C2×C4semidirect
31. C2×C3×C5×D5C2×C4semidirect
36. C3⋊#1044D4semidirect
40. C5×Dic15C2×C4semidirect
81. #1201AGL(1,𝔽5)semidirect
88. C2×C5×D5C4×D3semidirect
105. C3×C52C4×D4semidirect
110. C2×C5×D3C2×AGL(1,𝔽5)semidirect
115. C2×C3×D5C2×AGL(1,𝔽5)semidirect
126. Dic15C2×AGL(1,𝔽5)semidirect
170. C52#67⋊C4semidirect
195. C2×D5D3×AGL(1,𝔽5)semidirect
212. C3×C5D4×AGL(1,𝔽5)semidirect
220. C2×D3D5⋊AGL(1,𝔽5)semidirect
256. C5#67⋊AGL(1,𝔽5)semidirect
270. C3#11717377semidirect

Maximal subgroups

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

subgroupindex
C2×D5⋊ΓL(1,𝔽16)2
C2×D3×#10442
#119254192
#119253612
#119253542
(C2×D3)⋊#10432
C3×#878312
#117173773
#67⋊AGL(1,𝔽5)5