Finite Group Databasebeta

← #72230564

Subgroups — #72230564

Order 2400.

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

normal subgroups only

This group has 436 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×#10597183C211yesmaximal, characteristic, complemented
2C2×D5⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3#11925420C211yesmaximal, characteristic
4#11925355C211yesmaximal, characteristic, complemented
5Dic3⋊#1044C211yesmaximal, characteristic, complemented
6Dic3×#1043C211yesmaximal, characteristic, complemented
7C3×#87831C211yesmaximal, characteristic, complemented
8#11717363—31nomaximal, Hall
9C2×D15⋊D5C2211yescharacteristic
10C2×C3×D52C2211yescharacteristic
11C2×C3⋊#1044C2211yescharacteristic
12C2×C3⋊#1043C2211yescharacteristic
13C2×C3×#1044C2211yescharacteristic
14C2×C3×#1043C2211yescharacteristic
15#10597183—22no—
16D5⋊ΓL(1,𝔽16)—22no—
17#10597121C412yescomplemented
18Dic3×C5⋊D5C2211yescharacteristic
19C5×#1202C412yescomplemented
20#67⋊AGL(1,𝔽5)—52nomaximal
21C2×D5⋊AGL(1,𝔽5)—31no—
22C2×D5⋊AGL(1,𝔽5)—31no—
23#87877—31no—
24C4⋊#1044—31no—
25C4×#1043—31no—
26#87831—31no—
27#87831D311yescharacteristic, complemented
28C2×C5×D15C2×C412yescomplemented
29C2×C3×C5⋊D5C2311yescharacteristic
30C2×C3×C5×D5C2×C412yescomplemented
31D15⋊D5—21no—
32D15⋊D5—21no—
33C3×D52—21no—
34C3×D52—21no—
35C3⋊#1044—21no—
36C3⋊#1043D412yescomplemented
37C3×#1044—21no—
38C3×#1043—21no—
39C3⋊#1039C2×C411yescharacteristic, complemented
40C52×Dic3C2×C411yescharacteristic, complemented
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—
47Dic3×AGL(1,𝔽5)—52no—
48Dic3×AGL(1,𝔽5)—102no—
49C2×D52—31no—
50C2×D52C2×D311yescharacteristic
51C2×#1044—31no—
52C2×#1044C2×D311yescharacteristic
53C2×#1043—31no—
54C2×#1043C2×D311yescharacteristic
55D5⋊AGL(1,𝔽5)—62no—
56D5⋊AGL(1,𝔽5)—62no—
57C4×C5⋊D5—31no—
58C5×D20—32no—
59#6037—32no—
60D4×AGL(1,𝔽5)—152no—
61C2×C3×C52C22×C411yescharacteristic
62C5×D15—22no—
63C3×C5⋊D5C2×D411yescharacteristic
64C3×C5⋊D5#4111yescharacteristic
65C3×C5×D5—22no—
66C22×C3×D5—52no—
67C2×D3×D5—52no—
68C2×ΓL(1,𝔽16)—52no—
69C2×ΓL(1,𝔽16)—52no—
70C2×ΓL(1,𝔽16)—104no—
71C2×ΓL(1,𝔽16)—102no—
72C2×C3×AGL(1,𝔽5)—52no—
73C2×C3×AGL(1,𝔽5)—52no—
74C2×C3×AGL(1,𝔽5)—102no—
75D3×AGL(1,𝔽5)—104no—
76#1220—52no—
77C5×#67—52no—
78#1202AGL(1,𝔽5)12yescomplemented
79#1201—52no—
80D5×Dic3—52no—
81D5×Dic3—102no—
82C2×C5⋊D5C22×D311yescharacteristic
83C2×C5×D5—32no—
84C2×C5×D5C4×D312yescomplemented
85D52—21no—
86D52—21no—
87D52—61no—
88D52—61no—
89#1044—21no—
90#1044—61no—
91#1043—21no—
92#1043—32no—
93C4×C52—31no—
94#1039—31no—
95#67⋊C4—251noHall
96C22×AGL(1,𝔽5)—152no—
97C22×AGL(1,𝔽5)—152no—
98D4×D5—152no—
99C22⋊AGL(1,𝔽5)—52no—
100C22⋊AGL(1,𝔽5)—152no—
101C4⋊AGL(1,𝔽5)—152no—
102C4×AGL(1,𝔽5)—152no—
103C4×AGL(1,𝔽5)—302no—
104C3×C52C4×D411yescharacteristic, complemented, Hall
105C22×C3×C5—52no—
106C2×D15C2×AGL(1,𝔽5)12yescomplemented
107C2×C5×D3—52no—
108C2×C3×D5—52no—
109C2×C3×D5—102no—
110C2×C3×D5—102no—
111C2×C3×D5—102no—
112C2×C3×D5C2×AGL(1,𝔽5)12yescomplemented
113D3×D5—102no—
114D3×D5—102no—
115ΓL(1,𝔽16)—54no—
116ΓL(1,𝔽16)—104no—
117ΓL(1,𝔽16)—102no—
118C3×AGL(1,𝔽5)—102no—
119C3×AGL(1,𝔽5)—102no—
120C3×AGL(1,𝔽5)—102no—
121C3×AGL(1,𝔽5)—102no—
122Dic15—52no—
123Dic15—102no—
124C5×Dic3—22no—
125C5×Dic3C2×AGL(1,𝔽5)12yescomplemented
126C2×C52C2×C4×D311yescharacteristic
127C5⋊D5D3×D411yescharacteristic
128C5⋊D5#23711yescharacteristic
129C5×D5—22no—
130C5×D5—62no—
131C2×#67—251no—
132C22×Dic3—251no—
133C2×C4×D3—251no—
134C3×C22⋊C4—251no—
135(C2×D3)⋊C4—251no—
136Dic3⋊C4—251no—
137C4×Dic3—251no—
138C22×D5—52no—
139C22×D5—152no—
140C2×AGL(1,𝔽5)—52no—
141C2×AGL(1,𝔽5)—52no—
142C2×AGL(1,𝔽5)—102no—
143C2×AGL(1,𝔽5)—152no—
144C2×AGL(1,𝔽5)—152no—
145C2×AGL(1,𝔽5)—302no—
146C2×AGL(1,𝔽5)—304no—
147C2×AGL(1,𝔽5)—304no—
148C5×D4—152no—
149C22⋊D5—152no—
150C22⋊D5—152no—
151D20—32no—
152C4×D5—152no—
153C4×D5—302no—
154C4×D4—751noSylow
155C2×C3×C5—22no—
156C2×C3×C5—102no—
157C2×C3×C5C22×AGL(1,𝔽5)12yes—
158D15—22no—
159C3×D5—22no—
160C3×D5—52no—
161C3×D5—52no—
162C3×D5—102no—
163C3×D5—102no—
164C5×D3—102no—
165C52#67⋊C411yescharacteristic, complemented, Sylow
166C23×C3—251no—
167C22×D3—251no—
168C2×C3×C4—251no—
169C2×C3×C4—251no—
170#67—52no—
171#67—252no—
172C2×Dic3—251no—
173C2×Dic3—251no—
174C2×Dic3—251no—
175C2×Dic3—502no—
176C4×D3—502no—
177C22×C5—52no—
178C22×C5—152no—
179C2×D5—32no—
180C2×D5—52no—
181C2×D5—102no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—302no—
185C2×D5—302no—
186C2×D5D3×AGL(1,𝔽5)12yescomplemented
187AGL(1,𝔽5)—102no—
188AGL(1,𝔽5)—102no—
189AGL(1,𝔽5)—102no—
190AGL(1,𝔽5)—102no—
191AGL(1,𝔽5)—154no—
192AGL(1,𝔽5)—304no—
193AGL(1,𝔽5)—302no—
194C4×C5—32no—
195C4×C5—62no—
196Dic5—152no—
197Dic5—302no—
198C2×D4—751no—
199C22×C4—751no—
200C22×C4—751no—
201C4⋊C4—751no—
202C22⋊C4—251no—
203C22⋊C4—751no—
204C42—751no—
205C3×C5—22no—
206C3×C5D4×AGL(1,𝔽5)12yescomplemented
207C22×C3—52no—
208C22×C3—251no—
209C22×C3—501no—
210C22×C3—501no—
211C2×D3—52no—
212C2×D3—501no—
213C2×D3—501no—
214C3×C4—501no—
215C3×C4—501no—
216Dic3—252no—
217Dic3—251no—
218Dic3—501no—
219Dic3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
220C2×C5—22no—
221C2×C5—102no—
222C2×C5—302no—
223C2×C5C2×D3×AGL(1,𝔽5)12yes—
224D5—22no—
225D5—52no—
226D5—52no—
227D5—62no—
228D5—102no—
229D5—102no—
230C23—251no—
231C23—751no—
232D4—152no—
233D4—752no—
234C2×C4—251no—
235C2×C4—251no—
236C2×C4—751no—
237C2×C4—751no—
238C2×C4—751no—
239C2×C4—1502no—
240C2×C4—1502no—
241C2×C3—102no—
242C2×C3—251no—
243C2×C3—251no—
244C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
245D3—102no—
246C5—22no—
247C5#67⋊AGL(1,𝔽5)12yescomplemented
248C22—52no—
249C22—152no—
250C22—251no—
251C22—501no—
252C22—501no—
253C22—1501no—
254C22—1501no—
255C4—31no—
256C4—501no—
257C4—501no—
258C4—752no—
259C4—751no—
260C4—1501no—
261C3#1171736311yescharacteristic, complemented, Sylow
262C2—102no—
263C2—251no—
264C2—251no—
265C2—302no—
266C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radical#11925420C2
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×#10597183C2semidirect
2. C2×D5⋊ΓL(1,𝔽16)C2semidirect
4. #11925355C2semidirect
5. Dic3⋊#1044C2semidirect
6. Dic3×#1043C2semidirect
7. C3×#87831C2semidirect
17. #10597121C4semidirect
19. C5×#1202C4semidirect
27. #87831D3semidirect
28. C2×C5×D15C2×C4semidirect
30. C2×C3×C5×D5C2×C4semidirect
36. C3⋊#1043D4semidirect
39. C3⋊#1039C2×C4semidirect
40. C52×Dic3C2×C4semidirect
78. #1202AGL(1,𝔽5)semidirect
84. C2×C5×D5C4×D3semidirect
104. C3×C52C4×D4semidirect
106. C2×D15C2×AGL(1,𝔽5)semidirect
112. C2×C3×D5C2×AGL(1,𝔽5)semidirect
125. C5×Dic3C2×AGL(1,𝔽5)semidirect
165. C52#67⋊C4semidirect
186. C2×D5D3×AGL(1,𝔽5)semidirect
206. C3×C5D4×AGL(1,𝔽5)semidirect
219. Dic3D5⋊AGL(1,𝔽5)semidirect
247. C5#67⋊AGL(1,𝔽5)semidirect
261. C3#11717363semidirect

Maximal subgroups

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

subgroupindex
C2×#105971832
C2×D5⋊ΓL(1,𝔽16)2
#119254202
#119253552
Dic3⋊#10442
Dic3×#10432
C3×#878312
#117173633
#67⋊AGL(1,𝔽5)5