Finite Group Databasebeta

← C2×C3×C5×#356 #14275165

Subgroups — C2×C3×C5×#356 #14275165

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×C5×#101C212yesmaximal
3C3×C5×#356C214yesmaximal, direct product factor
4C2×C5×#356C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic
6C22×C3×C5×Q8C2211yescharacteristic
7C2×C3×C5×M16—22no—
8C3×C5×C4⋊Q8—22no—
9C3×C5×C4⋊Q8C412yes—
10C3×C5×C4⋊Q8C2212yes—
11C2×C3×C5×C4⋊C4—22no—
12C2×C3×C42×C5C411yescharacteristic
13C3×C5×#101—22no—
14C3×C5×#101C2214yes—
15C2×C3×#356C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×C4⋊Q8C2×C311yescharacteristic
17C2×C5×#101C2×C312yes—
18C5×#356C2×C314yesdirect product factor
19C2×C3×C5×Q8—21no—
20C2×C3×C5×Q8—41no—
21C2×C3×C5×Q8—41no—
22C2×C3×C5×Q8—41no—
23C2×C3×C5×Q8—41no—
24C2×C3×C5×Q8C2×C412yes—
25C2×C3×C5×Q8C2×C411yescharacteristic
26C2×C3×C5×Q8D412yes—
27C2×C3×C5×Q8C2311yescharacteristic
28C22×C3×C4×C5—21no—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×C4×C5D412yes—
32C3×C5×M16—24no—
33C3×C5×M16—42no—
34C2×C3×C5×C8—42no—
35C3×C5×C4⋊C4—24no—
36C3×C5×C4⋊C4—24no—
37C3×C42×C5—21no—
38C3×C42×C5C2×C412yes—
39C2×C3×C4⋊Q8C2×C511yescharacteristic
40C2×C3×#101C2×C512yes—
41C3×#356C2×C514yesdirect product factor
42C22×C5×Q8C3×C411yescharacteristic
43C22×C5×Q8C22×C311yescharacteristic
44C2×C5×M16—22no—
45C5×C4⋊Q8—22no—
46C5×C4⋊Q8C3×C412yes—
47C5×C4⋊Q8C22×C312yes—
48C2×C5×C4⋊C4—22no—
49C2×C42×C5C3×C411yescharacteristic
50C5×#101—22no—
51C5×#101C22×C314yes—
52C2×#356C3×C511yescharacteristic, direct product factor, Sylow
53C23×C3×C5C22⋊C411yescharacteristic
54C3×C5×Q8—41no—
55C3×C5×Q8—42no—
56C3×C5×Q8—41no—
57C3×C5×Q8—42no—
58C3×C5×Q8—42no—
59C2×C3×C4×C5—24no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5—21no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5—22no—
66C2×C3×C4×C5—41no—
67C2×C3×C4×C5—41no—
68C2×C3×C4×C5C22⋊C411yescharacteristic
69C2×C3×C4×C5C22⋊C412yes—
70C2×C3×C4×C5C22×C411yescharacteristic
71C2×C3×C4×C5C2×D412yes—
72C3×C5×C8—44no—
73C22×C3×Q8C4×C511yescharacteristic
74C22×C3×Q8C22×C511yescharacteristic
75C2×C3×M16—22no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8C4×C512yes—
78C3×C4⋊Q8C22×C512yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C42C4×C511yescharacteristic
81C3×#101—22no—
82C3×#101C22×C514yes—
83C2×C5×Q8—21no—
84C2×C5×Q8—41no—
85C2×C5×Q8—41no—
86C2×C5×Q8—41no—
87C2×C5×Q8—41no—
88C2×C5×Q8C2×C3×C412yes—
89C2×C5×Q8C2×C3×C411yescharacteristic
90C2×C5×Q8C3×D412yes—
91C2×C5×Q8C23×C311yescharacteristic
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5C2×C3×C411yescharacteristic
95C22×C4×C5C3×D412yes—
96C5×M16—24no—
97C5×M16—42no—
98C2×C5×C8—42no—
99C5×C4⋊C4—24no—
100C5×C4⋊C4—24no—
101C42×C5—21no—
102C42×C5C2×C3×C412yes—
103C2×C4⋊Q8C2×C3×C511yescharacteristic
104C2×#101C2×C3×C512yes—
105#356C2×C3×C514yesdirect product factor
106C22×C3×C5—22no—
107C22×C3×C5#9911yescharacteristic
108C22×C3×C5#9911yescharacteristic
109C22×C3×C5C2×C22⋊C411yescharacteristic
110C3×C4×C5—24no—
111C3×C4×C5—22no—
112C3×C4×C5—22no—
113C3×C4×C5—21no—
114C3×C4×C5—21no—
115C3×C4×C5—42no—
116C2×C3×Q8—21no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8C2×C4×C512yes—
122C2×C3×Q8C2×C4×C511yescharacteristic
123C2×C3×Q8C5×D412yes—
124C2×C3×Q8C23×C511yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4C2×C4×C511yescharacteristic
128C22×C3×C4C5×D412yes—
129C3×M16—24no—
130C3×M16—42no—
131C2×C3×C8—42no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—24no—
134C3×C42—21no—
135C3×C42C2×C4×C512yes—
136C23×C5C3×C22⋊C411yescharacteristic
137C5×Q8—41no—
138C5×Q8—42no—
139C5×Q8—41no—
140C5×Q8—42no—
141C5×Q8—42no—
142C2×C4×C5—24no—
143C2×C4×C5—22no—
144C2×C4×C5—22no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—22no—
148C2×C4×C5—22no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5C3×C22⋊C411yescharacteristic
152C2×C4×C5C3×C22⋊C412yes—
153C2×C4×C5C22×C3×C411yescharacteristic
154C2×C4×C5C2×C3×D412yes—
155C5×C8—44no—
156C22×Q8C3×C4×C511yescharacteristic
157C22×Q8C22×C3×C511yescharacteristic
158C2×M16—22no—
159C4⋊Q8—22no—
160C4⋊Q8C3×C4×C512yes—
161C4⋊Q8C22×C3×C512yes—
162C2×C4⋊C4—22no—
163C2×C42C3×C4×C511yescharacteristic
164#101—22no—
165#101C22×C3×C514yes—
166C2×C3×C5—21no—
167C2×C3×C5—21no—
168C2×C3×C5#35612yesdirect product factor
169C2×C3×C5C2×#9911yescharacteristic
170C23×C3C5×C22⋊C411yescharacteristic
171C3×Q8—41no—
172C3×Q8—42no—
173C3×Q8—41no—
174C3×Q8—42no—
175C3×Q8—42no—
176C2×C3×C4—24no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4C5×C22⋊C411yescharacteristic
186C2×C3×C4C5×C22⋊C412yes—
187C2×C3×C4C22×C4×C511yescharacteristic
188C2×C3×C4C2×C5×D412yes—
189C3×C8—44no—
190C22×C5—22no—
191C22×C5C3×#9911yescharacteristic
192C22×C5C3×#9911yescharacteristic
193C22×C5C2×C3×C22⋊C411yescharacteristic
194C4×C5—24no—
195C4×C5—22no—
196C4×C5—22no—
197C4×C5—21no—
198C4×C5—21no—
199C4×C5—42no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C3×C4×C512yes—
206C2×Q8C2×C3×C4×C511yescharacteristic
207C2×Q8C3×C5×D412yes—
208C2×Q8C23×C3×C511yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C2×C3×C4×C511yescharacteristic
212C22×C4C3×C5×D412yes—
213M16—24no—
214M16—42no—
215C2×C8—42no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C42—21no—
219C42C2×C3×C4×C512yes—
220C3×C5C2×#35611yescharacteristic, direct product factor, Hall
221C22×C3—22no—
222C22×C3C5×#9911yescharacteristic
223C22×C3C5×#9911yescharacteristic
224C22×C3C2×C5×C22⋊C411yescharacteristic
225C3×C4—24no—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—42no—
231C2×C5—21no—
232C2×C5—21no—
233C2×C5C3×#35612yesdirect product factor
234C2×C5C2×C3×#9911yescharacteristic
235C23C3×C5×C22⋊C411yescharacteristic
236Q8—41no—
237Q8—42no—
238Q8—41no—
239Q8—42no—
240Q8—42no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4C3×C5×C22⋊C411yescharacteristic
251C2×C4C3×C5×C22⋊C412yes—
252C2×C4C22×C3×C4×C511yescharacteristic
253C2×C4C2×C3×C5×D412yes—
254C8—44no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3C5×#35612yesdirect product factor
258C2×C3C2×C5×#9911yescharacteristic
259C5C2×C3×#35611yescharacteristic, direct product factor, Sylow
260C22—22no—
261C22C3×C5×#9911yescharacteristic
262C22C3×C5×#9911yescharacteristic
263C22C2×C3×C5×C22⋊C411yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—22no—
268C4—21no—
269C4—42no—
270C3C2×C5×#35611yescharacteristic, direct product factor, Sylow
271C2—21no—
272C2—21no—
273C2C3×C5×#35612yesdirect product factor
274C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC22×C3×C5#99
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC2×Q8C23×C3×C5
prime radicalC23×C3×C5C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×C5×#356C2direct (N × H)
4. C2×C5×#356C3direct (N × H)
15. C2×C3×#356C5direct (N × H)
18. C5×#356C2×C3direct (N × H)
41. C3×#356C2×C5direct (N × H)
52. C2×#356C3×C5direct (N × H)
105. #356C2×C3×C5direct (N × H)
168. C2×C3×C5#356direct (N × H)
220. C3×C5C2×#356direct (N × H)
233. C2×C5C3×#356direct (N × H)
257. C2×C3C5×#356direct (N × H)
259. C5C2×C3×#356direct (N × H)
270. C3C2×C5×#356direct (N × H)
273. C2C3×C5×#356direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C4⋊Q82
C2×C3×C5×#1012
C3×C5×#3562
C2×C5×#3563
C2×C3×#3565