Finite Group Databasebeta

← #11754314

Subgroups — #11754314

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#709C211yesmaximal, characteristic, complemented
2C2×#6773C211yesmaximal, characteristic, complemented
3#91819C211yesmaximal, characteristic, complemented
4C5×#924C211yesmaximal, characteristic, complemented
5#91550C212yesmaximal, complemented
6#91533C212yesmaximal, complemented
7#91450C211yesmaximal, characteristic, complemented
8D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#91411C212yesmaximal, complemented
10#91409C212yesmaximal, complemented
11#91359C211yesmaximal, characteristic, complemented
12#65963—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2212yes—
15C2×D60C2211yescharacteristic
16C2×C5×Dic6C2211yescharacteristic
17C2×C3×C22⋊D5C2211yescharacteristic
18C2×C3×C22⋊D5C2211yescharacteristic
19C3×#709—24no—
20C2×C3×D20C2211yescharacteristic
21C2×C3×C4×D5C2211yescharacteristic
22C2×C3×C4×D5C2211yescharacteristic
23C2×C3×Dic10C2211yescharacteristic
24C2×#1202C2212yes—
25C2×#1202C2212yes—
26C2×#1200C2212yes—
27C2×D5×Dic3C2212yes—
28#6773—24no—
29#6714C2212yes—
30C4⋊Dic15C2211yescharacteristic
31C5×#217C2212yes—
32C5×C4⋊Dic3C2211yescharacteristic
33C5×Dic3⋊C4C2212yes—
34#6669C2212yes—
35#6668C2212yes—
36#6667C2212yes—
37#6665C2212yes—
38#5833—51nomaximal, Hall
39C2×#713—31no—
40C2×#709D311yescharacteristic, complemented
41C5×C22⋊Q8—31no—
42#4092—31no—
43#4076—31no—
44#4042—32no—
45#4041—32no—
46#4041—31no—
47D5×C4⋊C4—31no—
48#4030—32no—
49#4030—32no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2312yes—
52C22×C3×D5C2311yescharacteristic
53C22×C3×D5C2311yescharacteristic
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5C2311yescharacteristic
56C2×C3×C4×C5C2311yescharacteristic
57#1220—42no—
58C2×Dic15C2312yes—
59D60—22no—
60C2×C5×Dic3C2312yes—
61C2×C5×Dic3C2312yes—
62C5×Dic6—22no—
63C3×C22⋊D5—22no—
64C3×C22⋊D5—22no—
65C2×C3×Dic5C2311yescharacteristic
66C2×C3×Dic5C2311yescharacteristic
67C3×D20—22no—
68C3×C4×D5—22no—
69C3×C4×D5D414yescomplemented
70C3×Dic10—22no—
71#1202—24no—
72#1202—42no—
73#1200—24no—
74D5×Dic3—42no—
75C2×C3×#41—51no—
76C2×#235—51no—
77#946—52no—
78#939—52no—
79#939—52no—
80#938—52no—
81#930—51no—
82#930—51no—
83C2×C4⋊Dic3—51no—
84#924—51no—
85#924D511yescharacteristic, complemented
86C2×C5×Q8—31no—
87C22×C4×C5C2×D311yescharacteristic
88C2×C22⋊D5—32no—
89C2×C22⋊D5C2×D311yescharacteristic
90C2×C22⋊D5C2×D311yescharacteristic
91#713—64no—
92#709—24no—
93C2×D20—32no—
94C2×D20—31no—
95C2×D20—32no—
96C2×D20C2×D311yescharacteristic
97C2×C4×D5—32no—
98C2×C4×D5—32no—
99C2×C4×D5C2×D311yescharacteristic
100C2×C4×D5C2×D311yescharacteristic
101C2×Dic10C2×D311yescharacteristic
102C5×C4⋊C4—32no—
103C5×C4⋊C4—31no—
104C5×C22⋊C4—32no—
105#685—32no—
106#685—32no—
107#685—32no—
108C4⋊Dic5—31no—
109#683—32no—
110#683—32no—
111#537—151noSylow
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—22no—
117C2×D15—22no—
118C2×D15—22no—
119C2×C3×D5—21no—
120C2×C3×D5—21no—
121C2×C3×D5—21no—
122C2×C3×D5—21no—
123C2×C3×D5—21no—
124C2×C3×D5C2×D412yes—
125C3×C4×C5—21no—
126C3×C4×C5C2×D412yes—
127Dic15—22no—
128C3×Dic5—21no—
129C3×Dic5C2×D412yes—
130C5×Dic3—22no—
131C5×Dic3—22no—
132C3×#41—104no—
133C2×C3×Q8—51no—
134C2×C3×D4—51no—
135C2×C3×D4—51no—
136C2×C3×D4—51no—
137C22×C3×C4—51no—
138C22×C3×C4—51no—
139C22×C3×C4C2×D511yescharacteristic
140C2×#67—52no—
141C2×#67—52no—
142C2×#67—52no—
143C22×Dic3—52no—
144#235—104no—
145C2×D12—51no—
146C2×C4×D3—52no—
147C2×Dic6C2×D511yescharacteristic
148#217—52no—
149#217C2×D512yes—
150(C2×D3)⋊C4—52no—
151(C2×D3)⋊C4—52no—
152C4⋊Dic3—51no—
153C4⋊Dic3—52no—
154C4⋊Dic3C2×D511yescharacteristic
155Dic3⋊C4—52no—
156Dic3⋊C4C2×D512yes—
157C23×C5C22×D311yescharacteristic
158C22×D5—32no—
159C22×D5C22×D311yescharacteristic
160C22×D5C22×D311yescharacteristic
161C5×Q8—62no—
162C2×C4×C5—22no—
163C2×C4×C5—32no—
164C2×C4×C5—32no—
165C2×C4×C5C22×D311yescharacteristic
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—22no—
168C22⋊D5—22no—
169C22⋊D5—122no—
170C2×Dic5—32no—
171C2×Dic5C22×D311yescharacteristic
172C2×Dic5C22×D311yescharacteristic
173D20—22no—
174D20—64no—
175D20—62no—
176D20—122no—
177C4×D5—22no—
178C4×D5—64no—
179C4×D5—122no—
180C4×D5#6714yescomplemented
181Dic10—22no—
182C2×#41—51no—
183C2×#41—151no—
184C22⋊Q8—31no—
185C22⋊Q8—152no—
186C22⋊Q8—151no—
187#121—152no—
188#121—152no—
189#121—151no—
190#121—152no—
191#121—151no—
192C2×C4⋊C4—151no—
193C2×C3×C5—41no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5H(5,𝔽2)11yescharacteristic
196C2×C3×C52−1+411yescharacteristic
197D15—42no—
198C3×D5—22no—
199C3×D5—41no—
200C23×C3—51no—
201C23×C3—51no—
202C23×C3C22×D511yescharacteristic
203C22×D3—52no—
204C3×Q8—102no—
205C3×D4—102no—
206C3×D4—102no—
207C3×D4—102no—
208C2×C3×C4—22no—
209C2×C3×C4—51no—
210C2×C3×C4—54no—
211C2×C3×C4—51no—
212C2×C3×C4—102no—
213C2×C3×C4C22×D511yescharacteristic
214C2×C3×C4C22×D511yescharacteristic
215#67—104no—
216#67—202no—
217#67—202no—
218C2×Dic3—52no—
219C2×Dic3—202no—
220C2×Dic3C22×D512yes—
221C2×Dic3C22×D512yes—
222D12—102no—
223C4×D3—104no—
224Dic6—22no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5C23×D311yescharacteristic
229C2×D5—21no—
230C2×D5—21no—
231C2×D5—21no—
232C2×D5—21no—
233C2×D5—21no—
234C2×D5—62no—
235C2×D5—62no—
236C2×D5—62no—
237C2×D5C2×#6712yes—
238C4×C5—21no—
239C4×C5—62no—
240C4×C5—62no—
241C4×C5C2×#6712yes—
242Dic5—21no—
243Dic5—62no—
244Dic5C2×#6712yes—
245#41—104no—
246#41—304no—
247C2×Q8—31no—
248C2×Q8—51no—
249C2×D4—51no—
250C2×D4—51no—
251C2×D4—51no—
252C2×D4—152no—
253C2×D4—152no—
254C2×D4—152no—
255C2×D4—151no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—152no—
259C22×C4—152no—
260C22×C4D3×D511yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—32no—
263C4⋊C4—152no—
264C4⋊C4—152no—
265C4⋊C4—151no—
266C22⋊C4—32no—
267C22⋊C4—152no—
268C22⋊C4—152no—
269C22⋊C4—152no—
270C3×C5#53711yescharacteristic, complemented, Hall
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—52no—
275C22×C3—101no—
276C22×C3—101no—
277C22×C3—101no—
278C22×C3—101no—
279C22×C3—101no—
280C22×C3C23×D511yescharacteristic
281C2×D3—102no—
282C2×D3—102no—
283C2×D3—102no—
284C3×C4—21no—
285C3×C4—52no—
286C3×C4—101no—
287C3×C4D4×D512yes—
288Dic3—22no—
289Dic3—22no—
290Dic3—102no—
291C2×C5—41no—
292C2×C5#100911yescharacteristic
293C2×C5#101011yescharacteristic
294C2×C5C22×#6711yescharacteristic
295D5—22no—
296D5—41no—
297D5—122no—
298C23—51no—
299C23—51no—
300C23—152no—
301C23C2×D3×D511yescharacteristic
302Q8—62no—
303Q8—102no—
304D4—102no—
305D4—102no—
306D4—102no—
307D4—304no—
308D4—302no—
309D4—602no—
310D4—602no—
311C2×C4—22no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—51no—
315C2×C4—54no—
316C2×C4—51no—
317C2×C4—102no—
318C2×C4—152no—
319C2×C4—304no—
320C2×C4—602no—
321C2×C4C2×D3×D511yescharacteristic
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—41no—
324C2×C3—102no—
325C2×C3—201no—
326C2×C3C2×D4×D511yescharacteristic
327C2×C3#414711yescharacteristic
328C2×C3#414911yescharacteristic
329D3—202no—
330C5#583311yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—52no—
335C22—101no—
336C22—101no—
337C22—101no—
338C22—101no—
339C22—101no—
340C22—302no—
341C22—302no—
342C22—302no—
343C22C22×D3×D511yescharacteristic
344C4—21no—
345C4—52no—
346C4—62no—
347C4—62no—
348C4—101no—
349C4—302no—
350C4D5×#6712yes—
351C3#6596311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—102no—
354C2—201no—
355C2—602no—
356C2#9200011yescharacteristic
357C2#9201911yescharacteristic
358C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#537
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#709C2semidirect
2. C2×#6773C2semidirect
3. #91819C2semidirect
4. C5×#924C2semidirect
5. #91550C2semidirect
6. #91533C2semidirect
7. #91450C2semidirect
8. D5×C4⋊Dic3C2semidirect
9. #91411C2semidirect
10. #91409C2semidirect
11. #91359C2semidirect
40. C2×#709D3semidirect
69. C3×C4×D5D4semidirect
85. #924D5semidirect
180. C4×D5#67semidirect
270. C3×C5#537semidirect
330. C5#5833semidirect
351. C3#65963semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7092
C2×#67732
#918192
C5×#9242
#915502
#915332
#914502
D5×C4⋊Dic32
#914112
#914092
#913592
#659633
#58335