Finite Group Databasebeta

← C3×2−1+6⋊C5 #14511522

Subgroups — C3×2−1+6⋊C5 #14511522

Order 1920.

1206 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 32 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
1C3×#585⋊C5C211yesmaximal, characteristic, complemented
2C2×C3×2−1+4⋊C5C212yesmaximal, complemented
32−1+6⋊C5C311yesmaximal, characteristic, direct product factor, Hall
4C3×2−1+4⋊C5C2211yescharacteristic, complemented
5C3×2−1+6C511yesmaximal, characteristic, complemented, Hall
6#585⋊C5C2×C311yescharacteristic, complemented
7C2×2−1+4⋊C5C2×C312yescomplemented
8C3×#585—51no—
9C3×#585—54no—
10C3×#585—52no—
11C3×#585C2×C511yescharacteristic, complemented
12C2×C3×2−1+4—51no—
13C2×C3×2−1+4—52no—
14C2×C3×2−1+4—52no—
15C2×C3×2−1+4C2×C512yescomplemented
162−1+4⋊C5C22×C311yescharacteristic, complemented
172−1+6C3×C511yescharacteristic, complemented, Sylow
18C3×C5×D4—161nomaximal
19C3×2−1+4—52no—
20C3×2−1+4—54no—
21C3×2−1+4—52no—
22C3×2−1+4—58no—
23C3×2−1+4—54no—
24C3×2−1+4—54no—
25C3×2−1+4—54no—
26C3×2−1+4—58no—
27C3×2−1+4—54no—
28C3×2−1+4—52no—
29C3×2−1+4—51no—
30C3×2−1+4C22×C511yescharacteristic
31C3×H(5,𝔽2)—58no—
32C3×H(5,𝔽2)—52no—
33C3×H(5,𝔽2)—54no—
34C3×H(5,𝔽2)—54no—
35C3×H(5,𝔽2)—52no—
36C3×H(5,𝔽2)—52no—
37C3×H(5,𝔽2)—52no—
38C2×C3×#41—51no—
39C2×C3×#41—54no—
40C2×C3×#41—52no—
41C2×C3×#41—54no—
42C2×C3×#41—58no—
43C2×C3×#41—58no—
44C2×C3×#41—54no—
45C2×C3×#41—51no—
46C2×C3×#41—52no—
47C2×C3×#41—52no—
48C2×C3×#41—58no—
49C2×C3×#41—54no—
50C2×C3×#41—52no—
51C2×C3×#41—54no—
52C22×C3×Q8—52no—
53C22×C3×Q8—52no—
54C22×C3×Q8—51no—
55C22×C3×Q8—54no—
56#585—51no—
57#585—54no—
58#585—52no—
59#585C2×C3×C511yescharacteristic, complemented
60C2×2−1+4—51no—
61C2×2−1+4—52no—
62C2×2−1+4—52no—
63C2×2−1+4C2×C3×C512yescomplemented
64C22×C3×C5—162no—
65C3×C4×C5—161no—
66C3×#41—54no—
67C3×#41—54no—
68C3×#41—58no—
69C3×#41—54no—
70C3×#41—52no—
71C3×#41—58no—
72C3×#41—58no—
73C3×#41—58no—
74C3×#41—58no—
75C3×#41—58no—
76C3×#41—54no—
77C3×#41—58no—
78C3×#41—54no—
79C3×#41—54no—
80C3×#41—52no—
81C3×#41—58no—
82C3×#41—58no—
83C3×#41—54no—
84C3×#41—58no—
85C3×#41—58no—
86C3×#41—52no—
87C3×#41—54no—
88C3×#41—54no—
89C3×#41—54no—
90C3×#41—54no—
91C3×#41—52no—
92C3×#41—52no—
93C3×#41—52no—
94C2×C3×Q8—51no—
95C2×C3×Q8—52no—
96C2×C3×Q8—54no—
97C2×C3×Q8—52no—
98C2×C3×Q8—52no—
99C2×C3×Q8—58no—
100C2×C3×Q8—54no—
101C2×C3×Q8—58no—
102C2×C3×Q8—58no—
103C2×C3×Q8—54no—
104C2×C3×Q8—51no—
105C2×C3×Q8—54no—
106C2×C3×Q8—52no—
107C2×C3×Q8—54no—
108C2×C3×D4—52no—
109C2×C3×D4—54no—
110C2×C3×D4—54no—
111C2×C3×D4—58no—
112C2×C3×D4—54no—
113C2×C3×D4—54no—
114C2×C3×D4—58no—
115C2×C3×D4—51no—
116C2×C3×D4—58no—
117C2×C3×D4—52no—
118C2×C3×D4—51no—
119C2×C3×D4—52no—
120C2×C3×D4—52no—
121C2×C3×D4—54no—
122C22×C3×C4—54no—
123C22×C3×C4—58no—
124C22×C3×C4—54no—
125C22×C3×C4—52no—
126C22×C3×C4—52no—
127C22×C3×C4—51no—
128C22×C3×C4—54no—
129C22×C3×C4—52no—
130C5×D4—161no—
1312−1+4—52no—
1322−1+4—54no—
1332−1+4—52no—
1342−1+4—58no—
1352−1+4—54no—
1362−1+4—54no—
1372−1+4—54no—
1382−1+4—58no—
1392−1+4—54no—
1402−1+4—52no—
1412−1+4—51no—
1422−1+4C22×C3×C511yescharacteristic
143H(5,𝔽2)—58no—
144H(5,𝔽2)—52no—
145H(5,𝔽2)—54no—
146H(5,𝔽2)—54no—
147H(5,𝔽2)—52no—
148H(5,𝔽2)—52no—
149H(5,𝔽2)—52no—
150C2×#41—51no—
151C2×#41—54no—
152C2×#41—52no—
153C2×#41—54no—
154C2×#41—58no—
155C2×#41—58no—
156C2×#41—54no—
157C2×#41—51no—
158C2×#41—52no—
159C2×#41—52no—
160C2×#41—58no—
161C2×#41—54no—
162C2×#41—52no—
163C2×#41—54no—
164C22×Q8—52no—
165C22×Q8—52no—
166C22×Q8—51no—
167C22×Q8—54no—
168C2×C3×C5—161no—
169C2×C3×C5—322no—
170C23×C3—51no—
171C23×C3—54no—
172C23×C3—52no—
173C23×C3—52no—
174C3×Q8—52no—
175C3×Q8—52no—
176C3×Q8—54no—
177C3×Q8—54no—
178C3×Q8—52no—
179C3×Q8—58no—
180C3×Q8—52no—
181C3×D4—52no—
182C3×D4—54no—
183C3×D4—52no—
184C3×D4—54no—
185C3×D4—54no—
186C3×D4—54no—
187C3×D4—58no—
188C3×D4—58no—
189C3×D4—52no—
190C3×D4—54no—
191C3×D4—51no—
192C3×D4C24⋊C511yescharacteristic
193C2×C3×C4—52no—
194C2×C3×C4—54no—
195C2×C3×C4—58no—
196C2×C3×C4—54no—
197C2×C3×C4—58no—
198C2×C3×C4—51no—
199C2×C3×C4—52no—
200C2×C3×C4—54no—
201C2×C3×C4—58no—
202C2×C3×C4—54no—
203C2×C3×C4—51no—
204C2×C3×C4—52no—
205C2×C3×C4—54no—
206C2×C3×C4—52no—
207C22×C5—162no—
208C4×C5—161no—
209#41—54no—
210#41—54no—
211#41—58no—
212#41—54no—
213#41—52no—
214#41—58no—
215#41—58no—
216#41—58no—
217#41—58no—
218#41—58no—
219#41—54no—
220#41—58no—
221#41—54no—
222#41—54no—
223#41—52no—
224#41—58no—
225#41—58no—
226#41—54no—
227#41—58no—
228#41—58no—
229#41—52no—
230#41—54no—
231#41—54no—
232#41—54no—
233#41—54no—
234#41—52no—
235#41—52no—
236#41—52no—
237C2×Q8—51no—
238C2×Q8—52no—
239C2×Q8—54no—
240C2×Q8—52no—
241C2×Q8—52no—
242C2×Q8—58no—
243C2×Q8—54no—
244C2×Q8—58no—
245C2×Q8—58no—
246C2×Q8—54no—
247C2×Q8—51no—
248C2×Q8—54no—
249C2×Q8—52no—
250C2×Q8—54no—
251C2×D4—52no—
252C2×D4—54no—
253C2×D4—54no—
254C2×D4—58no—
255C2×D4—54no—
256C2×D4—54no—
257C2×D4—58no—
258C2×D4—51no—
259C2×D4—58no—
260C2×D4—52no—
261C2×D4—51no—
262C2×D4—52no—
263C2×D4—52no—
264C2×D4—54no—
265C22×C4—54no—
266C22×C4—58no—
267C22×C4—54no—
268C22×C4—52no—
269C22×C4—52no—
270C22×C4—51no—
271C22×C4—54no—
272C22×C4—52no—
273C3×C5—161noHall
274C22×C3—52no—
275C22×C3—52no—
276C22×C3—51no—
277C22×C3—202no—
278C22×C3—202no—
279C22×C3—201no—
280C22×C3—204no—
281C22×C3C2×C24⋊C512yes—
282C3×C4—51no—
283C3×C4—54no—
284C3×C4—52no—
285C3×C4C2×C24⋊C511yescharacteristic
286C2×C5—161no—
287C2×C5—322no—
288C23—51no—
289C23—54no—
290C23—52no—
291C23—52no—
292Q8—52no—
293Q8—52no—
294Q8—54no—
295Q8—54no—
296Q8—52no—
297Q8—58no—
298Q8—52no—
299D4—52no—
300D4—54no—
301D4—52no—
302D4—54no—
303D4—54no—
304D4—54no—
305D4—58no—
306D4—58no—
307D4—52no—
308D4—54no—
309D4—51no—
310D4C3×C24⋊C511yescharacteristic
311C2×C4—52no—
312C2×C4—54no—
313C2×C4—58no—
314C2×C4—54no—
315C2×C4—58no—
316C2×C4—51no—
317C2×C4—52no—
318C2×C4—54no—
319C2×C4—58no—
320C2×C4—54no—
321C2×C4—51no—
322C2×C4—52no—
323C2×C4—54no—
324C2×C4—52no—
325C2×C3—22no—
326C2×C3—102no—
327C2×C3—101no—
328C2×C3—102no—
329C2×C3C22×C24⋊C511yescharacteristic
330C5—161noSylow
331C22—52no—
332C22—52no—
333C22—51no—
334C22—202no—
335C22—202no—
336C22—204no—
337C22—201no—
338C22C2×C3×C24⋊C512yes—
339C4—51no—
340C4—54no—
341C4—52no—
342C4C2×C3×C24⋊C511yescharacteristic
343C32−1+6⋊C511yescharacteristic, direct product factor, Sylow
344C2—22no—
345C2—101no—
346C2—102no—
347C2—102no—
348C2C22×C3×C24⋊C511yescharacteristic
subgroupG/H
center, socleC2×C3C22×C24⋊C5
commutator, cosocle, hypocenter2−1+4C22×C3×C5
FrattiniC2C22×C3×C24⋊C5
FittingC3×2−1+6C5
hypercenterC3×D4C24⋊C5
LCSG ▷ 2−1+4
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#585⋊C5C2semidirect
2. C2×C3×2−1+4⋊C5C2semidirect
3. 2−1+6⋊C5C3direct (N × H)
4. C3×2−1+4⋊C5C22semidirect
5. C3×2−1+6C5semidirect
6. #585⋊C5C2×C3semidirect
7. C2×2−1+4⋊C5C2×C3semidirect
11. C3×#585C2×C5semidirect
15. C2×C3×2−1+4C2×C5semidirect
16. 2−1+4⋊C5C22×C3semidirect
17. 2−1+6C3×C5semidirect
59. #585C2×C3×C5semidirect
63. C2×2−1+4C2×C3×C5semidirect
343. C32−1+6⋊C5direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#585⋊C52
C2×C3×2−1+4⋊C52
2−1+6⋊C53
C3×2−1+65
C3×C5×D416