Finite Group Databasebeta

← #14282272

Subgroups — #14282272

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6772C211yesmaximal, characteristic
2#11752340C211yesmaximal, characteristic, complemented
3C3×C4×#700C211yesmaximal, characteristic, complemented
4#11746495C212yesmaximal, complemented
5#11746489C212yesmaximal, complemented
6#10604406—31nomaximal, Hall
7C2×#6772C411yescharacteristic
8C2×#6758—21no—
9C2×C3×#700—21no—
10C4×Dic30C411yescharacteristic
11C4×C5×D12C411yescharacteristic
12C3×C42×D5C2211yescharacteristic
13#91440—21no—
14C4×#1201—21no—
15Dic5×D12C2211yescharacteristic
16#91398—21no—
17C4×D3×Dic5—21no—
18#91358C411yescharacteristic
19#91333C2211yescharacteristic
20C4⋊(C3⋊#170)—21no—
21C4⋊(C3⋊#170)C2212yes—
22C3×#4002—21no—
23C3×C4×#170C2212yes—
24#69854—51nomaximal, Hall
25C4×#711—31no—
26#65681—31no—
27C4×#700D311yescharacteristic, complemented
28D4⋊#170—32no—
29#64872—32no—
30C2×Dic30C2×C411yescharacteristic
31C2×C5×D12C2×C411yescharacteristic
32C2×C4×C5×D3—21no—
33C2×C3×C4×D5C2×C411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×#1201—21no—
37C2×D3×Dic5—21no—
38C2×D3×Dic5—21no—
39#6772—22no—
40#6772—22no—
41C2×C3⋊#170—22no—
42#6758—42no—
43C2×C3×#170—22no—
44C3×#700—24no—
45C3×C42×C5C2×C411yescharacteristic
46C4⋊Dic15C2×C411yescharacteristic
47#6712—21no—
48C4×Dic15—21no—
49C5×(C2×D3)⋊C4—21no—
50C5×C4⋊Dic3C2×C411yescharacteristic
51C3×C4×Dic5D411yescharacteristic
52C3×C4×Dic5D411yescharacteristic
53C3×C4×Dic5C2311yescharacteristic
54#6669—21no—
55(C2×D3)⋊Dic5—21no—
56#6665—21no—
57Dic3×Dic5—21no—
58Dic3×Dic5—21no—
59C4×#235—51no—
60C3×C4×M16—51no—
61#4924—51no—
62#4459—52no—
63#4456—52no—
64C2×#711—31no—
65C2×#700—21no—
66C2×#700—61no—
67C4×C5×D4—31no—
68D4×Dic5—31no—
69C4×C22⋊D5—61no—
70C2×C4×Dic5—61no—
71#4038—31no—
72#4033—31no—
73#4027—61no—
74#4023—61no—
75C42×D5C2×D311yescharacteristic
76C4×Dic10—31no—
77#4002—21no—
78C4⋊#170—32no—
79C4⋊#170—61no—
80C4×#170C2×D312yes—
81#1489—151noSylow
82C22×C5×D3—21no—
83C22×C3×D5C22⋊C411yescharacteristic
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22⋊C411yescharacteristic
86C2×C3×C4×C5C22×C411yescharacteristic
87C2×Dic15—21no—
88C2×Dic15—21no—
89Dic30—21no—
90Dic30M1612yescomplemented
91C2×C5×Dic3—21no—
92C5×D12—21no—
93C5×D12M1612yescomplemented
94C4×C5×D3—42no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C2×D411yescharacteristic
97C2×C3×Dic5C2×D411yescharacteristic
98C3×C4×D5—22no—
99C3×C4×D5—21no—
100C3×C4×D5—21no—
101C3×C4×D5—21no—
102C3×C4×D5—21no—
103#1201—24no—
104D3×Dic5—24no—
105D3×Dic5—41no—
106D3×Dic5—41no—
107C3⋊#170—42no—
108C3×#170—24no—
109C2×#235—51no—
110C2×C3×M16—101no—
111C2×C3×C42—51no—
112C4×#67—101no—
113C4×#67—101no—
114#926—51no—
115C2×#208—101no—
116C4×D12—51no—
117C4×D12AGL(1,𝔽5)11yescharacteristic
118#872—101no—
119C42×D3—101no—
120C4×Dic6—51no—
121C4×Dic6—51no—
122C3×#97—101no—
123C3×C4×C8—52no—
124C4⋊(C3⋊C8)—52no—
125C4⋊(C3⋊C8)—101no—
126C2×C5×D4—31no—
127C22×C4×C5—61no—
128C2×C22⋊D5—61no—
129C22×Dic5—61no—
130C22×Dic5—61no—
131#711—62no—
132#711—62no—
133C2×C4×D5C4×D311yescharacteristic
134C2×C4×D5D1211yescharacteristic
135C2×C4×D5#6711yescharacteristic
136C2×Dic10—31no—
137C2×#170—22no—
138C2×#170—62no—
139#700—24no—
140#700—122no—
141C5×C4⋊C4—31no—
142C5×C22⋊C4—61no—
143C42×C5C4×D311yescharacteristic
144C22⋊Dic5—61no—
145#685—61no—
146C4⋊Dic5—31no—
147#683—61no—
148#683—61no—
149C4×Dic5—61no—
150C4×Dic5—61no—
151C4×Dic5—61no—
152C4×Dic5D1211yescharacteristic
153C4×Dic5#6711yescharacteristic
154C4×Dic5C22×D311yescharacteristic
155C4×#41—151no—
156#423—151no—
157C4×M16—51no—
158Q8⋊C8—152no—
159D4⋊C8—152no—
160C22×C3×C5C2×C22⋊C411yescharacteristic
161C2×C5×D3—22no—
162C2×C5×D3—41no—
163C2×C5×D3—41no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C3×C4×C5—21no—
168C3×C4×C5#10412yes—
169C3×C4×C5C2×M1611yescharacteristic
170C3×C4×C5C2×M1611yescharacteristic
171Dic15—22no—
172Dic15—41no—
173C3×Dic5—21no—
174C3×Dic5—21no—
175C3×Dic5AGL(1,ℤ/8)11yescharacteristic
176C3×Dic5#13711yescharacteristic
177C5×Dic3—41no—
178C22×C3×C4—51no—
179C22×C3×C4—51no—
180C22×C3×C4—51no—
181C2×#67—101no—
182#235—102no—
183#235—102no—
184C2×D12C2×AGL(1,𝔽5)11yescharacteristic
185C2×C4×D3—21no—
186C2×C4×D3—101no—
187C2×C4×D3—101no—
188C2×Dic6—51no—
189C3×M16—104no—
190C2×C3×C8—102no—
191C3×C42—51no—
192C3×C42—51no—
193C3×C42—51no—
194C3×C42C2×AGL(1,𝔽5)11yescharacteristic
195#217—101no—
196(C2×D3)⋊C4—21no—
197(C2×D3)⋊C4—101no—
198C4⋊Dic3—51no—
199C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
200Dic3⋊C4—101no—
201Dic3⋊C4—101no—
202C4×Dic3—101no—
203C4×Dic3—101no—
204C4×Dic3—101no—
205#208—202no—
206C2×C3⋊C8—102no—
207C23×C5—61no—
208C22×D5(C2×D3)⋊C411yescharacteristic
209C5×D4—32no—
210C5×D4—61no—
211C2×C4×C5—61no—
212C2×C4×C5—122no—
213C2×C4×C5(C2×D3)⋊C411yescharacteristic
214C2×C4×C5(C2×D3)⋊C411yescharacteristic
215C2×C4×C5C2×C4×D311yescharacteristic
216C22⋊D5—64no—
217C2×Dic5—64no—
218C2×Dic5—61no—
219C2×Dic5—61no—
220C2×Dic5—121no—
221C2×Dic5—121no—
222C2×Dic5(C2×D3)⋊C411yescharacteristic
223C2×Dic5C2×D1211yescharacteristic
224C2×Dic5C2×#6711yescharacteristic
225C4×D5—22no—
226C4×D5—21no—
227C4×D5—21no—
228C4×D5—21no—
229C4×D5—21no—
230Dic10—32no—
231Dic10—61no—
232#170—24no—
233#170—122no—
234C2×#41—151no—
235C2×M16—101no—
236C2×M16—301no—
237C4×Q8—151no—
238C4×Q8—151no—
239C4×D4—31no—
240C4×D4—151no—
241C4×D4—301no—
242C4×D4—301no—
243#117—151no—
244#117—301no—
245C2×C42—51no—
246C2×C42—301no—
247C4⋊C8—152no—
248C4⋊C8—301no—
249#97—101no—
250C4×C8—52no—
251C2×C3×C5#40711yescharacteristic
252C2×C3×C5#41711yescharacteristic
253C2×C3×C5C2×#10411yescharacteristic
254C3×D5—41no—
255C5×D3—42no—
256C23×C3—51no—
257C22×D3—21no—
258C2×C3×C4—51no—
259C2×C3×C4—51no—
260C2×C3×C4—51no—
261C2×C3×C4—101no—
262C2×C3×C4—101no—
263C2×C3×C4—101no—
264C2×C3×C4—101no—
265C2×C3×C4—102no—
266C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
267C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
268C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
269#67—104no—
270C2×Dic3—21no—
271C2×Dic3—101no—
272C2×Dic3—101no—
273D12—21no—
274D12#70012yescomplemented
275C4×D3—42no—
276C4×D3—104no—
277C4×D3—201no—
278C4×D3—201no—
279Dic6—52no—
280Dic6—101no—
281C3×C8—104no—
282C3⋊C8—202no—
283C22×C5—62no—
284C22×C5—121no—
285C22×C5—121no—
286C22×C5C2×(C2×D3)⋊C411yescharacteristic
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—21no—
290C4×C5—21no—
291C4×C5—121no—
292C4×C5#80512yes—
293C4×C5D3×M1611yescharacteristic
294C4×C5D3×M1611yescharacteristic
295Dic5—21no—
296Dic5—21no—
297Dic5—62no—
298Dic5—121no—
299Dic5#90811yescharacteristic
300Dic5#90911yescharacteristic
301#41—302no—
302#41—302no—
303C2×Q8—151no—
304C2×D4—31no—
305C2×D4—301no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—61no—
310C22×C4—301no—
311C22×C4—301no—
312M16—104no—
313M16—602no—
314C2×C8—102no—
315C2×C8—302no—
316C4⋊C4—31no—
317C4⋊C4—151no—
318C4⋊C4—301no—
319C4⋊C4—301no—
320C22⋊C4—61no—
321C22⋊C4—301no—
322C22⋊C4—301no—
323C42—51no—
324C42—51no—
325C42—51no—
326C42—301no—
327C42—301no—
328C42—301no—
329C42D3×AGL(1,𝔽5)11yescharacteristic
330C3×C5#148911yescharacteristic, complemented, Hall
331C22×C3—101no—
332C22×C3—101no—
333C22×C3—101no—
334C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
335C2×D3—22no—
336C2×D3—41no—
337C2×D3—41no—
338C3×C4—21no—
339C3×C4—51no—
340C3×C4—51no—
341C3×C4—101no—
342C3×C4—101no—
343C3×C4#400812yes—
344C3×C4C2×#70011yescharacteristic
345C3×C4D5⋊M1611yescharacteristic
346Dic3—41no—
347Dic3—102no—
348Dic3—201no—
349C2×C5—122no—
350C2×C5C2×#80511yescharacteristic
351C2×C5#512611yescharacteristic
352C2×C5#513311yescharacteristic
353D5—41no—
354C23—51no—
355C23—61no—
356Q8—152no—
357Q8—301no—
358D4—32no—
359D4—61no—
360D4—304no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—51no—
364C2×C4—61no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—102no—
370C2×C4—122no—
371C2×C4—304no—
372C2×C4—301no—
373C2×C4—301no—
374C2×C4—601no—
375C2×C4—601no—
376C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
377C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
378C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
379C8—104no—
380C8—602no—
381C2×C3—201no—
382C2×C3#6575611yescharacteristic
383C2×C3C2×#400811yescharacteristic
384C2×C3#6579311yescharacteristic
385D3—42no—
386C5#6985411yescharacteristic, complemented, Sylow
387C22—62no—
388C22—101no—
389C22—101no—
390C22—101no—
391C22—121no—
392C22—121no—
393C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
394C4—21no—
395C4—51no—
396C4—51no—
397C4—101no—
398C4—101no—
399C4—121no—
400C4—302no—
401C4—601no—
402C4#9115612yes—
403C4D3×#70011yescharacteristic
404C4#9191511yescharacteristic
405C3#1060440611yescharacteristic, complemented, Sylow
406C2—122no—
407C2—201no—
408C2#1175170811yescharacteristic
409C2C2×#9115611yescharacteristic
410C2#1175180611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#6772C4
cosocleC3×C4×Dic5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1489
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752340C2semidirect
3. C3×C4×#700C2semidirect
4. #11746495C2semidirect
5. #11746489C2semidirect
27. C4×#700D3semidirect
90. Dic30M16semidirect
93. C5×D12M16semidirect
274. D12#700semidirect
330. C3×C5#1489semidirect
386. C5#69854semidirect
405. C3#10604406semidirect

Maximal subgroups

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

subgroupindex
C4×#67722
#117523402
C3×C4×#7002
#117464952
#117464892
#106044063
#698545