Finite Group Databasebeta

← #14297122

Subgroups — #14297122

Order 1920.

1236 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C22×#6772C211yesmaximal, characteristic
2#11752415C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C2×#91156C214yesmaximal, complemented
5#10604930—31nomaximal, Hall
6C22×Dic30C411yescharacteristic, complemented
7C22×C5×D12C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C22×#1201—21no—
10C22×D3×Dic5—21no—
11C2×#6772—21no—
12C2×#6772—21no—
13C2×#6772—28no—
14C2×#6772C412yescomplemented
15C2×#6772C2212yes—
16C2×#6758C2212yescomplemented
17C2×C3×C22⋊AGL(1,𝔽5)—21no—
18C2×C3×C4×AGL(1,𝔽5)C2212yescomplemented
19#91241—21no—
20#91232—21no—
21C3×(C2×AGL(1,𝔽5))⋊C4—21no—
22#91156—28no—
23#69454—51nomaximal, Hall
24C22×#711—31no—
25C2×#4008—34no—
26C4×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
27#65756—31no—
28C23×C5×D3—21no—
29C23×C3×D5D411yescharacteristic
30C22×C3×AGL(1,𝔽5)—22no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×Dic15—21no—
33C2×Dic30—21no—
34C2×Dic30—24no—
35C2×Dic30C2×C412yes—
36C2×C5×D12—21no—
37C2×C5×D12—24no—
38C2×C5×D12C2×C412yes—
39C22×C3×Dic5D411yescharacteristic
40C2×C3×C4×D5—24no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×#1201—28no—
46C2×#1201—22no—
47C2×#1201—22no—
48C2×D3×Dic5—28no—
49C2×D3×Dic5—22no—
50C2×D3×Dic5—22no—
51#6772—28no—
52#6772—28no—
53#6772—24no—
54#6772—24no—
55#6772—24no—
56#6772D418yes—
57C2×C3⋊#170—22no—
58#6758—24no—
59C3×C22⋊AGL(1,𝔽5)—24no—
60C3×C4×AGL(1,𝔽5)—24no—
61C22×#235—51no—
62C3×C4×C22⋊C4—51no—
63#5207—51no—
64C2×#805—54no—
65C22×C5×D4—31no—
66C22×C22⋊D5—61no—
67C23×Dic5—61no—
68C2×#711—32no—
69C2×#711—32no—
70C2×#711—68no—
71C2×#711—61no—
72C2×#711—61no—
73C22×C4×D5C2×D311yescharacteristic
74C22×Dic10—31no—
75C2×C22⋊AGL(1,𝔽5)—21no—
76C2×C4×AGL(1,𝔽5)C2×D312yescomplemented
77C2×#700—32no—
78C22⋊#170—61no—
79#4008—68no—
80(C2×AGL(1,𝔽5))⋊C4—21no—
81#4003—61no—
82#1789—151noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C5×D3—24no—
85C22×C5×D3—22no—
86C22×C5×D3—42no—
87C22×C5×D3—42no—
88C22×C3×D5—22no—
89C22×C3×D5—22no—
90C22×C3×D5—41no—
91C22×C3×D5C22⋊C411yescharacteristic
92C22×C3×D5C2×D411yescharacteristic
93C2×C3×AGL(1,𝔽5)—24no—
94C2×C3×AGL(1,𝔽5)—42no—
95C2×C3×AGL(1,𝔽5)—42no—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×Dic15—24no—
100C2×Dic15—22no—
101Dic30—24no—
102Dic30—22no—
103Dic30C22⋊C414yescomplemented
104C5×D12—24no—
105C5×D12—22no—
106C5×D12C22⋊C414yescomplemented
107C2×C3×Dic5—22no—
108C2×C3×Dic5C22⋊C411yescharacteristic
109C2×C3×Dic5C2×D411yescharacteristic
110C3×C4×D5—22no—
111C3×C4×D5—22no—
112C3×C4×D5—22no—
113C3×C4×D5C2×D414yes—
114#1201—28no—
115#1201—28no—
116D3×Dic5—28no—
117D3×Dic5—28no—
118C3⋊#170—42no—
119C23×C3×C4—51no—
120C22×#67—101no—
121C2×#235—52no—
122C2×#235—52no—
123C2×#235—108no—
124C2×#235—101no—
125C2×#235—101no—
126C22×D12AGL(1,𝔽5)11yescharacteristic, complemented
127C22×C4×D3—101no—
128C22×Dic6—51no—
129C2×C3×C22⋊C4—101no—
130C2×C3×C42—52no—
131C2×#208—52no—
132C3×#95—101no—
133#830—101no—
134#830—101no—
135#805—108no—
136C24×C5—61no—
137C23×D5#6711yescharacteristic
138C22×AGL(1,𝔽5)—22no—
139C2×C5×D4—32no—
140C2×C5×D4—64no—
141C2×C5×D4—61no—
142C22×C4×C5C4×D311yescharacteristic
143C2×C22⋊D5—68no—
144C2×C22⋊D5—62no—
145C2×C22⋊D5—62no—
146C22×Dic5—68no—
147C22×Dic5—61no—
148C22×Dic5—62no—
149C22×Dic5—62no—
150C22×Dic5D1211yescharacteristic
151#711—38no—
152#711—64no—
153#711—64no—
154#711—68no—
155#711—68no—
156#711—64no—
157C2×C4×D5—24no—
158C2×C4×D5C4×D311yescharacteristic
159C2×C4×D5D1211yescharacteristic
160C2×C4×D5#6711yescharacteristic
161C2×C4×D5C22×D311yescharacteristic
162C2×Dic10—32no—
163C2×Dic10—64no—
164C2×Dic10—61no—
165C22⋊AGL(1,𝔽5)—24no—
166C2×#170—62no—
167C4×AGL(1,𝔽5)—24no—
168#700—64no—
169C22×#41—151no—
170C2×#104—154no—
171#407—151no—
172C4×C22⋊C4—51no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5#10412yescomplemented
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C2×C5×D3—24no—
178C2×C5×D3—48no—
179C2×C5×D3—42no—
180C2×C5×D3—42no—
181C2×C3×D5—21no—
182C2×C3×D5—21no—
183C2×C3×D5—21no—
184C2×C3×D5—21no—
185C2×C3×D5—21no—
186C2×C3×D5—44no—
187C2×C3×D5#12012yes—
188C3×AGL(1,𝔽5)—44no—
189C3×C4×C5—21no—
190C3×C4×C5C2×C22⋊C412yes—
191Dic15—24no—
192C3×Dic5—21no—
193C3×Dic5#12012yes—
194C24×C3—51no—
195C23×D3—21no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—104no—
202C22×C3×C4—102no—
203C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
204C2×#67—102no—
205C2×#67—102no—
206C2×#67—108no—
207C22×Dic3—101no—
208#235—58no—
209#235—104no—
210#235—104no—
211#235—108no—
212#235—108no—
213#235—104no—
214C2×D12—21no—
215C2×D12—24no—
216C2×D12C2×AGL(1,𝔽5)12yes—
217C2×C4×D3—102no—
218C2×C4×D3—102no—
219C2×C4×D3—108no—
220C2×Dic6—52no—
221C2×Dic6—104no—
222C2×Dic6—101no—
223C3×C22⋊C4—104no—
224C3×C42—104no—
225#208—104no—
226C2×C3⋊C8—102no—
227C23×C5—64no—
228C23×C5—62no—
229C23×C5—122no—
230C23×C5—122no—
231C23×C5(C2×D3)⋊C411yescharacteristic
232C22×D5—22no—
233C22×D5—22no—
234C22×D5—41no—
235C22×D5(C2×D3)⋊C411yescharacteristic
236C22×D5C2×#6711yescharacteristic
237C2×AGL(1,𝔽5)—24no—
238C2×AGL(1,𝔽5)—42no—
239C2×AGL(1,𝔽5)—42no—
240C5×D4—34no—
241C5×D4—64no—
242C5×D4—62no—
243C2×C4×C5—22no—
244C2×C4×C5(C2×D3)⋊C411yescharacteristic
245C2×C4×C5C2×C4×D311yescharacteristic
246C22⋊D5—68no—
247C22⋊D5—68no—
248C2×Dic5—22no—
249C2×Dic5—68no—
250C2×Dic5—64no—
251C2×Dic5—62no—
252C2×Dic5—68no—
253C2×Dic5(C2×D3)⋊C411yescharacteristic
254C2×Dic5C2×D1211yescharacteristic
255C4×D5—22no—
256C4×D5—22no—
257C4×D5—22no—
258C4×D5D3×D414yes—
259Dic10—34no—
260Dic10—64no—
261Dic10—62no—
262#170—122no—
263C2×#41—152no—
264C2×#41—152no—
265C2×#41—301no—
266C2×#41—301no—
267C2×#41—308no—
268C22×Q8—151no—
269C22×D4—31no—
270C22×D4—301no—
271C23×C4—51no—
272C23×C4—301no—
273C2×M16—152no—
274C2×C22⋊C4—101no—
275C2×C42—52no—
276#104—308no—
277C22⋊C8—301no—
278C22⋊C8—301no—
279#95—101no—
280C2×C3×C5—22no—
281C2×C3×C5#37911yescharacteristic
282C2×C3×C5C2×#10411yescharacteristic
283C2×C3×C5#42111yescharacteristic
284C3×D5—22no—
285C3×D5—41no—
286C5×D3—44no—
287C23×C3—51no—
288C23×C3—51no—
289C23×C3—102no—
290C23×C3—102no—
291C23×C3—201no—
292C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
293C22×D3—24no—
294C22×D3—22no—
295C22×D3—42no—
296C22×D3—42no—
297C2×C3×C4—22no—
298C2×C3×C4—51no—
299C2×C3×C4—54no—
300C2×C3×C4—51no—
301C2×C3×C4—102no—
302C2×C3×C4—102no—
303C2×C3×C4—102no—
304C2×C3×C4—102no—
305C2×C3×C4—104no—
306C2×C3×C4—202no—
307C2×C3×C4—202no—
308C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
309C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
310#67—108no—
311#67—108no—
312C2×Dic3—104no—
313C2×Dic3—102no—
314D12—24no—
315D12—22no—
316D12C22⋊AGL(1,𝔽5)14yescomplemented
317C4×D3—108no—
318C4×D3—108no—
319Dic6—54no—
320Dic6—104no—
321Dic6—102no—
322C3⋊C8—202no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—64no—
326C22×C5—128no—
327C22×C5—122no—
328C22×C5—122no—
329C22×C5#80512yescomplemented
330C22×C5C2×(C2×D3)⋊C411yescharacteristic
331C2×D5—21no—
332C2×D5—21no—
333C2×D5—21no—
334C2×D5—21no—
335C2×D5—21no—
336C2×D5—44no—
337C2×D5#93712yes—
338AGL(1,𝔽5)—44no—
339C4×C5—21no—
340C4×C5D3×C22⋊C412yes—
341Dic5—21no—
342Dic5—64no—
343Dic5#88212yes—
344C24—51no—
345C24—61no—
346#41—158no—
347#41—304no—
348#41—304no—
349#41—304no—
350#41—308no—
351#41—308no—
352C2×Q8—152no—
353C2×Q8—304no—
354C2×Q8—301no—
355C2×D4—32no—
356C2×D4—64no—
357C2×D4—61no—
358C2×D4—302no—
359C2×D4—302no—
360C2×D4—308no—
361C22×C4—51no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—104no—
367C22×C4—102no—
368C22×C4—302no—
369C22×C4—302no—
370C22×C4—308no—
371C22×C4—301no—
372C22×C4D3×AGL(1,𝔽5)11yescharacteristic
373M16—304no—
374C2×C8—302no—
375C22⋊C4—104no—
376C42—104no—
377C3×C5#178911yescharacteristic, complemented, Hall
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—52no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3—101no—
384C22×C3—101no—
385C22×C3—101no—
386C22×C3—204no—
387C22×C3#400812yescomplemented
388C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
389C2×D3—24no—
390C2×D3—48no—
391C2×D3—42no—
392C2×D3—42no—
393C3×C4—21no—
394C3×C4—52no—
395C3×C4—101no—
396C3×C4—204no—
397C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
398Dic3—104no—
399C2×C5—22no—
400C2×C5—124no—
401C2×C5C2×#80511yescharacteristic
402C2×C5#495511yescharacteristic
403C2×C5#500511yescharacteristic
404D5—22no—
405D5—41no—
406C23—51no—
407C23—51no—
408C23—64no—
409C23—62no—
410C23—102no—
411C23—102no—
412C23—122no—
413C23—122no—
414C23—201no—
415C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
416Q8—154no—
417Q8—304no—
418Q8—302no—
419D4—34no—
420D4—64no—
421D4—62no—
422D4—308no—
423D4—308no—
424C2×C4—22no—
425C2×C4—51no—
426C2×C4—54no—
427C2×C4—51no—
428C2×C4—102no—
429C2×C4—102no—
430C2×C4—102no—
431C2×C4—102no—
432C2×C4—104no—
433C2×C4—202no—
434C2×C4—202no—
435C2×C4—308no—
436C2×C4—308no—
437C2×C4—304no—
438C2×C4—302no—
439C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
440C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
441C8—602no—
442C2×C3—22no—
443C2×C3—102no—
444C2×C3—201no—
445C2×C3C2×#400811yescharacteristic
446C2×C3#6576911yescharacteristic
447C2×C3#6580011yescharacteristic
448D3—44no—
449C5#6945411yescharacteristic, complemented, Sylow
450C22—21no—
451C22—21no—
452C22—52no—
453C22—64no—
454C22—101no—
455C22—101no—
456C22—101no—
457C22—101no—
458C22—101no—
459C22—128no—
460C22—122no—
461C22—122no—
462C22—204no—
463C22#9115612yescomplemented
464C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
465C4—21no—
466C4—52no—
467C4—101no—
468C4—204no—
469C4—304no—
470C4D3×C22⋊AGL(1,𝔽5)12yes—
471C3#1060493011yescharacteristic, complemented, Sylow
472C2—22no—
473C2—102no—
474C2—124no—
475C2—201no—
476C2C2×#9115611yescharacteristic
477C2#1175171911yescharacteristic
478C2#1175202511yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#6772C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1789
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752415C2semidirect
3. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
4. C2×#91156C2semidirect
6. C22×Dic30C4semidirect
7. C22×C5×D12C4semidirect
14. C2×#6772C4semidirect
16. C2×#6758C22semidirect
18. C2×C3×C4×AGL(1,𝔽5)C22semidirect
26. C4×C22⋊AGL(1,𝔽5)D3semidirect
76. C2×C4×AGL(1,𝔽5)C2×D3semidirect
103. Dic30C22⋊C4semidirect
106. C5×D12C22⋊C4semidirect
126. C22×D12AGL(1,𝔽5)semidirect
175. C22×C3×C5#104semidirect
316. D12C22⋊AGL(1,𝔽5)semidirect
329. C22×C5#805semidirect
377. C3×C5#1789semidirect
387. C22×C3#4008semidirect
449. C5#69454semidirect
463. C22#91156semidirect
471. C3#10604930semidirect

Maximal subgroups

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

subgroupindex
C22×#67722
#117524152
C3×C4×C22⋊AGL(1,𝔽5)2
C2×#911562
#106049303
#694545