Finite Group Databasebeta

← #14282141

Subgroups — #14282141

Order 1920.

496 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 108 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
1Q8×#1200C211yesmaximal, characteristic
2#11751922C211yesmaximal, characteristic
3#11751797C211yesmaximal, characteristic
4#11746801C211yesmaximal, characteristic, complemented
5C3×Q8⋊#170C211yesmaximal, characteristic, complemented
6#11746495C211yesmaximal, characteristic, complemented
7#11746494C211yesmaximal, characteristic, complemented
8#10604439—31nomaximal, Hall
9C2×Q8×D15C411yescharacteristic
10C2×#6740—21no—
11C5×Q8×Dic3C411yescharacteristic
12C3×Q8×Dic5C2211yescharacteristic
13#91437—21no—
14#91437C411yescharacteristic
15C4×#1200—21no—
16C4×#1200C2211yescharacteristic
17#91340—21no—
18#91340C411yescharacteristic
19#91333—21no—
20#91333C2211yescharacteristic
21C4⋊(C3⋊#170)C2211yescharacteristic
22C4×C3⋊#170C2211yescharacteristic
23C3×C4⋊#170C2211yescharacteristic
24C3×C4×#170C2211yescharacteristic
25Dic3⋊#170—21no—
26Dic3⋊#170—21no—
27#91171—21no—
28#68236—51nomaximal, Hall
29C4×Q8×D5—31no—
30#65681—31no—
31#65680—31no—
32Q8⋊#170—31no—
33Q8⋊#170D311yescharacteristic, complemented
34#64872—31no—
35#64872—31no—
36C2×C3×C5×Q8C2×C411yescharacteristic
37Q8×D15—22no—
38Q8×D15—22no—
39C2×C4×D15—21no—
40C2×C4×D15C2×C411yescharacteristic
41C2×Dic30—21no—
42C2×Dic30C2×C411yescharacteristic
43C2×#1200—21no—
44C2×#1200D411yescharacteristic
45C2×#1200D411yescharacteristic
46C2×C3⋊#170—21no—
47C2×C3⋊#170—21no—
48C2×C3×#170—21no—
49C2×C3×#170—21no—
50#6740—42no—
51C5×C4⋊Dic3—21no—
52C5×C4⋊Dic3C2×C411yescharacteristic
53C4×C5×Dic3—21no—
54C4×C5×Dic3C2×C411yescharacteristic
55C3×C4⋊Dic5—21no—
56C3×C4⋊Dic5C2×C411yescharacteristic
57C3×C4×Dic5—21no—
58C3×C4×Dic5C2311yescharacteristic
59#6669—21no—
60#6669—21no—
61#6669—21no—
62Dic3×Dic5—21no—
63Dic3×Dic5—21no—
64Dic3×Dic5D411yescharacteristic
65Dic3×Dic5D411yescharacteristic
66C4×D3×Q8—51no—
67#4837—51no—
68#4685—51no—
69C3×Q8⋊C8—51no—
70#4535—51no—
71#4484—51no—
72#4456—51no—
73C2×Q8×D5—31no—
74C2×#700—61no—
75C4×C5×Q8—31no—
76Q8×Dic5C2×D311yescharacteristic
77D5×C4⋊C4—31no—
78D5×C4⋊C4—61no—
79#4033—31no—
80#4033—61no—
81C42×D5—31no—
82C42×D5—61no—
83C4×Dic10—31no—
84C4×Dic10—61no—
85#4003—61no—
86C4⋊#170—31no—
87C4⋊#170—61no—
88C4⋊#170—61no—
89C4⋊#170C2×D311yescharacteristic
90C4×#170—31no—
91C4×#170C2×D311yescharacteristic
92#1487—151noSylow
93C22×D15C22⋊C411yescharacteristic
94C3×C5×Q8—21no—
95C3×C5×Q8M1612yescomplemented
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×Dic15—21no—
99C2×Dic15C22⋊C411yescharacteristic
100C4×D15—24no—
101C4×D15—21no—
102C4×D15—21no—
103Dic30—24no—
104Dic30—21no—
105Dic30M1612yescomplemented
106C2×C5×Dic3—21no—
107C2×C5×Dic3C22⋊C411yescharacteristic
108C2×C5×Dic3C22⋊C411yescharacteristic
109C2×C3×Dic5—21no—
110C2×C3×Dic5C2×D411yescharacteristic
111C2×C3×Dic5C2×D411yescharacteristic
112#1200—41no—
113#1200—42no—
114#1200SD1612yes—
115#1200Q1612yes—
116C3⋊#170—41no—
117C3⋊#170—41no—
118C3×#170—41no—
119C3×#170—41no—
120C2×D3×Q8—51no—
121C3×C4×Q8—51no—
122Q8×Dic3AGL(1,𝔽5)11yescharacteristic
123C2×ΓL(1,𝔽25)—101no—
124D3×C4⋊C4—51no—
125D3×C4⋊C4—101no—
126#887—51no—
127#887—101no—
128C42×D3—51no—
129C42×D3—101no—
130C4×Dic6—51no—
131C4×Dic6—101no—
132C3×C4⋊C8—51no—
133C3×C4×C8—51no—
134(C2×D3)⋊C8—101no—
135Dic3⋊C8—101no—
136Dic3⋊C8—101no—
137C4⋊(C3⋊C8)—51no—
138C4×C3⋊C8—51no—
139C2×C5×Q8C4×D311yescharacteristic
140Q8×D5—62no—
141Q8×D5—62no—
142C2×C4×D5—31no—
143C2×C4×D5—31no—
144C2×C4×D5—31no—
145C2×C4×D5—61no—
146C2×C4×D5—61no—
147C2×Dic10—31no—
148C2×Dic10—61no—
149C2×#170—21no—
150C2×#170—21no—
151C2×#170—61no—
152C2×#170—61no—
153#700—122no—
154C5×C4⋊C4—31no—
155C5×C4⋊C4—61no—
156C42×C5—31no—
157C42×C5—61no—
158C4⋊Dic5—21no—
159C4⋊Dic5C4×D311yescharacteristic
160#683—61no—
161#683—61no—
162#683—61no—
163C4×Dic5—21no—
164C4×Dic5—31no—
165C4×Dic5—31no—
166C4×Dic5—61no—
167C4×Dic5—61no—
168C4×Dic5C22×D311yescharacteristic
169C2×C4×Q8—151no—
170#431—151no—
171#423—151no—
172Q8⋊C8—51no—
173Q8⋊C8—151no—
174Q8⋊C8—151no—
175Q8⋊C8—151no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C2×D15—21no—
178C2×D15—21no—
179C2×D15#10312yes—
180C3×C4×C5—22no—
181C3×C4×C5C2×M1611yescharacteristic
182C3×C4×C5C2×M1611yescharacteristic
183Dic15—21no—
184Dic15—22no—
185C3×Dic5—21no—
186C3×Dic5—41no—
187C3×Dic5C2×SD1611yescharacteristic
188C3×Dic5C2×Q1611yescharacteristic
189C5×Dic3—21no—
190C5×Dic3—41no—
191C5×Dic3#10312yes—
192C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
193D3×Q8—102no—
194D3×Q8—102no—
195C2×C4×D3—51no—
196C2×C4×D3—51no—
197C2×C4×D3—51no—
198C2×C4×D3—101no—
199C2×C4×D3—101no—
200C2×Dic6—51no—
201C2×Dic6—101no—
202C2×C3×C8—101no—
203C2×C3×C8—101no—
204C3×C4⋊C4—51no—
205C3×C4⋊C4—101no—
206C3×C42—51no—
207C3×C42—101no—
208C4⋊Dic3—21no—
209C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
210Dic3⋊C4—101no—
211Dic3⋊C4—101no—
212Dic3⋊C4—101no—
213C4×Dic3—21no—
214C4×Dic3—51no—
215C4×Dic3—51no—
216C4×Dic3—101no—
217C4×Dic3—101no—
218C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
219C2×C3⋊C8—101no—
220C2×C3⋊C8—101no—
221ΓL(1,𝔽25)—202no—
222C22×D5—31no—
223C5×Q8—21no—
224C5×Q8ΓL(1,𝔽25)12yescomplemented
225C2×C4×C5—21no—
226C2×C4×C5—31no—
227C2×C4×C5—31no—
228C2×C4×C5—61no—
229C2×C4×C5C2×C4×D311yescharacteristic
230C2×Dic5—21no—
231C2×Dic5—31no—
232C2×Dic5—61no—
233C2×Dic5D3×D411yescharacteristic
234C2×Dic5D3×D411yescharacteristic
235C4×D5—32no—
236C4×D5—32no—
237C4×D5—64no—
238C4×D5—61no—
239C4×D5—61no—
240C4×D5—121no—
241C4×D5—122no—
242Dic10—32no—
243Dic10—64no—
244Dic10—61no—
245#170—41no—
246#170—41no—
247#170—121no—
248#170—121no—
249C22×Q8—151no—
250C2×M16—301no—
251C4×Q8—31no—
252C4×Q8—51no—
253C4×Q8—151no—
254C4×Q8—151no—
255C4×Q8—301no—
256C4×Q8—301no—
257C2×C4⋊C4—151no—
258C2×C4⋊C4—301no—
259C2×C42—151no—
260C2×C42—301no—
261C4⋊C8—51no—
262C4⋊C8—151no—
263C4⋊C8—301no—
264C4⋊C8—301no—
265C22⋊C8—301no—
266C4×C8—51no—
267C4×C8—151no—
268C2×C3×C5#40711yescharacteristic
269C2×C3×C5C2×#10311yescharacteristic
270C2×C3×C5#42111yescharacteristic
271D15—22no—
272C22×D3—51no—
273C3×Q8—21no—
274C3×Q8#70012yescomplemented
275C2×C3×C4—21no—
276C2×C3×C4—51no—
277C2×C3×C4—51no—
278C2×C3×C4—101no—
279C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
280C2×Dic3—21no—
281C2×Dic3—51no—
282C2×Dic3—101no—
283C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
284C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
285C4×D3—52no—
286C4×D3—52no—
287C4×D3—104no—
288C4×D3—101no—
289C4×D3—101no—
290C4×D3—201no—
291C4×D3—202no—
292Dic6—52no—
293Dic6—104no—
294Dic6—101no—
295C3×C8—201no—
296C3×C8—201no—
297C3⋊C8—201no—
298C3⋊C8—201no—
299C22×C5D3×C22⋊C411yescharacteristic
300C2×D5—32no—
301C2×D5—61no—
302C2×D5—61no—
303C4×C5—22no—
304C4×C5—32no—
305C4×C5—61no—
306C4×C5—121no—
307C4×C5C2×ΓL(1,𝔽25)11yescharacteristic
308C4×C5D3×M1611yescharacteristic
309Dic5—21no—
310Dic5—41no—
311Dic5—62no—
312Dic5—61no—
313Dic5D3×SD1611yescharacteristic
314Dic5D3×Q1611yescharacteristic
315C2×Q8—151no—
316C2×Q8—301no—
317C2×Q8—302no—
318C2×Q8—302no—
319C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
320C22×C4—151no—
321C22×C4—151no—
322C22×C4—151no—
323C22×C4—301no—
324C22×C4—301no—
325M16—602no—
326C2×C8—101no—
327C2×C8—101no—
328C2×C8—301no—
329C2×C8—301no—
330C4⋊C4—31no—
331C4⋊C4—51no—
332C4⋊C4—61no—
333C4⋊C4—101no—
334C4⋊C4—301no—
335C4⋊C4—301no—
336C4⋊C4—301no—
337C42—31no—
338C42—51no—
339C42—61no—
340C42—101no—
341C42—151no—
342C42—151no—
343C42—301no—
344C42—301no—
345C3×C5#148711yescharacteristic, complemented, Hall
346C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
347C2×D3—52no—
348C2×D3—101no—
349C2×D3—101no—
350C3×C4—22no—
351C3×C4—51no—
352C3×C4—51no—
353C3×C4—101no—
354C3×C4—201no—
355C3×C4C2×#70011yescharacteristic
356C3×C4D5⋊M1611yescharacteristic
357Dic3—21no—
358Dic3—41no—
359Dic3—102no—
360Dic3—101no—
361Dic3#400912yes—
362C2×C5#472611yescharacteristic
363C2×C5D3×#10311yescharacteristic
364C2×C5#482111yescharacteristic
365D5—62no—
366C23—151no—
367Q8—21no—
368Q8—152no—
369Q8—304no—
370Q8—301no—
371Q8#674012yescomplemented
372C2×C4—21no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—61no—
378C2×C4—101no—
379C2×C4—152no—
380C2×C4—152no—
381C2×C4—151no—
382C2×C4—304no—
383C2×C4—301no—
384C2×C4—301no—
385C2×C4—301no—
386C2×C4—601no—
387C2×C4—602no—
388C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
389C8—201no—
390C8—201no—
391C8—601no—
392C8—601no—
393C2×C3#6575611yescharacteristic
394C2×C3C2×#400911yescharacteristic
395C2×C3#6579511yescharacteristic
396D3—102no—
397C5#6823611yescharacteristic, complemented, Sylow
398C22—152no—
399C22—301no—
400C22—301no—
401C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
402C4—22no—
403C4—32no—
404C4—51no—
405C4—51no—
406C4—61no—
407C4—101no—
408C4—121no—
409C4—201no—
410C4—301no—
411C4—302no—
412C4#9193111yescharacteristic
413C4C2×#674011yescharacteristic
414C3#1060443911yescharacteristic, complemented, Sylow
415C2—302no—
416C2D3×#400911yescharacteristic
417C2#1175197311yescharacteristic
418C2#1175205211yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×D15C22⋊C4
cosocleC3×C4×Dic5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1487
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11746801C2semidirect
5. C3×Q8⋊#170C2semidirect
6. #11746495C2semidirect
7. #11746494C2semidirect
33. Q8⋊#170D3semidirect
95. C3×C5×Q8M16semidirect
105. Dic30M16semidirect
224. C5×Q8ΓL(1,𝔽25)semidirect
274. C3×Q8#700semidirect
345. C3×C5#1487semidirect
371. Q8#6740semidirect
397. C5#68236semidirect
414. C3#10604439semidirect

Maximal subgroups

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

subgroupindex
Q8×#12002
#117519222
#117517972
#117468012
C3×Q8⋊#1702
#117464952
#117464942
#106044393
#682365