Finite Group Databasebeta

← #14299554

Subgroups — #14299554

Order 1920.

556 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 126 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×C3×#700C211yesmaximal, characteristic
2#11752415C211yesmaximal, characteristic
3C2×#91161C212yesmaximal, complemented
4D5×#830C211yesmaximal, characteristic
5#11746532C212yesmaximal, complemented
6#10603916—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6758C2211yescharacteristic
9C2×#6758C2211yescharacteristic
10C22×C3×#170—21no—
11C2×C3×#700—21no—
12C2×C3×#700—24no—
13C2×C3×#700C2212yes—
14C2×D5×C3⋊C8C2212yes—
15#91241—21no—
16#91232—21no—
17#91161—24no—
18#91114C411yescharacteristic
19C5×#830C411yescharacteristic
20#90955C412yes—
21#72001—51nomaximal, Hall
22C22×#700D311yescharacteristic
23#65756—31no—
24C2×#3995—32no—
25D5×C22⋊C8—31no—
26#64896—32no—
27C23×C3×D5Q811yescharacteristic
28C22×C3×C4×C5C2×C411yescharacteristic
29C22×C3×Dic5D411yescharacteristic
30C2×C3×C4×D5—22no—
31C2×C3×C4×D5—22no—
32C2×C3×C4×D5C2×C411yescharacteristic
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5Q811yescharacteristic
35C2×C3×C4×D5C2311yescharacteristic
36C2×C3⋊#170—21no—
37C2×C3⋊#170—21no—
38#6758—22no—
39#6758—22no—
40C2×C3×#170—24no—
41C2×C3×#170—22no—
42C3×#700—24no—
43C3×#700—22no—
44C3×#700D414yes—
45C2×C3⋊#168C2×C412yes—
46C2×C5×C3⋊C8C2×C412yes—
47D5×C3⋊C8—22no—
48D5×C3⋊C8—22no—
49C22×C3×M16—51no—
50#5207—51no—
51C2×#830—51no—
52C2×#822—52no—
53#4558—52no—
54C22×C4×D5C2×D311yescharacteristic
55C22×#170—21no—
56C2×#700—21no—
57C2×#700—24no—
58C2×#700—31no—
59C2×#700—31no—
60C2×#700C2×D312yes—
61C2×C8×D5—32no—
62C22⋊#170—61no—
63#4003—61no—
64#3995—64no—
65C5×C22⋊C8—31no—
66C22⋊#168—31no—
67#3952—32no—
68#1855—151noSylow
69C23×C3×C5C4⋊C411yescharacteristic
70C22×C3×D5—22no—
71C22×C3×D5—22no—
72C22×C3×D5—41no—
73C22×C3×D5C4⋊C411yescharacteristic
74C22×C3×D5C2×Q811yescharacteristic
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5C4⋊C411yescharacteristic
77C2×C3×C4×C5C22×C411yescharacteristic
78C2×C3×Dic5—21no—
79C2×C3×Dic5—21no—
80C2×C3×Dic5C4⋊C411yescharacteristic
81C2×C3×Dic5C2×D411yescharacteristic
82C3×C4×D5—22no—
83C3×C4×D5—22no—
84C3×C4×D5—21no—
85C3×C4×D5—21no—
86C3×C4×D5C2×D412yes—
87C3×C4×D5#4112yes—
88C3⋊#170—41no—
89C3⋊#170—41no—
90C3×#170—24no—
91C3⋊#168—22no—
92C5×C3⋊C8—22no—
93C23×C3×C4—51no—
94C2×C3×M16—52no—
95C2×C3×M16—104no—
96C2×C3×M16—101no—
97C22×C3×C8—101no—
98C2×#208—51no—
99C2×#208—51no—
100C22×C3⋊C8—52no—
101#830—51no—
102#830—52no—
103#830—101no—
104#830—101no—
105#830AGL(1,𝔽5)11yescharacteristic
106#822—104no—
107C23×D5Dic611yescharacteristic
108C22×C4×C5C4×D311yescharacteristic
109C22×Dic5#6711yescharacteristic
110C2×C4×D5—22no—
111C2×C4×D5—22no—
112C2×C4×D5Dic611yescharacteristic
113C2×C4×D5C4×D311yescharacteristic
114C2×C4×D5#6711yescharacteristic
115C2×C4×D5C22×D311yescharacteristic
116C2×#170—24no—
117C2×#170—22no—
118C2×#170—61no—
119C2×#170—61no—
120#700—24no—
121#700—22no—
122#700—62no—
123#700—62no—
124#700#6714yes—
125C2×C5×C8—32no—
126C2×#168—32no—
127C8×D5—62no—
128C8×D5—62no—
129C22×M16—51no—
130C2×#108—152no—
131#407—151no—
132C2×C22⋊C8—151no—
133#341—152no—
134C22×C3×C5—21no—
135C22×C3×C5—21no—
136C22×C3×C5#10812yescomplemented
137C22×C3×C5C2×C4⋊C411yescharacteristic
138C2×C3×D5—21no—
139C2×C3×D5—21no—
140C2×C3×D5—21no—
141C2×C3×D5—21no—
142C2×C3×D5—21no—
143C2×C3×D5—44no—
144C2×C3×D5C22⋊Q812yes—
145C3×C4×C5—21no—
146C3×C4×C5C4×D412yes—
147C3×Dic5—21no—
148C3×Dic5#12011yescharacteristic
149C3×Dic5#12311yescharacteristic
150C24×C3—51no—
151C22×C3×C4—51no—
152C22×C3×C4—51no—
153C22×C3×C4—51no—
154C22×C3×C4—51no—
155C22×C3×C4—51no—
156C22×C3×C4—102no—
157C22×C3×C4—102no—
158C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
159C3×M16—54no—
160C3×M16—102no—
161C3×M16—104no—
162C2×C3×C8—102no—
163C2×C3×C8—104no—
164#208—102no—
165#208—102no—
166C2×C3⋊C8—52no—
167C2×C3⋊C8—102no—
168C2×C3⋊C8—102no—
169C2×C3⋊C8—101no—
170C2×C3⋊C8—101no—
171C2×C3⋊C8C2×AGL(1,𝔽5)12yes—
172C23×C5Dic3⋊C411yescharacteristic
173C22×D5—22no—
174C22×D5—22no—
175C22×D5—41no—
176C22×D5Dic3⋊C411yescharacteristic
177C22×D5C2×Dic611yescharacteristic
178C2×C4×C5—22no—
179C2×C4×C5Dic3⋊C411yescharacteristic
180C2×C4×C5C2×C4×D311yescharacteristic
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5Dic3⋊C411yescharacteristic
184C2×Dic5C2×#6711yescharacteristic
185C4×D5—22no—
186C4×D5—22no—
187C4×D5—21no—
188C4×D5—21no—
189C4×D5#23512yes—
190C4×D5C2×#6712yes—
191#170—24no—
192#170—121no—
193#170—121no—
194C5×C8—62no—
195#168—62no—
196C23×C4—51no—
197C2×M16—52no—
198C2×M16—104no—
199C2×M16—101no—
200C2×M16—151no—
201C2×M16—151no—
202C22×C8—101no—
203C22×C8—152no—
204#108—304no—
205C22⋊C8—31no—
206C22⋊C8—152no—
207C22⋊C8—151no—
208C22⋊C8—301no—
209C22⋊C8—301no—
210C2×C3×C5—22no—
211C2×C3×C5#38511yescharacteristic
212C2×C3×C5C2×#10811yescharacteristic
213C2×C3×C5#43011yescharacteristic
214C3×D5—22no—
215C3×D5—41no—
216C23×C3—51no—
217C23×C3—51no—
218C23×C3—102no—
219C23×C3—102no—
220C23×C3—201no—
221C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic
222C2×C3×C4—22no—
223C2×C3×C4—51no—
224C2×C3×C4—52no—
225C2×C3×C4—52no—
226C2×C3×C4—51no—
227C2×C3×C4—101no—
228C2×C3×C4—101no—
229C2×C3×C4—102no—
230C2×C3×C4—101no—
231C2×C3×C4—101no—
232C2×C3×C4—102no—
233C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
234C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
235C3×C8—104no—
236C3⋊C8—22no—
237C3⋊C8—102no—
238C3⋊C8—201no—
239C3⋊C8—201no—
240C22×C5—21no—
241C22×C5—21no—
242C22×C5#82212yescomplemented
243C22×C5C2×Dic3⋊C411yescharacteristic
244C2×D5—21no—
245C2×D5—21no—
246C2×D5—21no—
247C2×D5—21no—
248C2×D5—21no—
249C2×D5—44no—
250C2×D5#92412yes—
251C4×C5—21no—
252C4×C5C4×#6712yes—
253Dic5—21no—
254Dic5#92911yescharacteristic
255Dic5#95311yescharacteristic
256C24—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—102no—
263C22×C4—102no—
264C22×C4D3×AGL(1,𝔽5)11yescharacteristic
265M16—54no—
266M16—102no—
267M16—104no—
268M16—302no—
269M16—302no—
270C2×C8—32no—
271C2×C8—102no—
272C2×C8—104no—
273C2×C8—152no—
274C2×C8—302no—
275C2×C8—302no—
276C2×C8—301no—
277C2×C8—301no—
278C3×C5#185511yescharacteristic, complemented, Hall
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—52no—
282C22×C3—101no—
283C22×C3—101no—
284C22×C3—101no—
285C22×C3—101no—
286C22×C3—101no—
287C22×C3—204no—
288C22×C3#399512yescomplemented
289C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
290C3×C4—21no—
291C3×C4—51no—
292C3×C4—51no—
293C3×C4—101no—
294C3×C4D4×AGL(1,𝔽5)12yes—
295C2×C5—22no—
296C2×C5C2×#82211yescharacteristic
297C2×C5#512411yescharacteristic
298C2×C5#521011yescharacteristic
299D5—22no—
300D5—41no—
301C23—51no—
302C23—51no—
303C23—102no—
304C23—102no—
305C23—201no—
306C23Dic3⋊AGL(1,𝔽5)11yescharacteristic
307C2×C4—22no—
308C2×C4—51no—
309C2×C4—52no—
310C2×C4—52no—
311C2×C4—51no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—102no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—102no—
318C2×C4Dic3⋊AGL(1,𝔽5)11yescharacteristic
319C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
320C8—62no—
321C8—104no—
322C8—302no—
323C8—601no—
324C8—601no—
325C2×C3—22no—
326C2×C3—102no—
327C2×C3—201no—
328C2×C3#6569911yescharacteristic
329C2×C3C2×#399511yescharacteristic
330C2×C3#6572911yescharacteristic
331C5#7200111yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—52no—
335C22—101no—
336C22—101no—
337C22—101no—
338C22—101no—
339C22—101no—
340C22—204no—
341C22#9116112yescomplemented
342C22C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
343C4—21no—
344C4—51no—
345C4—51no—
346C4—101no—
347C4AGL(1,𝔽5)×#6712yes—
348C3#1060391611yescharacteristic, complemented, Sylow
349C2—22no—
350C2—102no—
351C2—201no—
352C2C2×#9116111yescharacteristic
353C2#1175173211yescharacteristic
354C2#1175202711yescharacteristic
subgroupG/H
centerC22C2×Dic3⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C4⋊C4
prime radicalC23×C3×D5Q8
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1855
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
3. C2×#91161C2semidirect
5. #11746532C2semidirect
136. C22×C3×C5#108semidirect
242. C22×C5#822semidirect
278. C3×C5#1855semidirect
288. C22×C3#3995semidirect
331. C5#72001semidirect
341. C22#91161semidirect
348. C3#10603916semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#7002
#117524152
C2×#911612
D5×#8302
#117465322
#106039163
#720015