Finite Group Databasebeta

← #14289801

Subgroups — #14289801

Order 1920.

456 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 104 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×D4×Dic10C211yesmaximal, characteristic
2#11751109C211yesmaximal, characteristic
3#11746921C211yesmaximal, characteristic, complemented
4#11746916C211yesmaximal, characteristic, complemented
5C5×D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#11745856C211yesmaximal, characteristic, complemented
7#11745841C211yesmaximal, characteristic, complemented
8#10598982—31nomaximal, Hall
9C22×C3×Dic10—21no—
10C3×C4×C5×D4C2211yescharacteristic
11#91758C2211yescharacteristic
12C3×D4×Dic5—21no—
13C3×#4070—21no—
14C3×#4034—21no—
15C3×#4024—22no—
16C3×#4015C2211yescharacteristic
17C3×C4×Dic10C2211yescharacteristic
18#91116—21no—
19#91092—21no—
20C4⋊(C3⋊#168)C2211yescharacteristic
21C5×C4⋊(C3⋊C8)C2211yescharacteristic
22C4×C5×C3⋊C8C2211yescharacteristic
23#90990—21no—
24#90974—22no—
25#90971—21no—
26#71915—51nomaximal, Hall
27D4×Dic10D311yescharacteristic
28#65297—31no—
29#65139—31no—
30#64970—31no—
31C5×D4⋊C8—31no—
32#64702—31no—
33#64676—31no—
34C2×C3×C5×D4D411yescharacteristic
35C22×C3×C4×C5—21no—
36C2×Dic30—21no—
37C22×C3×Dic5—22no—
38C2×C3×Dic10—22no—
39C2×C3×Dic10—21no—
40C2×C3×Dic10—42no—
41C2×C3×Dic10—42no—
42C2×C3×Dic10D411yescharacteristic
43C3×C5×C4⋊C4D411yescharacteristic
44C3×C5×C22⋊C4—21no—
45C3×C42×C5C2311yescharacteristic
46C4⋊Dic15—21no—
47C4⋊Dic15—21no—
48C2×C3⋊#168—21no—
49C2×C5×C3⋊C8—21no—
50C2×C5×C3⋊C8—21no—
51C3×C22⋊Dic5—22no—
52C3×C4⋊Dic5—21no—
53C3×C4⋊Dic5D412yes—
54C3×C4⋊Dic5D411yescharacteristic
55C3×#683—22no—
56C3×#683—21no—
57C3×C4×Dic5—21no—
58C3×D4×Q8—51no—
59#5093—51no—
60#5021—51no—
61#5008—51no—
62#4547—51no—
63#4535—51no—
64D4⋊(C3⋊C8)D511yescharacteristic
65C22×Dic10—21no—
66C4×C5×D4C2×D311yescharacteristic
67D4×Dic5—21no—
68#4070—21no—
69#4034—21no—
70#4024—22no—
71#4015—31no—
72#4015C2×D311yescharacteristic
73C4×Dic10C2×D311yescharacteristic
74C5×C4⋊C8—31no—
75C4×C5×C8—31no—
76#3964—61no—
77#3950—61no—
78#3948—61no—
79#3948—62no—
80#3940—61no—
81C4⋊#168—31no—
82#1637—151noSylow
83C23×C3×C5—21no—
84C3×C5×D4—21no—
85C3×C5×D4Q1612yes—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×Dic15—41no—
93Dic30—42no—
94C2×C3×Dic5—21no—
95C2×C3×Dic5—21no—
96C2×C3×Dic5—22no—
97C2×C3×Dic5—44no—
98C3×Dic10—21no—
99C3×Dic10—44no—
100C3×Dic10—41no—
101C3×Dic10—41no—
102C3×Dic10SD1612yes—
103C3⋊#168—41no—
104C5×C3⋊C8—41no—
105C5×C3⋊C8—41no—
106C22×C3×Q8—101no—
107C3×C4⋊Q8—51no—
108C3×C4⋊Q8—101no—
109C3×C22⋊Q8—102no—
110C3×C22⋊Q8—101no—
111C3×C4×Q8—51no—
112C3×C4×D4—101no—
113C3×C4×D4C2×D511yescharacteristic
114#869—51no—
115#835—101no—
116#832—101no—
117#810—102no—
118#808—101no—
119#807—101no—
120C4⋊(C3⋊C8)—51no—
121C4⋊(C3⋊C8)C2×D511yescharacteristic
122C4×C3⋊C8C2×D511yescharacteristic
123C2×C5×D4#6711yescharacteristic
124C22×C4×C5—21no—
125C22×Dic5—22no—
126C2×Dic10—22no—
127C2×Dic10—21no—
128C2×Dic10—42no—
129C2×Dic10—42no—
130C2×Dic10—61no—
131C2×Dic10#6711yescharacteristic
132C2×C5×C8—61no—
133C2×C5×C8—61no—
134C5×C4⋊C4#6711yescharacteristic
135C5×C22⋊C4—21no—
136C42×C5C22×D311yescharacteristic
137C22⋊Dic5—22no—
138C4⋊Dic5—21no—
139C4⋊Dic5—61no—
140C4⋊Dic5—61no—
141C4⋊Dic5#6712yes—
142C4⋊Dic5#6711yescharacteristic
143#683—22no—
144#683—21no—
145C4×Dic5—21no—
146C2×#168—61no—
147D4×Q8—51no—
148#487—151no—
149#477—151no—
150#476—151no—
151#332—151no—
152Q8⋊C8—151no—
153D4⋊C8—31no—
154C22×C3×C5—22no—
155C22×C3×C5—41no—
156C22×C3×C5—41no—
157C22×C3×C5#12011yescharacteristic
158C3×C4×C5—21no—
159C3×C4×C5—41no—
160C3×C4×C5C2×SD1611yescharacteristic
161C3×C4×C5C2×Q1611yescharacteristic
162C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
163C3×C4×C5#13711yescharacteristic
164Dic15—81no—
165C3×Dic5—22no—
166C3×Dic5—42no—
167C3×Dic5—41no—
168C2×C3×Q8—51no—
169C2×C3×Q8—102no—
170C2×C3×Q8—101no—
171C2×C3×Q8—202no—
172C2×C3×Q8—202no—
173C2×C3×D4D2011yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—102no—
176C2×Dic6—101no—
177C3×C4⋊C4—51no—
178C3×C4⋊C4—52no—
179C3×C4⋊C4—102no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4—101no—
182C3×C4⋊C4D2011yescharacteristic
183C3×C22⋊C4—21no—
184C3×C22⋊C4—102no—
185C3×C42—101no—
186C3×C42C22×D511yescharacteristic
187C4⋊Dic3—101no—
188C4⋊Dic3—101no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—101no—
192C23×C5—21no—
193C5×D4—21no—
194C5×D4#21612yes—
195C2×C4×C5—21no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5C2×#6711yescharacteristic
199C2×C4×C5C2×#6711yescharacteristic
200C2×C4×C5C2×#6711yescharacteristic
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—22no—
204C2×Dic5—44no—
205C2×Dic5—121no—
206Dic10—21no—
207Dic10—44no—
208Dic10—41no—
209Dic10—41no—
210Dic10—122no—
211Dic10#21412yes—
212C5×C8—121no—
213C5×C8—121no—
214#168—121no—
215C22×Q8—101no—
216C4⋊Q8—51no—
217C4⋊Q8—101no—
218C4⋊Q8—151no—
219C22⋊Q8—102no—
220C22⋊Q8—101no—
221C4×Q8—51no—
222C4×D4—101no—
223C4×D4D3×D511yescharacteristic
224#107—301no—
225#106—301no—
226C4⋊C8—31no—
227C4⋊C8—151no—
228#103—302no—
229#103—301no—
230#102—301no—
231C4×C8—31no—
232C2×C3×C5—42no—
233C2×C3×C5#45011yescharacteristic
234C2×C3×C5#45111yescharacteristic
235C2×C3×C5#45411yescharacteristic
236C23×C3—21no—
237C3×Q8—52no—
238C3×Q8—101no—
239C3×Q8—204no—
240C3×Q8—201no—
241C3×Q8—201no—
242C3×D4—21no—
243C3×D4Dic2012yes—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—101no—
248C2×C3×C4—101no—
249C2×C3×C4—102no—
250C2×C3×C4—204no—
251C2×C3×C4C2×D2011yescharacteristic
252C2×C3×C4D4×D511yescharacteristic
253C2×C3×C4D4×D511yescharacteristic
254C2×Dic3—201no—
255Dic6—202no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—201no—
259C22×C5—22no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5#95311yescharacteristic
263C4×C5—21no—
264C4×C5—41no—
265C4×C5#93211yescharacteristic
266C4×C5C2×#21411yescharacteristic
267C4×C5#94211yescharacteristic
268C4×C5C2×#21611yescharacteristic
269Dic5—22no—
270Dic5—42no—
271Dic5—41no—
272Dic5—241no—
273C2×Q8—51no—
274C2×Q8—102no—
275C2×Q8—101no—
276C2×Q8—202no—
277C2×Q8—202no—
278C2×Q8—301no—
279C2×D4#120211yescharacteristic
280C22×C4—21no—
281C22×C4—102no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—301no—
285C4⋊C4—52no—
286C4⋊C4—51no—
287C4⋊C4—102no—
288C4⋊C4—101no—
289C4⋊C4—101no—
290C4⋊C4—301no—
291C4⋊C4—301no—
292C4⋊C4#120211yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—102no—
295C42—101no—
296C42C2×D3×D511yescharacteristic
297C3×C5#163711yescharacteristic, complemented, Hall
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3#402811yescharacteristic
302C3×C4—21no—
303C3×C4—41no—
304C3×C4—102no—
305C3×C4—202no—
306C3×C4—201no—
307C3×C4C2×Dic2011yescharacteristic
308C3×C4#405511yescharacteristic
309C3×C4#405711yescharacteristic
310C3×C4D5×SD1611yescharacteristic
311Dic3—401no—
312C2×C5—42no—
313C2×C5#521811yescharacteristic
314C2×C5#522811yescharacteristic
315C2×C5#524511yescharacteristic
316C23—21no—
317Q8—52no—
318Q8—101no—
319Q8—204no—
320Q8—201no—
321Q8—201no—
322Q8—602no—
323D4—21no—
324D4#666212yes—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—102no—
331C2×C4—204no—
332C2×C4—601no—
333C2×C4C2×#120211yescharacteristic
334C2×C4D5×#6711yescharacteristic
335C2×C4D5×#6711yescharacteristic
336C8—121no—
337C8—121no—
338C8—601no—
339C2×C3—42no—
340C2×C3#6502811yescharacteristic
341C2×C3#6506511yescharacteristic
342C2×C3#6511611yescharacteristic
343C5#7191511yescharacteristic, complemented, Sylow
344C22—22no—
345C22—41no—
346C22—41no—
347C22#9156411yescharacteristic
348C4—21no—
349C4—41no—
350C4—102no—
351C4—201no—
352C4—202no—
353C4—1201no—
354C4#9131511yescharacteristic
355C4C2×#666211yescharacteristic
356C4D5×#21411yescharacteristic
357C4#9148911yescharacteristic
358C3#1059898211yescharacteristic, complemented, Sylow
359C2—42no—
360C2#1174855611yescharacteristic
361C2#1174904511yescharacteristic
362C2#1174945711yescharacteristic
subgroupG/H
centerC22#91564
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5#120
prime radicalC2×C3×C5×D4D4
hypercenterC4×D4D3×D5
hypocenterC3×C5#1637
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #11746921C2semidirect
4. #11746916C2semidirect
6. #11745856C2semidirect
7. #11745841C2semidirect
297. C3×C5#1637semidirect
343. C5#71915semidirect
358. C3#10598982semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Dic102
#117511092
#117469212
#117469162
C5×D4⋊(C3⋊C8)2
#117458562
#117458412
#105989823
#719155