Finite Group Databasebeta

← #14284910

Subgroups — #14284910

Order 1920.

480 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 110 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
1D15×C4⋊Q8C211yesmaximal, characteristic
2#11747015C211yesmaximal, characteristic
3#11746859C211yesmaximal, characteristic
4C5×#4546C211yesmaximal, characteristic, complemented
5C3×#64766C211yesmaximal, characteristic, complemented
6#11745863C211yesmaximal, characteristic, complemented
7#11745863C211yesmaximal, characteristic, complemented
8#10599276—31nomaximal, Hall
9C2×Q8×D15C412yes—
10C3×C5×C4⋊Q8C2211yescharacteristic
11#91833C412yes—
12C4⋊C4×D15—21no—
13C4⋊C4×D15—21no—
14#91777—21no—
15#91777—21no—
16C42×D15C2211yescharacteristic
17#91758C2211yescharacteristic
18C2×#6651—21no—
19C5×C4⋊(C3⋊C8)C2211yescharacteristic
20C5×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊#168C2211yescharacteristic
22C3×C4⋊#168C2211yescharacteristic
23#90983—21no—
24#90957—21no—
25#90951—21no—
26#68483—51nomaximal, Hall
27D5×C4⋊Q8—31no—
28#65126—31no—
29#65122—31no—
30C5×#333—31no—
31#64766D311yescharacteristic, complemented
32#64701—31no—
33#64701—31no—
34C2×C3×C5×Q8C2×C412yes—
35Q8×D15—44no—
36C2×C4×D15—22no—
37C2×C4×D15C2×C411yescharacteristic
38C2×C4×D15D411yescharacteristic
39C2×C4×D15D411yescharacteristic
40C2×Dic30—22no—
41C2×Dic30C2×C412yes—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C42×C5C2311yescharacteristic
45C4⋊Dic15—21no—
46C4⋊Dic15—21no—
47#6712—22no—
48#6712—22no—
49C4×Dic15C2×C411yescharacteristic
50C4×Dic15D411yescharacteristic
51C4×Dic15D411yescharacteristic
52C2×C5×C3⋊C8—21no—
53C2×C5×C3⋊C8—21no—
54C2×C3×#168—21no—
55C2×C3×#168—21no—
56#6651—42no—
57D3×C4⋊Q8—51no—
58#4837—51no—
59#4833—51no—
60C3×#333—51no—
61#4546D511yescharacteristic, complemented
62#4481—51no—
63#4481—51no—
64C2×Q8×D5—32no—
65C5×C4⋊Q8C2×D311yescharacteristic
66#4090—32no—
67C2×#676—61no—
68D5×C4⋊C4—61no—
69D5×C4⋊C4—61no—
70#4034—61no—
71#4034—61no—
72C42×D5—31no—
73#4015—31no—
74C5×C4⋊C8—31no—
75C5×C4⋊C8—31no—
76#3952—61no—
77#3946—61no—
78C8×Dic5—61no—
79C4⋊#168C2×D311yescharacteristic
80C4⋊#168C2×D311yescharacteristic
81#1548—151noSylow
82C22×D15C22⋊C411yescharacteristic
83C3×C5×Q8—42no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5C22×C411yescharacteristic
86C2×C3×C4×C5C2×D411yescharacteristic
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×Dic15—22no—
89C2×Dic15C22⋊C411yescharacteristic
90C2×Dic15C22⋊C411yescharacteristic
91C2×Dic15C22⋊C411yescharacteristic
92C4×D15—21no—
93C4×D15—21no—
94C4×D15—22no—
95C4×D15—44no—
96C4×D15SD1612yes—
97C4×D15Q1612yes—
98Dic30—44no—
99Dic30—42no—
100C3×#168—41no—
101C3×#168—41no—
102C5×C3⋊C8—41no—
103C5×C3⋊C8—41no—
104C2×D3×Q8—52no—
105C3×C4⋊Q8C2×D511yescharacteristic
106#944—52no—
107C2×ΓL(1,𝔽25)—101no—
108D3×C4⋊C4—101no—
109D3×C4⋊C4—101no—
110#888—101no—
111#888—101no—
112C42×D3—51no—
113#869—51no—
114C3×C4⋊C8—51no—
115C3×C4⋊C8—51no—
116(C2×D3)⋊C8—101no—
117Dic3⋊C8—101no—
118C8×Dic3—101no—
119C4⋊(C3⋊C8)C2×D511yescharacteristic
120C4⋊(C3⋊C8)C2×D511yescharacteristic
121C2×C5×Q8C4×D312yes—
122Q8×D5—124no—
123C2×C4×D5—31no—
124C2×C4×D5—31no—
125C2×C4×D5—31no—
126C2×C4×D5—62no—
127C2×Dic10—32no—
128C2×Dic10—62no—
129C2×C5×C8—61no—
130C2×C5×C8—61no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C42×C5C22×D311yescharacteristic
134C4⋊Dic5—61no—
135C4⋊Dic5—61no—
136#683—62no—
137#683—62no—
138C4×Dic5—31no—
139C4×Dic5—31no—
140C4×Dic5—31no—
141C2×#168—21no—
142C2×#168—21no—
143#676—122no—
144C2×C4⋊Q8—151no—
145#431—151no—
146#423—151no—
147#333—31no—
148#333—51no—
149#333—151no—
150#333—151no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×D15—21no—
153C2×D15—21no—
154C2×D15#10312yes—
155C3×C4×C5—21no—
156C3×C4×C5—42no—
157C3×C4×C5C2×SD1611yescharacteristic
158C3×C4×C5C2×Q1611yescharacteristic
159C3×C4×C5#13711yescharacteristic
160C3×C4×C5#13711yescharacteristic
161Dic15—21no—
162Dic15—42no—
163Dic15#10012yes—
164Dic15#10312yes—
165C2×C3×Q8C4×D512yes—
166D3×Q8—204no—
167C2×C4×D3—51no—
168C2×C4×D3—51no—
169C2×C4×D3—51no—
170C2×C4×D3—102no—
171C2×Dic6—52no—
172C2×Dic6—102no—
173C2×C3×C8—101no—
174C2×C3×C8—101no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C42C22×D511yescharacteristic
178C4⋊Dic3—101no—
179C4⋊Dic3—101no—
180Dic3⋊C4—102no—
181Dic3⋊C4—102no—
182C4×Dic3—51no—
183C4×Dic3—51no—
184C4×Dic3—51no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187ΓL(1,𝔽25)—202no—
188C22×D5—31no—
189C5×Q8—42no—
190C2×C4×C5—22no—
191C2×C4×C5C2×C4×D311yescharacteristic
192C2×C4×C5D3×D411yescharacteristic
193C2×C4×C5D3×D411yescharacteristic
194C2×Dic5—31no—
195C2×Dic5—31no—
196C2×Dic5—31no—
197C2×Dic5—62no—
198C4×D5—32no—
199C4×D5—32no—
200C4×D5—62no—
201C4×D5—61no—
202C4×D5—61no—
203C4×D5—124no—
204Dic10—124no—
205Dic10—122no—
206C5×C8—121no—
207C5×C8—121no—
208#168—41no—
209#168—41no—
210C22×Q8—152no—
211C2×M16—301no—
212C4⋊Q8—152no—
213C4⋊Q8—151no—
214C4⋊Q8—301no—
215C4⋊Q8—301no—
216C4⋊Q8D3×D511yescharacteristic
217C2×C4⋊C4—301no—
218C2×C4⋊C4—301no—
219C2×C42—151no—
220C4⋊C8—31no—
221C4⋊C8—31no—
222C4⋊C8—51no—
223C4⋊C8—51no—
224C4⋊C8—301no—
225C22⋊C8—301no—
226C4×C8—301no—
227C2×C3×C5C2×#10011yescharacteristic
228C2×C3×C5C2×#10311yescharacteristic
229C2×C3×C5#41911yescharacteristic
230D15—22no—
231C22×D3—51no—
232C3×Q8—42no—
233C2×C3×C4—22no—
234C2×C3×C4C2×C4×D511yescharacteristic
235C2×C3×C4D4×D511yescharacteristic
236C2×C3×C4D4×D511yescharacteristic
237C2×Dic3—51no—
238C2×Dic3—51no—
239C2×Dic3—51no—
240C2×Dic3—102no—
241C4×D3—52no—
242C4×D3—52no—
243C4×D3—102no—
244C4×D3—101no—
245C4×D3—101no—
246C4×D3—204no—
247Dic6—204no—
248Dic6—202no—
249C3×C8—201no—
250C3×C8—201no—
251C3⋊C8—41no—
252C3⋊C8—41no—
253C22×C5D3×C22⋊C411yescharacteristic
254C2×D5—32no—
255C2×D5—61no—
256C2×D5—61no—
257C4×C5—21no—
258C4×C5—42no—
259C4×C5D3×SD1611yescharacteristic
260C4×C5#91511yescharacteristic
261C4×C5D3×Q1611yescharacteristic
262C4×C5#91811yescharacteristic
263Dic5—32no—
264Dic5—32no—
265Dic5—61no—
266Dic5—122no—
267C2×Q8—152no—
268C2×Q8—302no—
269C2×Q8—604no—
270C2×Q8#120012yes—
271C22×C4—151no—
272C22×C4—151no—
273C22×C4—151no—
274C22×C4—302no—
275M16—602no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—101no—
279C2×C8—101no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—302no—
283C4⋊C4—302no—
284C4⋊C4—301no—
285C4⋊C4—301no—
286C42—151no—
287C42—151no—
288C42—151no—
289C42C2×D3×D511yescharacteristic
290C3×C5#154811yescharacteristic, complemented, Hall
291C22×C3D5×C22⋊C411yescharacteristic
292C2×D3—52no—
293C2×D3—101no—
294C2×D3—101no—
295C3×C4—21no—
296C3×C4—42no—
297C3×C4D5×SD1611yescharacteristic
298C3×C4#406111yescharacteristic
299C3×C4D5×Q1611yescharacteristic
300C3×C4#406411yescharacteristic
301Dic3—52no—
302Dic3—52no—
303Dic3—101no—
304Dic3—202no—
305C2×C5D3×#10011yescharacteristic
306C2×C5D3×#10311yescharacteristic
307C2×C5#480211yescharacteristic
308D5—62no—
309C23—151no—
310Q8—42no—
311Q8—602no—
312Q8—604no—
313C2×C4—22no—
314C2×C4—151no—
315C2×C4—151no—
316C2×C4—152no—
317C2×C4—152no—
318C2×C4—151no—
319C2×C4—302no—
320C2×C4—301no—
321C2×C4—301no—
322C2×C4—302no—
323C2×C4—604no—
324C2×C4C2×#120011yescharacteristic
325C2×C4#679011yescharacteristic
326C2×C4#679011yescharacteristic
327C8—121no—
328C8—121no—
329C8—201no—
330C8—201no—
331C2×C3D5×#10011yescharacteristic
332C2×C3D5×#10311yescharacteristic
333C2×C3#6509111yescharacteristic
334D3—102no—
335C5#6848311yescharacteristic, complemented, Sylow
336C22—152no—
337C22—301no—
338C22—301no—
339C22#9156111yescharacteristic
340C4—21no—
341C4—42no—
342C4—152no—
343C4—152no—
344C4—301no—
345C4—602no—
346C4#9150511yescharacteristic
347C4#9151111yescharacteristic
348C4#9152211yescharacteristic
349C4#9152411yescharacteristic
350C3#1059927611yescharacteristic, complemented, Sylow
351C2—302no—
352C2#1174912711yescharacteristic
353C2#1174920311yescharacteristic
354C2#1174921811yescharacteristic
subgroupG/H
centerC22#91561
commutatorC2×C3×C4×C5C22×C4
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×D15C22⋊C4
cosocleC3×C42×C5C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#1548
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C5×#4546C2semidirect
5. C3×#64766C2semidirect
6. #11745863C2semidirect
7. #11745863C2semidirect
31. #64766D3semidirect
61. #4546D5semidirect
290. C3×C5#1548semidirect
335. C5#68483semidirect
350. C3#10599276semidirect

Maximal subgroups

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

subgroupindex
D15×C4⋊Q82
#117470152
#117468592
C5×#45462
C3×#647662
#117458632
#105992763
#684835