Finite Group Databasebeta

← C5×#69168 #14316980

Subgroups — C5×#69168 #14316980

Order 1920.

472 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C5×#5734C211yesmaximal, characteristic, complemented
2C5×#5095C211yesmaximal, characteristic, complemented
3C5×#4753C212yesmaximal, complemented
4C3×C5×#356C211yesmaximal, characteristic, complemented
5C5×#4477C212yesmaximal, complemented
6C5×#2204—31nomaximal, Hall
7C5×#1007C2211yescharacteristic
8C3×C5×C4⋊Q8C2211yescharacteristic
9C5×#947—21no—
10C5×#947C2212yes—
11C5×#942—21no—
12C5×#908—22no—
13C5×#895—22no—
14C5×#892—22no—
15C4×C5×D12—21no—
16C3×C5×#101C2212yes—
17C5×#836C2211yescharacteristic
18C5×#825—22no—
19C5×#805—21no—
20#69168C511yesmaximal, characteristic, direct product factor, Hall
21C5×#555—31no—
22C5×#456—31no—
23C5×#454—32no—
24C5×#356D311yescharacteristic, complemented
25C5×#355—32no—
26C2×C3×C5×Q8D411yescharacteristic
27C2×C3×C5×Q8C2311yescharacteristic
28C5×#239—41no—
29C5×D3×Q8—41no—
30C5×#235—22no—
31C5×#235—21no—
32C2×C5×D12—21no—
33C2×C5×D12—21no—
34C2×C5×D12D412yes—
35C2×C4×C5×D3—22no—
36C3×C5×M16—22no—
37C3×C5×C4⋊C4—22no—
38C3×C42×C5D411yescharacteristic
39C5×#216—41no—
40C5×#215—41no—
41C5×(C2×D3)⋊C4—22no—
42C5×(C2×D3)⋊C4—22no—
43C5×(C2×D3)⋊C4—22no—
44C5×C4⋊Dic3—21no—
45C4×C5×Dic3—21no—
46C4×C5×Dic3D412yes—
47C5×#208—21no—
48C5×D24—42no—
49C5×#204—42no—
50#5734C2×C511yescharacteristic, complemented
51#5095C2×C511yescharacteristic, complemented
52#4753C2×C512yescomplemented
53C3×#356C2×C511yescharacteristic, complemented
54#4477C2×C512yescomplemented
55C5×2−1+4—31no—
56C5×#137—61no—
57C5×AGL(1,ℤ/8)—62no—
58C5×C4⋊Q8C2×D311yescharacteristic
59C5×#124—32no—
60C5×#124—61no—
61C5×#121—62no—
62C4×C5×D4—61no—
63C5×#117—62no—
64C5×#104—62no—
65C5×#104—61no—
66C5×#101—31no—
67C5×#101C2×D312yes—
68#2204—31noSylow
69C22×C5×D3—22no—
70C3×C5×Q8—41no—
71C3×C5×Q8—41no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5C2×D411yescharacteristic
75C2×C3×C4×C5C2×D412yes—
76C5×#67—41no—
77C2×C5×Dic3—22no—
78C5×D12—22no—
79C5×D12—21no—
80C5×D12—42no—
81C5×D12—42no—
82C5×D12—81no—
83C4×C5×D3—44no—
84C4×C5×D3—41no—
85C4×C5×D3—42no—
86C5×Dic6—22no—
87C5×Dic6—21no—
88C3×C5×C8—42no—
89C5×C3⋊C8—41no—
90#1007C22×C511yescharacteristic
91C3×C4⋊Q8C22×C511yescharacteristic
92#947—21no—
93#947C22×C512yes—
94#942—21no—
95#908—22no—
96#895—22no—
97#892—22no—
98C4×D12—21no—
99C3×#101C22×C512yes—
100#836C22×C511yescharacteristic
101#825—22no—
102#805—21no—
103C5×#41—61no—
104C5×#41—62no—
105C5×#41—121no—
106C2×C5×Q8—121no—
107C2×C5×Q8D1211yescharacteristic
108C2×C5×Q8C22×D311yescharacteristic
109C2×C5×D4—32no—
110C2×C5×D4—61no—
111C2×C5×D4—61no—
112C22×C4×C5—62no—
113C5×Q16—121no—
114C5×SD16—122no—
115C5×SD16—121no—
116C5×D8—122no—
117C5×M16—22no—
118C5×M16—61no—
119C5×C4⋊C4—22no—
120C5×C4⋊C4—61no—
121C5×C22⋊C4—62no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—62no—
124C42×C5—32no—
125C42×C5—61no—
126C42×C5D1211yescharacteristic
127#555—31no—
128#456—31no—
129#454—32no—
130#356C5×D311yescharacteristic, complemented
131#355—32no—
132C22×C3×C5#12011yescharacteristic
133C2×C5×D3—42no—
134C2×C5×D3—42no—
135C2×C5×D3—41no—
136C2×C5×D3—42no—
137C3×C4×C5—22no—
138C3×C4×C5—21no—
139C3×C4×C5—22no—
140C3×C4×C5—41no—
141C5×Dic3—42no—
142C5×Dic3—41no—
143C2×C3×Q8C5×D411yescharacteristic
144C2×C3×Q8C23×C511yescharacteristic
145#239—41no—
146D3×Q8—41no—
147#235—21no—
148#235—22no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12C5×D412yes—
152C2×C4×D3—22no—
153C3×M16—22no—
154C3×C4⋊C4—22no—
155C3×C42C5×D411yescharacteristic
156#216—41no—
157#215—41no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—22no—
161C4⋊Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3C5×D412yes—
164#208—21no—
165D24—42no—
166#204—42no—
167C23×C5—62no—
168C5×Q8—41no—
169C5×Q8—41no—
170C5×Q8—62no—
171C5×Q8—61no—
172C5×D4—62no—
173C5×D4—61no—
174C5×D4—122no—
175C5×D4—122no—
176C5×D4—121no—
177C5×D4—241no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—62no—
181C2×C4×C5—124no—
182C2×C4×C5—122no—
183C2×C4×C5—121no—
184C2×C4×C5C2×D1211yescharacteristic
185C2×C4×C5D3×D412yes—
186C5×C8—42no—
187C5×C8—121no—
1882−1+4—31no—
189#137—61no—
190AGL(1,ℤ/8)—62no—
191C4⋊Q8C2×C5×D311yescharacteristic
192#124—32no—
193#124—61no—
194#121—62no—
195C4×D4—61no—
196#117—62no—
197#104—62no—
198#104—61no—
199#101—31no—
200#101C2×C5×D312yes—
201C2×C3×C5—21no—
202C2×C3×C5B(4,𝔽2)11yescharacteristic
203C5×D3—81no—
204C5×D3—82no—
205C22×D3—22no—
206C3×Q8—41no—
207C3×Q8—41no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4C2×C5×D411yescharacteristic
211C2×C3×C4C2×C5×D412yes—
212#67—41no—
213C2×Dic3—22no—
214D12—22no—
215D12—21no—
216D12—42no—
217D12—42no—
218D12—81no—
219C4×D3—44no—
220C4×D3—41no—
221C4×D3—42no—
222Dic6—22no—
223Dic6—21no—
224C3×C8—42no—
225C3⋊C8—41no—
226C22×C5—122no—
227C22×C5—122no—
228C22×C5—121no—
229C22×C5—122no—
230C22×C5#88211yescharacteristic
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—21no—
234C4×C5—41no—
235C4×C5—122no—
236C4×C5—121no—
237#41—61no—
238#41—62no—
239#41—121no—
240C2×Q8—121no—
241C2×Q8C5×D1211yescharacteristic
242C2×Q8C22×C5×D311yescharacteristic
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—62no—
247Q16—121no—
248SD16—122no—
249SD16—121no—
250D8—122no—
251M16—22no—
252M16—61no—
253C4⋊C4—22no—
254C4⋊C4—61no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C22⋊C4—62no—
258C42—32no—
259C42—61no—
260C42C5×D1211yescharacteristic
261C3×C5#220411yescharacteristic, complemented, Hall
262C22×C3C5×#12011yescharacteristic
263C2×D3—42no—
264C2×D3—42no—
265C2×D3—41no—
266C2×D3—42no—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—41no—
271Dic3—42no—
272Dic3—41no—
273C2×C5—21no—
274C2×C5—242no—
275C2×C5—241no—
276C2×C5#474111yescharacteristic
277C23—62no—
278Q8—41no—
279Q8—41no—
280Q8—62no—
281Q8—61no—
282D4—62no—
283D4—61no—
284D4—122no—
285D4—122no—
286D4—121no—
287D4—241no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—62no—
291C2×C4—124no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4C2×C5×D1211yescharacteristic
295C2×C4C5×D3×D412yes—
296C8—42no—
297C8—121no—
298C2×C3—21no—
299C2×C3C5×B(4,𝔽2)11yescharacteristic
300D3—82no—
301D3—81no—
302C5#6916811yescharacteristic, direct product factor, Sylow
303C22—122no—
304C22—122no—
305C22—121no—
306C22—122no—
307C22C5×#88211yescharacteristic
308C4—22no—
309C4—22no—
310C4—21no—
311C4—41no—
312C4—122no—
313C4—121no—
314C3C5×#220411yescharacteristic, complemented, Sylow
315C2—21no—
316C2—242no—
317C2—241no—
318C2C5×#474111yescharacteristic
subgroupG/H
centerC2×C5#4741
commutator, cosocleC2×C3×Q8C23×C5
FrattiniC2×Q8C22×C5×D3
FittingC3×C5×#356C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#356D3
hypocenterC3C5×#2204
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×Q8 ◁ C5×#356

Complements

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

NH ≅ G/Ntype
1. C5×#5734C2semidirect
2. C5×#5095C2semidirect
3. C5×#4753C2semidirect
4. C3×C5×#356C2semidirect
5. C5×#4477C2semidirect
20. #69168C5direct (N × H)
24. C5×#356D3semidirect
50. #5734C2×C5semidirect
51. #5095C2×C5semidirect
52. #4753C2×C5semidirect
53. C3×#356C2×C5semidirect
54. #4477C2×C5semidirect
130. #356C5×D3semidirect
261. C3×C5#2204semidirect
302. C5#69168direct (N × H)
314. C3C5×#2204semidirect

Maximal subgroups

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

subgroupindex
C5×#57342
C5×#50952
C5×#47532
C3×C5×#3562
C5×#44772
C5×#22043
#691685