Finite Group Databasebeta

← C5×#12225 #11934945

Subgroups — C5×#12225 #11934945

Order 1280.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2809C211yesmaximal, characteristic
2C5×#2028C211yesmaximal, characteristic
3C5×#1617C212yesmaximal
4C5×#1353C212yesmaximal
5C5×#1319C211yesmaximal, characteristic
6C5×#544C2211yescharacteristic
7C5×#484—22no—
8C5×#419—21no—
9C5×#400C2212yes—
10C5×#398—22no—
11C5×#391—21no—
12C5×#391C2211yescharacteristic
13C5×#389—21no—
14C5×#387—22no—
15C5×#343C2211yescharacteristic
16C5×#339—21no—
17C5×#328—22no—
18C5×#324C2212yes—
19#12225C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8D411yescharacteristic
21C2×C5×M16—21no—
22C5×C4⋊Q8—41no—
23C5×#125—41no—
24C5×C22⋊Q8—22no—
25C5×#117—21no—
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4D412yes—
31C2×C5×C4⋊C4C2311yescharacteristic
32C2×C42×C5—21no—
33C2×C42×C5D411yescharacteristic
34C5×#107—42no—
35C5×#103—41no—
36C5×#103—41no—
37C5×#103—42no—
38C5×C22⋊C8—22no—
39C5×#95—22no—
40C5×#95—22no—
41C5×#95—22no—
42C5×#95—22no—
43C5×#95D412yes—
44#2809C2×C511yescharacteristic
45#2028C2×C511yescharacteristic
46#1617C2×C512yes—
47#1353C2×C512yes—
48#1319C2×C511yescharacteristic
49C2×C5×Q8—21no—
50C2×C5×Q8—22no—
51C2×C5×Q8—41no—
52C2×C5×Q8—41no—
53C2×C5×Q8Q1612yes—
54C22×C4×C5—21no—
55C22×C4×C5—21no—
56C22×C4×C5—22no—
57C22×C4×C5—22no—
58C22×C4×C5C2×D411yescharacteristic
59C22×C4×C5C2×D412yes—
60C5×M16—41no—
61C5×M16—41no—
62C2×C5×C8—41no—
63C2×C5×C8—42no—
64C5×C4⋊C4—21no—
65C5×C4⋊C4—21no—
66C5×C4⋊C4—22no—
67C5×C4⋊C4—41no—
68C5×C4⋊C4—42no—
69C5×C4⋊C4—44no—
70C5×C4⋊C4—42no—
71C5×C4⋊C4—42no—
72C5×C4⋊C4—81no—
73C5×C4⋊C4—81no—
74C5×C22⋊C4—41no—
75C42×C5—41no—
76C42×C5—41no—
77C42×C5—81no—
78#544C22×C511yescharacteristic
79#484—22no—
80#419—21no—
81#400C22×C512yes—
82#398—22no—
83#391—21no—
84#391C22×C511yescharacteristic
85#389—21no—
86#387—22no—
87#343C22×C511yescharacteristic
88#339—21no—
89#328—22no—
90#324C22×C512yes—
91C23×C5#12011yescharacteristic
92C5×Q8—42no—
93C5×Q8—81no—
94C5×Q8—82no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—41no—
101C2×C4×C5—41no—
102C2×C4×C5—41no—
103C2×C4×C5—41no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—42no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—42no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—42no—
113C2×C4×C5C2×Q1611yescharacteristic
114C2×C4×C5#13711yescharacteristic
115C5×C8—81no—
116C5×C8—82no—
117C22×Q8C5×D411yescharacteristic
118C2×M16—21no—
119C4⋊Q8—41no—
120#125—41no—
121C22⋊Q8—22no—
122#117—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C5×D412yes—
128C2×C4⋊C4C23×C511yescharacteristic
129C2×C42—21no—
130C2×C42C5×D411yescharacteristic
131#107—42no—
132#103—41no—
133#103—41no—
134#103—42no—
135C22⋊C8—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95C5×D412yes—
141C22×C5—21no—
142C22×C5—21no—
143C22×C5#45111yescharacteristic
144C22×C5#45511yescharacteristic
145C22×C5B(4,𝔽2)11yescharacteristic
146C4×C5—21no—
147C4×C5—21no—
148C4×C5—42no—
149C4×C5—42no—
150C4×C5—42no—
151C4×C5—82no—
152C4×C5—81no—
153C4×C5—81no—
154C4×C5—82no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C5×Q1612yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×C5×D411yescharacteristic
165C22×C4C2×C5×D412yes—
166M16—41no—
167M16—41no—
168C2×C8—42no—
169C2×C8—41no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C42—41no—
182C42—41no—
183C42—81no—
184C2×C5—21no—
185C2×C5—21no—
186C2×C5#160211yescharacteristic
187C2×C5#220111yescharacteristic
188C2×C5#220411yescharacteristic
189C23C5×#12011yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4C2×C5×Q1611yescharacteristic
212C2×C4C5×#13711yescharacteristic
213C8—81no—
214C8—82no—
215C5#1222511yescharacteristic, direct product factor, Sylow
216C22—21no—
217C22—21no—
218C22C5×#45111yescharacteristic
219C22C5×#45511yescharacteristic
220C22C5×B(4,𝔽2)11yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2C5×#160211yescharacteristic
233C2C5×#220111yescharacteristic
234C2C5×#220411yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC23×C5#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12225C5direct (N × H)
215. C5#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28092
C5×#20282
C5×#16172
C5×#13532
C5×#13192
#122255