Finite Group Databasebeta

← #12008628

Subgroups — #12008628

Order 1280.

356 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#10610388C211yesmaximal, characteristic
2#10599886C212yesmaximal, complemented
3C5×#1315C211yesmaximal, characteristic, complemented
4#10598180C211yesmaximal, characteristic, complemented
5#10598014C212yesmaximal, complemented
6#66166C411yescharacteristic
7C2×#4094C411yescharacteristic
8C2×#4053C411yescharacteristic
9C5×#408C2211yescharacteristic
10#65407C411yescharacteristic
11#65399C2211yescharacteristic
12C2×C8×Dic5C2211yescharacteristic
13#65018—22no—
14#65014—22no—
15C5×C22⋊C16C2212yes—
16C22⋊#672C2212yes—
17#64722—22no—
18#64721—22no—
19#11491—51nomaximal, Sylow
20C2×C5×#41C2×C411yescharacteristic
21C2×#709C2×C411yescharacteristic
22C2×C5×M16C2×C411yescharacteristic
23C22×C5×C8C2311yescharacteristic
24#4094—42no—
25#4093C811yescharacteristic
26#4090C811yescharacteristic
27#4086C811yescharacteristic
28#4085—22no—
29#4082C811yescharacteristic
30C4×C22⋊D5—22no—
31C2×C4×Dic5C2×C411yescharacteristic
32C2×#681—21no—
33C2×#681C2×C411yescharacteristic
34C22×#168—21no—
35C22×#168C2×C411yescharacteristic
36#4053—42no—
37C2×#676—21no—
38C2×C8×D5—21no—
39C2×C5×C16—22no—
40C5×C22⋊C8—22no—
41#3952—22no—
42#3946—22no—
43C8×Dic5D412yes—
44C8×Dic5D412yes—
45C2×#672—22no—
46#2931—51no—
47#2163—52no—
48#1315—51no—
49#1315—52no—
50#1315D511yescharacteristic, complemented
51C5×#41—41no—
52C5×#41—41no—
53C2×C5×Q8C2×C811yescharacteristic
54C2×C5×D4—21no—
55C2×C5×D4C2×C811yescharacteristic
56C22×C4×C5—21no—
57C22×C4×C5C22×C411yescharacteristic
58C2×C22⋊D5—21no—
59C2×C22⋊D5—22no—
60C22×Dic5M1612yes—
61#709—41no—
62#709—41no—
63C2×D20C2×C811yescharacteristic
64C2×C4×D5—21no—
65C2×Dic10C2×C811yescharacteristic
66C5×M16—41no—
67C2×C5×C8—21no—
68C2×C5×C8—41no—
69C2×C5×C8C2×D412yes—
70C22⋊Dic5—22no—
71#685—22no—
72#683—22no—
73C4×Dic5C22⋊C412yes—
74C4×Dic5C2×C811yescharacteristic
75C4×Dic5C2×C811yescharacteristic
76#681—42no—
77#681—41no—
78C2×#168—22no—
79C2×#168—21no—
80C2×#168—42no—
81C2×#168C22⋊C411yescharacteristic
82C2×#168C22⋊C411yescharacteristic
83#676—42no—
84C8×D5—42no—
85C5×C16—42no—
86#672—42no—
87C2×#131—51no—
88C2×#131—51no—
89#532—51no—
90#436—102no—
91C8×D4—102no—
92#423—51no—
93#408—51no—
94#408C2×D511yescharacteristic
95C2×C4×C8—51no—
96C4⋊C16—102no—
97C22⋊C16—52no—
98C22⋊C16C2×D512yes—
99#346—102no—
100C23×C5—21no—
101C23×C5C2×M1611yescharacteristic
102C22×D5—21no—
103C5×Q8—41no—
104C5×D4—41no—
105C5×D4—42no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—41no—
110C2×C4×C5—41no—
111C2×C4×C5C2×C22⋊C411yescharacteristic
112C2×C4×C5C22×C811yescharacteristic
113C22⋊D5—44no—
114C22⋊D5—42no—
115C2×Dic5—21no—
116C2×Dic5—42no—
117C2×Dic5C22⋊C812yes—
118C2×Dic5C22⋊C812yes—
119D20—41no—
120C4×D5—41no—
121C4×D5—41no—
122Dic10—41no—
123C5×C8—22no—
124C5×C8—41no—
125#168—21no—
126#168—41no—
127#168#10012yes—
128C2×#41—51no—
129C2×#41C4×D511yescharacteristic
130#131—202no—
131#131—202no—
132C2×M16—51no—
133C2×M16—101no—
134C2×M16—101no—
135C2×M16C4×D511yescharacteristic
136C22×C8—51no—
137C22×C8—101no—
138C22×C8—101no—
139C22×C8C22×D511yescharacteristic
140C4⋊Q8—51no—
141#127—51no—
142#124—51no—
143#124—51no—
144#121—102no—
145C4×D4—102no—
146C2×C42—51no—
147C2×C16—22no—
148C2×C16—102no—
149C4⋊C8—102no—
150C22⋊C8—22no—
151C22⋊C8—102no—
152C4×C8—52no—
153C4×C8—52no—
154C22×C5—22no—
155C22×C5—21no—
156C22×C5—42no—
157C22×C5—41no—
158C22×C5C2×C22⋊C811yescharacteristic
159C2×D5—42no—
160C2×D5—41no—
161C4×C5—21no—
162C4×C5—41no—
163C4×C5#41011yescharacteristic
164C4×C5C2×#10011yescharacteristic
165Dic5—22no—
166Dic5—22no—
167Dic5—41no—
168#41—41no—
169#41—41no—
170#41—201no—
171#41—201no—
172C2×Q8—51no—
173C2×Q8C8×D511yescharacteristic
174C2×D4—21no—
175C2×D4—51no—
176C2×D4—102no—
177C2×D4—101no—
178C2×D4C8×D511yescharacteristic
179C22×C4—21no—
180C22×C4—52no—
181C22×C4—101no—
182C22×C4C2×C4×D511yescharacteristic
183M16—41no—
184M16—202no—
185M16—202no—
186M16—201no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—51no—
190C2×C8—51no—
191C2×C8—102no—
192C2×C8—101no—
193C2×C8—202no—
194C2×C8—202no—
195C2×C8D4×D512yes—
196C4⋊C4—102no—
197C22⋊C4—102no—
198C22⋊C4—102no—
199C42—52no—
200C42—51no—
201C42—51no—
202C16—42no—
203C16—202no—
204C2×C5—41no—
205C2×C5—81no—
206C2×C5#146011yescharacteristic
207C2×C5#211712yes—
208D5—81no—
209C23—21no—
210C23—101no—
211C23D5×M1611yescharacteristic
212Q8—41no—
213Q8—201no—
214D4—41no—
215D4—42no—
216D4—201no—
217D4—204no—
218D4—202no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—52no—
225C2×C4—52no—
226C2×C4—101no—
227C2×C4—202no—
228C2×C4—201no—
229C2×C4—201no—
230C2×C4D5×C22⋊C411yescharacteristic
231C2×C4C2×C8×D511yescharacteristic
232C8—22no—
233C8—41no—
234C8—52no—
235C8—101no—
236C8—201no—
237C5#1149111yescharacteristic, complemented, Sylow
238C22—21no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—201no—
243C22—202no—
244C22D5×C22⋊C811yescharacteristic
245C4—21no—
246C4—41no—
247C4—102no—
248C4—102no—
249C4—201no—
250C4#6502911yescharacteristic
251C4D5×#10011yescharacteristic
252C2—41no—
253C2—81no—
254C2—401no—
255C2#1059889411yescharacteristic
256C2#1059991812yes—
subgroupG/H
centerC2×C4D5×C22⋊C4
commutatorC2×C4×C5C22×C8
FrattiniC22×C8C22×D5
FittingC5×#1315C2
socleC22×C5C2×C22⋊C8
prime radical#66166C4
cosocleC22×C5×C8C23
hypercenter#1315D5
hypocenterC5#11491
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #10599886C2semidirect
3. C5×#1315C2semidirect
4. #10598180C2semidirect
5. #10598014C2semidirect
50. #1315D5semidirect
237. C5#11491semidirect

Maximal subgroups

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

subgroupindex
#106103882
#105998862
C5×#13152
#105981802
#105980142
#114915