Finite Group Databasebeta

← #20284523

Subgroups — #20284523

Order 6784.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4⋊Q8×D53C211yesmaximal, characteristic, complemented
2C4×#11719947C211yesmaximal, characteristic
3#16810917C212yesmaximal, complemented
4#16810911C212yesmaximal, complemented
5#16810909C211yesmaximal, characteristic, complemented
6#16810905C211yesmaximal, characteristic, complemented
7#16810903C211yesmaximal, characteristic, complemented
8#16810664C212yesmaximal, complemented
9#16810645C211yesmaximal, characteristic, complemented
10#16810621C212yesmaximal
11C53×#474C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×Q8×D53—21no—
14C2×#11719912C2212yes—
15#13133217C2211yescharacteristic
16#13133216—21no—
17#13133216C2212yes—
18#13133215—21no—
19C4×Dic106C2211yescharacteristic
20#13133182C2211yescharacteristic
21D4×Dic53C2211yescharacteristic
22C4×C22⋊D53—21no—
23#13133177—21no—
24#13133175C2211yescharacteristic
25#13133173—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53C2212yes—
28C2×C4×Dic53—21no—
29C42×D53C2211yescharacteristic
30#13133150C2212yes—
31#13133149C2212yes—
32#13133148C2212yes—
33C4⋊#88322C2211yescharacteristic
34#13133146C2212yes—
35#13133142C2212yes—
36#13133140C2212yes—
37#13133138C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×D4×C53C2211yescharacteristic
40#107×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#102×C53C2212yes—
43C2×Q8×C53—21no—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C2×C22⋊D53—21no—
47C22×Dic53—21no—
48C22×Dic53—21no—
49Q8×D53—44no—
50#11719947—22no—
51#11719947Q814yescomplemented
52C2×C4×D53—22no—
53C2×C4×D53D411yescharacteristic
54C2×C4×D53D411yescharacteristic
55C2×C4×D53C2311yescharacteristic
56C2×Dic106—21no—
57C2×Dic106—22no—
58C2×Dic106C2311yescharacteristic
59C2×C8×C53C2312yes—
60C4⋊C4×C53—21no—
61C4⋊C4×C53C2312yes—
62C4⋊C4×C53C2311yescharacteristic
63C22⋊C4×C53—21no—
64C42×C53C2311yescharacteristic
65C22⋊Dic53—21no—
66#11719921—21no—
67C4⋊Dic53—21no—
68C4⋊Dic53C2312yes—
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—22no—
72#11719919—22no—
73#11719919—21no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53D411yescharacteristic
78C4×Dic53D411yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81#11719912—24no—
82C23×C53—21no—
83C22×D53C2×D411yescharacteristic
84Q8×C53—42no—
85D4×C53—21no—
86D4×C53C2×Q812yescomplemented
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—42no—
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—24no—
94C2×Dic53—21no—
95C2×Dic53—24no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—42no—
99C2×Dic53C2×D411yescharacteristic
100C2×Dic53C2×D411yescharacteristic
101C2×Dic53C2×D411yescharacteristic
102C4×D53—22no—
103C4×D53—22no—
104C4×D53—44no—
105C4×D53C2×Q812yes—
106C4×D53#4112yes—
107Dic106—21no—
108Dic106—44no—
109Dic106—42no—
110Dic106C2×Q812yescomplemented
111C8×C53—22no—
112#88322—22no—
113C22×C53—22no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53C22×D411yescharacteristic
117C2×D53—21no—
118C2×D53—21no—
119C2×D53C22⋊Q812yes—
120C4×C53—21no—
121C4×C53—42no—
122C4×C53—41no—
123C4×C53AGL(1,ℤ/8)12yes—
124C4×C53C22×Q811yescharacteristic
125C4×C53C2×#4111yescharacteristic
126Dic53—22no—
127Dic53—21no—
128Dic53—42no—
129Dic53—41no—
130Dic53C22⋊Q812yes—
131Dic53#13712yes—
132#3079—531nomaximal, Sylow
133C2×C53—42no—
134C2×C53C2×C22⋊Q811yescharacteristic
135C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C53C2×#13711yescharacteristic
137D53—22no—
138C2×C4⋊Q8—531no—
139C4×#41—531no—
140#477—532no—
141#476—532no—
142#475—531no—
143#475—531no—
144#474—531no—
145#474D5311yescharacteristic, complemented
146#428—532no—
147#423—531no—
148#417—532no—
149C53#307911yescharacteristic, complemented, Sylow
150C2×#41—531no—
151C22×Q8—1061no—
152C2×M16—532no—
153C4⋊Q8—532no—
154C4⋊Q8—531no—
155C4⋊Q8—1061no—
156C4⋊Q8—1061no—
157C4⋊Q8C2×D5311yescharacteristic
158C4×Q8—531no—
159C4×Q8—531no—
160C4×D4—531no—
161C4×D4—1061no—
162C4×D4—1061no—
163C4×D4C2×D5311yescharacteristic
164#117—531no—
165#117—1061no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1061no—
168C2×C42—531no—
169C2×C42—1061no—
170#107—532no—
171#107C2×D5312yes—
172#106—532no—
173#106—532no—
174C4⋊C8—532no—
175C4⋊C8—531no—
176C4⋊C8C2×D5311yescharacteristic
177#103—532no—
178#103—532no—
179#102—532no—
180#102C2×D5312yes—
181#41—534no—
182#41—1062no—
183C2×Q8—21no—
184C2×Q8—531no—
185C2×Q8—1062no—
186C2×Q8—1061no—
187C2×Q8—2124no—
188C2×D4—1061no—
189C2×D4C22×D5311yescharacteristic
190C22×C4—21no—
191C22×C4—531no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—1062no—
195C22×C4—1061no—
196C22×C4—1061no—
197M16—1064no—
198C2×C8—532no—
199C2×C8C22×D5312yes—
200C4⋊C4—21no—
201C4⋊C4—532no—
202C4⋊C4—531no—
203C4⋊C4—1062no—
204C4⋊C4—1062no—
205C4⋊C4—1061no—
206C4⋊C4—1061no—
207C4⋊C4—1061no—
208C4⋊C4C22×D5312yes—
209C4⋊C4C22×D5311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—1061no—
212C22⋊C4—1061no—
213C42—531no—
214C42—531no—
215C42—531no—
216C42—1061no—
217C42—1061no—
218C42—1061no—
219C42C22×D5311yescharacteristic
220C23—21no—
221C23—531no—
222Q8—42no—
223Q8—532no—
224Q8—1061no—
225Q8—2124no—
226Q8—2122no—
227D4—21no—
228D4—1064no—
229D4Q8×D5312yescomplemented
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—531no—
234C2×C4—531no—
235C2×C4—532no—
236C2×C4—532no—
237C2×C4—531no—
238C2×C4—1062no—
239C2×C4—1064no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—1062no—
244C2×C4—2124no—
245C2×C4—2122no—
246C2×C4D4×D5311yescharacteristic
247C2×C4D4×D5311yescharacteristic
248C2×C4C23×D5311yescharacteristic
249C8—22no—
250C8—1062no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—532no—
255C22—1061no—
256C22—1061no—
257C22C2×D4×D5311yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—532no—
262C4—532no—
263C4—1062no—
264C4—1061no—
265C4—2122no—
266C4—2121no—
267C4#1313324112yes—
268C4C2×Q8×D5311yescharacteristic
269C4#41×D5311yescharacteristic
270C2—42no—
271C2—1062no—
272C2C22⋊Q8×D5311yescharacteristic
273C2C2×#1313324111yescharacteristic
274C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3079
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D53C2semidirect
3. #16810917C2semidirect
4. #16810911C2semidirect
5. #16810909C2semidirect
6. #16810905C2semidirect
7. #16810903C2semidirect
8. #16810664C2semidirect
9. #16810645C2semidirect
11. C53×#474C2semidirect
51. #11719947Q8semidirect
86. D4×C53C2×Q8semidirect
110. Dic106C2×Q8semidirect
145. #474D53semidirect
149. C53#3079semidirect
229. D4Q8×D53semidirect