Finite Group Databasebeta

← #36362852

Subgroups — #36362852

Order 22144.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19149726C211yesmaximal, characteristic, complemented
2C2×#19149704C211yesmaximal, characteristic, complemented
3C22×#14583815C211yesmaximal, characteristic, complemented
4#26012060C211yesmaximal, characteristic, complemented
5#26011637C214yesmaximal, complemented
6#26011631C214yesmaximal, complemented
7#26011499C211yesmaximal, characteristic, complemented
8#26011457C211yesmaximal, characteristic, complemented
9#416×C173C211yesmaximal, characteristic, complemented
10#19149726—24no—
11#19149726—24no—
12C2×#14583826—21no—
13C2×#14583826C2211yescharacteristic
14C2×#14583825—22no—
15C2×#14583825C2212yescomplemented
16C22×Dic346—21no—
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2212yescomplemented
19C22×D692C2211yescharacteristic
20#19149704—24no—
21C2×#14583815—24no—
22C2×#14583815C2214yes—
23C2×#14583813C2212yes—
24C2×#14583811C2212yes—
25C2×#14583810C2211yescharacteristic
26#19149676—21no—
27#19149667—21no—
28#19149654C2214yes—
29#19149649—22no—
30#19149602C2214yescomplemented
31#19149591C2214yescomplemented
32C2×#41×C173C2211yescharacteristic
33C22×C8×C173C2211yescharacteristic
34#117×C173C2211yescharacteristic
35#103×C173C2212yescomplemented
36#102×C173C2212yescomplemented
37C23×D173—21no—
38#14583826—28no—
39#14583826—22no—
40#14583826—22no—
41#14583825—216no—
42#14583825—24no—
43#14583825—24no—
44C2×Dic346—28no—
45C2×Dic346—22no—
46C2×Dic346—21no—
47C2×Dic346—22no—
48C2×Dic346D411yescharacteristic
49C2×Dic346D414yescomplemented
50C2×Dic346C2312yes—
51Q8×D173—24no—
52Q8×D173—24no—
53C2×D692—24no—
54C2×D692—22no—
55C2×D692—22no—
56C2×D692—21no—
57C2×D692D411yescharacteristic
58C2×D692C2312yes—
59C2×C22⋊D173—22no—
60C2×C22⋊D173D412yes—
61C22×Dic173—22no—
62C22×Dic173—21no—
63C2×C4×D173—22no—
64C2×C4×D173C2312yes—
65#14583815—28no—
66#14583813—24no—
67#14583811—24no—
68#14583810—22no—
69C2×#13063436C2312yes—
70#14583803—24no—
71#41×C173—22no—
72#41×C173D414yescomplemented
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—21no—
75C2×D4×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78C2×C8×C173—22no—
79C2×C8×C173C2312yes—
80C4⋊C4×C173C2312yes—
81C22⋊C4×C173—21no—
82C42×C173—21no—
83C22×D173—22no—
84C22×D173—42no—
85C22×D173—82no—
86C22×D173C2×D412yes—
87Dic346—28no—
88Dic346—22no—
89Dic346—24no—
90Dic346—22no—
91Dic346C2×D414yes—
92D692—22no—
93D692—22no—
94D692—22no—
95D692—42no—
96D692—84no—
97C22⋊D173—28no—
98C22⋊D173—24no—
99C2×Dic173—28no—
100C2×Dic173—24no—
101C2×Dic173—22no—
102C2×Dic173C2×D412yes—
103C4×D173—28no—
104C4×D173—22no—
105C4×D173—22no—
106#13063436—22no—
107C23×C173—21no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—21no—
110Q8×C173C2×D412yes—
111D4×C173—24no—
112D4×C173—21no—
113D4×C173C2×D412yes—
114C2×C4×C173—21no—
115C2×C4×C173—24no—
116C2×C4×C173—22no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2411yescharacteristic
121C8×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—42no—
125C22×C173#12012yes—
126C22×C173C22×D411yescharacteristic
127C2×D173—22no—
128C2×D173—24no—
129C2×D173—42no—
130C2×D173—44no—
131C2×D173—84no—
132C4×C173—22no—
133C4×C173—42no—
134C4×C173#12012yes—
135C4×C173C22×D411yescharacteristic
136C4×C173C22×D411yescharacteristic
137Dic173—24no—
138Dic173—22no—
139C2×C173—22no—
140C2×C173—42no—
141C2×C173C2×#12011yescharacteristic
142C2×C173#57712yes—
143D173—42no—
144D173—82no—
145C173#301011yescharacteristic, complemented, Sylow
146#3010—1731nomaximal, Sylow
147C2×2−1+4—1731no—
148C2×AGL(1,ℤ/8)—1731no—
149C22×SD16—1731no—
150#535—1731no—
151#449—1734no—
152#448—1734no—
153#419—1731no—
154#416D17311yescharacteristic, complemented
155#408—1731no—
1562−1+4—3464no—
1572−1+4—3464no—
158C2×#41—3461no—
159C2×#41—3462no—
160C2×#41—1732no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8—3461no—
164C22×Q8—1732no—
165C22×Q8—1731no—
166C22×D4—1731no—
167AGL(1,ℤ/8)—3464no—
168C2×SD16—3464no—
169C2×SD16—1732no—
170C2×SD16—1734no—
171C2×D8—1732no—
172C2×M16—1731no—
173C22×C8C2×D17311yescharacteristic
174#127—3461no—
175#124—3461no—
176#121—1734no—
177#120—3462no—
178#117C2×D17311yescharacteristic
179#103—1734no—
180#103C2×D17312yescomplemented
181#102C2×D17312yescomplemented
182C22⋊C8—1734no—
183C24—3461no—
184#41—22no—
185#41—3468no—
186#41—34616no—
187#41—3464no—
188#41—3464no—
189#41—3462no—
190#41—3462no—
191#41D69214yescomplemented
192C2×Q8—3468no—
193C2×Q8—3462no—
194C2×Q8—3464no—
195C2×Q8—3464no—
196C2×Q8—3461no—
197C2×Q8—3462no—
198C2×Q8—1734no—
199C2×Q8—1732no—
200C2×Q8—1731no—
201C2×Q8C22×D17311yescharacteristic
202C2×D4—21no—
203C2×D4—3464no—
204C2×D4—3462no—
205C2×D4—3462no—
206C2×D4—3461no—
207C2×D4—3462no—
208C2×D4—1732no—
209C2×D4—1731no—
210C2×D4—1732no—
211C2×D4C22×D17311yescharacteristic
212C22×C4—21no—
213C22×C4—3462no—
214C22×C4—3461no—
215C22×C4—3462no—
216C22×C4—1732no—
217C22×C4C22×D17311yescharacteristic
218SD16—3468no—
219SD16—3464no—
220D8—3464no—
221M16—3462no—
222C2×C8—22no—
223C2×C8—1732no—
224C2×C8C22×D17312yes—
225C4⋊C4C22×D17312yes—
226C22⋊C4—21no—
227C22⋊C4—3464no—
228C42—21no—
229C23—21no—
230C23—3462no—
231C23—13842no—
232C23—1732no—
233C23—6922no—
234C23D4×D17311yescharacteristic
235Q8—21no—
236Q8—3468no—
237Q8—3464no—
238Q8—3462no—
239Q8—3462no—
240Q8—1734no—
241Q8C2×D69212yes—
242D4—24no—
243D4—21no—
244D4—3468no—
245D4—3464no—
246D4—3462no—
247D4—3462no—
248D4—3462no—
249D4—13844no—
250D4—6922no—
251D4C2×D69212yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—3468no—
256C2×C4—3468no—
257C2×C4—3464no—
258C2×C4—3462no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—1732no—
262C2×C4D4×D17311yescharacteristic
263C2×C4D4×D17312yes—
264C2×C4C2×D69212yes—
265C2×C4C23×D17311yescharacteristic
266C8—22no—
267C8—3462no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—3462no—
272C22—3464no—
273C22—13844no—
274C22—6922no—
275C22—6924no—
276C22#1914964912yes—
277C22C2×D4×D17311yescharacteristic
278C4—22no—
279C4—42no—
280C4—3464no—
281C4—3462no—
282C4#1914964912yes—
283C4C2×D4×D17311yescharacteristic
284C4C22×D69211yescharacteristic
285C2—22no—
286C2—42no—
287C2—13842no—
288C2—6922no—
289C2C2×#1914964911yescharacteristic
290C2#2601229812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
hypercenter#416D173
hypocenterC173#3010
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#19149726C2semidirect
2. C2×#19149704C2semidirect
3. C22×#14583815C2semidirect
4. #26012060C2semidirect
5. #26011637C2semidirect
6. #26011631C2semidirect
7. #26011499C2semidirect
8. #26011457C2semidirect
9. #416×C173C2semidirect
15. C2×#14583825C22semidirect
18. C2×Q8×D173C22semidirect
30. #19149602C22semidirect
31. #19149591C22semidirect
35. #103×C173C22semidirect
36. #102×C173C22semidirect
49. C2×Dic346D4semidirect
72. #41×C173D4semidirect
145. C173#3010semidirect
154. #416D173semidirect
180. #103C2×D173semidirect
181. #102C2×D173semidirect
191. #41D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497262
C2×#191497042
C22×#145838152
#260120602
#260116372
#260116312
#260114992
#260114572
#416×C1732
#3010173