Finite Group Databasebeta

← #36364367

Subgroups — #36364367

Order 22144.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#19149655C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic
3#26011825C212yesmaximal, complemented
4#26011805C211yesmaximal, characteristic, complemented
5#26011802C211yesmaximal, characteristic, complemented
6#26011740C212yesmaximal, complemented
7#26011734C212yesmaximal, complemented
8#26011640C212yesmaximal, complemented
9#26011630C211yesmaximal, characteristic, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#448×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173—21no—
13C2×D4×D173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15C2×#14583813C2212yescomplemented
16C2×#14583809C2212yescomplemented
17#19149655—21no—
18#19149655—21no—
19#19149655C2212yes—
20#19149655C2212yescomplemented
21#19149654C2211yescharacteristic
22#19149653C2211yescharacteristic
23#19149652—21no—
24#19149651—21no—
25#19149650C2211yescharacteristic
26C2×C4⋊Dic173C2211yescharacteristic, complemented
27C2×#14583805—21no—
28C4⋊C4×D173C2211yescharacteristic
29C2×#14583803—21no—
30C22⋊C4×D173—21no—
31#19149613C2212yescomplemented
32#19149612C2212yescomplemented
33#19149603C2212yescomplemented
34#19149601C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic, complemented
37#19149591C2212yes—
38C22×Q8×C173C2211yescharacteristic, complemented
39C2×SD16×C173C2212yescomplemented
40#121×C173C2211yescharacteristic
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic, complemented
43C23×D173—21no—
44C23×D173D411yescharacteristic
45C2×D692—21no—
46C2×D692C2311yescharacteristic
47C2×C22⋊D173—21no—
48C2×C22⋊D173—21no—
49C2×C22⋊D173—21no—
50D4×D173—42no—
51D4×D173—44no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—21no—
55C2×C4×D173—42no—
56C2×C4×D173D411yescharacteristic
57C2×C4×D173D411yescharacteristic
58C2×C4×D173C2311yescharacteristic
59C2×C4×D173C2311yescharacteristic
60#14583813—42no—
61#14583809—42no—
62C2×#13063436C2312yes—
63C4⋊Dic173—21no—
64C4⋊Dic173C2312yes—
65C4⋊Dic173C2311yescharacteristic
66#14583805—21no—
67#14583805—22no—
68#14583805—22no—
69C22⋊Dic173—21no—
70#14583803—22no—
71#14583803—22no—
72#14583803—21no—
73#14583803—21no—
74C2×Q8×C173—21no—
75C2×Q8×C173—42no—
76C2×Q8×C173C2312yes—
77C2×D4×C173—21no—
78C2×D4×C173C2311yescharacteristic
79C22×C4×C173—21no—
80C22×C4×C173C2311yescharacteristic
81SD16×C173—42no—
82C2×C8×C173C2312yes—
83C4⋊C4×C173C2311yescharacteristic
84C22⋊C4×C173—21no—
85C22×D173—22no—
86C22×D173—22no—
87C22×D173—21no—
88C22×D173—41no—
89C22×D173—42no—
90C22×D173—42no—
91C22×D173—42no—
92C22×D173C2×D411yescharacteristic
93C22×D173C2×D411yescharacteristic
94D692—22no—
95D692—42no—
96C22⋊D173—42no—
97C22⋊D173—42no—
98C22⋊D173—42no—
99C2×Dic173—21no—
100C2×Dic173—22no—
101C2×Dic173—41no—
102C2×Dic173—42no—
103C2×Dic173C2×D411yescharacteristic
104C2×Dic173C2×D411yescharacteristic
105C4×D173—22no—
106C4×D173—22no—
107C4×D173—41no—
108C4×D173—41no—
109C4×D173—42no—
110#13063436—22no—
111C23×C173—21no—
112C23×C173C2×D411yescharacteristic
113Q8×C173—22no—
114Q8×C173—22no—
115Q8×C173—42no—
116D4×C173—22no—
117D4×C173—42no—
118C2×C4×C173—22no—
119C2×C4×C173—21no—
120C2×C4×C173—22no—
121C2×C4×C173—42no—
122C2×C4×C173C2×D411yescharacteristic
123C2×C4×C173C2411yescharacteristic
124C8×C173—22no—
125C22×C173—21no—
126C22×C173—21no—
127C22×C173—41no—
128C22×C173—41no—
129C22×C173—41no—
130C22×C173AGL(1,ℤ/8)12yes—
131C22×C173C22×D411yescharacteristic
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—21no—
135C2×D173—21no—
136C2×D173—21no—
137C2×D173—44no—
138C2×D173—41no—
139C2×D173—41no—
140C2×D173—41no—
141C2×D173—84no—
142C2×D173#13712yes—
143C4×C173—21no—
144C4×C173—42no—
145C4×C173—41no—
146C4×C173H(5,𝔽2)12yes—
147Dic173—21no—
148Dic173—21no—
149Dic173—41no—
150Dic173—42no—
151C2×C173—22no—
152C2×C173—81no—
153C2×C173#53411yescharacteristic
154C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C173C2×#13711yescharacteristic
156D173—22no—
157D173—41no—
158D173—81no—
159C173#319311yescharacteristic, complemented, Sylow
160#3193—1731nomaximal, Sylow
161C2×C22⋊Q8—1731no—
162C2×#121—1731no—
163#484—1732no—
164#480—1731no—
165#480—1731no—
166#468—1732no—
167#468—1732no—
168#450—1732no—
169#448—1731no—
170#448D17311yescharacteristic, complemented
171#407—1731no—
172C22×Q8C2×D17311yescharacteristic, complemented
173C22×D4—3461no—
174C22×D4—1731no—
175C23×C4—1731no—
176C2×SD16—1732no—
177C2×SD16C2×D17312yescomplemented
178C2×M16—1732no—
179C22⋊Q8—3461no—
180C22⋊Q8—3461no—
181C22⋊Q8—1732no—
182C22⋊Q8—1732no—
183#121—3461no—
184#121—3461no—
185#121—1731no—
186#121—1731no—
187#121—1731no—
188#121C2×D17311yescharacteristic
189C2×C4⋊C4—3461no—
190C2×C4⋊C4—1731no—
191C2×C4⋊C4—1731no—
192C2×C22⋊C4—3461no—
193C2×C22⋊C4—3461no—
194#107—1732no—
195#107—1732no—
196#103—1732no—
197#103C2×D17312yescomplemented
198#102—1732no—
199#102—1732no—
200C22⋊C8—1731no—
201C22⋊C8—1732no—
202C22⋊C8C2×D17311yescharacteristic, complemented
203C24—3461no—
204C24—1731no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D17312yes—
208C2×D4—21no—
209C2×D4—3461no—
210C2×D4—3461no—
211C2×D4—3461no—
212C2×D4—3461no—
213C2×D4—1731no—
214C2×D4—6922no—
215C2×D4—6924no—
216C2×D4C22×D17311yescharacteristic
217C22×C4—21no—
218C22×C4—3461no—
219C22×C4—3461no—
220C22×C4—1731no—
221C22×C4—1731no—
222C22×C4—1731no—
223C22×C4—1731no—
224C22×C4—1731no—
225C22×C4—6922no—
226C22×C4C22×D17311yescharacteristic
227SD16—42no—
228SD16—6922no—
229M16—6922no—
230C2×C8—1732no—
231C2×C8C22×D17312yes—
232C4⋊C4—3462no—
233C4⋊C4—3462no—
234C4⋊C4—3461no—
235C4⋊C4—3461no—
236C4⋊C4—1732no—
237C4⋊C4—1731no—
238C4⋊C4C22×D17311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—3462no—
241C22⋊C4—3462no—
242C22⋊C4—3461no—
243C22⋊C4—3461no—
244C22⋊C4—3461no—
245C23—21no—
246C23—3462no—
247C23—3462no—
248C23—3461no—
249C23—1731no—
250C23—1731no—
251C23—6921no—
252C23—6922no—
253C23—6922no—
254C23—6922no—
255C23D4×D17311yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—3462no—
262D4—6922no—
263D4—6922no—
264D4—6922no—
265D4—6922no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3462no—
271C2×C4—3462no—
272C2×C4—3462no—
273C2×C4—3461no—
274C2×C4—1731no—
275C2×C4—1731no—
276C2×C4—6921no—
277C2×C4—6921no—
278C2×C4—6922no—
279C2×C4—6921no—
280C2×C4—6922no—
281C2×C4D4×D17311yescharacteristic
282C2×C4C23×D17311yescharacteristic
283C8—22no—
284C8—3462no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—3461no—
291C22—3461no—
292C22—3461no—
293C22—3461no—
294C22—3461no—
295C22—13844no—
296C22—1732no—
297C22—6924no—
298C22—6921no—
299C22—6921no—
300C22—6921no—
301C22#1914970612yes—
302C22C2×D4×D17311yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—3461no—
307C4—3461no—
308C4—6921no—
309C4—6922no—
310C4#1914972412yes—
311C2—22no—
312C2—81no—
313C2—3462no—
314C2—13841no—
315C2—6921no—
316C2#2601205111yescharacteristic
317C2C2×#1914970611yescharacteristic
318C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#448D173
hypocenterC173#3193
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19149655C2semidirect
3. #26011825C2semidirect
4. #26011805C2semidirect
5. #26011802C2semidirect
6. #26011740C2semidirect
7. #26011734C2semidirect
8. #26011640C2semidirect
9. #26011630C2semidirect
10. #26011453C2semidirect
11. #448×C173C2semidirect
15. C2×#14583813C22semidirect
16. C2×#14583809C22semidirect
20. #19149655C22semidirect
26. C2×C4⋊Dic173C22semidirect
31. #19149613C22semidirect
32. #19149612C22semidirect
33. #19149603C22semidirect
34. #19149601C22semidirect
35. #19149600C22semidirect
36. C22⋊#13063436C22semidirect
38. C22×Q8×C173C22semidirect
39. C2×SD16×C173C22semidirect
41. #103×C173C22semidirect
42. C22⋊C8×C173C22semidirect
159. C173#3193semidirect
170. #448D173semidirect
172. C22×Q8C2×D173semidirect
177. C2×SD16C2×D173semidirect
197. #103C2×D173semidirect
202. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496552
#121×D1732
#260118252
#260118052
#260118022
#260117402
#260117342
#260116402
#260116302
#260114532
#448×C1732
#3193173