Finite Group Databasebeta

← #36364025

Subgroups — #36364025

Order 22144.

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

normal subgroups only

This group has 572 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×#19149712C211yesmaximal, characteristic, complemented
2C2×#19149709C211yesmaximal, characteristic, complemented
3C2×#19149705C211yesmaximal, characteristic, complemented
4#26012037C211yesmaximal, characteristic, complemented
5C2×#19149666C211yesmaximal, characteristic, complemented
6#26011882C212yesmaximal, complemented
7#26011879C212yesmaximal, complemented
8#26011874C212yesmaximal, complemented
9#26011872C212yesmaximal, complemented
10M16×Dic173C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14#19149712—24no—
15#19149709—24no—
16#19149705—24no—
17C2×Dic692C2212yescomplemented
18C2×#14583816C2212yescomplemented
19C2×#14583815C2212yescomplemented
20C2×#14583814C2212yes—
21C2×#14583814C2212yescomplemented
22C2×#14583811C2212yes—
23C2×#14583810C2211yescharacteristic
24#19149677C2211yescharacteristic, complemented
25#19149675C2211yescharacteristic, complemented
26#19149668C2211yescharacteristic, complemented
27#19149666—22no—
28#19149666C2212yes—
29#19149666C2212yes—
30#19149666C2211yescharacteristic, complemented
31#19149651—22no—
32C4×C22⋊D173—22no—
33C2×C22⋊Dic173—22no—
34C2×C4×Dic173C2211yescharacteristic
35#19149589C2212yescomplemented
36C8×Dic173C2212yescomplemented
37C2×#41×C173C2211yescharacteristic
38C22×D4×C173C2211yescharacteristic
39AGL(1,ℤ/8)×C173—24no—
40C2×SD16×C173C2212yes—
41C2×D8×C173C2212yes—
42C2×M16×C173C2211yescharacteristic
43#14583826—22no—
44#14583826—22no—
45C2×Dic346—21no—
46C2×Dic346—22no—
47C2×Dic346—22no—
48C2×Dic346C2312yes—
49C2×Dic346C2311yescharacteristic
50C2×D692C2311yescharacteristic
51C2×C22⋊D173—22no—
52C2×C22⋊D173—21no—
53C22×Dic173—21no—
54C22×Dic173D412yes—
55C2×C4×D173—21no—
56Dic692—24no—
57#14583816—24no—
58#14583815—24no—
59#14583814—24no—
60#14583814—24no—
61#14583811—24no—
62#14583810D414yescomplemented
63C2×#13063436C2312yes—
64#14583805—22no—
65C22⋊Dic173—24no—
66C22⋊Dic173—24no—
67C22⋊Dic173—22no—
68#14583803—22no—
69C4×Dic173D412yes—
70C4×Dic173C2311yescharacteristic
71C4×Dic173C2311yescharacteristic
72C24×C173—21no—
73#41×C173—22no—
74#41×C173—22no—
75C2×Q8×C173C2311yescharacteristic
76C2×D4×C173—21no—
77C2×D4×C173—22no—
78C2×D4×C173—22no—
79C2×D4×C173—21no—
80C2×D4×C173C2311yescharacteristic
81C2×D4×C173C2312yes—
82C22×C4×C173—21no—
83C22×C4×C173C2311yescharacteristic
84SD16×C173—24no—
85D8×C173—24no—
86M16×C173D414yescomplemented
87C2×C8×C173C2312yes—
88C22×D173—21no—
89Dic346—21no—
90Dic346—22no—
91Dic346—22no—
92Dic346—21no—
93Dic346—42no—
94D692—21no—
95D692—21no—
96C22⋊D173—44no—
97C22⋊D173—42no—
98C2×Dic173—21no—
99C2×Dic173—22no—
100C2×Dic173—42no—
101C2×Dic173—42no—
102C2×Dic173C2×D412yes—
103C2×Dic173C2×D412yes—
104C4×D173—42no—
105#13063436C2×D414yes—
106C23×C173—21no—
107C23×C173—22no—
108C23×C173—44no—
109C23×C173—42no—
110C23×C173C2×D411yescharacteristic
111Q8×C173—21no—
112Q8×C173—21no—
113D4×C173—22no—
114D4×C173—21no—
115D4×C173—21no—
116D4×C173—22no—
117D4×C173—42no—
118D4×C173—42no—
119C2×C4×C173—21no—
120C2×C4×C173—42no—
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2×D412yes—
124C2×C4×C173C2411yescharacteristic
125C8×C173C2×D414yes—
126C22×C173—22no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173—44no—
130C22×C173—42no—
131C22×C173—84no—
132C22×C173#12712yes—
133C22×C173C22×D411yescharacteristic
134C2×D173—41no—
135C2×D173—42no—
136C4×C173—41no—
137C4×C173#12712yes—
138C4×C173C22×D411yescharacteristic
139C4×C173C22×D411yescharacteristic
140Dic173—22no—
141Dic173—22no—
142Dic173—41no—
143Dic173—42no—
144C2×C173—22no—
145C2×C173—81no—
146C2×C173—82no—
147C2×C173C2×#12711yescharacteristic
148C2×C173#57512yes—
149D173—81no—
150C173#315311yescharacteristic, complemented, Sylow
151#3153—1731nomaximal, Sylow
152C2×#137—1731no—
153C2×#137—1731no—
154C2×AGL(1,ℤ/8)—1731no—
155C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
156#532—1731no—
157C2×#124—1731no—
158#497—1732no—
159#496—1732no—
160#495—1732no—
161#495—1732no—
162C4×M16—1731no—
163C2×#41—1731no—
164C2×#41C2×D17311yescharacteristic
165C22×Q8—1731no—
166C22×D4C2×D17311yescharacteristic
167#137—3464no—
168#137—3464no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3464no—
171C2×Q16—1732no—
172C2×Q16—1732no—
173C2×SD16—1732no—
174C2×SD16—1732no—
175C2×SD16—1732no—
176C2×SD16C2×D17312yes—
177C2×D8—1732no—
178C2×D8C2×D17312yes—
179C2×M16—1731no—
180C2×M16C2×D17311yescharacteristic
181C4⋊Q8—1731no—
182#127—1731no—
183#124—3462no—
184#124—1731no—
185#124—1732no—
186#124—1732no—
187#124—1731no—
188#121—3462no—
189C4×D4—3462no—
190C2×C22⋊C4—3462no—
191C2×C42—1731no—
192#97—1732no—
193C4×C8—1732no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3462no—
198#41—3462no—
199C2×Q8—3461no—
200C2×Q8—3462no—
201C2×Q8—3462no—
202C2×Q8—1732no—
203C2×Q8—1731no—
204C2×Q8C22×D17311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3461no—
210C2×D4—3462no—
211C2×D4—1731no—
212C2×D4C22×D17312yes—
213C2×D4C22×D17311yescharacteristic
214C22×C4—21no—
215C22×C4—3461no—
216C22×C4—3461no—
217C22×C4—1732no—
218C22×C4C22×D17311yescharacteristic
219Q16—3464no—
220Q16—3464no—
221SD16—24no—
222SD16—3464no—
223SD16—3464no—
224SD16—3464no—
225D8—24no—
226D8—3464no—
227M16—1734no—
228M16C22⋊D17314yescomplemented
229C2×C8—1732no—
230C2×C8C22×D17312yes—
231C4⋊C4—3462no—
232C22⋊C4—3462no—
233C22⋊C4—3464no—
234C22⋊C4—3464no—
235C22⋊C4—3462no—
236C42—1732no—
237C42—1731no—
238C42—1731no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3461no—
244C23D4×D17311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3461no—
248Q8—3462no—
249Q8—3462no—
250Q8—3461no—
251Q8—6922no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3461no—
259D4—3461no—
260D4—6924no—
261D4—6922no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3462no—
265C2×C4—3461no—
266C2×C4—1732no—
267C2×C4—1732no—
268C2×C4—6922no—
269C2×C4—6922no—
270C2×C4—6922no—
271C2×C4D4×D17311yescharacteristic
272C2×C4D4×D17312yes—
273C2×C4C2×C22⋊D17312yes—
274C2×C4C23×D17311yescharacteristic
275C8—1734no—
276C8C2×C22⋊D17314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—6921no—
284C22—6922no—
285C22#1914967512yes—
286C22C2×D4×D17311yescharacteristic
287C4—41no—
288C4—3462no—
289C4—3462no—
290C4—6921no—
291C4—6922no—
292C4#1914967512yes—
293C4C2×D4×D17311yescharacteristic
294C4C22×C22⋊D17311yescharacteristic
295C2—22no—
296C2—81no—
297C2—82no—
298C2—13841no—
299C2C2×#1914967511yescharacteristic
300C2#2601228712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3153
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×#19149709C2semidirect
3. C2×#19149705C2semidirect
4. #26012037C2semidirect
5. C2×#19149666C2semidirect
6. #26011882C2semidirect
7. #26011879C2semidirect
8. #26011874C2semidirect
9. #26011872C2semidirect
10. M16×Dic173C2semidirect
11. C2×AGL(1,ℤ/8)×C173C2semidirect
17. C2×Dic692C22semidirect
18. C2×#14583816C22semidirect
19. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
24. #19149677C22semidirect
25. #19149675C22semidirect
26. #19149668C22semidirect
30. #19149666C22semidirect
35. #19149589C22semidirect
36. C8×Dic173C22semidirect
62. #14583810D4semidirect
86. M16×C173D4semidirect
150. C173#3153semidirect
155. C2×AGL(1,ℤ/8)D173semidirect
228. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191497092
C2×#191497052
#260120372
C2×#191496662
#260118822
#260118792
#260118742
#260118722
M16×Dic1732
C2×AGL(1,ℤ/8)×C1732
#3153173