Finite Group Databasebeta

← #36364011

Subgroups — #36364011

Order 22144.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19149707C211yesmaximal, characteristic, complemented
2C2×#19149705C211yesmaximal, characteristic, complemented
3C2×#19149704C211yesmaximal, characteristic, complemented
4#26012037C211yesmaximal, characteristic, complemented
5C2×#19149675C211yesmaximal, characteristic, complemented
6#26011879C212yesmaximal, complemented
7#26011877C212yesmaximal, complemented
8#26011866C212yesmaximal, complemented
9#26011862C212yesmaximal, complemented
10M16×Dic173C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×D692C2211yescharacteristic
14C22×C22⋊D173—22no—
15#19149707—24no—
16#19149705—24no—
17#19149704—24no—
18C2×#14583815C2212yescomplemented
19C2×#14583814C2212yescomplemented
20C2×#14583813C2212yescomplemented
21C2×D1384C2212yescomplemented
22C2×#14583811C2212yescomplemented
23C2×#14583811C2212yescomplemented
24C2×#14583810C2211yescharacteristic
25#19149677C2211yescharacteristic, complemented
26#19149675—22no—
27#19149675C2212yes—
28#19149675C2212yes—
29#19149675C2211yescharacteristic, complemented
30#19149668C2211yescharacteristic, complemented
31#19149666C2211yescharacteristic, complemented
32#19149651—22no—
33C4×C22⋊D173—22no—
34C2×C4×Dic173C2211yescharacteristic
35#19149589C2212yescomplemented
36C8×Dic173C2212yescomplemented
37C2×#41×C173C2211yescharacteristic
38C22×D4×C173C2211yescharacteristic
39AGL(1,ℤ/8)×C173—24no—
40C2×SD16×C173C2212yescomplemented
41C2×D8×C173C2212yescomplemented
42C2×M16×C173C2211yescharacteristic
43C23×D173—21no—
44#14583826—22no—
45#14583826—22no—
46C2×Dic346C2311yescharacteristic
47C2×D692—22no—
48C2×D692—22no—
49C2×D692—21no—
50C2×D692C2311yescharacteristic
51C2×D692C2312yes—
52C2×C22⋊D173—22no—
53C2×C22⋊D173—24no—
54C2×C22⋊D173—24no—
55C2×C22⋊D173—21no—
56C2×C22⋊D173—48no—
57C22×Dic173D412yes—
58C2×C4×D173—21no—
59#14583815—24no—
60#14583814—24no—
61#14583813—24no—
62D1384—24no—
63#14583811—24no—
64#14583811—24no—
65#14583810D414yescomplemented
66C2×#13063436C2312yes—
67#14583805—22no—
68C22⋊Dic173—22no—
69#14583803—22no—
70C4×Dic173D412yes—
71C4×Dic173C2311yescharacteristic
72C4×Dic173C2311yescharacteristic
73C24×C173—21no—
74#41×C173—22no—
75#41×C173—22no—
76C2×Q8×C173C2311yescharacteristic
77C2×D4×C173—21no—
78C2×D4×C173—22no—
79C2×D4×C173—22no—
80C2×D4×C173—21no—
81C2×D4×C173C2311yescharacteristic
82C2×D4×C173C2312yes—
83C22×C4×C173—21no—
84C22×C4×C173C2311yescharacteristic
85SD16×C173—24no—
86D8×C173—24no—
87M16×C173D414yescomplemented
88C2×C8×C173C2312yes—
89C22×D173—21no—
90C22×D173—22no—
91C22×D173—44no—
92C22×D173—42no—
93Dic346—21no—
94Dic346—21no—
95D692—21no—
96D692—22no—
97D692—21no—
98D692—22no—
99D692—42no—
100C22⋊D173—48no—
101C22⋊D173—42no—
102C22⋊D173—44no—
103C22⋊D173—48no—
104C2×Dic173—24no—
105C2×Dic173—21no—
106C2×Dic173C2×D412yes—
107C2×Dic173C2×D412yes—
108C4×D173—42no—
109#13063436C2×D414yescomplemented
110C23×C173—22no—
111C23×C173—21no—
112C23×C173—44no—
113C23×C173—42no—
114C23×C173C2×D411yescharacteristic
115Q8×C173—21no—
116Q8×C173—21no—
117D4×C173—21no—
118D4×C173—22no—
119D4×C173—21no—
120D4×C173—22no—
121D4×C173—42no—
122D4×C173—42no—
123C2×C4×C173—21no—
124C2×C4×C173—42no—
125C2×C4×C173C2×D411yescharacteristic
126C2×C4×C173C2×D412yes—
127C2×C4×C173C2×D412yes—
128C2×C4×C173C2411yescharacteristic
129C8×C173C2×D414yescomplemented
130C22×C173—22no—
131C22×C173—41no—
132C22×C173—42no—
133C22×C173—44no—
134C22×C173—42no—
135C22×C173—84no—
136C22×C173#12712yes—
137C22×C173C22×D411yescharacteristic
138C2×D173—41no—
139C2×D173—42no—
140C2×D173—44no—
141C2×D173—42no—
142C2×D173—84no—
143C4×C173—41no—
144C4×C173#12712yes—
145C4×C173C22×D411yescharacteristic
146C4×C173C22×D411yescharacteristic
147Dic173—22no—
148Dic173—41no—
149Dic173AGL(1,ℤ/8)14yescomplemented
150C2×C173—22no—
151C2×C173—81no—
152C2×C173—82no—
153C2×C173C2×#12711yescharacteristic
154C2×C173C2×AGL(1,ℤ/8)12yes—
155D173—81no—
156D173—82no—
157C173#315111yescharacteristic, complemented, Sylow
158#3151—1731nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1731no—
160C2×AGL(1,ℤ/8)—1731no—
161C2×AGL(1,ℤ/8)—1731no—
162C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
163#532—1731no—
164C2×#127—1731no—
165#496—1732no—
166#496—1732no—
167#493—1732no—
168#492—1732no—
169C4×M16—1731no—
170C2×#41—1731no—
171C2×#41C2×D17311yescharacteristic
172C22×D4—3462no—
173C22×D4—1731no—
174C22×D4C2×D17311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3464no—
177AGL(1,ℤ/8)—3464no—
178AGL(1,ℤ/8)—3464no—
179C2×SD16—1732no—
180C2×SD16—1732no—
181C2×SD16—1732no—
182C2×SD16C2×D17312yescomplemented
183C2×D8—1732no—
184C2×D8—1732no—
185C2×D8—1732no—
186C2×D8C2×D17312yescomplemented
187C2×M16—1731no—
188C2×M16C2×D17311yescharacteristic
189C4⋊Q8—1731no—
190#127—3462no—
191#127—1732no—
192#127—1732no—
193#127—1731no—
194#124—1731no—
195#124—1731no—
196#121—3462no—
197C4×D4—3462no—
198C2×C42—1731no—
199#97—1732no—
200C4×C8—1732no—
201C24—21no—
202C24—3461no—
203#41—22no—
204#41—22no—
205#41—3462no—
206#41—3462no—
207C2×Q8—1731no—
208C2×Q8C22×D17311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3461no—
214C2×D4—3462no—
215C2×D4—3464no—
216C2×D4—3464no—
217C2×D4—3462no—
218C2×D4—3461no—
219C2×D4—3462no—
220C2×D4—1731no—
221C2×D4—1732no—
222C2×D4—6928no—
223C2×D4C22×D17311yescharacteristic
224C2×D4C22×D17312yes—
225C22×C4—21no—
226C22×C4—3461no—
227C22×C4—1732no—
228C22×C4C22×D17311yescharacteristic
229SD16—24no—
230SD16—3464no—
231SD16—3464no—
232SD16—3464no—
233D8—24no—
234D8—3464no—
235D8—3464no—
236D8—3464no—
237M16—1734no—
238M16C22⋊D17314yescomplemented
239C2×C8—1732no—
240C2×C8C22×D17312yes—
241C4⋊C4—3462no—
242C22⋊C4—3462no—
243C22⋊C4—3462no—
244C42—1732no—
245C42—1731no—
246C42—1731no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3461no—
252C23—3462no—
253C23—6924no—
254C23—6922no—
255C23D4×D17311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3461no—
259Q8—3461no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3461no—
267D4—3462no—
268D4—3461no—
269D4—3462no—
270D4—6928no—
271D4—6924no—
272D4—6928no—
273D4—6922no—
274D4—6922no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3464no—
278C2×C4—3461no—
279C2×C4—1732no—
280C2×C4—1732no—
281C2×C4—6922no—
282C2×C4D4×D17311yescharacteristic
283C2×C4D4×D17312yes—
284C2×C4C2×C22⋊D17312yes—
285C2×C4C23×D17311yescharacteristic
286C8—1734no—
287C8C2×C22⋊D17314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—13844no—
295C22—6924no—
296C22—6922no—
297C22—6921no—
298C22—6922no—
299C22#1914967512yes—
300C22C2×D4×D17311yescharacteristic
301C4—41no—
302C4—3462no—
303C4—1734no—
304C4—6921no—
305C4#1914967512yes—
306C4C2×D4×D17311yescharacteristic
307C4C22×C22⋊D17311yescharacteristic
308C2—22no—
309C2—81no—
310C2—82no—
311C2—13842no—
312C2—13841no—
313C2C2×#1914967511yescharacteristic
314C2AGL(1,ℤ/8)×D17312yes—
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#3151
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×#19149707C2semidirect
2. C2×#19149705C2semidirect
3. C2×#19149704C2semidirect
4. #26012037C2semidirect
5. C2×#19149675C2semidirect
6. #26011879C2semidirect
7. #26011877C2semidirect
8. #26011866C2semidirect
9. #26011862C2semidirect
10. M16×Dic173C2semidirect
11. C2×AGL(1,ℤ/8)×C173C2semidirect
18. C2×#14583815C22semidirect
19. C2×#14583814C22semidirect
20. C2×#14583813C22semidirect
21. C2×D1384C22semidirect
22. C2×#14583811C22semidirect
23. C2×#14583811C22semidirect
25. #19149677C22semidirect
29. #19149675C22semidirect
30. #19149668C22semidirect
31. #19149666C22semidirect
35. #19149589C22semidirect
36. C8×Dic173C22semidirect
40. C2×SD16×C173C22semidirect
41. C2×D8×C173C22semidirect
65. #14583810D4semidirect
87. M16×C173D4semidirect
109. #13063436C2×D4semidirect
129. C8×C173C2×D4semidirect
149. Dic173AGL(1,ℤ/8)semidirect
157. C173#3151semidirect
162. C2×AGL(1,ℤ/8)D173semidirect
182. C2×SD16C2×D173semidirect
186. C2×D8C2×D173semidirect
238. M16C22⋊D173semidirect
287. C8C2×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C2×#191497052
C2×#191497042
#260120372
C2×#191496752
#260118792
#260118772
#260118662
#260118622
M16×Dic1732
C2×AGL(1,ℤ/8)×C1732
#3151173