Finite Group Databasebeta

← #36363180

Subgroups — #36363180

Order 22144.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19149705C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3#26012003C211yesmaximal, characteristic, complemented
4C2×#19149650C211yesmaximal, characteristic, complemented
5#26011743C212yesmaximal, complemented
6#26011736C212yesmaximal, complemented
7#26011726C212yesmaximal, complemented
8#26011719C212yesmaximal, complemented
9#26011534C211yesmaximal, characteristic, complemented
10#26011493C211yesmaximal, characteristic, complemented
11#418×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×C22⋊D173—21no—
14C22×C4×D173C2211yescharacteristic
15#19149705—24no—
16C2×#14583814C2212yescomplemented
17C2×#14583811C2212yescomplemented
18C2×#14583810C2211yescharacteristic
19C2×#14583809C2212yescomplemented
20M16×D173—24no—
21C2×C8×D173C2212yescomplemented
22#19149663—21no—
23#19149659C2211yescharacteristic, complemented
24#19149659C2211yescharacteristic, complemented
25#19149654C2211yescharacteristic, complemented
26#19149654C2211yescharacteristic, complemented
27#19149650—21no—
28#19149650—21no—
29#19149650C2212yes—
30#19149650C2212yes—
31#19149649—21no—
32C4×D692—21no—
33C4×D692—21no—
34C2×C4⋊Dic173C2211yescharacteristic
35C2×C22⋊Dic173—21no—
36#19149612C2212yescomplemented
37#19149610C2212yescomplemented
38#19149604C2212yescomplemented
39#19149601C2212yescomplemented
40C22×D4×C173C2211yescharacteristic
41C2×M16×C173C2211yescharacteristic
42#117×C173C2211yescharacteristic
43#102×C173C2212yescomplemented
44#102×C173C2212yescomplemented
45C23×D173D411yescharacteristic
46#14583826—22no—
47#14583826—22no—
48C2×Dic346C2311yescharacteristic
49C2×D692—21no—
50C2×D692—21no—
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54C2×C22⋊D173—21no—
55C2×C22⋊D173—44no—
56C22×Dic173—21no—
57C22×Dic173D411yescharacteristic
58C2×C4×D173—22no—
59C2×C4×D173—21no—
60C2×C4×D173—22no—
61C2×C4×D173D411yescharacteristic
62C2×C4×D173D411yescharacteristic
63C2×C4×D173C2311yescharacteristic
64C2×C4×D173C2311yescharacteristic
65#14583814—24no—
66#14583811—24no—
67#14583810D414yescomplemented
68#14583809—24no—
69C2×#13063436C2312yes—
70C8×D173—24no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173C2312yes—
75C22⋊Dic173—22no—
76C22⋊Dic173—22no—
77#14583803—21no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81C24×C173—21no—
82C2×D4×C173—22no—
83C2×D4×C173—21no—
84C2×D4×C173—22no—
85C2×D4×C173C2312yes—
86C22×C4×C173C2311yescharacteristic
87M16×C173—22no—
88C2×C8×C173C2312yes—
89C4⋊C4×C173C2311yescharacteristic
90C4⋊C4×C173C2311yescharacteristic
91C22⋊C4×C173—21no—
92C42×C173—21no—
93C22×D173—22no—
94C22×D173—22no—
95C22×D173—21no—
96C22×D173—41no—
97C22×D173C2×D411yescharacteristic
98C22×D173C2×D411yescharacteristic
99Dic346—21no—
100Dic346—21no—
101D692—21no—
102D692—21no—
103D692—42no—
104D692—42no—
105C22⋊D173—44no—
106C22⋊D173—42no—
107C22⋊D173—44no—
108C2×Dic173—22no—
109C2×Dic173—21no—
110C2×Dic173—22no—
111C2×Dic173—42no—
112C2×Dic173C2×D411yescharacteristic
113C2×Dic173C2×D411yescharacteristic
114C4×D173—22no—
115C4×D173—22no—
116C4×D173—21no—
117C4×D173—21no—
118C4×D173—21no—
119C4×D173—21no—
120C4×D173—42no—
121#13063436C2×D414yescomplemented
122C23×C173—22no—
123C23×C173—42no—
124C23×C173—42no—
125C23×C173—42no—
126C23×C173C2×D411yescharacteristic
127D4×C173—22no—
128D4×C173—22no—
129D4×C173—42no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173C2×D411yescharacteristic
133C2×C4×C173C2×D412yes—
134C2×C4×C173#4112yes—
135C2×C4×C173C2411yescharacteristic
136C8×C173—22no—
137C22×C173—21no—
138C22×C173—21no—
139C22×C173—42no—
140C22×C173—42no—
141C22×C173—42no—
142C22×C173—84no—
143C22×C173#12112yes—
144C22×C173C22×D411yescharacteristic
145C2×D173—21no—
146C2×D173—21no—
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—44no—
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173AGL(1,ℤ/8)12yescomplemented
155C4×C173—41no—
156C4×C173—41no—
157C4×C173#12112yes—
158C4×C173C22×D411yescharacteristic
159C4×C173C2×#4111yescharacteristic
160Dic173—21no—
161Dic173—21no—
162Dic173—41no—
163Dic173—42no—
164C2×C173—22no—
165C2×C173—82no—
166C2×C173C2×#12111yescharacteristic
167C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C173#57511yescharacteristic
169D173—22no—
170D173—41no—
171D173—81no—
172C173#305511yescharacteristic, complemented, Sylow
173#3055—1731nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1731no—
175C22×M16—1731no—
176#525—1731no—
177C2×#121—1731no—
178#469—1732no—
179#468—1732no—
180#466—1732no—
181#465—1732no—
182#428—1731no—
183#418D17311yescharacteristic, complemented
184#417—1731no—
185C2×#41—1731no—
186C22×D4—3461no—
187C22×D4C2×D17311yescharacteristic
188C23×C4—1731no—
189AGL(1,ℤ/8)—3464no—
190C2×SD16—1732no—
191C2×D8—1732no—
192C2×M16—3464no—
193C2×M16—1731no—
194C2×M16—1732no—
195C2×M16C2×D17311yescharacteristic
196C22×C8—1732no—
197#123—3461no—
198C22⋊Q8—1731no—
199C22⋊Q8—1731no—
200#121—3461no—
201#121—3461no—
202#121—1732no—
203#121—1732no—
204#121—1731no—
205#121—1731no—
206#120—3461no—
207C4×D4—3461no—
208C4×D4—3461no—
209#117C2×D17311yescharacteristic
210C2×C4⋊C4—1731no—
211C2×C22⋊C4—3461no—
212#107—1732no—
213#106—1732no—
214#103—1732no—
215#102—1732no—
216#102C2×D17312yescomplemented
217#102C2×D17312yescomplemented
218C24—21no—
219C24—1731no—
220#41—3462no—
221#41—3462no—
222C2×Q8—1731no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3461no—
227C2×D4—3461no—
228C2×D4—3462no—
229C2×D4—3462no—
230C2×D4—3461no—
231C2×D4—1731no—
232C2×D4—6924no—
233C2×D4C22×D17312yes—
234C22×C4—3462no—
235C22×C4—3462no—
236C22×C4—3461no—
237C22×C4—3461no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4C22×D17311yescharacteristic
244SD16—3464no—
245D8—3464no—
246M16—22no—
247M16—3464no—
248M16—1734no—
249C2×C8—3464no—
250C2×C8—1732no—
251C2×C8C22×D17312yes—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—1732no—
256C4⋊C4C22×D17311yescharacteristic
257C4⋊C4C22×D17311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3461no—
260C22⋊C4—3461no—
261C22⋊C4—3461no—
262C22⋊C4—3461no—
263C22⋊C4—3462no—
264C22⋊C4—3462no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3462no—
271C23—3462no—
272C23—3461no—
273C23—1731no—
274C23—1731no—
275C23—6921no—
276C23D4×D17311yescharacteristic
277Q8—3461no—
278Q8—3461no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3461no—
283D4—3461no—
284D4—6922no—
285D4—6922no—
286D4—6924no—
287D4—6924no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3462no—
292C2×C4—3462no—
293C2×C4—3462no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3462no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—6922no—
303C2×C4—6922no—
304C2×C4D4×D17311yescharacteristic
305C2×C4D4×D17312yes—
306C2×C4#1458382612yes—
307C2×C4C23×D17311yescharacteristic
308C8—22no—
309C8—1734no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3461no—
317C22—3461no—
318C22—3461no—
319C22—3461no—
320C22—3461no—
321C22—1732no—
322C22—6924no—
323C22—6921no—
324C22—6921no—
325C22—6921no—
326C22#1914965212yes—
327C22C2×D4×D17311yescharacteristic
328C4—41no—
329C4—41no—
330C4—3461no—
331C4—3461no—
332C4—6921no—
333C4—6922no—
334C4#1914965212yes—
335C4C2×D4×D17311yescharacteristic
336C4C2×#1458382611yescharacteristic
337C2—22no—
338C2—82no—
339C2—3462no—
340C2—13841no—
341C2—6921no—
342C2C2×#1914965211yescharacteristic
343C2AGL(1,ℤ/8)×D17311yescharacteristic
344C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#418×C173C2
socleC22×C173C22×D4
hypercenter#418D173
hypocenterC173#3055
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×M16×D173C2semidirect
3. #26012003C2semidirect
4. C2×#19149650C2semidirect
5. #26011743C2semidirect
6. #26011736C2semidirect
7. #26011726C2semidirect
8. #26011719C2semidirect
9. #26011534C2semidirect
10. #26011493C2semidirect
11. #418×C173C2semidirect
16. C2×#14583814C22semidirect
17. C2×#14583811C22semidirect
19. C2×#14583809C22semidirect
21. C2×C8×D173C22semidirect
23. #19149659C22semidirect
24. #19149659C22semidirect
25. #19149654C22semidirect
26. #19149654C22semidirect
36. #19149612C22semidirect
37. #19149610C22semidirect
38. #19149604C22semidirect
39. #19149601C22semidirect
43. #102×C173C22semidirect
44. #102×C173C22semidirect
67. #14583810D4semidirect
121. #13063436C2×D4semidirect
154. C2×D173AGL(1,ℤ/8)semidirect
172. C173#3055semidirect
183. #418D173semidirect
216. #102C2×D173semidirect
217. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×M16×D1732
#260120032
C2×#191496502
#260117432
#260117362
#260117262
#260117192
#260115342
#260114932
#418×C1732
#3055173