Finite Group Databasebeta

← #36364325

Subgroups — #36364325

Order 22144.

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

normal subgroups only

This group has 472 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
1#26012001C211yesmaximal, characteristic, complemented
2C2×#19149650C211yesmaximal, characteristic, complemented
3#26011827C211yesmaximal, characteristic, complemented
4#26011813C212yesmaximal, complemented
5#26011807C211yesmaximal, characteristic, complemented
6#26011724C212yesmaximal, complemented
7#26011719C212yesmaximal, complemented
8#26011642C211yesmaximal, characteristic, complemented
9#26011633C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#447×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic, complemented
13C22×C22⋊D173—21no—
14C22×C4×D173C2211yescharacteristic
15C2×#14583814C2212yescomplemented
16C2×C8×D173C2212yescomplemented
17#19149661—21no—
18#19149659C2211yescharacteristic, complemented
19#19149658C2211yescharacteristic
20#19149650—21no—
21#19149650—21no—
22#19149650C2212yes—
23#19149650C2212yes—
24#19149650C2211yescharacteristic, complemented
25#19149647—21no—
26D4×Dic173—21no—
27#19149639—21no—
28#19149637C2211yescharacteristic, complemented
29C2×C4⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173—21no—
31#19149613C2212yescomplemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149600C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yescomplemented
38C22×D4×C173C2211yescharacteristic
39C2×D8×C173C2212yescomplemented
40#121×C173C2211yescharacteristic
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583825—42no—
45C2×Dic346C2311yescharacteristic
46C2×C22⋊D173—21no—
47C2×C22⋊D173—21no—
48C2×C22⋊D173—22no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—44no—
51C22×Dic173—21no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—42no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59#14583814—42no—
60C2×#13063436C2312yes—
61C8×D173—42no—
62C4⋊Dic173—21no—
63C4⋊Dic173—21no—
64C4⋊Dic173C2311yescharacteristic
65C4⋊Dic173C2312yes—
66#14583805—21no—
67C22⋊Dic173—22no—
68C22⋊Dic173—22no—
69C22⋊Dic173—21no—
70C22⋊Dic173—21no—
71#14583803—21no—
72#14583803—21no—
73C4×Dic173—21no—
74C24×C173—21no—
75C2×D4×C173—21no—
76C2×D4×C173—21no—
77C2×D4×C173—42no—
78C2×D4×C173C2311yescharacteristic
79C2×D4×C173C2312yes—
80C22×C4×C173C2311yescharacteristic
81D8×C173—42no—
82C2×C8×C173C2312yes—
83C4⋊C4×C173C2311yescharacteristic
84C22⋊C4×C173—21no—
85C22×D173—22no—
86C22×D173—22no—
87C22×D173—41no—
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90Dic346—22no—
91C22⋊D173—44no—
92C22⋊D173—42no—
93C22⋊D173—42no—
94C22⋊D173—44no—
95C2×Dic173—22no—
96C2×Dic173—21no—
97C2×Dic173—21no—
98C2×Dic173—41no—
99C2×Dic173—42no—
100C2×Dic173—42no—
101C2×Dic173C2×D411yescharacteristic
102C2×Dic173C2×D411yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—41no—
106C4×D173—41no—
107#13063436—22no—
108C23×C173—21no—
109C23×C173—22no—
110C23×C173—42no—
111C23×C173—42no—
112C23×C173—42no—
113C23×C173C2×D411yescharacteristic
114D4×C173—22no—
115D4×C173—22no—
116D4×C173—22no—
117D4×C173—42no—
118D4×C173—42no—
119C2×C4×C173—22no—
120C2×C4×C173—21no—
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2411yescharacteristic
123C8×C173—22no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—42no—
127C22×C173—41no—
128C22×C173—41no—
129C22×C173—42no—
130C22×C173—42no—
131C22×C173—41no—
132C22×C173—84no—
133C22×C173#13512yes—
134C22×C173C22×D411yescharacteristic
135C2×D173—21no—
136C2×D173—21no—
137C2×D173—21no—
138C2×D173—21no—
139C2×D173—21no—
140C2×D173—44no—
141C2×D173AGL(1,ℤ/8)12yes—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173H(5,𝔽2)12yes—
145Dic173—21no—
146Dic173—21no—
147Dic173—41no—
148Dic173—42no—
149Dic173—41no—
150C2×C173—22no—
151C2×C173—81no—
152C2×C173—82no—
153C2×C173#53411yescharacteristic
154C2×C173C2×#13511yescharacteristic
155C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
156D173—22no—
157D173—41no—
158C173#318811yescharacteristic, complemented, Sylow
159#3188—1731nomaximal, Sylow
160#525—1731no—
161C2×#121—1731no—
162#484—1731no—
163#482—1732no—
164#481—1731no—
165#466—1732no—
166#465—1732no—
167#450—1731no—
168#449—1732no—
169#447D17311yescharacteristic, complemented
170C2×C22⋊C8—1731no—
171C2×#41—1731no—
172C22×D4—3461no—
173C22×D4C2×D17311yescharacteristic
174C23×C4—1731no—
175C2×SD16—1732no—
176C2×D8C2×D17312yescomplemented
177C22×C8—1732no—
178#123—3461no—
179C22⋊Q8—1731no—
180C22⋊Q8—1731no—
181#121—3461no—
182#121—3461no—
183#121—1732no—
184#121—1732no—
185#121—1731no—
186#121C2×D17311yescharacteristic
187#120—3461no—
188C4×D4—3461no—
189C4×D4—3461no—
190#117—1731no—
191C2×C4⋊C4—1731no—
192C2×C22⋊C4—3461no—
193#107—1732no—
194#106—1732no—
195#103—1732no—
196#102—1732no—
197#102—1732no—
198#102C2×D17312yescomplemented
199C22⋊C8—1731no—
200C22⋊C8—1732no—
201C22⋊C8C2×D17311yescharacteristic
202C24—21no—
203C24—1731no—
204#41—6922no—
205C2×Q8—1731no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—3461no—
210C2×D4—3461no—
211C2×D4—3462no—
212C2×D4—3462no—
213C2×D4—6924no—
214C2×D4C22×D17311yescharacteristic
215C2×D4C22×D17312yes—
216C22×C4—3461no—
217C22×C4—3461no—
218C22×C4—1731no—
219C22×C4—1731no—
220C22×C4—1731no—
221C22×C4—1731no—
222C22×C4—1731no—
223C22×C4—6922no—
224C22×C4C22×D17311yescharacteristic
225SD16—6922no—
226D8—42no—
227C2×C8—1732no—
228C2×C8—6922no—
229C2×C8C22×D17312yes—
230C4⋊C4—3461no—
231C4⋊C4—3461no—
232C4⋊C4—3461no—
233C4⋊C4—1731no—
234C4⋊C4—1732no—
235C4⋊C4C22×D17311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3461no—
238C22⋊C4—3461no—
239C22⋊C4—3461no—
240C22⋊C4—3462no—
241C22⋊C4—3462no—
242C22⋊C4—3461no—
243C42—3461no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—3462no—
250C23—3462no—
251C23—1731no—
252C23—1731no—
253C23—6921no—
254C23D4×D17311yescharacteristic
255Q8—3462no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—6924no—
262D4—6922no—
263D4—6922no—
264D4—6924no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3462no—
268C2×C4—3462no—
269C2×C4—3461no—
270C2×C4—3461no—
271C2×C4—3462no—
272C2×C4—1731no—
273C2×C4—1731no—
274C2×C4—6921no—
275C2×C4—6921no—
276C2×C4—6921no—
277C2×C4—6922no—
278C2×C4—6922no—
279C2×C4D4×D17311yescharacteristic
280C2×C4C23×D17311yescharacteristic
281C8—22no—
282C8—3462no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—84no—
292C22—3461no—
293C22—3461no—
294C22—3461no—
295C22—3461no—
296C22—3461no—
297C22—1732no—
298C22—6924no—
299C22#1914969812yes—
300C22C2×D4×D17311yescharacteristic
301C4—21no—
302C4—41no—
303C4—3461no—
304C4—3461no—
305C4—6922no—
306C4—6921no—
307C4—6921no—
308C4#1914972412yes—
309C2—22no—
310C2—81no—
311C2—82no—
312C2—3462no—
313C2—6921no—
314C2#2601205111yescharacteristic
315C2C2×#1914969811yescharacteristic
316C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3188
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012001C2semidirect
2. C2×#19149650C2semidirect
3. #26011827C2semidirect
4. #26011813C2semidirect
5. #26011807C2semidirect
6. #26011724C2semidirect
7. #26011719C2semidirect
8. #26011642C2semidirect
9. #26011633C2semidirect
10. C22⋊C8×D173C2semidirect
11. #447×C173C2semidirect
12. C2×#14583825C22semidirect
15. C2×#14583814C22semidirect
16. C2×C8×D173C22semidirect
18. #19149659C22semidirect
24. #19149650C22semidirect
28. #19149637C22semidirect
31. #19149613C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149600C22semidirect
35. #19149600C22semidirect
37. #19149591C22semidirect
39. C2×D8×C173C22semidirect
41. #102×C173C22semidirect
158. C173#3188semidirect
169. #447D173semidirect
176. C2×D8C2×D173semidirect
198. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
C2×#191496502
#260118272
#260118132
#260118072
#260117242
#260117192
#260116422
#260116332
C22⋊C8×D1732
#447×C1732
#3188173