Finite Group Databasebeta

← #36364312

Subgroups — #36364312

Order 22144.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#19149650C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic, complemented
3#26011805C211yesmaximal, characteristic, complemented
4#26011803C212yesmaximal, complemented
5#26011802C211yesmaximal, characteristic, complemented
6#26011726C212yesmaximal, complemented
7#26011724C212yesmaximal, complemented
8#26011625C211yesmaximal, characteristic, complemented
9#26011623C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#447×C173C211yesmaximal, characteristic, complemented
12C22×C22⋊D173—21no—
13C2×D4×D173—21no—
14C2×D4×D173C2211yescharacteristic, complemented
15C22×C4×D173C2211yescharacteristic
16C2×#14583811C2212yescomplemented
17C2×C8×D173C2212yescomplemented
18#19149654C2211yescharacteristic, complemented
19#19149653C2211yescharacteristic
20#19149652—21no—
21#19149651—21no—
22#19149650—21no—
23#19149650—21no—
24#19149650C2212yes—
25#19149650C2211yescharacteristic, complemented
26#19149650C2212yescomplemented
27C2×C4⋊Dic173C2211yescharacteristic, complemented
28C4⋊C4×D173C2211yescharacteristic, complemented
29C2×C22⋊Dic173—21no—
30C22⋊C4×D173—21no—
31#19149613C2212yescomplemented
32#19149612C2212yescomplemented
33#19149601C2212yescomplemented
34#19149600C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic, complemented
37#19149591C2212yescomplemented
38C22×D4×C173C2211yescharacteristic, complemented
39C2×D8×C173C2212yescomplemented
40#121×C173C2211yescharacteristic
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic, complemented
43C23×D173—21no—
44C23×D173D411yescharacteristic
45C2×D692—21no—
46C2×D692C2311yescharacteristic
47C2×C22⋊D173—22no—
48C2×C22⋊D173—22no—
49C2×C22⋊D173—21no—
50C2×C22⋊D173—21no—
51C2×C22⋊D173—21no—
52C2×C22⋊D173—44no—
53D4×D173—42no—
54D4×D173—44no—
55C22×Dic173—21no—
56C22×Dic173D411yescharacteristic
57C2×C4×D173—21no—
58C2×C4×D173—42no—
59C2×C4×D173D411yescharacteristic
60C2×C4×D173D411yescharacteristic
61C2×C4×D173C2311yescharacteristic
62C2×C4×D173C2311yescharacteristic
63#14583811—42no—
64C2×#13063436C2312yes—
65C8×D173—42no—
66C4⋊Dic173—21no—
67C4⋊Dic173C2311yescharacteristic
68C4⋊Dic173C2312yes—
69#14583805—21no—
70C22⋊Dic173—21no—
71C22⋊Dic173—22no—
72C22⋊Dic173—22no—
73#14583803—21no—
74#14583803—21no—
75C24×C173—21no—
76C2×D4×C173—21no—
77C2×D4×C173—21no—
78C2×D4×C173—42no—
79C2×D4×C173C2311yescharacteristic
80C2×D4×C173C2312yes—
81C22×C4×C173C2311yescharacteristic
82D8×C173—42no—
83C2×C8×C173C2312yes—
84C4⋊C4×C173C2311yescharacteristic
85C22⋊C4×C173—21no—
86C22×D173—21no—
87C22×D173—41no—
88C22×D173—41no—
89C22×D173—42no—
90C22×D173—42no—
91C22×D173—42no—
92C22×D173D814yescomplemented
93C22×D173C2×D411yescharacteristic
94C22×D173C2×D411yescharacteristic
95D692—22no—
96D692—42no—
97C22⋊D173—42no—
98C22⋊D173—42no—
99C22⋊D173—42no—
100C22⋊D173—44no—
101C22⋊D173—44no—
102C2×Dic173—21no—
103C2×Dic173—22no—
104C2×Dic173—41no—
105C2×Dic173—42no—
106C2×Dic173C2×D411yescharacteristic
107C2×Dic173C2×D411yescharacteristic
108C4×D173—22no—
109C4×D173—22no—
110C4×D173—41no—
111C4×D173—41no—
112C4×D173—42no—
113#13063436—22no—
114C23×C173—21no—
115C23×C173—22no—
116C23×C173—42no—
117C23×C173—42no—
118C23×C173—42no—
119C23×C173C2×D411yescharacteristic
120D4×C173—22no—
121D4×C173—22no—
122D4×C173—22no—
123D4×C173—42no—
124D4×C173—42no—
125C2×C4×C173—22no—
126C2×C4×C173—21no—
127C2×C4×C173C2×D411yescharacteristic
128C2×C4×C173C2411yescharacteristic
129C8×C173—22no—
130C22×C173—21no—
131C22×C173—21no—
132C22×C173—42no—
133C22×C173—41no—
134C22×C173—41no—
135C22×C173—42no—
136C22×C173—42no—
137C22×C173—41no—
138C22×C173—84no—
139C22×C173C2×D812yes—
140C22×C173C22×D411yescharacteristic
141C2×D173—21no—
142C2×D173—21no—
143C2×D173—21no—
144C2×D173—21no—
145C2×D173—44no—
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C2×D173—84no—
150C2×D173C2×D812yes—
151C2×D173C2×D812yes—
152C4×C173—21no—
153C4×C173—41no—
154C4×C173H(5,𝔽2)12yes—
155Dic173—21no—
156Dic173—21no—
157Dic173—41no—
158Dic173—42no—
159C2×C173—22no—
160C2×C173—81no—
161C2×C173—82no—
162C2×C173#53411yescharacteristic
163C2×C173C22×D811yescharacteristic
164C2×C173#57511yescharacteristic
165D173—22no—
166D173—22no—
167D173—81no—
168C173#318611yescharacteristic, complemented, Sylow
169#3186—1731nomaximal, Sylow
170C2×#121—1731no—
171C2×#121—1731no—
172#480—1731no—
173#480—1732no—
174#480—1731no—
175#466—1732no—
176#466—1732no—
177#447—1731no—
178#447—1732no—
179#447D17311yescharacteristic, complemented
180C2×C22⋊C8—1731no—
181C22×D4—3461no—
182C22×D4—3461no—
183C22×D4—1731no—
184C22×D4C2×D17311yescharacteristic, complemented
185C23×C4—1731no—
186C2×D8—1732no—
187C2×D8C2×D17312yescomplemented
188C22×C8—1732no—
189#121—3461no—
190#121—3461no—
191#121—3461no—
192#121—3461no—
193#121—1732no—
194#121—1731no—
195#121—1732no—
196#121—1731no—
197#121—1731no—
198#121C2×D17311yescharacteristic
199C2×C4⋊C4—1731no—
200C2×C4⋊C4—1731no—
201C2×C22⋊C4—3461no—
202C2×C22⋊C4—3461no—
203#107—1732no—
204#107—1732no—
205#102—1732no—
206#102—1732no—
207#102—1732no—
208#102C2×D17312yescomplemented
209C22⋊C8—1731no—
210C22⋊C8—1732no—
211C22⋊C8C2×D17311yescharacteristic, complemented
212C24—21no—
213C24—3461no—
214C24—1731no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—42no—
218C2×D4—3461no—
219C2×D4—3461no—
220C2×D4—3462no—
221C2×D4—3462no—
222C2×D4—3461no—
223C2×D4—3461no—
224C2×D4—1731no—
225C2×D4—6922no—
226C2×D4—6924no—
227C2×D4—6924no—
228C2×D4C22×D17311yescharacteristic
229C2×D4C22×D17312yes—
230C22×C4—3461no—
231C22×C4—3461no—
232C22×C4—1731no—
233C22×C4—1731no—
234C22×C4—1731no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4—6922no—
238C22×C4C22×D17311yescharacteristic
239D8—42no—
240D8—6922no—
241C2×C8—1732no—
242C2×C8—6922no—
243C2×C8C22×D17312yes—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—1731no—
247C4⋊C4—1732no—
248C4⋊C4C22×D17311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—3461no—
251C22⋊C4—3461no—
252C22⋊C4—3462no—
253C22⋊C4—3462no—
254C22⋊C4—3461no—
255C23—21no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—3461no—
261C23—1731no—
262C23—1731no—
263C23—1734no—
264C23—6921no—
265C23—6921no—
266C23—6922no—
267C23—6922no—
268C23—6922no—
269C23D4×D17311yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—3462no—
276D4—6922no—
277D4—6922no—
278D4—6922no—
279D4—6924no—
280D4—6924no—
281D4—6922no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—3462no—
285C2×C4—3462no—
286C2×C4—3461no—
287C2×C4—3462no—
288C2×C4—1731no—
289C2×C4—1731no—
290C2×C4—6921no—
291C2×C4—6921no—
292C2×C4—6922no—
293C2×C4—6921no—
294C2×C4—6922no—
295C2×C4D4×D17311yescharacteristic
296C2×C4C23×D17311yescharacteristic
297C8—22no—
298C8—3462no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—84no—
308C22—3461no—
309C22—3461no—
310C22—3461no—
311C22—3461no—
312C22—13844no—
313C22—1732no—
314C22—1732no—
315C22—6924no—
316C22—6921no—
317C22—6921no—
318C22—6921no—
319C22D8×D17312yes—
320C22C2×D4×D17311yescharacteristic
321C4—21no—
322C4—41no—
323C4—3461no—
324C4—3461no—
325C4—6921no—
326C4—6922no—
327C4#1914972412yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—3462no—
332C2—3462no—
333C2—13841no—
334C2#2601205111yescharacteristic
335C2C2×D8×D17311yescharacteristic
336C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3186
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. C2×#19149650C2semidirect
2. #121×D173C2semidirect
3. #26011805C2semidirect
4. #26011803C2semidirect
5. #26011802C2semidirect
6. #26011726C2semidirect
7. #26011724C2semidirect
8. #26011625C2semidirect
9. #26011623C2semidirect
10. C22⋊C8×D173C2semidirect
11. #447×C173C2semidirect
14. C2×D4×D173C22semidirect
16. C2×#14583811C22semidirect
17. C2×C8×D173C22semidirect
18. #19149654C22semidirect
25. #19149650C22semidirect
26. #19149650C22semidirect
27. C2×C4⋊Dic173C22semidirect
28. C4⋊C4×D173C22semidirect
31. #19149613C22semidirect
32. #19149612C22semidirect
33. #19149601C22semidirect
34. #19149600C22semidirect
35. #19149600C22semidirect
36. C22⋊#13063436C22semidirect
37. #19149591C22semidirect
38. C22×D4×C173C22semidirect
39. C2×D8×C173C22semidirect
41. #102×C173C22semidirect
42. C22⋊C8×C173C22semidirect
92. C22×D173D8semidirect
168. C173#3186semidirect
179. #447D173semidirect
184. C22×D4C2×D173semidirect
187. C2×D8C2×D173semidirect
208. #102C2×D173semidirect
211. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496502
#121×D1732
#260118052
#260118032
#260118022
#260117262
#260117242
#260116252
#260116232
C22⋊C8×D1732
#447×C1732
#3186173