Finite Group Databasebeta

← #36364313

Subgroups — #36364313

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×#19149654C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic, complemented
3#26011806C212yesmaximal, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011728C212yesmaximal, complemented
6#26011726C212yesmaximal, complemented
7#26011627C211yesmaximal, characteristic, complemented
8#26011626C212yesmaximal, complemented
9#26011624C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#480×C173C211yesmaximal, characteristic, complemented
12C22×D692—21no—
13C22×D692C2211yescharacteristic, complemented
14C2×D4×D173—21no—
15C2×D4×D173C2211yescharacteristic, complemented
16C22×C4×D173C2211yescharacteristic
17C2×D1384C2212yescomplemented
18C2×#14583811C2212yescomplemented
19C2×C8×D173C2212yescomplemented
20#19149654—21no—
21#19149654—21no—
22#19149654C2212yes—
23#19149654C2211yescharacteristic, complemented
24#19149654C2212yescomplemented
25#19149653C2211yescharacteristic
26#19149652—21no—
27#19149651—21no—
28#19149650C2211yescharacteristic, complemented
29C4⋊C4×D173C2211yescharacteristic, complemented
30C2×#14583803—21no—
31C22⋊C4×D173—21no—
32#19149612C2212yescomplemented
33#19149601C2212yescomplemented
34#19149599C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic, complemented
37#19149591C2212yescomplemented
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic, complemented
40#107×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic, complemented
43C23×D173—21no—
44C23×D173—21no—
45C23×D173D411yescharacteristic
46C2×D692—21no—
47C2×D692—22no—
48C2×D692—22no—
49C2×D692—21no—
50C2×D692—42no—
51C2×D692—44no—
52C2×D692C2311yescharacteristic
53C2×D692C2312yes—
54C2×C22⋊D173—21no—
55C2×C22⋊D173—21no—
56C2×C22⋊D173—21no—
57D4×D173—42no—
58D4×D173—44no—
59C22×Dic173D411yescharacteristic
60C2×C4×D173—21no—
61C2×C4×D173—42no—
62C2×C4×D173D411yescharacteristic
63C2×C4×D173D411yescharacteristic
64C2×C4×D173C2311yescharacteristic
65C2×C4×D173C2311yescharacteristic
66D1384—42no—
67#14583811—42no—
68C2×#13063436C2312yes—
69C8×D173—42no—
70C4⋊Dic173C2311yescharacteristic
71#14583805—21no—
72C22⋊Dic173—21no—
73#14583803—21no—
74#14583803—22no—
75#14583803—22no—
76#14583803—21no—
77C2×D4×C173—21no—
78C2×D4×C173C2311yescharacteristic
79C22×C4×C173—21no—
80C22×C4×C173C2311yescharacteristic
81C2×C8×C173C2312yes—
82C4⋊C4×C173—21no—
83C4⋊C4×C173C2311yescharacteristic
84C4⋊C4×C173C2312yes—
85C22⋊C4×C173—21no—
86C22×D173—21no—
87C22×D173—22no—
88C22×D173—41no—
89C22×D173—41no—
90C22×D173—42no—
91C22×D173—42no—
92C22×D173—42no—
93C22×D173—42no—
94C22×D173—42no—
95C22×D173—42no—
96C22×D173D814yescomplemented
97C22×D173C2×D411yescharacteristic
98C22×D173C2×D411yescharacteristic
99D692—22no—
100D692—22no—
101D692—22no—
102D692—42no—
103D692—42no—
104D692—44no—
105D692—44no—
106C22⋊D173—42no—
107C22⋊D173—42no—
108C22⋊D173—42no—
109C2×Dic173—21no—
110C2×Dic173—41no—
111C2×Dic173C2×D411yescharacteristic
112C2×Dic173C2×D411yescharacteristic
113C4×D173—22no—
114C4×D173—22no—
115C4×D173—41no—
116C4×D173—41no—
117C4×D173—42no—
118#13063436—22no—
119C23×C173—21no—
120C23×C173C2×D411yescharacteristic
121D4×C173—22no—
122D4×C173—42no—
123C2×C4×C173—22no—
124C2×C4×C173—21no—
125C2×C4×C173—22no—
126C2×C4×C173—42no—
127C2×C4×C173C2×D411yescharacteristic
128C2×C4×C173C2411yescharacteristic
129C8×C173—22no—
130C22×C173—21no—
131C22×C173—21no—
132C22×C173—41no—
133C22×C173—41no—
134C22×C173—41no—
135C22×C173C2×D812yes—
136C22×C173C22×D411yescharacteristic
137C2×D173—21no—
138C2×D173—21no—
139C2×D173—21no—
140C2×D173—21no—
141C2×D173—44no—
142C2×D173—41no—
143C2×D173—42no—
144C2×D173—41no—
145C2×D173—42no—
146C2×D173—41no—
147C2×D173—42no—
148C2×D173—84no—
149C2×D173—84no—
150C2×D173C2×D812yes—
151C2×D173C2×D812yes—
152C4×C173—21no—
153C4×C173—42no—
154C4×C173—41no—
155C4×C173H(5,𝔽2)12yes—
156Dic173—21no—
157Dic173—21no—
158Dic173—41no—
159C2×C173—22no—
160C2×C173—81no—
161C2×C173#53411yescharacteristic
162C2×C173C22×D811yescharacteristic
163C2×C173#57511yescharacteristic
164D173—22no—
165D173—22no—
166D173—81no—
167D173—82no—
168C173#318611yescharacteristic, complemented, Sylow
169#3186—1731nomaximal, Sylow
170C2×#121—1731no—
171C2×#121—1731no—
172#480—1731no—
173#480—1732no—
174#480D17311yescharacteristic, complemented
175#466—1732no—
176#466—1732no—
177#447—1731no—
178#447—1732no—
179#447—1731no—
180C2×C22⋊C8—1731no—
181C22×D4—3461no—
182C22×D4—3461no—
183C22×D4—1731no—
184C22×D4—1731no—
185C23×C4—1731no—
186C2×D8—1732no—
187C2×D8—1732no—
188C22×C8—1732no—
189#121—3461no—
190#121—3461no—
191#121—3461no—
192#121—3461no—
193#121—1731no—
194#121—1731no—
195#121—1732no—
196#121—1731no—
197#121—1732no—
198#121C2×D17311yescharacteristic
199C2×C4⋊C4—1731no—
200C2×C4⋊C4C2×D17311yescharacteristic, complemented
201C2×C22⋊C4—3461no—
202C2×C22⋊C4—3461no—
203#107—1732no—
204#107C2×D17312yescomplemented
205#102—1732no—
206#102—1732no—
207#102—1732no—
208#102C2×D17312yescomplemented
209C22⋊C8—1731no—
210C22⋊C8—1732no—
211C22⋊C8C2×D17311yescharacteristic, complemented
212C24—3461no—
213C24—3461no—
214C24—1731no—
215C2×D4—21no—
216C2×D4—3461no—
217C2×D4—3462no—
218C2×D4—3462no—
219C2×D4—3461no—
220C2×D4—3461no—
221C2×D4—3461no—
222C2×D4—3461no—
223C2×D4—1731no—
224C2×D4—1732no—
225C2×D4—6922no—
226C2×D4—6924no—
227C2×D4—6922no—
228C2×D4—6924no—
229C2×D4C22×D17311yescharacteristic
230C22×C4—21no—
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—6922no—
240D8—6922no—
241C2×C8—1732no—
242C2×C8—6922no—
243C2×C8C22×D17312yes—
244C4⋊C4—21no—
245C4⋊C4—3461no—
246C4⋊C4—1731no—
247C4⋊C4C22×D17311yescharacteristic
248C4⋊C4C22×D17312yes—
249C22⋊C4—21no—
250C22⋊C4—3461no—
251C22⋊C4—3462no—
252C22⋊C4—3462no—
253C22⋊C4—3461no—
254C22⋊C4—3461no—
255C23—21no—
256C23—3461no—
257C23—3462no—
258C23—1731no—
259C23—1731no—
260C23—1734no—
261C23—6921no—
262C23—6921no—
263C23—6922no—
264C23—6922no—
265C23—6922no—
266C23—6922no—
267C23—6922no—
268C23—6922no—
269C23D4×D17311yescharacteristic
270D4—22no—
271D4—42no—
272D4—3462no—
273D4—3462no—
274D4—3462no—
275D4—6922no—
276D4—6924no—
277D4—6924no—
278D4—6922no—
279D4—6922no—
280D4—6922no—
281D4—6922no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—3462no—
287C2×C4—3462no—
288C2×C4—3461no—
289C2×C4—1731no—
290C2×C4—1731no—
291C2×C4—6921no—
292C2×C4—6921no—
293C2×C4—6922no—
294C2×C4—6921no—
295C2×C4D4×D17311yescharacteristic
296C2×C4C23×D17311yescharacteristic
297C8—22no—
298C8—3462no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—3461no—
305C22—3461no—
306C22—3461no—
307C22—3461no—
308C22—13844no—
309C22—13844no—
310C22—1732no—
311C22—1732no—
312C22—6924no—
313C22—6922no—
314C22—6921no—
315C22—6922no—
316C22—6921no—
317C22—6922no—
318C22—6921no—
319C22D8×D17312yes—
320C22C2×D4×D17311yescharacteristic
321C4—21no—
322C4—42no—
323C4—41no—
324C4—3461no—
325C4—3461no—
326C4—6921no—
327C4#1914972512yes—
328C2—22no—
329C2—81no—
330C2—3462no—
331C2—3462no—
332C2—13842no—
333C2—13841no—
334C2#2601205211yescharacteristic
335C2C2×D8×D17311yescharacteristic
336C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
hypercenter#480D173
hypocenterC173#3186
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. C2×#19149654C2semidirect
2. #121×D173C2semidirect
3. #26011806C2semidirect
4. #26011804C2semidirect
5. #26011728C2semidirect
6. #26011726C2semidirect
7. #26011627C2semidirect
8. #26011626C2semidirect
9. #26011624C2semidirect
10. C22⋊C8×D173C2semidirect
11. #480×C173C2semidirect
13. C22×D692C22semidirect
15. C2×D4×D173C22semidirect
17. C2×D1384C22semidirect
18. C2×#14583811C22semidirect
19. C2×C8×D173C22semidirect
23. #19149654C22semidirect
24. #19149654C22semidirect
28. #19149650C22semidirect
29. C4⋊C4×D173C22semidirect
32. #19149612C22semidirect
33. #19149601C22semidirect
34. #19149599C22semidirect
35. #19149599C22semidirect
36. C22⋊#13063436C22semidirect
37. #19149591C22semidirect
39. C2×C4⋊C4×C173C22semidirect
40. #107×C173C22semidirect
41. #102×C173C22semidirect
42. C22⋊C8×C173C22semidirect
96. C22×D173D8semidirect
168. C173#3186semidirect
174. #480D173semidirect
200. C2×C4⋊C4C2×D173semidirect
204. #107C2×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×#191496542
#121×D1732
#260118062
#260118042
#260117282
#260117262
#260116272
#260116262
#260116242
C22⋊C8×D1732
#480×C1732
#3186173