Finite Group Databasebeta

← #36365377

Subgroups — #36365377

Order 22144.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#26012145C211yesmaximal, characteristic, complemented
2D4×D692C211yesmaximal, characteristic, complemented
3#26011838C211yesmaximal, characteristic, complemented
4#26011805C212yesmaximal, complemented
5#26011772C211yesmaximal, characteristic, complemented
6#26011727C212yesmaximal, complemented
7#26011687C211yesmaximal, characteristic, complemented
8#26011627C212yesmaximal, complemented
9C4×D1384C211yesmaximal, characteristic, complemented
10C2×#19149601C212yesmaximal, complemented
11C8×D4×C173C211yesmaximal, characteristic, complemented
12C2×#14583825—21no—
13C22×D692C2212yes—
14C2×D4×D173—21no—
15C2×D1384C2211yescharacteristic
16C2×D1384C2212yescomplemented
17#19149679C2211yescharacteristic, complemented
18#19149676C2211yescharacteristic
19#19149663—22no—
20#19149659—21no—
21#19149654—21no—
22#19149653C2212yes—
23#19149653C2212yescomplemented
24#19149649—22no—
25C4×D692C2211yescharacteristic
26C4×D692C2211yescharacteristic, complemented
27C2×C4⋊Dic173C2212yescomplemented
28#19149613C2212yescomplemented
29#19149613C2211yescharacteristic, complemented
30#19149601C2211yescharacteristic
31#19149601C2212yes—
32#19149601C2212yescomplemented
33#19149601C2211yescharacteristic, complemented
34#19149601C2212yescomplemented
35#19149601C2212yescomplemented
36C22×C8×C173C2212yescomplemented
37C4×D4×C173C2211yescharacteristic
38C4⋊C8×C173C2211yescharacteristic, complemented
39C22⋊C8×C173C2212yescomplemented
40C4×C8×C173C2211yescharacteristic, complemented
41C23×D173—22no—
42#14583825—44no—
43C2×Dic346—21no—
44C2×D692—24no—
45C2×D692—22no—
46C2×D692—21no—
47C2×D692—21no—
48C2×D692—42no—
49C2×D692C2312yes—
50C2×D692C2311yescharacteristic
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54D4×D173—44no—
55C22×Dic173—22no—
56C2×C4×D173—21no—
57C2×C4×D173—21no—
58D1384—22no—
59D1384—42no—
60C4⋊Dic173—22no—
61C4⋊Dic173—21no—
62C4⋊Dic173C2311yescharacteristic
63C4⋊Dic173C2312yes—
64C4⋊Dic173C2311yescharacteristic
65#14583803—21no—
66#14583803—21no—
67#14583803—22no—
68#14583803—22no—
69C2×D4×C173D411yescharacteristic
70C22×C4×C173C2312yes—
71C2×C8×C173—42no—
72C2×C8×C173C2311yescharacteristic
73C2×C8×C173C2312yes—
74C2×C8×C173C2311yescharacteristic
75C4⋊C4×C173D411yescharacteristic
76C22⋊C4×C173D412yes—
77C42×C173C2311yescharacteristic
78C22×D173—21no—
79C22×D173—21no—
80C22×D173—22no—
81C22×D173—44no—
82C22×D173—44no—
83C22×D173—82no—
84Dic346—42no—
85D692—21no—
86D692—22no—
87D692—21no—
88D692—21no—
89D692—22no—
90D692—44no—
91D692—42no—
92D692—81no—
93D692#4112yescomplemented
94C22⋊D173—44no—
95C22⋊D173—44no—
96C2×Dic173—21no—
97C2×Dic173—21no—
98C2×Dic173—22no—
99C2×Dic173—44no—
100C4×D173—42no—
101C4×D173—42no—
102C23×C173C2×D412yes—
103D4×C173D814yescomplemented
104C2×C4×C173—22no—
105C2×C4×C173—22no—
106C2×C4×C173C2×D411yescharacteristic
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2411yescharacteristic
110C8×C173—21no—
111C8×C173—22no—
112C8×C173—21no—
113C22×C173—22no—
114C22×C173—22no—
115C22×C173C2×D814yes—
116C22×C173C22×D411yescharacteristic
117C2×D173—22no—
118C2×D173—41no—
119C2×D173—41no—
120C2×D173—42no—
121C2×D173—41no—
122C2×D173—41no—
123C2×D173—42no—
124C2×D173—42no—
125C2×D173—41no—
126C2×D173—88no—
127C4×C173—22no—
128C4×C173—21no—
129C4×C173C2×D812yes—
130C4×C173C2×#4111yescharacteristic
131C4×C173H(5,𝔽2)11yescharacteristic
132Dic173—41no—
133Dic173—41no—
134Dic173—42no—
135C2×C173—24no—
136C2×C173#54611yescharacteristic
137C2×C173C22×D811yescharacteristic
138C2×C173#57711yescharacteristic
139D173—42no—
140D173—81no—
141D173—82no—
142C173#329411yescharacteristic, complemented, Sylow
143#3294—1731nomaximal, Sylow
144#547—1731no—
145D42—1731no—
146#486—1731no—
147#480—1732no—
148#474—1731no—
149#466—1732no—
150#459—1731no—
151#447—1732no—
152C4×D8—1731no—
153C8×D4D17311yescharacteristic, complemented
154C2×#102—1732no—
155C2×#41—3461no—
156C22×D4—3461no—
157C22×D4—1732no—
158C2×D8—1731no—
159C2×D8—1732no—
160C22×C8C2×D17312yescomplemented
161C4⋊Q8—1731no—
162#127—1731no—
163#123—3462no—
164C22⋊Q8—3461no—
165#121—3461no—
166#121—1732no—
167#121—1732no—
168#120—3462no—
169C4×D4—1731no—
170C4×D4—1731no—
171C4×D4C2×D17311yescharacteristic
172C2×C4⋊C4—1732no—
173#107—1731no—
174#107—1732no—
175C4⋊C8C2×D17311yescharacteristic, complemented
176#102—1731no—
177#102—1732no—
178#102—1732no—
179#102—1732no—
180#102—1731no—
181#102—1732no—
182C22⋊C8C2×D17312yescomplemented
183C4×C8C2×D17311yescharacteristic, complemented
184C24—3462no—
185#41—6924no—
186C2×Q8—3461no—
187C2×D4—3464no—
188C2×D4—3462no—
189C2×D4—3462no—
190C2×D4—3461no—
191C2×D4—3462no—
192C2×D4—3461no—
193C2×D4—1732no—
194C2×D4—1731no—
195C2×D4—1731no—
196C2×D4—6922no—
197C2×D4—6924no—
198C2×D4D69211yescharacteristic
199C22×C4—3462no—
200C22×C4—3461no—
201C22×C4—3461no—
202C22×C4C22×D17312yes—
203D8—3462no—
204D8—6922no—
205C2×C8—42no—
206C2×C8C22×D17311yescharacteristic
207C2×C8C22×D17311yescharacteristic
208C2×C8C22×D17312yes—
209C4⋊C4—3462no—
210C4⋊C4—3461no—
211C4⋊C4—1731no—
212C4⋊C4—1732no—
213C4⋊C4—1731no—
214C4⋊C4D69211yescharacteristic
215C22⋊C4—3461no—
216C22⋊C4—3461no—
217C22⋊C4—3462no—
218C22⋊C4—3462no—
219C22⋊C4D69212yes—
220C42C22×D17311yescharacteristic
221C23—3461no—
222C23—3461no—
223C23—3462no—
224C23—13842no—
225C23—6924no—
226C23—6924no—
227C23C2×D69212yes—
228Q8—6922no—
229D4—3461no—
230D4—3462no—
231D4—3461no—
232D4—3461no—
233D4—3462no—
234D4—13841no—
235D4—1732no—
236D4—6924no—
237D4—6924no—
238D4—6924no—
239D4—6922no—
240D4D138414yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3461no—
244C2×C4—3461no—
245C2×C4—3462no—
246C2×C4—6924no—
247C2×C4—6922no—
248C2×C4—6922no—
249C2×C4C2×D69211yescharacteristic
250C2×C4C2×D69212yes—
251C2×C4C2×D69211yescharacteristic
252C2×C4C23×D17311yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—3462no—
259C22—13848no—
260C22—6921no—
261C22—6922no—
262C22—6922no—
263C22—6921no—
264C22—6922no—
265C22—6921no—
266C22—6921no—
267C22—6921no—
268C22C2×D138414yes—
269C22C22×D69211yescharacteristic
270C4—21no—
271C4—22no—
272C4—6921no—
273C4—6921no—
274C4—6922no—
275C4C2×D138412yes—
276C4#41×D17311yescharacteristic
277C4#1914972411yescharacteristic
278C2—24no—
279C2—13842no—
280C2—13841no—
281C2—6922no—
282C2#2601213611yescharacteristic
283C2C22×D138411yescharacteristic
284C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC8×D4×C173C2
socleC22×C173C22×D4
hypercenterC8×D4D173
hypocenterC173#3294
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26012145C2semidirect
2. D4×D692C2semidirect
3. #26011838C2semidirect
4. #26011805C2semidirect
5. #26011772C2semidirect
6. #26011727C2semidirect
7. #26011687C2semidirect
8. #26011627C2semidirect
9. C4×D1384C2semidirect
10. C2×#19149601C2semidirect
11. C8×D4×C173C2semidirect
16. C2×D1384C22semidirect
17. #19149679C22semidirect
23. #19149653C22semidirect
26. C4×D692C22semidirect
27. C2×C4⋊Dic173C22semidirect
28. #19149613C22semidirect
29. #19149613C22semidirect
32. #19149601C22semidirect
33. #19149601C22semidirect
34. #19149601C22semidirect
35. #19149601C22semidirect
36. C22×C8×C173C22semidirect
38. C4⋊C8×C173C22semidirect
39. C22⋊C8×C173C22semidirect
40. C4×C8×C173C22semidirect
93. D692#41semidirect
103. D4×C173D8semidirect
142. C173#3294semidirect
153. C8×D4D173semidirect
160. C22×C8C2×D173semidirect
175. C4⋊C8C2×D173semidirect
182. C22⋊C8C2×D173semidirect
183. C4×C8C2×D173semidirect
240. D4D1384semidirect

Maximal subgroups

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

subgroupindex
#260121452
D4×D6922
#260118382
#260118052
#260117722
#260117272
#260116872
#260116272
C4×D13842
C2×#191496012
C8×D4×C1732
#3294173