Finite Group Databasebeta

← #36365378

Subgroups — #36365378

Order 22144.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#26012146C211yesmaximal, characteristic, complemented
2D4×D692C211yesmaximal, characteristic, complemented
3#26011838C211yesmaximal, characteristic, complemented
4#26011806C212yesmaximal, complemented
5#26011771C211yesmaximal, characteristic, complemented
6#26011726C212yesmaximal, complemented
7#26011686C211yesmaximal, characteristic, complemented
8#26011623C212yesmaximal, complemented
9C8×D692C211yesmaximal, characteristic, complemented
10#102×D173C212yesmaximal, complemented
11C4×D8×C173C211yesmaximal, characteristic, complemented
12C2×#14583826—21no—
13C22×D692—21no—
14C2×D4×D173C2212yes—
15C2×#14583811C2212yescomplemented
16C2×C8×D173C2212yescomplemented
17#19149679C2211yescharacteristic, complemented
18#19149676C2211yescharacteristic
19#19149661—22no—
20#19149658—21no—
21#19149654C2212yes—
22#19149653—21no—
23#19149650C2212yescomplemented
24#19149649—22no—
25C4×D692C2211yescharacteristic
26C4⋊C4×D173C2212yescomplemented
27#19149612C2212yescomplemented
28C4⋊#13063436C2211yescharacteristic, complemented
29#19149601C2212yescomplemented
30#19149601C2212yescomplemented
31#19149600C2212yes—
32#19149599C2212yescomplemented
33#19149591C2212yescomplemented
34C2×D8×C173C2211yescharacteristic
35C4×D4×C173C2211yescharacteristic
36C4×D4×C173C2211yescharacteristic, complemented
37#107×C173C2211yescharacteristic, complemented
38#102×C173C2211yescharacteristic
39#102×C173C2211yescharacteristic, complemented
40C4×C8×C173C2211yescharacteristic, complemented
41C23×D173—22no—
42#14583826—44no—
43C2×Dic346—21no—
44C2×D692—22no—
45C2×D692—21no—
46C2×D692—21no—
47C2×D692—44no—
48C2×D692D411yescharacteristic
49C2×D692C2312yes—
50C2×C22⋊D173—22no—
51C2×C22⋊D173—22no—
52D4×D173—24no—
53D4×D173—42no—
54C2×C4×D173—22no—
55C2×C4×D173C2312yes—
56#14583811—42no—
57C2×#13063436C2312yes—
58C8×D173—42no—
59C4⋊Dic173—21no—
60C4⋊Dic173D411yescharacteristic
61C4⋊Dic173C2312yes—
62#14583805—22no—
63C22⋊Dic173—22no—
64#14583803—22no—
65#14583803D412yes—
66C2×D4×C173C2311yescharacteristic
67C2×D4×C173C2311yescharacteristic
68C22×C4×C173—21no—
69C22×C4×C173—21no—
70D8×C173—22no—
71C2×C8×C173C2311yescharacteristic
72C2×C8×C173C2311yescharacteristic
73C4⋊C4×C173C2311yescharacteristic
74C4⋊C4×C173C2311yescharacteristic
75C22⋊C4×C173—21no—
76C22⋊C4×C173—21no—
77C42×C173C2311yescharacteristic
78C22×D173—22no—
79C22×D173—44no—
80C22×D173—44no—
81C22×D173—82no—
82C22×D173C2×D412yes—
83Dic346—42no—
84D692—22no—
85D692—22no—
86D692—44no—
87D692—42no—
88D692—81no—
89D692D814yescomplemented
90C22⋊D173—44no—
91C22⋊D173—44no—
92C2×Dic173—22no—
93C2×Dic173C2×D412yes—
94C4×D173—22no—
95C4×D173—22no—
96C4×D173—44no—
97#13063436—22no—
98C23×C173—21no—
99C23×C173—21no—
100D4×C173—21no—
101D4×C173—21no—
102D4×C173—21no—
103D4×C173#4112yescomplemented
104C2×C4×C173—21no—
105C2×C4×C173—21no—
106C2×C4×C173—42no—
107C2×C4×C173—42no—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2411yescharacteristic
111C8×C173—21no—
112C8×C173—21no—
113C22×C173—22no—
114C22×C173—41no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173—41no—
119C22×C173C22×D411yescharacteristic
120C2×D173—22no—
121C2×D173—22no—
122C2×D173—42no—
123C2×D173—42no—
124C2×D173—42no—
125C2×D173—88no—
126C2×D173C2×D814yes—
127C4×C173—21no—
128C4×C173—41no—
129C4×C173—41no—
130C4×C173C2×D812yes—
131C4×C173C2×#4111yescharacteristic
132C4×C173H(5,𝔽2)11yescharacteristic
133Dic173—22no—
134Dic173—42no—
135C2×C173—42no—
136C2×C173—81no—
137C2×C173#54611yescharacteristic
138C2×C173C22×D811yescharacteristic
139C2×C173#57711yescharacteristic
140D173—24no—
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×D8D17311yescharacteristic, complemented
153C8×D4—1731no—
154C2×#102—1732no—
155C2×#41—3461no—
156C22×D4—3461no—
157C22×D4—1732no—
158C2×D8—1732no—
159C2×D8C2×D17311yescharacteristic
160C22×C8—1732no—
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×D4C2×D17311yescharacteristic
171C4×D4C2×D17311yescharacteristic, complemented
172C2×C4⋊C4—1732no—
173#107—1732no—
174#107C2×D17311yescharacteristic, complemented
175C4⋊C8—1731no—
176#102—1732no—
177#102—1732no—
178#102—1732no—
179#102—1732no—
180#102C2×D17311yescharacteristic
181#102C2×D17311yescharacteristic, complemented
182C22⋊C8—1732no—
183C4×C8C2×D17311yescharacteristic, complemented
184C24—3462no—
185#41—6924no—
186C2×Q8—3461no—
187C2×D4—3462no—
188C2×D4—3464no—
189C2×D4—3462no—
190C2×D4—3462no—
191C2×D4—3461no—
192C2×D4—3461no—
193C2×D4—1731no—
194C2×D4—1732no—
195C2×D4—6924no—
196C2×D4—6922no—
197C2×D4C22×D17311yescharacteristic
198C2×D4C22×D17311yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—3462no—
202C22×C4—1732no—
203D8—22no—
204D8—6922no—
205C2×C8—1732no—
206C2×C8—6922no—
207C2×C8C22×D17311yescharacteristic
208C2×C8C22×D17311yescharacteristic
209C4⋊C4—3462no—
210C4⋊C4—3461no—
211C4⋊C4—1732no—
212C4⋊C4—1731no—
213C4⋊C4C22×D17311yescharacteristic
214C4⋊C4C22×D17311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3462no—
218C22⋊C4—3462no—
219C22⋊C4—1732no—
220C42C22×D17311yescharacteristic
221C23—21no—
222C23—21no—
223C23—3462no—
224C23—13842no—
225C23—1732no—
226C23—6924no—
227C23—6924no—
228Q8—6922no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—3462no—
233D4—3462no—
234D4—13841no—
235D4—1734no—
236D4—6924no—
237D4—6924no—
238D4—6924no—
239D4—6922no—
240D4#1458382612yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—3462no—
246C2×C4—3462no—
247C2×C4—3462no—
248C2×C4—1732no—
249C2×C4—6924no—
250C2×C4D4×D17311yescharacteristic
251C2×C4D4×D17311yescharacteristic
252C2×C4C23×D17311yescharacteristic
253C8—21no—
254C8—21no—
255C8—3462no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—3462no—
263C22—3462no—
264C22—13848no—
265C22—1734no—
266C22—6922no—
267C22—6922no—
268C22—6922no—
269C22C2×D4×D17311yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—3462no—
274C4—6922no—
275C4D8×D17312yes—
276C4C2×#1458382611yescharacteristic
277C4#1914972511yescharacteristic
278C2—42no—
279C2—81no—
280C2—3464no—
281C2—13842no—
282C2#2601213711yescharacteristic
283C2C2×D8×D17311yescharacteristic
284C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×D8×C173C2
socleC22×C173C22×D4
hypercenterC4×D8D173
hypocenterC173#3294
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #26012146C2semidirect
2. D4×D692C2semidirect
3. #26011838C2semidirect
4. #26011806C2semidirect
5. #26011771C2semidirect
6. #26011726C2semidirect
7. #26011686C2semidirect
8. #26011623C2semidirect
9. C8×D692C2semidirect
10. #102×D173C2semidirect
11. C4×D8×C173C2semidirect
15. C2×#14583811C22semidirect
16. C2×C8×D173C22semidirect
17. #19149679C22semidirect
23. #19149650C22semidirect
26. C4⋊C4×D173C22semidirect
27. #19149612C22semidirect
28. C4⋊#13063436C22semidirect
29. #19149601C22semidirect
30. #19149601C22semidirect
32. #19149599C22semidirect
33. #19149591C22semidirect
36. C4×D4×C173C22semidirect
37. #107×C173C22semidirect
39. #102×C173C22semidirect
40. C4×C8×C173C22semidirect
89. D692D8semidirect
103. D4×C173#41semidirect
142. C173#3294semidirect
152. C4×D8D173semidirect
171. C4×D4C2×D173semidirect
174. #107C2×D173semidirect
181. #102C2×D173semidirect
183. C4×C8C2×D173semidirect
240. D4#14583826semidirect

Maximal subgroups

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

subgroupindex
#260121462
D4×D6922
#260118382
#260118062
#260117712
#260117262
#260116862
#260116232
C8×D6922
#102×D1732
C4×D8×C1732
#3294173