Finite Group Databasebeta

← #36365370

Subgroups — #36365370

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
1D4×D692C211yesmaximal, characteristic, complemented
2#26011839C211yesmaximal, characteristic, complemented
3#26011804C212yesmaximal, complemented
4#26011768C211yesmaximal, characteristic, complemented
5#26011725C212yesmaximal, complemented
6#26011686C211yesmaximal, characteristic, complemented
7#26011624C212yesmaximal, complemented
8C4×#14583811C211yesmaximal, characteristic, complemented
9D4×#13063436C211yesmaximal, characteristic, complemented
10C2×#19149599C212yesmaximal, complemented
11#547×C173C211yesmaximal, characteristic, complemented
12C22×D692C2212yes—
13C2×D4×D173—21no—
14C2×#14583811C2211yescharacteristic
15C2×#14583811C2212yescomplemented
16C22×#13063436C2212yescomplemented
17#19149676C2211yescharacteristic
18#19149654—21no—
19#19149653C2212yes—
20#19149649—22no—
21C4×D692C2211yescharacteristic
22#19149612C2212yescomplemented
23#19149612C2211yescharacteristic, complemented
24C4⋊#13063436C2211yescharacteristic, complemented
25#19149600C2211yescharacteristic, complemented
26#19149599C2211yescharacteristic
27#19149599C2212yes—
28#19149599C2212yescomplemented
29#19149599C2212yescomplemented
30#19149599C2212yescomplemented
31C22⋊#13063436C2212yescomplemented
32C4×#13063436C2211yescharacteristic, complemented
33C2×#41×C173—21no—
34C4⋊Q8×C173C2211yescharacteristic, complemented
35#123×C173—22no—
36C22⋊Q8×C173—21no—
37#121×C173C2212yescomplemented
38C4×D4×C173C2211yescharacteristic
39C4×D4×C173C2211yescharacteristic, complemented
40C2×C4⋊C4×C173C2212yescomplemented
41C23×D173—22no—
42C2×D692—24no—
43C2×D692—22no—
44C2×D692—21no—
45C2×D692—21no—
46C2×D692—42no—
47C2×D692C2312yes—
48C2×D692C2311yescharacteristic
49C2×C22⋊D173—22no—
50D4×D173—44no—
51C2×C4×D173—21no—
52#14583811—22no—
53#14583811—42no—
54C2×#13063436—42no—
55C2×#13063436C2311yescharacteristic
56C2×#13063436C2311yescharacteristic
57C2×#13063436C2312yes—
58C4⋊Dic173C2311yescharacteristic
59#14583803—21no—
60#14583803—22no—
61#41×C173—44no—
62C2×Q8×C173—21no—
63C2×D4×C173—22no—
64C2×D4×C173D411yescharacteristic
65C2×D4×C173C2311yescharacteristic
66C22×C4×C173—22no—
67C22×C4×C173—21no—
68C22×C4×C173C2312yes—
69C4⋊C4×C173—21no—
70C4⋊C4×C173—22no—
71C4⋊C4×C173D411yescharacteristic
72C4⋊C4×C173C2312yes—
73C4⋊C4×C173C2311yescharacteristic
74C22⋊C4×C173—21no—
75C22⋊C4×C173—22no—
76C22⋊C4×C173D412yes—
77C42×C173C2311yescharacteristic
78C22×D173—21no—
79C22×D173—22no—
80C22×D173—44no—
81C22×D173—44no—
82C22×D173—82no—
83D692—22no—
84D692—21no—
85D692—22no—
86D692—44no—
87D692—42no—
88D692—81no—
89D692#4112yescomplemented
90C22⋊D173—44no—
91C2×Dic173—21no—
92C4×D173—42no—
93#13063436—21no—
94#13063436—21no—
95#13063436—22no—
96C23×C173—21no—
97C23×C173C2×D412yes—
98Q8×C173—42no—
99D4×C173—21no—
100D4×C173—21no—
101D4×C173—44no—
102D4×C173D814yescomplemented
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—21no—
106C2×C4×C173—22no—
107C2×C4×C173—44no—
108C2×C4×C173—42no—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2×D412yes—
111C2×C4×C173C2×D411yescharacteristic
112C2×C4×C173C2411yescharacteristic
113C22×C173—22no—
114C22×C173—22no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173C2×D814yes—
119C22×C173C22×D411yescharacteristic
120C2×D173—22no—
121C2×D173—41no—
122C2×D173—41no—
123C2×D173—42no—
124C2×D173—42no—
125C2×D173—42no—
126C2×D173—88no—
127C4×C173—22no—
128C4×C173—21no—
129C4×C173—41no—
130C4×C173—42no—
131C4×C173C2×D812yes—
132C4×C173C2×#4111yescharacteristic
133C4×C173H(5,𝔽2)11yescharacteristic
134Dic173—41no—
135C2×C173—24no—
136C2×C173—81no—
137C2×C173#54611yescharacteristic
138C2×C173C22×D811yescharacteristic
139C2×C173#57711yescharacteristic
140D173—42no—
141D173—82no—
142C173#329411yescharacteristic, complemented, Sylow
143#3294—1731nomaximal, Sylow
144#547D17311yescharacteristic, complemented
145D42—1731no—
146#486—1731no—
147#480—1732no—
148#474—1731no—
149#466—1732no—
150#459—1731no—
151#447—1732no—
152C4×D8—1731no—
153C8×D4—1731no—
154C2×#102—1732no—
155C2×#41—21no—
156C22×D4—3461no—
157C22×D4—1732no—
158C2×D8—1731no—
159C2×D8—1732no—
160C22×C8—1732no—
161C4⋊Q8C2×D17311yescharacteristic, complemented
162#127—1731no—
163#123—22no—
164C22⋊Q8—21no—
165#121—3461no—
166#121—1732no—
167#121C2×D17312yescomplemented
168#120—3462no—
169C4×D4—1731no—
170C4×D4C2×D17311yescharacteristic
171C4×D4C2×D17311yescharacteristic, complemented
172C2×C4⋊C4C2×D17312yescomplemented
173#107—1731no—
174#107—1732no—
175C4⋊C8—1731no—
176#102—1731no—
177#102—1731no—
178#102—1732no—
179#102—1732no—
180#102—1732no—
181#102—1732no—
182C22⋊C8—1732no—
183C4×C8—1731no—
184C24—3462no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—3464no—
189C2×D4—3462no—
190C2×D4—3461no—
191C2×D4—3462no—
192C2×D4—3461no—
193C2×D4—1732no—
194C2×D4—1731no—
195C2×D4—6922no—
196C2×D4—6924no—
197C2×D4C22⋊D17311yescharacteristic
198C2×D4C22×D17311yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—3461no—
202C22×C4C22×D17312yes—
203D8—3462no—
204D8—6922no—
205C2×C8—1731no—
206C2×C8—1731no—
207C2×C8—1732no—
208C2×C8—6922no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1731no—
212C4⋊C4C22⋊D17311yescharacteristic
213C4⋊C4C22×D17311yescharacteristic
214C4⋊C4C22×D17312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—3461no—
218C22⋊C4—3462no—
219C22⋊C4C22⋊D17312yes—
220C42C22×D17311yescharacteristic
221C23—21no—
222C23—3461no—
223C23—3462no—
224C23—13842no—
225C23—6924no—
226C23—6924no—
227C23C2×C22⋊D17312yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—3462no—
233D4—3461no—
234D4—3462no—
235D4—13841no—
236D4—1732no—
237D4—6924no—
238D4—6924no—
239D4—6922no—
240D4#1458381114yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—3461no—
248C2×C4—6922no—
249C2×C4C2×C22⋊D17311yescharacteristic
250C2×C4C2×C22⋊D17312yes—
251C2×C4C2×C22⋊D17311yescharacteristic
252C2×C4C23×D17311yescharacteristic
253C8—3461no—
254C8—3461no—
255C8—3462no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3462no—
262C22—13848no—
263C22—6922no—
264C22—6922no—
265C22—6922no—
266C22—6921no—
267C22—6921no—
268C22C2×#1458381114yes—
269C22C22×C22⋊D17311yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—6921no—
275C4C2×#1458381112yes—
276C4#41×D17311yescharacteristic
277C4#1914972511yescharacteristic
278C2—24no—
279C2—81no—
280C2—13842no—
281C2—6922no—
282C2#2601212911yescharacteristic
283C2C22×#1458381111yescharacteristic
284C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3294
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D692C2semidirect
2. #26011839C2semidirect
3. #26011804C2semidirect
4. #26011768C2semidirect
5. #26011725C2semidirect
6. #26011686C2semidirect
7. #26011624C2semidirect
8. C4×#14583811C2semidirect
9. D4×#13063436C2semidirect
10. C2×#19149599C2semidirect
11. #547×C173C2semidirect
15. C2×#14583811C22semidirect
16. C22×#13063436C22semidirect
22. #19149612C22semidirect
23. #19149612C22semidirect
24. C4⋊#13063436C22semidirect
25. #19149600C22semidirect
28. #19149599C22semidirect
29. #19149599C22semidirect
30. #19149599C22semidirect
31. C22⋊#13063436C22semidirect
32. C4×#13063436C22semidirect
34. C4⋊Q8×C173C22semidirect
37. #121×C173C22semidirect
39. C4×D4×C173C22semidirect
40. C2×C4⋊C4×C173C22semidirect
89. D692#41semidirect
102. D4×C173D8semidirect
142. C173#3294semidirect
144. #547D173semidirect
161. C4⋊Q8C2×D173semidirect
167. #121C2×D173semidirect
171. C4×D4C2×D173semidirect
172. C2×C4⋊C4C2×D173semidirect
240. D4#14583811semidirect

Maximal subgroups

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

subgroupindex
D4×D6922
#260118392
#260118042
#260117682
#260117252
#260116862
#260116242
C4×#145838112
D4×#130634362
C2×#191495992
#547×C1732
#3294173