Finite Group Databasebeta

← #36366698

Subgroups — #36366698

Order 22144.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19149702C211yesmaximal, characteristic, complemented
2#135×D173C211yesmaximal, characteristic, complemented
3#26011933C212yesmaximal, complemented
4#26011932C211yesmaximal, characteristic, complemented
5#26011929C211yesmaximal, characteristic, complemented
6#26011928C212yesmaximal, complemented
7#26011926C211yesmaximal, characteristic, complemented
8#26011924C212yesmaximal, complemented
9#26011922C212yesmaximal, complemented
10M32×D173C211yesmaximal, characteristic, complemented
11#510×C173C211yesmaximal, characteristic, complemented
12C2×#14583827—21no—
13#41×D173—21no—
14#19149702—21no—
15#19149702—21no—
16#19149702C2212yes—
17#19149702C2212yescomplemented
18#19149702C2211yescharacteristic, complemented
19#19149701C2211yescharacteristic
20#19149700—21no—
21#19149699—21no—
22#19149698C2211yescharacteristic, complemented
23Q16×D173C2211yescharacteristic, complemented
24C2×#14583813—21no—
25SD16×D173—21no—
26C2×D1384C2211yescharacteristic, complemented
27D8×D173C2211yescharacteristic, complemented
28C2×C8×D173C2211yescharacteristic
29#19149625C2212yescomplemented
30#19149624C2212yescomplemented
31#19149622C2212yescomplemented
32#19149622C2212yescomplemented
33D2768C2212yescomplemented
34#19149620C2212yescomplemented
35#19149619C2211yescharacteristic, complemented
36#19149618C2212yescomplemented
37C16×D173C2212yescomplemented
38#135×C173C2211yescharacteristic
39C2×Q16×C173C2211yescharacteristic, complemented
40Q32×C173C2212yescomplemented
41SD32×C173C2212yescomplemented
42M32×C173C2211yescharacteristic, complemented
43#14583827—21no—
44#14583827—22no—
45#14583827—22no—
46#14583827—41no—
47#14583827—41no—
48#14583826—21no—
49#14583826—41no—
50#14583825—21no—
51#14583825—41no—
52Q8×D173—21no—
53C2×D692—21no—
54C2×D692—41no—
55D4×D173—21no—
56D4×D173—41no—
57C2×C4×D173—41no—
58C2×C4×D173—41no—
59C2×C4×D173D411yescharacteristic
60Dic692C2311yescharacteristic
61#14583816—21no—
62#14583815—21no—
63#14583814—21no—
64#14583813—21no—
65#14583813—22no—
66#14583813—22no—
67D1384—21no—
68D1384C2312yes—
69D1384C2311yescharacteristic
70#14583811—21no—
71C2×#13063436D411yescharacteristic
72C8×D173D411yescharacteristic
73C8×D173D411yescharacteristic
74C8×D173C2311yescharacteristic
75C8×D173C2311yescharacteristic
76#14583801C2312yes—
77#41×C173—21no—
78C2×Q8×C173—21no—
79Q16×C173—21no—
80Q16×C173C2312yes—
81Q16×C173C2311yescharacteristic
82SD16×C173—21no—
83D8×C173C2311yescharacteristic
84C2×C8×C173C2311yescharacteristic
85C16×C173C2312yes—
86C22×D173—41no—
87C22×D173—41no—
88C22×D173D811yescharacteristic
89Dic346—21no—
90Dic346—41no—
91D692—21no—
92D692—22no—
93D692—41no—
94D692—42no—
95D692—42no—
96D692—41no—
97C22⋊D173—41no—
98C22⋊D173—41no—
99C22⋊D173—41no—
100C2×Dic173—41no—
101C2×Dic173D811yescharacteristic
102C4×D173—41no—
103C4×D173—41no—
104C4×D173—41no—
105C4×D173—42no—
106C4×D173—42no—
107C4×D173D811yescharacteristic
108C4×D173D811yescharacteristic
109C4×D173C2×D411yescharacteristic
110C4×D173C2×D411yescharacteristic
111#13063436C2×D411yescharacteristic
112#13063436C2×D411yescharacteristic
113Q8×C173—22no—
114Q8×C173—21no—
115Q8×C173—41no—
116D4×C173—21no—
117D4×C173—41no—
118C2×C4×C173—41no—
119C2×C4×C173—41no—
120C2×C4×C173C2×D411yescharacteristic
121C8×C173C2×D411yescharacteristic
122C8×C173C2411yescharacteristic
123C22×C173—41no—
124C22×C173C2×D811yescharacteristic
125C2×D173—41no—
126C2×D173—41no—
127C2×D173—42no—
128C2×D173—82no—
129C2×D173—82no—
130C2×D173C2×D811yescharacteristic
131C2×D173C2×D811yescharacteristic
132C4×C173—41no—
133C4×C173—42no—
134C4×C173C2×D811yescharacteristic
135C4×C173C22×D411yescharacteristic
136Dic173—41no—
137Dic173C2×D811yescharacteristic
138Dic173C2×D811yescharacteristic
139C2×C173—21no—
140C2×C173—81no—
141C2×C173C22×D811yescharacteristic
142D173—21no—
143D173—21no—
144D173—81no—
145D173—82no—
146C173#341411yescharacteristic, complemented, Sylow
147#3414—1731nomaximal, Sylow
148C2×#135—1731no—
149C2×#135—1731no—
150#510—1731no—
151#510—1732no—
152#510D17311yescharacteristic, complemented
153#509—1731no—
154#509—1732no—
155#509—1731no—
156#508—1732no—
157#508—1732no—
158C2×M32—1731no—
159C2×#41—3461no—
160C2×#41—3461no—
161#135—3461no—
162#135—3461no—
163#135—3461no—
164#135—3461no—
165#135—1732no—
166#135—1731no—
167#135—1732no—
168#135—1731no—
169#135—1731no—
170#135C2×D17311yescharacteristic
171C2×Q16—1731no—
172C2×Q16C2×D17311yescharacteristic, complemented
173C2×SD16—3461no—
174C2×SD16—3461no—
175C2×D8—1731no—
176C2×D8—1731no—
177C22×C8—1731no—
178Q32—1732no—
179Q32C2×D17312yescomplemented
180SD32—1732no—
181SD32—1732no—
182SD32—1732no—
183SD32C2×D17312yescomplemented
184D16—1732no—
185D16—1732no—
186M32—1731no—
187M32—1732no—
188M32C2×D17311yescharacteristic, complemented
189C2×C16—1732no—
190#41—21no—
191#41—3461no—
192#41—3462no—
193#41—3461no—
194#41—3462no—
195#41—3461no—
196#41—6921no—
197#41—6921no—
198#41—6921no—
199#41—6921no—
200C2×Q8—21no—
201C2×Q8—3461no—
202C2×D4—3461no—
203C2×D4—3461no—
204C2×D4—6921no—
205C2×D4—6921no—
206C22×C4—1731no—
207C22×C4—6921no—
208C22×C4—6921no—
209Q16—21no—
210Q16—3461no—
211Q16—1731no—
212Q16C22×D17311yescharacteristic
213Q16C22×D17312yes—
214SD16—21no—
215SD16—3461no—
216SD16—3462no—
217SD16—3461no—
218SD16—3462no—
219SD16—3461no—
220D8—3461no—
221D8—3461no—
222D8—1732no—
223D8—1731no—
224D8C22×D17311yescharacteristic
225C2×C8—1731no—
226C2×C8—1731no—
227C2×C8—1731no—
228C2×C8—1731no—
229C2×C8—1731no—
230C2×C8C22×D17311yescharacteristic
231C16—1732no—
232C16C22×D17312yes—
233C23—1731no—
234C23—6921no—
235C23—6921no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—3461no—
240Q8—6921no—
241D4—21no—
242D4—41no—
243D4—3461no—
244D4—3462no—
245D4—6921no—
246D4—6922no—
247D4—6922no—
248D4—6921no—
249D4—6921no—
250D4—6921no—
251D4—6921no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1731no—
255C2×C4—1731no—
256C2×C4—1731no—
257C2×C4—1731no—
258C2×C4—1731no—
259C2×C4—6921no—
260C2×C4—6922no—
261C2×C4—6922no—
262C2×C4—6921no—
263C2×C4—6921no—
264C2×C4—6921no—
265C2×C4D4×D17311yescharacteristic
266C8—1731no—
267C8—1731no—
268C8D4×D17311yescharacteristic
269C8C23×D17311yescharacteristic
270C22—41no—
271C22—13842no—
272C22—13842no—
273C22—1731no—
274C22—1731no—
275C22—6921no—
276C22—6922no—
277C22—6921no—
278C22D8×D17311yescharacteristic
279C4—41no—
280C4—42no—
281C4—1731no—
282C4—1731no—
283C4—6921no—
284C4D8×D17311yescharacteristic
285C4C2×D4×D17311yescharacteristic
286C2—21no—
287C2—81no—
288C2—3461no—
289C2—3461no—
290C2—13842no—
291C2—13841no—
292C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC2C2×D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#510×C173C2
socleC2×C173C22×D8
hypercenter#510D173
hypocenterC173#3414
LCSG ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#19149702C2semidirect
2. #135×D173C2semidirect
3. #26011933C2semidirect
4. #26011932C2semidirect
5. #26011929C2semidirect
6. #26011928C2semidirect
7. #26011926C2semidirect
8. #26011924C2semidirect
9. #26011922C2semidirect
10. M32×D173C2semidirect
11. #510×C173C2semidirect
17. #19149702C22semidirect
18. #19149702C22semidirect
22. #19149698C22semidirect
23. Q16×D173C22semidirect
26. C2×D1384C22semidirect
27. D8×D173C22semidirect
29. #19149625C22semidirect
30. #19149624C22semidirect
31. #19149622C22semidirect
32. #19149622C22semidirect
33. D2768C22semidirect
34. #19149620C22semidirect
35. #19149619C22semidirect
36. #19149618C22semidirect
37. C16×D173C22semidirect
39. C2×Q16×C173C22semidirect
40. Q32×C173C22semidirect
41. SD32×C173C22semidirect
42. M32×C173C22semidirect
146. C173#3414semidirect
152. #510D173semidirect
172. C2×Q16C2×D173semidirect
179. Q32C2×D173semidirect
183. SD32C2×D173semidirect
188. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497022
#135×D1732
#260119332
#260119322
#260119292
#260119282
#260119262
#260119242
#260119222
M32×D1732
#510×C1732
#3414173