Finite Group Databasebeta

← #36366708

Subgroups — #36366708

Order 22144.

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

normal subgroups only

This group has 360 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
1#26012304C211yesmaximal, characteristic, complemented
2#26012293C211yesmaximal, characteristic, complemented
3#26011933C212yesmaximal, complemented
4#26011929C211yesmaximal, characteristic, complemented
5#26011928C212yesmaximal, complemented
6#26011924C212yesmaximal, complemented
7#26011921C211yesmaximal, characteristic, complemented
8C2×#19149622C211yesmaximal, characteristic, complemented
9SD32×D173C212yesmaximal, complemented
10#26011909C211yesmaximal, characteristic, complemented
11#510×C173C211yesmaximal, characteristic, complemented
12#19149727—21no—
13#19149725—21no—
14#19149711—22no—
15#19149710—21no—
16#19149706—22no—
17#19149704—21no—
18#19149702C2212yes—
19#19149702C2212yescomplemented
20#19149701C2211yescharacteristic, complemented
21Q16×D173C2212yescomplemented
22C2×D1384C2211yescharacteristic
23D8×D173C2212yes—
24#19149685C2211yescharacteristic
25#19149625C2211yescharacteristic, complemented
26#19149624C2212yescomplemented
27#19149623C2211yescharacteristic, complemented
28#19149622C2212yescomplemented
29#19149622C2212yescomplemented
30#19149622C2211yescharacteristic, complemented
31D2768C2212yescomplemented
32#19149620C2211yescharacteristic, complemented
33#19149618C2212yescomplemented
34C2×#14583801C2211yescharacteristic, complemented
35C16×D173C2212yescomplemented
36#135×C173C2211yescharacteristic
37C2×Q16×C173C2211yescharacteristic, complemented
38Q32×C173C2212yescomplemented
39SD32×C173C2212yescomplemented
40M32×C173C2211yescharacteristic, complemented
41#14583827—22no—
42#14583827—22no—
43#14583827—41no—
44#14583826—21no—
45#14583826—41no—
46#14583826—42no—
47#14583826D411yescharacteristic
48Q8×D173—22no—
49Q8×D173—41no—
50C2×D692—21no—
51C2×D692—41no—
52D4×D173—22no—
53D4×D173—42no—
54Dic692C2311yescharacteristic
55#14583816—22no—
56#14583815—21no—
57#14583813—22no—
58#14583813—22no—
59D1384—21no—
60D1384C2311yescharacteristic
61D1384C2312yes—
62#14583811—22no—
63#14583810D411yescharacteristic
64#14583809D412yes—
65C8×D173C2312yes—
66#14583801C2311yescharacteristic
67#14583801C2311yescharacteristic
68#41×C173—21no—
69C2×Q8×C173—21no—
70Q16×C173—21no—
71Q16×C173C2312yes—
72Q16×C173C2311yescharacteristic
73SD16×C173—21no—
74D8×C173C2311yescharacteristic
75C2×C8×C173C2311yescharacteristic
76C16×C173C2312yes—
77C22×D173—41no—
78C22×D173—42no—
79Dic346—21no—
80Dic346—42no—
81Dic346D811yescharacteristic
82D692—22no—
83D692—21no—
84D692—42no—
85D692—42no—
86D692—41no—
87D692D811yescharacteristic
88C22⋊D173—41no—
89C22⋊D173—42no—
90C22⋊D173D812yes—
91C4×D173—42no—
92C4×D173—41no—
93C4×D173—42no—
94C4×D173—42no—
95C4×D173C2×D412yes—
96#13063436C2×D412yes—
97Q8×C173—21no—
98Q8×C173—22no—
99Q8×C173—41no—
100D4×C173—21no—
101D4×C173—41no—
102C2×C4×C173—41no—
103C2×C4×C173—41no—
104C2×C4×C173C2×D411yescharacteristic
105C8×C173C2×D411yescharacteristic
106C8×C173C2411yescharacteristic
107C22×C173—41no—
108C22×C173C2×D811yescharacteristic
109C2×D173—41no—
110C2×D173—42no—
111C2×D173—162no—
112C2×D173—161no—
113C2×D173C2×D812yes—
114C4×C173—41no—
115C4×C173—42no—
116C4×C173C2×D811yescharacteristic
117C4×C173C22×D411yescharacteristic
118Dic173—41no—
119Dic173C2×D812yes—
120C2×C173—21no—
121C2×C173—81no—
122C2×C173C22×D811yescharacteristic
123D173—22no—
124D173—81no—
125D173—82no—
126C173#341611yescharacteristic, complemented, Sylow
127#3416—1731nomaximal, Sylow
128#578—1731no—
129#576—1731no—
130#510—1732no—
131#510D17311yescharacteristic, complemented
132#509—1732no—
133#509—1731no—
134#508—1731no—
135#508—1732no—
136C2×SD32—1731no—
137C2×SD32—1732no—
138#504—1731no—
1392−1+4—3461no—
140H(5,𝔽2)—3461no—
141#137—3462no—
142#137—3461no—
143AGL(1,ℤ/8)—3462no—
144AGL(1,ℤ/8)—3461no—
145#135—1732no—
146#135—1732no—
147#135—1731no—
148#135C2×D17311yescharacteristic
149C2×Q16—1732no—
150C2×Q16C2×D17311yescharacteristic, complemented
151C2×D8—1732no—
152C2×D8—1731no—
153#131—1731no—
154Q32—1731no—
155Q32C2×D17312yescomplemented
156SD32—1732no—
157SD32—1732no—
158SD32—1731no—
159SD32—1731no—
160SD32—1732no—
161SD32C2×D17312yescomplemented
162D16—1731no—
163D16—1732no—
164M32—1732no—
165M32C2×D17311yescharacteristic, complemented
166C2×C16—1731no—
167C2×C16—1732no—
168#41—21no—
169#41—3462no—
170#41—3462no—
171#41—3461no—
172#41—1731no—
173#41—6922no—
174#41—6921no—
175#41—6921no—
176C2×Q8—21no—
177C2×Q8—3462no—
178C2×Q8—6921no—
179C2×D4—3462no—
180C2×D4—3461no—
181C2×D4—6921no—
182C2×D4—6922no—
183Q16—21no—
184Q16—3462no—
185Q16—1731no—
186Q16C22×D17312yes—
187Q16C22×D17311yescharacteristic
188SD16—21no—
189SD16—3462no—
190SD16—3462no—
191SD16—3461no—
192D8—3462no—
193D8—3461no—
194D8—1731no—
195D8—1732no—
196D8C22×D17311yescharacteristic
197M16—1731no—
198M16—1732no—
199C2×C8—1732no—
200C2×C8C22×D17311yescharacteristic
201C16—1731no—
202C16—1731no—
203C16C22×D17312yes—
204C23—6922no—
205C23—6921no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—3461no—
210Q8—1731no—
211Q8—6922no—
212D4—21no—
213D4—41no—
214D4—3462no—
215D4—3461no—
216D4—1732no—
217D4—1731no—
218D4—6922no—
219D4—6922no—
220D4—6922no—
221D4—6921no—
222D4—6921no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1732no—
226C2×C4—6922no—
227C2×C4—6922no—
228C2×C4—6922no—
229C2×C4—6921no—
230C2×C4D4×D17311yescharacteristic
231C8—1732no—
232C8D4×D17311yescharacteristic
233C8C23×D17311yescharacteristic
234C22—41no—
235C22—1732no—
236C22—6921no—
237C22—6922no—
238C22—27681no—
239C22—27682no—
240C22D8×D17311yescharacteristic
241C4—41no—
242C4—42no—
243C4—1732no—
244C4—6921no—
245C4D8×D17311yescharacteristic
246C4C2×D4×D17311yescharacteristic
247C2—21no—
248C2—81no—
249C2—3462no—
250C2—13841no—
251C2—13842no—
252C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC2C2×D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#510×C173C2
socleC2×C173C22×D8
hypercenter#510D173
hypocenterC173#3416
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. #26012304C2semidirect
2. #26012293C2semidirect
3. #26011933C2semidirect
4. #26011929C2semidirect
5. #26011928C2semidirect
6. #26011924C2semidirect
7. #26011921C2semidirect
8. C2×#19149622C2semidirect
9. SD32×D173C2semidirect
10. #26011909C2semidirect
11. #510×C173C2semidirect
19. #19149702C22semidirect
20. #19149701C22semidirect
21. Q16×D173C22semidirect
25. #19149625C22semidirect
26. #19149624C22semidirect
27. #19149623C22semidirect
28. #19149622C22semidirect
29. #19149622C22semidirect
30. #19149622C22semidirect
31. D2768C22semidirect
32. #19149620C22semidirect
33. #19149618C22semidirect
34. C2×#14583801C22semidirect
35. C16×D173C22semidirect
37. C2×Q16×C173C22semidirect
38. Q32×C173C22semidirect
39. SD32×C173C22semidirect
40. M32×C173C22semidirect
126. C173#3416semidirect
131. #510D173semidirect
150. C2×Q16C2×D173semidirect
155. Q32C2×D173semidirect
161. SD32C2×D173semidirect
165. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123042
#260122932
#260119332
#260119292
#260119282
#260119242
#260119212
C2×#191496222
SD32×D1732
#260119092
#510×C1732
#3416173