Finite Group Databasebeta

← #36366706

Subgroups — #36366706

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#26012305C211yesmaximal, characteristic, complemented
2#26012293C211yesmaximal, characteristic, complemented
3#26011933C212yesmaximal, complemented
4#26011931C211yesmaximal, characteristic, complemented
5#26011926C211yesmaximal, characteristic, complemented
6#26011925C212yesmaximal, complemented
7#26011922C212yesmaximal, complemented
8C2×#19149624C211yesmaximal, characteristic, complemented
9SD32×D173C212yesmaximal, complemented
10#26011907C211yesmaximal, characteristic, complemented
11#508×C173C211yesmaximal, characteristic, complemented
12#19149726—21no—
13#19149725—21no—
14#19149709—21no—
15#19149708—22no—
16#19149706—22no—
17#19149704—21no—
18#19149702C2212yes—
19#19149698C2212yescomplemented
20C2×Dic692C2211yescharacteristic, complemented
21Q16×D173C2212yescomplemented
22C2×D1384C2211yescharacteristic
23D8×D173C2212yes—
24#19149685C2211yescharacteristic
25#19149625C2212yescomplemented
26#19149624C2212yescomplemented
27#19149624C2212yescomplemented
28#19149623C2212yescomplemented
29#19149622C2212yescomplemented
30#19149620C2212yescomplemented
31#19149619C2211yescharacteristic, complemented
32#19149618C2212yescomplemented
33C16×D173C2212yescomplemented
34#135×C173C2211yescharacteristic
35#135×C173C2211yescharacteristic, complemented
36Q32×C173C2211yescharacteristic, complemented
37SD32×C173C2211yescharacteristic, complemented
38SD32×C173C2211yescharacteristic, complemented
39D16×C173C2211yescharacteristic, complemented
40C2×C16×C173C2211yescharacteristic, complemented
41#14583827—22no—
42#14583826—41no—
43#14583826—41no—
44#14583826D411yescharacteristic
45#14583825—22no—
46#14583825—42no—
47C2×Dic346—21no—
48C2×Dic346—41no—
49Q8×D173—22no—
50C2×D692—21no—
51C2×D692—41no—
52D4×D173—22no—
53D4×D173—42no—
54Dic692—21no—
55Dic692C2312yes—
56#14583816—22no—
57#14583814—22no—
58#14583813—22no—
59D1384—21no—
60D1384C2312yes—
61#14583811—22no—
62#14583810D411yescharacteristic
63#14583809D412yes—
64C8×D173C2312yes—
65#14583801C2312yes—
66#41×C173—21no—
67#41×C173—21no—
68Q16×C173C2311yescharacteristic
69Q16×C173C2311yescharacteristic
70SD16×C173—21no—
71SD16×C173—21no—
72D8×C173C2311yescharacteristic
73D8×C173C2311yescharacteristic
74C2×C8×C173C2311yescharacteristic
75C16×C173C2311yescharacteristic
76C16×C173C2311yescharacteristic
77C22×D173—41no—
78C22×D173—42no—
79Dic346—22no—
80Dic346—41no—
81Dic346—42no—
82Dic346D811yescharacteristic
83D692—22no—
84D692—42no—
85D692—41no—
86D692D811yescharacteristic
87C22⋊D173—42no—
88C22⋊D173—42no—
89C22⋊D173D812yes—
90C2×Dic173—41no—
91C2×Dic173—42no—
92C4×D173—42no—
93C4×D173—42no—
94C4×D173C2×D412yes—
95#13063436C2×D412yes—
96Q8×C173—21no—
97Q8×C173—21no—
98D4×C173—21no—
99D4×C173—21no—
100D4×C173—41no—
101D4×C173—41no—
102C2×C4×C173—41no—
103C2×C4×C173—41no—
104C2×C4×C173C2×D411yescharacteristic
105C8×C173C2×D411yescharacteristic
106C8×C173C2411yescharacteristic
107C22×C173—41no—
108C22×C173—41no—
109C22×C173C2×D811yescharacteristic
110C2×D173—42no—
111C2×D173—162no—
112C2×D173—161no—
113C2×D173C2×D812yes—
114C4×C173—41no—
115C4×C173—41no—
116C4×C173C2×D811yescharacteristic
117C4×C173C22×D411yescharacteristic
118Dic173—42no—
119Dic173C2×D812yes—
120C2×C173—21no—
121C2×C173—81no—
122C2×C173—81no—
123C2×C173C22×D811yescharacteristic
124D173—22no—
125D173—82no—
126C173#341611yescharacteristic, complemented, Sylow
127#3416—1731nomaximal, Sylow
128#578—1731no—
129#576—1731no—
130#510—1731no—
131#510—1732no—
132#509—1731no—
133#509—1732no—
134#508—1732no—
135#508D17311yescharacteristic, complemented
136C2×SD32—1732no—
137C2×SD32—1731no—
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#135C2×D17311yescharacteristic
148#135C2×D17311yescharacteristic, complemented
149C2×Q16—1732no—
150C2×Q16—1731no—
151C2×D8—1732no—
152C2×D8—1731no—
153#131—1731no—
154Q32—1732no—
155Q32C2×D17311yescharacteristic, complemented
156SD32—1732no—
157SD32—1732no—
158SD32—1732no—
159SD32—1732no—
160SD32C2×D17311yescharacteristic, complemented
161SD32C2×D17311yescharacteristic, complemented
162D16—1732no—
163D16C2×D17311yescharacteristic, complemented
164M32—1731no—
165M32—1732no—
166C2×C16—1732no—
167C2×C16C2×D17311yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—3462no—
171#41—3462no—
172#41—1731no—
173#41—6921no—
174#41—6921no—
175#41—6922no—
176C2×Q8—3462no—
177C2×Q8—3461no—
178C2×Q8—6921no—
179C2×D4—3462no—
180C2×D4—3461no—
181C2×D4—6921no—
182C2×D4—6922no—
183Q16—3462no—
184Q16—3461no—
185Q16—1732no—
186Q16C22×D17311yescharacteristic
187Q16C22×D17311yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—3462no—
191SD16—3462no—
192D8—3462no—
193D8—3461no—
194D8—1732no—
195D8C22×D17311yescharacteristic
196D8C22×D17311yescharacteristic
197M16—1731no—
198M16—1732no—
199C2×C8—1732no—
200C2×C8C22×D17311yescharacteristic
201C16—1732no—
202C16C22×D17311yescharacteristic
203C16C22×D17311yescharacteristic
204C23—6922no—
205C23—6921no—
206Q8—21no—
207Q8—21no—
208Q8—3462no—
209Q8—1731no—
210Q8—6922no—
211Q8—6921no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3462no—
217D4—1732no—
218D4—1731no—
219D4—6922no—
220D4—6922no—
221D4—6922no—
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—41no—
236C22—1732no—
237C22—6922no—
238C22—27681no—
239C22—27682no—
240C22D8×D17311yescharacteristic
241C4—41no—
242C4—41no—
243C4—1732no—
244C4—6922no—
245C4D8×D17311yescharacteristic
246C4C2×D4×D17311yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—3462no—
251C2—13842no—
252C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC2C2×D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#508×C173C2
socleC2×C173C22×D8
hypercenter#508D173
hypocenterC173#3416
LCSG ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #26012305C2semidirect
2. #26012293C2semidirect
3. #26011933C2semidirect
4. #26011931C2semidirect
5. #26011926C2semidirect
6. #26011925C2semidirect
7. #26011922C2semidirect
8. C2×#19149624C2semidirect
9. SD32×D173C2semidirect
10. #26011907C2semidirect
11. #508×C173C2semidirect
19. #19149698C22semidirect
20. C2×Dic692C22semidirect
21. Q16×D173C22semidirect
25. #19149625C22semidirect
26. #19149624C22semidirect
27. #19149624C22semidirect
28. #19149623C22semidirect
29. #19149622C22semidirect
30. #19149620C22semidirect
31. #19149619C22semidirect
32. #19149618C22semidirect
33. C16×D173C22semidirect
35. #135×C173C22semidirect
36. Q32×C173C22semidirect
37. SD32×C173C22semidirect
38. SD32×C173C22semidirect
39. D16×C173C22semidirect
40. C2×C16×C173C22semidirect
126. C173#3416semidirect
135. #508D173semidirect
148. #135C2×D173semidirect
155. Q32C2×D173semidirect
160. SD32C2×D173semidirect
161. SD32C2×D173semidirect
163. D16C2×D173semidirect
167. C2×C16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123052
#260122932
#260119332
#260119312
#260119262
#260119252
#260119222
C2×#191496242
SD32×D1732
#260119072
#508×C1732
#3416173