Finite Group Databasebeta

← #36365340

Subgroups — #36365340

Order 22144.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012306C212yesmaximal, complemented
2#26012301C212yesmaximal, complemented
3#26011886C211yesmaximal, characteristic, complemented
4#26011765C212yesmaximal, complemented
5#26011752C212yesmaximal, complemented
6#26011675C212yesmaximal, complemented
7#26011668C211yesmaximal, characteristic, complemented
8#26011610C212yesmaximal, complemented
9#454×C173C211yesmaximal, characteristic, complemented
10#19149726C2212yes—
11#19149724C2211yescharacteristic
12#19149712C2212yes—
13#19149710—22no—
14#19149710C2212yes—
15#19149709—22no—
16#19149709C2212yescomplemented
17#19149708—22no—
18#19149703—22no—
19#19149699—22no—
20#19149699C2212yes—
21#19149698—22no—
22C2×#14583816—22no—
23C2×#14583816C2212yescomplemented
24C2×#14583814C2212yes—
25SD16×D173—22no—
26#19149687C2212yes—
27#19149686C2212yescomplemented
28#19149679C2211yescharacteristic, complemented
29#19149614C2212yescomplemented
30#19149607C2212yes—
31#19149597C2212yescomplemented
32#19149596C2211yescharacteristic
33#19149590C2211yescharacteristic, complemented
342−1+4×C173C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C173C2212yes—
36#124×C173C2211yescharacteristic
37#104×C173C2212yescomplemented
38#101×C173C2211yescharacteristic, complemented
39#14583826—22no—
40#14583826C2312yes—
41#14583825—22no—
42#14583825—22no—
43#14583825—42no—
44C2×Dic346—22no—
45C2×Dic346C2312yes—
46Q8×D173—22no—
47C2×C22⋊D173—41no—
48D4×D173—22no—
49Dic692—22no—
50#14583816—22no—
51#14583816—22no—
52#14583816—22no—
53#14583816—22no—
54#14583816D412yes—
55#14583815—22no—
56#14583814—22no—
57#14583814—22no—
58#14583814—22no—
59#14583814D412yes—
60#14583813—22no—
61#14583813D412yes—
62#14583811D412yes—
63#14583810—22no—
64#14583810C2312yes—
65#14583809—22no—
66C2×#13063436—22no—
67C2×#13063436C2312yes—
68C8×D173—22no—
69C4⋊Dic173—41no—
70#41×C173—22no—
71#41×C173—41no—
72#41×C173C2312yes—
73C2×Q8×C173—22no—
74C2×Q8×C173C2311yescharacteristic
75C2×D4×C173C2311yescharacteristic
76SD16×C173—22no—
77D8×C173—22no—
78M16×C173C2312yes—
79C22⋊C4×C173—41no—
80C42×C173C2311yescharacteristic
81C22×D173—41no—
82Dic346—22no—
83Dic346—22no—
84Dic346—42no—
85Dic346C2×D412yes—
86D692C2×D412yes—
87C22⋊D173—22no—
88C22⋊D173—42no—
89C22⋊D173—42no—
90C2×Dic173—22no—
91C2×Dic173—41no—
92C2×Dic173—42no—
93C4×D173—22no—
94C4×D173—42no—
95#13063436—22no—
96#13063436C2×D412yes—
97#13063436C2×D412yes—
98C23×C173—21no—
99Q8×C173—22no—
100Q8×C173—41no—
101Q8×C173C2×D412yes—
102D4×C173—22no—
103D4×C173—22no—
104D4×C173—41no—
105D4×C173C2×D412yes—
106C2×C4×C173—22no—
107C2×C4×C173—21no—
108C2×C4×C173—21no—
109C2×C4×C173—42no—
110C2×C4×C173C2411yescharacteristic
111C8×C173—22no—
112C22×C173—22no—
113C22×C173—41no—
114C22×C173—81no—
115C22×C173H(5,𝔽2)11yescharacteristic
116C2×D173—22no—
117C2×D173—161no—
118C4×C173—22no—
119C4×C173—41no—
120C4×C173—41no—
121C4×C173C22×D412yes—
122Dic173—22no—
123Dic173—42no—
124C2×C173—21no—
125C2×C173—42no—
126C2×C173—81no—
127C2×C173D4211yescharacteristic
128D173—42no—
129C173#328911yescharacteristic, complemented, Sylow
130#3289—1731nomaximal, Sylow
131#578—1732no—
132#577—1732no—
133#497—1731no—
134#473—1732no—
135#471—1732no—
136#456—1732no—
137#455—1731no—
138#454D17311yescharacteristic, complemented
139#444—1732no—
1402−1+4—1732no—
1412−1+4C2×D17311yescharacteristic, complemented
142H(5,𝔽2)—1731no—
143#137—3462no—
144#137—3462no—
145#137—3462no—
146#137—1732no—
147#137—1732no—
148#137—1732no—
149AGL(1,ℤ/8)—3462no—
150AGL(1,ℤ/8)C2×D17312yes—
151#135—3462no—
152#135—3462no—
153#135—1732no—
154C2×Q16—3462no—
155C2×Q16—1732no—
156C2×SD16—3462no—
157C2×SD16—1732no—
158#131—1732no—
159#131—1732no—
160C4⋊Q8—1731no—
161#124C2×D17311yescharacteristic
162#108—1732no—
163#104—1732no—
164#104C2×D17312yescomplemented
165#101—1732no—
166#101C2×D17311yescharacteristic, complemented
167#100—1731no—
168#97—1731no—
169#41—22no—
170#41—41no—
171#41—3462no—
172#41—3462no—
173#41—3462no—
174#41—1732no—
175#41—6922no—
176#41C22×D17312yes—
177C2×Q8—22no—
178C2×Q8—3462no—
179C2×Q8—3462no—
180C2×Q8—1732no—
181C2×Q8C22×D17311yescharacteristic
182C2×D4—3462no—
183C2×D4—6921no—
184C2×D4C22×D17311yescharacteristic
185Q16—3462no—
186Q16—3462no—
187Q16—3462no—
188Q16—3462no—
189Q16—3462no—
190Q16—1732no—
191SD16—22no—
192SD16—3462no—
193SD16—3462no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—1732no—
198SD16—1732no—
199D8—22no—
200D8—1732no—
201M16—3462no—
202M16—3462no—
203M16—1732no—
204M16C22×D17312yes—
205C2×C8—3462no—
206C2×C8—3462no—
207C2×C8—1732no—
208C4⋊C4—6921no—
209C22⋊C4—41no—
210C42C22×D17311yescharacteristic
211C23—21no—
212C23—6921no—
213Q8—22no—
214Q8—41no—
215Q8—3462no—
216Q8—3462no—
217Q8—1732no—
218Q8—6922no—
219Q8D4×D17312yes—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—3462no—
224D4—1732no—
225D4—6922no—
226D4—6922no—
227D4D4×D17312yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—3462no—
233C2×C4—3462no—
234C2×C4—6922no—
235C2×C4—6921no—
236C2×C4—6922no—
237C2×C4C23×D17311yescharacteristic
238C8—22no—
239C8—3462no—
240C8—1732no—
241C8—1732no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—3462no—
246C22—27681no—
247C22#1914972411yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—3462no—
252C4—6922no—
253C4C2×D4×D17312yes—
254C2—21no—
255C2—42no—
256C2—81no—
257C2—6922no—
258C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#454×C173C2
socleC2×C173D42
hypercenter#454D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012306C2semidirect
2. #26012301C2semidirect
3. #26011886C2semidirect
4. #26011765C2semidirect
5. #26011752C2semidirect
6. #26011675C2semidirect
7. #26011668C2semidirect
8. #26011610C2semidirect
9. #454×C173C2semidirect
16. #19149709C22semidirect
23. C2×#14583816C22semidirect
27. #19149686C22semidirect
28. #19149679C22semidirect
29. #19149614C22semidirect
31. #19149597C22semidirect
33. #19149590C22semidirect
34. 2−1+4×C173C22semidirect
37. #104×C173C22semidirect
38. #101×C173C22semidirect
129. C173#3289semidirect
138. #454D173semidirect
141. 2−1+4C2×D173semidirect
164. #104C2×D173semidirect
166. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123062
#260123012
#260118862
#260117652
#260117522
#260116752
#260116682
#260116102
#454×C1732
#3289173