Finite Group Databasebeta

← #36365338

Subgroups — #36365338

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#26012304C212yesmaximal, complemented
2#26012297C212yesmaximal, complemented
3#26011763C212yesmaximal, complemented
4#26011751C212yesmaximal, complemented
5#26011673C212yesmaximal, complemented
6#26011668C211yesmaximal, characteristic, complemented
7#26011663C211yesmaximal, characteristic, complemented
8#26011614C212yesmaximal, complemented
9#497×C173C211yesmaximal, characteristic, complemented
10#19149727C2212yes—
11#19149727C2211yescharacteristic, complemented
12#19149724C2211yescharacteristic
13#19149711—22no—
14#19149711—22no—
15#19149710C2212yes—
16#19149710C2212yes—
17#19149710C2212yescomplemented
18#19149708—22no—
19#19149706—22no—
20#19149705C2212yes—
21#19149702—22no—
22#19149701C2212yes—
23#19149700—22no—
24Q16×D173—22no—
25SD16×D173—22no—
26#19149685C2212yes—
27#19149685C2212yescomplemented
28#19149615C2212yescomplemented
29#19149607C2212yes—
30#19149607C2212yescomplemented
31#19149598C2211yescharacteristic, complemented
32#19149598C2212yescomplemented
33#19149596C2211yescharacteristic
34C2×Q16×C173C2212yescomplemented
35C2×SD16×C173C2212yes—
36C4⋊Q8×C173C2211yescharacteristic, complemented
37#124×C173C2211yescharacteristic
38#97×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583827—22no—
41#14583827—22no—
42#14583826—42no—
43#14583826—41no—
44#14583826C2312yes—
45#14583826C2312yes—
46#14583825—22no—
47Q8×D173—22no—
48Q8×D173—22no—
49Q8×D173—22no—
50C2×C22⋊D173—41no—
51D4×D173—22no—
52Dic692D412yes—
53#14583816—22no—
54#14583816—22no—
55#14583816—22no—
56#14583815D412yes—
57#14583815D412yes—
58#14583814—22no—
59#14583813—22no—
60#14583813—22no—
61#14583813—22no—
62D1384D412yes—
63#14583811—22no—
64#14583810C2312yes—
65#14583810C2312yes—
66#14583809—22no—
67#14583809—22no—
68C8×D173—22no—
69C8×D173—22no—
70C2×Q8×C173C2312yes—
71C2×Q8×C173C2311yescharacteristic
72C2×D4×C173C2311yescharacteristic
73Q16×C173—22no—
74Q16×C173—22no—
75SD16×C173—22no—
76SD16×C173—22no—
77C2×C8×C173C2312yes—
78C4⋊C4×C173—41no—
79C22⋊C4×C173—41no—
80C42×C173C2311yescharacteristic
81C22×D173—41no—
82Dic346—41no—
83Dic346—42no—
84Dic346C2×D412yes—
85Dic346C2×D412yes—
86D692—41no—
87D692—42no—
88D692C2×D412yes—
89D692C2×D412yes—
90C22⋊D173—22no—
91C22⋊D173—22no—
92C22⋊D173—42no—
93C2×Dic173—41no—
94C4×D173—22no—
95C4×D173—22no—
96C4×D173—42no—
97C4×D173—42no—
98C4×D173—42no—
99#13063436—22no—
100#13063436—22no—
101C23×C173—21no—
102Q8×C173—22no—
103Q8×C173—22no—
104Q8×C173—22no—
105D4×C173—22no—
106C2×C4×C173—21no—
107C2×C4×C173—22no—
108C2×C4×C173—21no—
109C2×C4×C173C2411yescharacteristic
110C8×C173C2×D412yes—
111C8×C173C2×D412yes—
112C22×C173—41no—
113C22×C173—81no—
114C22×C173H(5,𝔽2)11yescharacteristic
115C2×D173—22no—
116C2×D173—22no—
117C2×D173—161no—
118C4×C173—41no—
119C4×C173—41no—
120C4×C173—42no—
121C4×C173C22×D412yes—
122Dic173—22no—
123Dic173—22no—
124C2×C173—21no—
125C2×C173—81no—
126C2×C173D4211yescharacteristic
127D173—42no—
128D173—42no—
129C173#328911yescharacteristic, complemented, Sylow
130#3289—1731nomaximal, Sylow
131#578—1732no—
132#577—1732no—
133#497D17311yescharacteristic, complemented
134#473—1732no—
135#471—1732no—
136#456—1732no—
137#455—1731no—
138#454—1731no—
139#444—1732no—
1402−1+4—1732no—
1412−1+4—1731no—
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)—1732no—
151#135—3462no—
152#135—3462no—
153#135—1732no—
154C2×Q16—3462no—
155C2×Q16C2×D17312yescomplemented
156C2×SD16—3462no—
157C2×SD16C2×D17312yes—
158#131—1732no—
159#131—1732no—
160C4⋊Q8C2×D17311yescharacteristic, complemented
161#124C2×D17311yescharacteristic
162#108—1732no—
163#104—1732no—
164#104—1732no—
165#101—1731no—
166#101—1732no—
167#100—1731no—
168#97C2×D17311yescharacteristic, complemented
169#41—3462no—
170#41—3462no—
171#41—3462no—
172#41—3462no—
173#41—1732no—
174#41—1732no—
175#41—6921no—
176#41—6922no—
177C2×Q8—3462no—
178C2×Q8—3462no—
179C2×Q8—3462no—
180C2×Q8C22×D17311yescharacteristic
181C2×Q8C22×D17312yes—
182C2×D4—3462no—
183C2×D4—6921no—
184C2×D4C22×D17311yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—3462no—
188Q16—3462no—
189Q16—3462no—
190Q16—1732no—
191SD16—22no—
192SD16—22no—
193SD16—3462no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—1732no—
198SD16—1732no—
199D8—3462no—
200D8—1732no—
201M16—3462no—
202M16—3462no—
203M16—1732no—
204M16—1732no—
205C2×C8—3462no—
206C2×C8—3462no—
207C2×C8C22×D17312yes—
208C4⋊C4—41no—
209C22⋊C4—41no—
210C42C22×D17311yescharacteristic
211C23—21no—
212C23—6921no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—1732no—
217Q8—1732no—
218Q8—6922no—
219Q8—6921no—
220D4—22no—
221D4—3462no—
222D4—3462no—
223D4—1732no—
224D4—1732no—
225D4—6921no—
226D4—6922no—
227D4—6922no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—3462no—
232C2×C4—3462no—
233C2×C4—6922no—
234C2×C4—6922no—
235C2×C4—6922no—
236C2×C4—6921no—
237C2×C4C23×D17311yescharacteristic
238C8—3462no—
239C8—3462no—
240C8D4×D17312yes—
241C8D4×D17312yes—
242C22—41no—
243C22—81no—
244C22—3462no—
245C22—3462no—
246C22—27681no—
247C22#1914972411yescharacteristic
248C4—41no—
249C4—41no—
250C4—42no—
251C4—3462no—
252C4—3462no—
253C4C2×D4×D17312yes—
254C2—21no—
255C2—81no—
256C2—6922no—
257C2—6922no—
258C2#2601212611yescharacteristic
subgroupG/H
centerC2#26012126
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC2×C173D42
hypercenter#497D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26012304C2semidirect
2. #26012297C2semidirect
3. #26011763C2semidirect
4. #26011751C2semidirect
5. #26011673C2semidirect
6. #26011668C2semidirect
7. #26011663C2semidirect
8. #26011614C2semidirect
9. #497×C173C2semidirect
11. #19149727C22semidirect
17. #19149710C22semidirect
27. #19149685C22semidirect
28. #19149615C22semidirect
30. #19149607C22semidirect
31. #19149598C22semidirect
32. #19149598C22semidirect
34. C2×Q16×C173C22semidirect
36. C4⋊Q8×C173C22semidirect
38. #97×C173C22semidirect
129. C173#3289semidirect
133. #497D173semidirect
155. C2×Q16C2×D173semidirect
160. C4⋊Q8C2×D173semidirect
168. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123042
#260122972
#260117632
#260117512
#260116732
#260116682
#260116632
#260116142
#497×C1732
#3289173