Finite Group Databasebeta

← #36362926

Subgroups — #36362926

Order 22144.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#19149727C211yesmaximal, characteristic, complemented
2#26012289C212yesmaximal, complemented
3#26011673C214yesmaximal, complemented
4#26011663C214yesmaximal, complemented
5#26011509C212yesmaximal, complemented
6C2×#19149598C211yesmaximal, characteristic, complemented
7#536×C173C211yesmaximal, characteristic, complemented
8#19149727—22no—
9#19149727—24no—
10#19149727C2214yes—
11C2×#14583827—22no—
12C2×#14583826—22no—
13C2×#14583826C2212yes—
14C2×Q8×D173—22no—
15#19149710C2214yescomplemented
16#19149709—22no—
17#19149705C2214yescomplemented
18#19149704—22no—
19#19149699—24no—
20C2×#14583810C2212yes—
21#19149607C2218yescomplemented
22#19149598C2214yescomplemented
23C2×#41×C173C2211yescharacteristic
24C22×Q8×C173C2211yescharacteristic
25C4⋊Q8×C173C2212yescomplemented
26#124×C173C2212yescomplemented
27#123×C173—22no—
28C22⋊Q8×C173—22no—
29#117×C173C2211yescharacteristic
30#14583827—28no—
31#14583827—28no—
32#14583826—28no—
33#14583826—28no—
34#14583826—22no—
35#14583826—22no—
36#14583826D414yes—
37#14583826C2314yes—
38C2×Dic346—22no—
39C2×Dic346D412yes—
40Q8×D173—28no—
41Q8×D173—28no—
42C2×D692—22no—
43C2×D692D412yes—
44C2×C22⋊D173—22no—
45C2×C4×D173—24no—
46C2×C4×D173—22no—
47#14583816—24no—
48#14583814—24no—
49#14583813—24no—
50#14583811—24no—
51#14583810—22no—
52#14583810C2314yes—
53C2×#13063436—22no—
54#41×C173—22no—
55#41×C173—22no—
56C2×Q8×C173—22no—
57C2×Q8×C173—22no—
58C2×Q8×C173D412yes—
59C2×Q8×C173C2311yescharacteristic
60C2×Q8×C173C2312yes—
61C2×D4×C173—21no—
62C2×D4×C173C2311yescharacteristic
63C22×C4×C173—21no—
64C22×C4×C173D412yes—
65C22×C4×C173C2311yescharacteristic
66C4⋊C4×C173—24no—
67C4⋊C4×C173—21no—
68C22⋊C4×C173—21no—
69C22⋊C4×C173—24no—
70C42×C173C2312yes—
71C22×D173—22no—
72Dic346—28no—
73Dic346—22no—
74Dic346C2×D414yes—
75D692—28no—
76D692—22no—
77D692C2×D414yes—
78C22⋊D173—24no—
79C22⋊D173—24no—
80C2×Dic173—22no—
81C4×D173—216no—
82C4×D173—24no—
83C4×D173—24no—
84#13063436—24no—
85C23×C173—21no—
86C23×C173C2×D411yescharacteristic
87Q8×C173—24no—
88Q8×C173—24no—
89Q8×C173—22no—
90D4×C173—22no—
91D4×C173—41no—
92D4×C173—41no—
93C2×C4×C173—22no—
94C2×C4×C173—22no—
95C2×C4×C173—22no—
96C2×C4×C173—21no—
97C2×C4×C173—42no—
98C2×C4×C173C2×D412yes—
99C2×C4×C173C2×D412yes—
100C2×C4×C173C2×D414yes—
101C2×C4×C173C2×D411yescharacteristic
102C2×C4×C173C2411yescharacteristic
103C22×C173—41no—
104C22×C173—42no—
105C22×C173—41no—
106C22×C173#12011yescharacteristic
107C22×C173#12011yescharacteristic
108C22×C173C22×D411yescharacteristic
109C2×D173—24no—
110C2×D173—44no—
111C4×C173—24no—
112C4×C173—42no—
113C4×C173—41no—
114C4×C173#12012yes—
115C4×C173C22×D412yes—
116Dic173—24no—
117C2×C173—21no—
118C2×C173—21no—
119C2×C173—21no—
120C2×C173—81no—
121C2×C173C2×#12011yescharacteristic
122D173—44no—
123C173#302011yescharacteristic, complemented, Sylow
124#3020—1731nomaximal, Sylow
125C2×2−1+4—1731no—
126#575—1732no—
127#536D17311yescharacteristic, complemented
128#456—1734no—
129#454—1734no—
130#421—1732no—
131C2×#101—1731no—
1322−1+4—3464no—
1332−1+4—3462no—
1342−1+4—1734no—
135C2×#41—3462no—
136C2×#41—3462no—
137C2×#41—1732no—
138C2×#41C2×D17311yescharacteristic
139C22×Q8—3462no—
140C22×Q8C2×D17311yescharacteristic
141#137—3462no—
142#137—1734no—
143AGL(1,ℤ/8)—3462no—
144AGL(1,ℤ/8)—1734no—
145#135—3464no—
146C2×M16—1732no—
147C4⋊Q8C2×D17312yescomplemented
148#124C2×D17312yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D17311yescharacteristic
152#104—1738no—
153#101—1734no—
154#41—22no—
155#41—22no—
156#41—3468no—
157#41—3468no—
158#41—3468no—
159#41—3462no—
160#41—3468no—
161#41—3462no—
162#41—1734no—
163#41—1734no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—3462no—
167C2×Q8—3468no—
168C2×Q8—3468no—
169C2×Q8—1732no—
170C2×Q8C22⋊D17312yes—
171C2×Q8C22×D17311yescharacteristic
172C2×Q8C22×D17312yes—
173C2×D4—21no—
174C2×D4—3462no—
175C2×D4—3462no—
176C2×D4—1732no—
177C2×D4C22×D17311yescharacteristic
178C22×C4—21no—
179C22×C4—3464no—
180C22×C4—3462no—
181C22×C4C22⋊D17312yes—
182C22×C4C22×D17311yescharacteristic
183Q16—3464no—
184SD16—3464no—
185SD16—3464no—
186D8—3464no—
187M16—3462no—
188M16—1734no—
189C2×C8—3462no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D17312yes—
195C23—21no—
196C23—3462no—
197C23C2×C22⋊D17311yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—3468no—
202Q8—3462no—
203Q8—1734no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—3468no—
208D4—3464no—
209D4—3464no—
210D4—3462no—
211D4—1734no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—34616no—
218C2×C4—3462no—
219C2×C4—3464no—
220C2×C4—3464no—
221C2×C4D4×D17312yes—
222C2×C4D4×D17312yes—
223C2×C4C2×C22⋊D17314yes—
224C2×C4C2×C22⋊D17311yescharacteristic
225C2×C4C23×D17311yescharacteristic
226C8—3464no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—3464no—
231C22—6924no—
232C22#1914964711yescharacteristic
233C22#1914964711yescharacteristic
234C22C22×C22⋊D17311yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—3464no—
239C4#1914964712yes—
240C4C2×D4×D17312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—6924no—
246C2C2×#1914964711yescharacteristic
subgroupG/H
centerC2C2×#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC2×C173C2×#120
hypercenter#536D173
hypocenterC173#3020
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#19149727C2semidirect
2. #26012289C2semidirect
3. #26011673C2semidirect
4. #26011663C2semidirect
5. #26011509C2semidirect
6. C2×#19149598C2semidirect
7. #536×C173C2semidirect
15. #19149710C22semidirect
17. #19149705C22semidirect
21. #19149607C22semidirect
22. #19149598C22semidirect
25. C4⋊Q8×C173C22semidirect
26. #124×C173C22semidirect
123. C173#3020semidirect
127. #536D173semidirect
147. C4⋊Q8C2×D173semidirect
148. #124C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497272
#260122892
#260116732
#260116632
#260115092
C2×#191495982
#536×C1732
#3020173