Finite Group Databasebeta

← #36364266

Subgroups — #36364266

Order 22144.

366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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#26012025C211yesmaximal, characteristic
2#26011897C211yesmaximal, characteristic, complemented
3#26011883C211yesmaximal, characteristic, complemented
4#26011859C211yesmaximal, characteristic, complemented
5#26011853C211yesmaximal, characteristic, complemented
6#26011730C212yesmaximal, complemented
7#26011722C212yesmaximal
8#26011596C212yesmaximal, complemented
9#26011589C212yesmaximal, complemented
10C2×#19149590C211yesmaximal, characteristic, complemented
11#533×C173C211yesmaximal, characteristic, complemented
12C2×#14583816C2212yes—
13C2×#14583813C2212yes—
14C22×#13063436C2212yes—
15#19149673—21no—
16#19149671C2211yescharacteristic
17#19149667C2211yescharacteristic
18#19149662—21no—
19#19149658C2211yescharacteristic
20#19149653C2211yescharacteristic
21C4×Dic346C2211yescharacteristic
22C4×D692C2211yescharacteristic
23C4×C22⋊D173—21no—
24#19149636—21no—
25#19149612C2212yes—
26#19149610C2212yes—
27#19149603C2212yes—
28#19149603C2212yes—
29#19149602C2212yes—
30#19149599C2212yes—
31#19149590C2212yes—
32#19149590C2211yescharacteristic
33#19149590C2211yescharacteristic
34C4⋊Q8×C173C2211yescharacteristic
35C4⋊Q8×C173C2211yescharacteristic
36#125×C173—21no—
37C22⋊Q8×C173—21no—
38C22⋊Q8×C173C2212yes—
39C4×Q8×C173—21no—
40C4×Q8×C173C2212yes—
41#117×C173—21no—
42C2×C42×C173C2211yescharacteristic
43C2×Dic346C2311yescharacteristic
44C2×D692C2311yescharacteristic
45C2×C22⋊D173—21no—
46C2×C4×D173—21no—
47#14583816—24no—
48#14583813—24no—
49C2×#13063436—24no—
50C2×#13063436C2312yes—
51C2×#13063436C2312yes—
52C4⋊Dic173C2311yescharacteristic
53C4⋊Dic173C2311yescharacteristic
54#14583805—21no—
55#14583805—21no—
56#14583805—21no—
57#14583805—21no—
58C22⋊Dic173—21no—
59#14583803—21no—
60#14583803—21no—
61#14583803—21no—
62#14583803—21no—
63C4×Dic173—21no—
64C2×Q8×C173—21no—
65C2×Q8×C173C2312yes—
66C22×C4×C173D411yescharacteristic
67C22×C4×C173D411yescharacteristic
68C22×C4×C173C2311yescharacteristic
69C4⋊C4×C173—21no—
70C4⋊C4×C173—21no—
71C4⋊C4×C173—21no—
72C4⋊C4×C173—22no—
73C4⋊C4×C173—22no—
74C4⋊C4×C173C2312yes—
75C22⋊C4×C173—22no—
76C42×C173—22no—
77C42×C173D411yescharacteristic
78C42×C173D411yescharacteristic
79C42×C173C2311yescharacteristic
80C42×C173C2311yescharacteristic
81C22×D173—21no—
82Dic346—22no—
83D692—22no—
84C22⋊D173—42no—
85C2×Dic173—21no—
86C2×Dic173—21no—
87C2×Dic173—21no—
88C4×D173—42no—
89#13063436—22no—
90#13063436#4114yes—
91C23×C173C2×D411yescharacteristic
92Q8×C173—22no—
93Q8×C173—22no—
94Q8×C173—42no—
95C2×C4×C173—22no—
96C2×C4×C173—22no—
97C2×C4×C173—22no—
98C2×C4×C173—22no—
99C2×C4×C173—41no—
100C2×C4×C173C2×D411yescharacteristic
101C2×C4×C173C2×D411yescharacteristic
102C2×C4×C173C2×D411yescharacteristic
103C2×C4×C173C2×D411yescharacteristic
104C2×C4×C173C2×D411yescharacteristic
105C2×C4×C173C2411yescharacteristic
106C22×C173—21no—
107C22×C173—21no—
108C22×C173#13712yes—
109C22×C173C22×D411yescharacteristic
110C2×D173—41no—
111C2×D173—41no—
112C2×D173—41no—
113C4×C173—21no—
114C4×C173—21no—
115C4×C173—21no—
116C4×C173—21no—
117C4×C173—21no—
118C4×C173—42no—
119C4×C173—42no—
120C4×C173C2×#4112yes—
121Dic173—41no—
122Dic173—41no—
123Dic173—41no—
124C2×C173—22no—
125C2×C173#53211yescharacteristic
126C2×C173C2×#13711yescharacteristic
127C2×C173#57511yescharacteristic
128D173—81no—
129C173#318111yescharacteristic, complemented, Sylow
130#3181—1731nomaximal, Sylow
131#533D17311yescharacteristic, complemented
132#529—1731no—
133#501—1731no—
134#497—1731no—
135#491—1731no—
136#489—1731no—
137#467—1732no—
138#465—1732no—
139#441—1732no—
140#440—1732no—
141C2×#97—1731no—
142C2×Q16—1732no—
143C2×SD16—1732no—
144C22×C8—1732no—
145C4⋊Q8C2×D17311yescharacteristic
146C4⋊Q8C2×D17311yescharacteristic
147#126—3461no—
148#125—21no—
149#125—1731no—
150#124—1731no—
151#123—3461no—
152C22⋊Q8—21no—
153C22⋊Q8—1731no—
154C22⋊Q8C2×D17312yes—
155#121—1731no—
156C4×Q8—21no—
157C4×Q8—1731no—
158C4×Q8C2×D17312yes—
159C4×D4—3461no—
160C4×D4—1731no—
161#117—21no—
162#117—3461no—
163C2×C42C2×D17311yescharacteristic
164#107—1732no—
165#106—1732no—
166#103—1732no—
167#103—1732no—
168#103—1732no—
169#102—1732no—
170#97—1732no—
171#97—1731no—
172#97—1731no—
173C2×Q8—21no—
174C2×Q8—1731no—
175C2×Q8C22×D17312yes—
176C2×D4—3461no—
177C2×D4—1731no—
178C22×C4—3461no—
179C22×C4C22⋊D17311yescharacteristic
180C22×C4C22⋊D17311yescharacteristic
181C22×C4C22×D17311yescharacteristic
182Q16—3464no—
183SD16—3464no—
184C2×C8—3464no—
185C2×C8—1732no—
186C2×C8—1732no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—3461no—
193C4⋊C4—3461no—
194C4⋊C4—3461no—
195C4⋊C4—3461no—
196C4⋊C4—1731no—
197C4⋊C4—1731no—
198C4⋊C4C22×D17312yes—
199C22⋊C4—22no—
200C22⋊C4—3461no—
201C22⋊C4—3461no—
202C22⋊C4—3461no—
203C22⋊C4—3461no—
204C22⋊C4—3461no—
205C42—22no—
206C42—3461no—
207C42C22⋊D17311yescharacteristic
208C42C22⋊D17311yescharacteristic
209C42C22×D17311yescharacteristic
210C42C22×D17311yescharacteristic
211C23—3461no—
212C23C2×C22⋊D17311yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—3462no—
217D4—3462no—
218D4—6922no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—3461no—
225C2×C4—3461no—
226C2×C4—3461no—
227C2×C4—6922no—
228C2×C4C2×C22⋊D17311yescharacteristic
229C2×C4C2×C22⋊D17311yescharacteristic
230C2×C4C2×C22⋊D17311yescharacteristic
231C2×C4C2×C22⋊D17311yescharacteristic
232C2×C4C2×C22⋊D17311yescharacteristic
233C2×C4C23×D17311yescharacteristic
234C8—3462no—
235C8—1734no—
236C22—21no—
237C22—21no—
238C22—6921no—
239C22—6921no—
240C22—6921no—
241C22#1914971012yes—
242C22C22×C22⋊D17311yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—6921no—
251C4—6921no—
252C4—6921no—
253C4#41×D17312yes—
254C2—22no—
255C2—13841no—
256C2#2601203711yescharacteristic
257C2C2×#1914971011yescharacteristic
258C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#533×C173C2
socleC22×C173C22×D4
prime radical#19149653C22
hypercenter#533D173
hypocenterC173#3181
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #26011897C2semidirect
3. #26011883C2semidirect
4. #26011859C2semidirect
5. #26011853C2semidirect
6. #26011730C2semidirect
8. #26011596C2semidirect
9. #26011589C2semidirect
10. C2×#19149590C2semidirect
11. #533×C173C2semidirect
129. C173#3181semidirect
131. #533D173semidirect

Maximal subgroups

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

subgroupindex
#260120252
#260118972
#260118832
#260118592
#260118532
#260117302
#260117222
#260115962
#260115892
C2×#191495902
#533×C1732
#3181173