Finite Group Databasebeta

← #36354589

Subgroups — #36354589

Order 22144.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#26012055C211yesmaximal, characteristic
2C2×#19149668C211yesmaximal, characteristic, complemented
3C2×#19149606C211yesmaximal, characteristic, complemented
4#26011500C211yesmaximal, characteristic, complemented
5C2×#19149595C211yesmaximal, characteristic, complemented
6#26011252C211yesmaximal, characteristic, complemented
7C2×#137×C173C211yesmaximal, characteristic, complemented
8C22×D692C2211yescharacteristic
9C2×#14583810C2211yescharacteristic
10#19149675—21no—
11#19149668—22no—
12#19149668—21no—
13#19149668—22no—
14#19149653—22no—
15#19149647—22no—
16#19149638C2211yescharacteristic
17C2×#14583803—22no—
18C2×C4×Dic173C2211yescharacteristic
19#19149606—22no—
20#19149603—22no—
21#19149595—22no—
22C2×#41×C173C2211yescharacteristic
23C22×Q8×C173C2211yescharacteristic
24#137×C173—22no—
25#137×C173—22no—
26C2×Q16×C173—21no—
27C2×SD16×C173—21no—
28C2×M16×C173C2211yescharacteristic
29C23×D173D412yes—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—21no—
33C2×D692—21no—
34C2×D692—21no—
35C2×D692—21no—
36C2×C22⋊D173—24no—
37C22×Dic173—21no—
38#14583810—21no—
39#14583810—21no—
40C2×#13063436—21no—
41C4⋊Dic173D412yes—
42C22⋊Dic173—21no—
43#14583803—28no—
44#14583803—24no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C4×Dic173D412yescomplemented
48#41×C173—21no—
49#41×C173—21no—
50#41×C173—21no—
51#41×C173—21no—
52C2×Q8×C173—21no—
53C2×Q8×C173—21no—
54C2×Q8×C173—21no—
55C2×Q8×C173—21no—
56C2×Q8×C173—21no—
57C2×Q8×C173D411yescharacteristic
58C2×Q8×C173D412yescomplemented
59C2×D4×C173—21no—
60C2×D4×C173D411yescharacteristic
61C22×C4×C173—21no—
62C22×C4×C173—21no—
63C22×C4×C173C2311yescharacteristic
64Q16×C173—24no—
65SD16×C173—24no—
66M16×C173—21no—
67M16×C173D412yescomplemented
68C2×C8×C173—21no—
69C22×D173—22no—
70C22×D173—22no—
71C22×D173—22no—
72C22×D173—44no—
73D692—41no—
74D692—41no—
75D692—41no—
76D692—41no—
77C22⋊D173—84no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—42no—
81#13063436—41no—
82C23×C173—21no—
83C23×C173C2×D411yescharacteristic
84Q8×C173—22no—
85Q8×C173—21no—
86Q8×C173—22no—
87Q8×C173—21no—
88Q8×C173—41no—
89Q8×C173—41no—
90D4×C173—21no—
91D4×C173—21no—
92D4×C173—41no—
93D4×C173—41no—
94C2×C4×C173—21no—
95C2×C4×C173—22no—
96C2×C4×C173—41no—
97C2×C4×C173—41no—
98C2×C4×C173—41no—
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×C173#4111yescharacteristic
106C8×C173—22no—
107C22×C173—22no—
108C22×C173—41no—
109C22×C173—42no—
110C22×C173#12011yescharacteristic
111C22×C173#12011yescharacteristic
112C22×C173#12111yescharacteristic
113C2×D173—44no—
114C2×D173—42no—
115C2×D173—44no—
116C2×D173—82no—
117C4×C173—41no—
118C4×C173—42no—
119C4×C173#12011yescharacteristic
120C4×C173#12111yescharacteristic
121C4×C173#12111yescharacteristic
122C4×C173#12711yescharacteristic
123Dic173—42no—
124Dic173—42no—
125C2×C173—21no—
126C2×C173—21no—
127C2×C173—81no—
128C2×C173#39211yescharacteristic
129C2×C173#45512yes—
130D173—82no—
131C173#200411yescharacteristic, complemented, Sylow
132#2004—1731nomaximal, Sylow
133C2×#137D17311yescharacteristic, complemented
134#535—1731no—
135C2×#124—1731no—
136C2×#104—1731no—
137#419—1731no—
138C2×#100—1731no—
139#339—1731no—
140C2×#41C2×D17311yescharacteristic
141C22×Q8C2×D17311yescharacteristic
142C22×D4—1731no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1731no—
148C2×M16C2×D17311yescharacteristic
149#127—3461no—
150#124—3461no—
151#124—3462no—
152#124—3462no—
153#121—3462no—
154#120—3462no—
155#117—1731no—
156C2×C22⋊C4—3462no—
157C2×C42—1731no—
158#104—3462no—
159#103—3462no—
160#100—3462no—
161C24—1732no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D17311yescharacteristic
172C2×Q8C22⋊D17312yescomplemented
173C2×D4—21no—
174C2×D4—3464no—
175C2×D4—3461no—
176C2×D4—3461no—
177C2×D4—3461no—
178C2×D4—3461no—
179C2×D4—3461no—
180C2×D4—3461no—
181C2×D4C22⋊D17311yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3461no—
185C22×C4C22×D17311yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—3461no—
190M16—3461no—
191M16C22⋊D17312yescomplemented
192C2×C8—21no—
193C2×C8—3461no—
194C4⋊C4—1732no—
195C22⋊C4—3468no—
196C22⋊C4—3464no—
197C22⋊C4—3461no—
198C42—3461no—
199C42—3461no—
200C42—1732no—
201C23—21no—
202C23—3462no—
203C23—3462no—
204C23—3462no—
205C23—6924no—
206C23D4×D17311yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—13844no—
218D4—6921no—
219D4—6921no—
220D4—6921no—
221D4—6921no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3462no—
229C2×C4—3462no—
230C2×C4—6922no—
231C2×C4D4×D17311yescharacteristic
232C2×C4D4×D17311yescharacteristic
233C2×C4C2×C22⋊D17311yescharacteristic
234C2×C4C2×C22⋊D17311yescharacteristic
235C2×C4C2×C22⋊D17311yescharacteristic
236C2×C4#1458382511yescharacteristic
237C8—22no—
238C8—6921no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—13842no—
243C22—6924no—
244C22—6922no—
245C22—6924no—
246C22#1914964711yescharacteristic
247C22#1914964711yescharacteristic
248C22#1914965011yescharacteristic
249C4—41no—
250C4—42no—
251C4—6922no—
252C4—6922no—
253C4#1914964811yescharacteristic
254C4#1914965111yescharacteristic
255C4#1914965111yescharacteristic
256C4#1914967511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—13842no—
261C2#2601139911yescharacteristic
262C2#2601166712yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#137×C173C2
socleC22×C173#120
prime radical#26012055C2
hypercenterC2×#137D173
hypocenterC173#2004
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19149668C2semidirect
3. C2×#19149606C2semidirect
4. #26011500C2semidirect
5. C2×#19149595C2semidirect
6. #26011252C2semidirect
7. C2×#137×C173C2semidirect
47. C4×Dic173D4semidirect
58. C2×Q8×C173D4semidirect
67. M16×C173D4semidirect
131. C173#2004semidirect
133. C2×#137D173semidirect
172. C2×Q8C22⋊D173semidirect
191. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
C2×#191496682
C2×#191496062
#260115002
C2×#191495952
#260112522
C2×#137×C1732
#2004173