Finite Group Databasebeta

← #36363072

Subgroups — #36363072

Order 22144.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19149705C211yesmaximal, characteristic
2#26012065C211yesmaximal, characteristic, complemented
3#26011966C211yesmaximal, characteristic, complemented
4#26011708C214yesmaximal, complemented
5#26011694C214yesmaximal, complemented
6#26011517C211yesmaximal, characteristic, complemented
7#26011500C211yesmaximal, characteristic, complemented
8#26011488C211yesmaximal, characteristic, complemented
9C22×SD16×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11#19149705—24no—
12C2×#14583814C2212yes—
13C2×#14583811C2212yes—
14C2×#14583810C2211yescharacteristic
15#19149677C2212yescomplemented
16#19149668C2212yescomplemented
17#19149662—22no—
18#19149655—22no—
19D4×Dic173—22no—
20D4×Dic173C2214yes—
21#19149638C2211yescharacteristic
22#19149638C2211yescharacteristic
23C2×C22⋊Dic173—22no—
24#19149608C2214yescomplemented
25#19149604C2212yescomplemented
26#19149603C2214yescomplemented
27#19149601C2212yescomplemented
28C22×Q8×C173C2211yescharacteristic
29C22×D4×C173C2211yescharacteristic
30C2×SD16×C173—24no—
31C2×SD16×C173C2214yes—
32C22×C8×C173C2211yescharacteristic
33#14583826—22no—
34#14583826—22no—
35C2×Dic346C2311yescharacteristic
36C2×D692C2311yescharacteristic
37C2×C22⋊D173—21no—
38C22×Dic173—24no—
39C2×C4×D173—21no—
40#14583814—24no—
41#14583811—24no—
42#14583810—22no—
43C2×#13063436C2312yes—
44C4⋊Dic173D412yes—
45C4⋊Dic173C2312yes—
46#14583805—24no—
47C22⋊Dic173—24no—
48C22⋊Dic173—21no—
49C22⋊Dic173D412yes—
50#14583803—24no—
51C4×Dic173—21no—
52C4×Dic173C2312yes—
53C24×C173—21no—
54C2×Q8×C173—21no—
55C2×Q8×C173—22no—
56C2×Q8×C173—22no—
57C2×Q8×C173C2312yes—
58C2×D4×C173—21no—
59C2×D4×C173—22no—
60C2×D4×C173D414yes—
61C2×D4×C173C2312yes—
62C22×C4×C173—21no—
63C22×C4×C173C2311yescharacteristic
64SD16×C173—28no—
65C2×C8×C173—22no—
66C2×C8×C173C2312yes—
67C22×D173—21no—
68Dic346—21no—
69Dic346—21no—
70D692—21no—
71D692—21no—
72C22⋊D173—42no—
73C2×Dic173—21no—
74C2×Dic173—22no—
75C2×Dic173—84no—
76C2×Dic173C2×D414yes—
77C4×D173—42no—
78#13063436—22no—
79C23×C173—22no—
80C23×C173—24no—
81C23×C173—82no—
82C23×C173C2×D411yescharacteristic
83Q8×C173—22no—
84Q8×C173—22no—
85Q8×C173—42no—
86D4×C173—24no—
87D4×C173—22no—
88D4×C173C2×D414yes—
89C2×C4×C173—22no—
90C2×C4×C173—42no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2×D412yes—
93C2×C4×C173#4112yes—
94C2×C4×C173C2411yescharacteristic
95C8×C173—22no—
96C22×C173—22no—
97C22×C173—24no—
98C22×C173—44no—
99C22×C173—84no—
100C22×C173#12112yes—
101C22×C173C22×D411yescharacteristic
102C2×D173—41no—
103C2×D173—42no—
104C4×C173—42no—
105C4×C173#12112yes—
106C4×C173C22×D411yescharacteristic
107C4×C173C2×#4111yescharacteristic
108Dic173—24no—
109Dic173—41no—
110Dic173—42no—
111C2×C173—22no—
112C2×C173—44no—
113C2×C173C2×#12111yescharacteristic
114C2×C173#57712yes—
115D173—81no—
116C173#304211yescharacteristic, complemented, Sylow
117#3042—1731nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1731no—
119C22×SD16D17311yescharacteristic, complemented
120#536—1731no—
121#518—1731no—
122#463—1734no—
123#461—1734no—
124#424—1731no—
125#419—1731no—
126#416—1731no—
127C2×#41—1731no—
128C22×Q8C2×D17311yescharacteristic
129C22×D4C2×D17311yescharacteristic
130AGL(1,ℤ/8)—3464no—
131C2×SD16—24no—
132C2×SD16—1732no—
133C2×SD16C2×D17314yes—
134C2×D8—1732no—
135C2×M16—1731no—
136C22×C8C2×D17311yescharacteristic
137C4⋊Q8—1732no—
138#124—1732no—
139#123—3462no—
140C22⋊Q8—3462no—
141C4×D4—3462no—
142C4×D4—1734no—
143#117—1731no—
144#117—1731no—
145C2×C22⋊C4—3462no—
146C4⋊C8—1734no—
147#103—1734no—
148#103—1732no—
149#102—1732no—
150C24—21no—
151#41—3462no—
152#41—3462no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1731no—
157C2×Q8C22×D17312yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—3461no—
161C2×D4—1731no—
162C2×D4C22⋊D17314yes—
163C2×D4C22×D17312yes—
164C22×C4—21no—
165C22×C4—3464no—
166C22×C4—3461no—
167C22×C4C22×D17311yescharacteristic
168SD16—28no—
169SD16—3464no—
170D8—3464no—
171M16—3462no—
172C2×C8—22no—
173C2×C8—1732no—
174C2×C8C22×D17312yes—
175C4⋊C4—3464no—
176C4⋊C4—1732no—
177C4⋊C4—1732no—
178C22⋊C4—3464no—
179C22⋊C4—3461no—
180C22⋊C4—3464no—
181C22⋊C4—1732no—
182C42—3461no—
183C42—1732no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—3461no—
188C23D4×D17311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3461no—
193Q8—3461no—
194D4—24no—
195D4—22no—
196D4—3461no—
197D4—3461no—
198D4—6922no—
199D4C2×C22⋊D17314yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—3462no—
203C2×C4—3461no—
204C2×C4—13844no—
205C2×C4—1734no—
206C2×C4—6922no—
207C2×C4D4×D17311yescharacteristic
208C2×C4C2×C22⋊D17312yes—
209C2×C4#1458382512yes—
210C2×C4C23×D17311yescharacteristic
211C8—22no—
212C8—3462no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—6921no—
218C22—6922no—
219C22#1914965112yes—
220C22C2×D4×D17311yescharacteristic
221C4—42no—
222C4—3464no—
223C4—6921no—
224C4—6922no—
225C4#1914965112yes—
226C4C22×C22⋊D17311yescharacteristic
227C4C2×#1458382511yescharacteristic
228C2—22no—
229C2—44no—
230C2—13841no—
231C2C2×#1914965111yescharacteristic
232C2#2601229712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×SD16×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149705C2
hypercenterC22×SD16D173
hypocenterC173#3042
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #26012065C2semidirect
3. #26011966C2semidirect
4. #26011708C2semidirect
5. #26011694C2semidirect
6. #26011517C2semidirect
7. #26011500C2semidirect
8. #26011488C2semidirect
9. C22×SD16×C173C2semidirect
15. #19149677C22semidirect
16. #19149668C22semidirect
24. #19149608C22semidirect
25. #19149604C22semidirect
26. #19149603C22semidirect
27. #19149601C22semidirect
116. C173#3042semidirect
119. C22×SD16D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260120652
#260119662
#260117082
#260116942
#260115172
#260115002
#260114882
C22×SD16×C1732
#3042173