Finite Group Databasebeta

← #36352087

Subgroups — #36352087

Order 22144.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012027C211yesmaximal, characteristic
2#26011545C211yesmaximal, characteristic, complemented
3#26011342C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011226C211yesmaximal, characteristic
6#26011224C211yesmaximal, characteristic
7#328×C173C211yesmaximal, characteristic, complemented
8#19149674—21no—
9#19149670C411yescharacteristic, complemented
10#19149668C411yescharacteristic, complemented
11#19149663—21no—
12#19149653C2211yescharacteristic
13Q8×Dic173C411yescharacteristic, complemented
14#19149641C411yescharacteristic, complemented
15#19149639—21no—
16#19149637—21no—
17C2×#14583805—21no—
18C2×C4×Dic173—21no—
19C2×C4×Dic173C2211yescharacteristic
20#19149608—21no—
21C22⋊#13063436C2211yescharacteristic
22#19149589—21no—
23#19149586C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×D692C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×#13063436—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173C2×C411yescharacteristic
37#14583805—24no—
38#14583805—21no—
39#14583803—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43C4×Dic173—21no—
44C4×Dic173—42no—
45C4×Dic173C2×C411yescharacteristic
46C4×Dic173C2×C411yescharacteristic
47C4×Dic173D411yescharacteristic
48C4×Dic173D411yescharacteristic
49C2×Q8×C173C2×C411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173C2×C411yescharacteristic
55C22⋊C4×C173—21no—
56C22×D173—21no—
57D692—41no—
58C22⋊D173—41no—
59C22⋊D173—41no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C22⋊C411yescharacteristic
69C2×Dic173C22⋊C411yescharacteristic
70C2×Dic173C22⋊C411yescharacteristic
71C4×D173—42no—
72#13063436—41no—
73C23×C173C2×D411yescharacteristic
74Q8×C173—41no—
75C2×C4×C173—22no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173C22×C411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C8×C173—41no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173C2×C22⋊C411yescharacteristic
87C22×C173AGL(1,ℤ/8)11yescharacteristic
88C22×C173#13711yescharacteristic
89C2×D173—41no—
90C2×D173—41no—
91C2×D173—41no—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—42no—
95C4×C173—41no—
96C4×C173—41no—
97Dic173—21no—
98Dic173—21no—
99Dic173—21no—
100Dic173—42no—
101Dic173—41no—
102Dic173#10412yes—
103C2×C173—21no—
104C2×C173—21no—
105C2×C173#41011yescharacteristic
106C2×C173#41711yescharacteristic
107C2×C173C2×#10411yescharacteristic
108D173—81no—
109C173#150911yescharacteristic, complemented, Sylow
110#1509—1731nomaximal, Sylow
111#529—1731no—
112#432—1731no—
113C4×C4⋊C4—1731no—
114#328—1731no—
115#328D17311yescharacteristic, complemented
116#327—1731no—
117#327—1731no—
118#126—3461no—
119#125—1731no—
120#124—1731no—
121#123—3461no—
122C22⋊Q8C2×D17311yescharacteristic
123#121—1731no—
124C4×Q8—1731no—
125C4×D4—3461no—
126C4×D4—1731no—
127#117—3461no—
128C2×C4⋊C4—3461no—
129C2×C42—3461no—
130C2×C42—1731no—
131C4⋊C8—3461no—
132C22⋊C8—1731no—
133C22⋊C8C2×D17311yescharacteristic
134#97—3461no—
135#95—1731no—
136#95C2×D17311yescharacteristic
137C2×Q8C4×D17311yescharacteristic
138C2×D4—3461no—
139C2×D4—1731no—
140C22×C4—21no—
141C22×C4—3461no—
142C22×C4—3461no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4C22×D17311yescharacteristic
146C2×C8—21no—
147C2×C8—3461no—
148C4⋊C4—21no—
149C4⋊C4—3464no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—1731no—
154C4⋊C4C4×D17311yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C22⋊C4—3461no—
160C42—3461no—
161C42—1731no—
162C42—1731no—
163C42—1731no—
164C42—1731no—
165C42—6922no—
166C23—3461no—
167C23D4×D17311yescharacteristic
168Q8—41no—
169D4—6921no—
170D4—6921no—
171D4—6921no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—3461no—
179C2×C4—3461no—
180C2×C4—3462no—
181C2×C4—3462no—
182C2×C4—3461no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—1731no—
186C2×C4—1731no—
187C2×C4—6922no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4C2×C4×D17311yescharacteristic
191C2×C4D4×D17311yescharacteristic
192C8—41no—
193C8—6921no—
194C22—21no—
195C22—21no—
196C22—6921no—
197C22—6921no—
198C22—6921no—
199C22C22⋊C4×D17311yescharacteristic
200C22#1914970611yescharacteristic
201C22#1914971111yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—3461no—
208C4—3461no—
209C4—3461no—
210C4—1732no—
211C4—6922no—
212C4—6921no—
213C2—21no—
214C2—21no—
215C2—13841no—
216C2#2601146311yescharacteristic
217C2#2601149111yescharacteristic
218C2#104×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173C2×C22⋊C4
prime radical#19149653C22
cosocleC22×C4×C173C23
hypercenter#328D173
hypocenterC173#1509
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011545C2semidirect
3. #26011342C2semidirect
4. #26011228C2semidirect
7. #328×C173C2semidirect
9. #19149670C4semidirect
10. #19149668C4semidirect
13. Q8×Dic173C4semidirect
14. #19149641C4semidirect
109. C173#1509semidirect
115. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120272
#260115452
#260113422
#260112282
#260112262
#260112242
#328×C1732
#1509173