Finite Group Databasebeta

← #36364406

Subgroups — #36364406

Order 22144.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012004C212yesmaximal, complemented
2#26011830C212yesmaximal, complemented
3#26011817C212yesmaximal, complemented
4#26011744C211yesmaximal, characteristic, complemented
5#26011739C212yesmaximal, complemented
6#26011652C212yesmaximal, complemented
7#26011639C212yesmaximal, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#470×C173C211yesmaximal, characteristic, complemented
10C2×#14583827C2212yes—
11C22×C4×D173C2211yescharacteristic
12C2×#14583813C2212yescomplemented
13C2×D1384C2211yescharacteristic, complemented
14C2×#14583810C2211yescharacteristic
15#19149664—22no—
16#19149655C2212yes—
17#19149654C2212yes—
18#19149653C2212yes—
19#19149649—22no—
20#19149641—22no—
21#19149639—22no—
22#19149637C2212yes—
23#19149612C2212yescomplemented
24#19149611C2211yescharacteristic, complemented
25#19149603C2212yescomplemented
26#19149601C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yes—
29#19149591C2212yes—
30C2×Q16×C173C2211yescharacteristic, complemented
31C2×M16×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#106×C173C2211yescharacteristic, complemented
34#103×C173C2212yescomplemented
35C23×D173D411yescharacteristic
36#14583827—42no—
37#14583827—42no—
38C2×D692—22no—
39C2×D692—22no—
40C2×D692C2312yes—
41C2×C22⋊D173—22no—
42C22×Dic173D411yescharacteristic
43C2×C4×D173—22no—
44C2×C4×D173—41no—
45C2×C4×D173—41no—
46C2×C4×D173D412yes—
47C2×C4×D173C2312yes—
48#14583813—42no—
49D1384—41no—
50#14583810—41no—
51C2×#13063436C2312yes—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54#14583803—22no—
55#14583803—22no—
56#14583803—22no—
57#14583803—22no—
58C4×Dic173—22no—
59C2×Q8×C173C2312yes—
60C22×C4×C173C2311yescharacteristic
61Q16×C173—41no—
62M16×C173—41no—
63C2×C8×C173C2311yescharacteristic
64C2×C8×C173C2311yescharacteristic
65C4⋊C4×C173—22no—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22×D173—22no—
69C22×D173—41no—
70C22×D173—42no—
71C22×D173C2×D412yes—
72D692—22no—
73D692—22no—
74D692—44no—
75D692—44no—
76C22⋊D173—44no—
77C2×Dic173—22no—
78C2×Dic173—41no—
79C2×Dic173C2×D412yes—
80C4×D173—22no—
81C4×D173—22no—
82C4×D173—42no—
83C4×D173—44no—
84#13063436—22no—
85C23×C173C2×D411yescharacteristic
86Q8×C173—22no—
87Q8×C173—22no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2411yescharacteristic
94C8×C173—21no—
95C8×C173—21no—
96C22×C173—21no—
97C22×C173—22no—
98C22×C173C22×D411yescharacteristic
99C2×D173—22no—
100C2×D173—22no—
101C2×D173—22no—
102C2×D173—44no—
103C2×D173—42no—
104C2×D173—42no—
105C2×D173—42no—
106C4×C173—21no—
107C4×C173—42no—
108C4×C173—42no—
109C4×C173H(5,𝔽2)11yescharacteristic
110C4×C173H(5,𝔽2)11yescharacteristic
111Dic173—22no—
112Dic173—42no—
113C2×C173—41no—
114C2×C173#53411yescharacteristic
115C2×C173#57512yes—
116D173—42no—
117D173—82no—
118C173#319711yescharacteristic, complemented, Sylow
119#3197—1731nomaximal, Sylow
120#525—1732no—
121#485—1732no—
122#482—1732no—
123#470D17311yescharacteristic, complemented
124#469—1731no—
125#468—1732no—
126#452—1732no—
127#449—1732no—
128#407—1731no—
129C2×#41—1732no—
130C23×C4—1731no—
131C2×Q16C2×D17311yescharacteristic, complemented
132C2×SD16—1732no—
133C2×D8—1731no—
134C2×M16—1731no—
135C2×M16C2×D17311yescharacteristic
136#123—3462no—
137C22⋊Q8—1732no—
138C22⋊Q8C2×D17312yes—
139#121—1732no—
140#121—1732no—
141#120—3462no—
142C4×D4—3462no—
143C4×D4—3462no—
144#117—1732no—
145#107—1732no—
146#106—1731no—
147#106C2×D17311yescharacteristic, complemented
148#103—1732no—
149#103C2×D17312yescomplemented
150#102—1732no—
151#102—1732no—
152C22⋊C8—1732no—
153C22⋊C8—1732no—
154C24—1731no—
155#41—6922no—
156#41—6922no—
157C2×Q8C22×D17312yes—
158C2×D4—3462no—
159C2×D4—3462no—
160C2×D4—3462no—
161C2×D4—1732no—
162C22×C4—3462no—
163C22×C4—1732no—
164C22×C4—1731no—
165C22×C4—1732no—
166C22×C4—6921no—
167C22×C4—6921no—
168C22×C4C22×D17311yescharacteristic
169Q16—41no—
170SD16—6922no—
171D8—6921no—
172M16—41no—
173M16—6921no—
174C2×C8—1732no—
175C2×C8C22×D17311yescharacteristic
176C2×C8C22×D17311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—3462no—
179C4⋊C4—1732no—
180C4⋊C4C22×D17312yes—
181C22⋊C4—22no—
182C22⋊C4—3462no—
183C22⋊C4—3462no—
184C22⋊C4—3462no—
185C22⋊C4—3462no—
186C42—3462no—
187C23—3462no—
188C23—1732no—
189C23—6921no—
190C23—6922no—
191C23D4×D17311yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—3462no—
195D4—3462no—
196D4—6924no—
197D4—6924no—
198D4—6924no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—3462no—
206C2×C4—1732no—
207C2×C4—6922no—
208C2×C4—6924no—
209C2×C4—6921no—
210C2×C4D4×D17311yescharacteristic
211C2×C4C23×D17311yescharacteristic
212C8—21no—
213C8—21no—
214C8—3462no—
215C22—22no—
216C22—21no—
217C22—3462no—
218C22—3462no—
219C22—3462no—
220C22—6924no—
221C22—6922no—
222C22—6922no—
223C22—6922no—
224C22C2×D4×D17311yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—3462no—
229C4—6922no—
230C4#1914972411yescharacteristic
231C4#1914972511yescharacteristic
232C2—41no—
233C2—13842no—
234C2—6922no—
235C2#2601204911yescharacteristic
236C2#2601229012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
hypercenter#470D173
hypocenterC173#3197
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #26012004C2semidirect
2. #26011830C2semidirect
3. #26011817C2semidirect
4. #26011744C2semidirect
5. #26011739C2semidirect
6. #26011652C2semidirect
7. #26011639C2semidirect
8. #26011454C2semidirect
9. #470×C173C2semidirect
12. C2×#14583813C22semidirect
13. C2×D1384C22semidirect
23. #19149612C22semidirect
24. #19149611C22semidirect
25. #19149603C22semidirect
26. #19149601C22semidirect
27. #19149599C22semidirect
30. C2×Q16×C173C22semidirect
33. #106×C173C22semidirect
34. #103×C173C22semidirect
118. C173#3197semidirect
123. #470D173semidirect
131. C2×Q16C2×D173semidirect
147. #106C2×D173semidirect
149. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
#260118302
#260118172
#260117442
#260117392
#260116522
#260116392
#260114542
#470×C1732
#3197173