Finite Group Databasebeta

← #36366975

Subgroups — #36366975

Order 22144.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#26012090C216yesmaximal, complemented
2#26012081C216yesmaximal, complemented
3#26012053C213yesmaximal, complemented
4#26012027C216yesmaximal, complemented
5#26012004C216yesmaximal, complemented
6C2×#19149639C213yesmaximal, complemented
7#543×C173C211yesmaximal, characteristic, complemented
8C2×#14583827C2212yescomplemented
9C22×C22⋊D173C2213yes—
10C23×Dic173C2211yescharacteristic
11C22×C4×D173C2213yescomplemented
12#19149674C22112yescomplemented
13#19149670C2216yescomplemented
14#19149668C2216yescomplemented
15#19149664C22112yescomplemented
16#19149663C22112yescomplemented
17#19149655C2216yescomplemented
18#19149654C2216yescomplemented
19#19149653C2216yes—
20#19149653C2216yes—
21#19149649C22112yescomplemented
22Q8×Dic173C2212yescomplemented
23#19149641C2216yescomplemented
24#19149639C22112yescomplemented
25#19149639C22112yescomplemented
26#19149637C2216yescomplemented
27C2×#14583805C2213yescomplemented
28C2×#14583803C2213yescomplemented
29C2×C4×Dic173C2213yescomplemented
30C22⋊Q8×C173C2216yescomplemented
31C22⋊Q8×C173C2216yescomplemented
32C2×C22⋊C4×C173C2213yes—
33C23×D173C2313yes—
34#14583827—44no—
35C2×D692C2316yes—
36C2×D692C2316yes—
37C2×C22⋊D173—212no—
38C2×C22⋊D173C23112yes—
39C22×Dic173—42no—
40C22×Dic173C2313yes—
41C22×Dic173C2313yes—
42C22×Dic173C2313yes—
43C2×C4×D173—46no—
44C2×C4×D173C2316yes—
45C2×C4×D173C2316yes—
46C4⋊Dic173C2316yes—
47C4⋊Dic173C2316yes—
48#14583805C23112yes—
49#14583803C23112yes—
50#14583803C23112yes—
51#14583803C23112yes—
52C4×Dic173C2316yes—
53C4×Dic173C2316yes—
54C24×C173C2311yescharacteristic
55C2×Q8×C173C2312yes—
56C2×Q8×C173C2312yes—
57C22×C4×C173C2313yes—
58C22×C4×C173C2313yes—
59C4⋊C4×C173C2316yes—
60C4⋊C4×C173C2316yes—
61C22⋊C4×C173C23112yes—
62C22×D173—26no—
63C22×D173—43no—
64C22×D173—43no—
65C22×D173C2416yes—
66D692—46no—
67D692—46no—
68C22⋊D173—224no—
69C2×Dic173—26no—
70C2×Dic173—26no—
71C2×Dic173—43no—
72C2×Dic173C2413yes—
73C2×Dic173C2416yes—
74C2×Dic173C2411yescharacteristic
75C4×D173—212no—
76C4×D173—46no—
77C23×C173—41no—
78C23×C173—41no—
79C23×C173#4114yescomplemented
80C23×C173C2413yes—
81Q8×C173—42no—
82Q8×C173—42no—
83C2×C4×C173—43no—
84C2×C4×C173—43no—
85C2×C4×C173C2416yes—
86C2×C4×C173C2416yes—
87C22×C173—23no—
88C22×C173—23no—
89C22×C173—44no—
90C22×C173C2×#4116yes—
91C22×C173C2511yescharacteristic
92C2×D173—26no—
93C2×D173—26no—
94C2×D173—26no—
95C2×D173—412no—
96C4×C173—26no—
97C4×C173—26no—
98Dic173—26no—
99Dic173—23no—
100Dic173H(5,𝔽2)12yes—
101C2×C173—26no—
102C2×C173C22×#4111yescharacteristic
103C2×C173C2×H(5,𝔽2)11yescharacteristic
104C2×C173#58511yescharacteristic
105D173—46no—
106C173#345911yescharacteristic, complemented, Sylow
107#3459—1731nomaximal, Sylow
108#543D17311yescharacteristic, complemented
109#539—1736no—
110#538—1736no—
111#534—1733no—
112#529—1736no—
113#525—1736no—
114C2×C4×D4—1733no—
115C2×#41—1732no—
116C22×D4—1733no—
117C23×C4—1731no—
118C23×C4—1733no—
119#126—17312no—
120#125—1736no—
121#124—1736no—
122#123—17312no—
123#123—17312no—
124C22⋊Q8—1736no—
125C22⋊Q8C2×D17316yescomplemented
126C22⋊Q8C2×D17316yescomplemented
127#121—1736no—
128#121—1736no—
129#121—1736no—
130#120—17312no—
131C4×Q8—1732no—
132C4×D4—1736no—
133C4×D4—17312no—
134C4×D4—17312no—
135#117—1736no—
136C2×C4⋊C4—1733no—
137C2×C22⋊C4—1733no—
138C2×C22⋊C4C2×D17313yes—
139C2×C42—1733no—
140C24—1733no—
141C24C22×D17311yescharacteristic
142#41—6924no—
143C2×Q8C22×D17312yes—
144C2×Q8C22×D17312yes—
145C2×D4—34612no—
146C2×D4—17312no—
147C2×D4—1736no—
148C2×D4—1736no—
149C22×C4—1733no—
150C22×C4—1733no—
151C22×C4—1736no—
152C22×C4—1733no—
153C22×C4—1736no—
154C22×C4—6926no—
155C22×C4—6922no—
156C22×C4C22×D17313yes—
157C22×C4C22×D17313yes—
158C4⋊C4—17312no—
159C4⋊C4—1736no—
160C4⋊C4—1736no—
161C4⋊C4C22×D17316yes—
162C4⋊C4C22×D17316yes—
163C22⋊C4—17312no—
164C22⋊C4—17312no—
165C22⋊C4—17312no—
166C22⋊C4C22×D173112yes—
167C42—1736no—
168C42—1736no—
169C23—41no—
170C23—41no—
171C23—3466no—
172C23—1736no—
173C23—6923no—
174C23—6923no—
175C23#1458382714yescomplemented
176C23C23×D17313yes—
177Q8—42no—
178Q8—42no—
179D4—34624no—
180D4—6926no—
181D4—6926no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—34612no—
185C2×C4—3466no—
186C2×C4—3466no—
187C2×C4—1736no—
188C2×C4—1733no—
189C2×C4—1731no—
190C2×C4—6926no—
191C2×C4—6923no—
192C2×C4C23×D17316yes—
193C2×C4C23×D17316yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—3466no—
198C22—3466no—
199C22—3466no—
200C22—69212no—
201C22C2×#1458382716yes—
202C22C24×D17311yescharacteristic
203C4—26no—
204C4—26no—
205C4—3466no—
206C4—3463no—
207C4—1732no—
208C2—26no—
209C2—6926no—
210C2C22×#1458382711yescharacteristic
211C2H(5,𝔽2)×D17311yescharacteristic
212C2#2601232611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#543×C173C2
hypercenter#543D173
hypocenterC173#3459
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #26012090C2semidirect
2. #26012081C2semidirect
3. #26012053C2semidirect
4. #26012027C2semidirect
5. #26012004C2semidirect
6. C2×#19149639C2semidirect
7. #543×C173C2semidirect
8. C2×#14583827C22semidirect
11. C22×C4×D173C22semidirect
12. #19149674C22semidirect
13. #19149670C22semidirect
14. #19149668C22semidirect
15. #19149664C22semidirect
16. #19149663C22semidirect
17. #19149655C22semidirect
18. #19149654C22semidirect
21. #19149649C22semidirect
22. Q8×Dic173C22semidirect
23. #19149641C22semidirect
24. #19149639C22semidirect
25. #19149639C22semidirect
26. #19149637C22semidirect
27. C2×#14583805C22semidirect
28. C2×#14583803C22semidirect
29. C2×C4×Dic173C22semidirect
30. C22⋊Q8×C173C22semidirect
31. C22⋊Q8×C173C22semidirect
79. C23×C173#41semidirect
106. C173#3459semidirect
108. #543D173semidirect
125. C22⋊Q8C2×D173semidirect
126. C22⋊Q8C2×D173semidirect
175. C23#14583827semidirect

Maximal subgroups

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

subgroupindex
#260120902
#260120812
#260120532
#260120272
#260120042
C2×#191496392
#543×C1732
#3459173