Finite Group Databasebeta

← #36366696

Subgroups — #36366696

Order 22144.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D173C212yesmaximal, complemented
2#26011933C212yesmaximal, complemented
3#26011930C212yesmaximal, complemented
4#26011928C212yesmaximal, complemented
5#26011925C212yesmaximal, complemented
6#26011923C211yesmaximal, characteristic, complemented
7#26011921C212yesmaximal, complemented
8C2×#19149618C211yesmaximal, characteristic, complemented
9#508×C173C211yesmaximal, characteristic, complemented
10#41×D173—22no—
11#19149702C2212yes—
12#19149701C2212yescomplemented
13#19149700—22no—
14#19149699—22no—
15#19149698C2212yes—
16Q16×D173C2212yescomplemented
17SD16×D173—22no—
18D8×D173C2212yescomplemented
19C2×C8×D173C2211yescharacteristic
20Dic1384C2211yescharacteristic, complemented
21#19149625C2212yescomplemented
22#19149624C2212yescomplemented
23#19149623C2212yescomplemented
24#19149622C2212yescomplemented
25D2768C2211yescharacteristic, complemented
26#19149620C2212yescomplemented
27#19149618C2212yescomplemented
28#19149618C2212yescomplemented
29C2×#14583801C2211yescharacteristic, complemented
30#135×C173C2212yescomplemented
31Q32×C173C2211yescharacteristic, complemented
32SD32×C173C2212yescomplemented
33D16×C173C2211yescharacteristic, complemented
34C2×C16×C173C2211yescharacteristic, complemented
35#14583827—22no—
36#14583826—22no—
37#14583826—42no—
38#14583825—22no—
39#14583825—42no—
40Q8×D173—22no—
41D4×D173—22no—
42D4×D173—42no—
43C2×C4×D173—42no—
44C2×C4×D173D411yescharacteristic
45Dic692C2312yes—
46#14583816—22no—
47#14583815—22no—
48#14583814—22no—
49#14583813—22no—
50D1384C2312yes—
51#14583811—22no—
52C2×#13063436D411yescharacteristic
53C8×D173D412yes—
54C8×D173C2312yes—
55#14583801C2312yes—
56#41×C173—22no—
57Q16×C173C2312yes—
58SD16×C173—22no—
59D8×C173C2312yes—
60C2×C8×C173C2311yescharacteristic
61C16×C173C2311yescharacteristic
62C16×C173C2311yescharacteristic
63C22×D173—42no—
64C22×D173D811yescharacteristic
65Dic346—22no—
66Dic346—42no—
67D692—22no—
68D692—42no—
69C22⋊D173—42no—
70C22⋊D173—42no—
71C22⋊D173—42no—
72C2×Dic173—42no—
73C2×Dic173D811yescharacteristic
74C4×D173—42no—
75C4×D173—42no—
76C4×D173—42no—
77C4×D173D812yes—
78C4×D173C2×D412yes—
79#13063436C2×D412yes—
80Q8×C173—22no—
81D4×C173—22no—
82D4×C173—42no—
83C2×C4×C173—42no—
84C2×C4×C173C2×D411yescharacteristic
85C8×C173C2×D411yescharacteristic
86C8×C173C2411yescharacteristic
87C22×C173—42no—
88C22×C173C2×D811yescharacteristic
89C2×D173—41no—
90C2×D173—42no—
91C2×D173—84no—
92C2×D173C2×D812yes—
93C4×C173—42no—
94C4×C173C2×D811yescharacteristic
95C4×C173C22×D411yescharacteristic
96Dic173—42no—
97Dic173C2×D812yes—
98C2×C173—21no—
99C2×C173—82no—
100C2×C173C22×D811yescharacteristic
101D173—22no—
102D173—82no—
103C173#341411yescharacteristic, complemented, Sylow
104#3414—1731nomaximal, Sylow
105C2×#135—1732no—
106#510—1732no—
107#510—1732no—
108#509—1732no—
109#509—1732no—
110#508—1732no—
111#508—1731no—
112#508D17311yescharacteristic, complemented
113C2×M32—1731no—
114C2×#41—3462no—
115#135—3462no—
116#135—3462no—
117#135—1732no—
118#135—1732no—
119#135—1732no—
120#135C2×D17312yescomplemented
121C2×Q16—1732no—
122C2×SD16—3462no—
123C2×D8—1732no—
124C22×C8—1731no—
125Q32—1732no—
126Q32—1731no—
127Q32C2×D17311yescharacteristic, complemented
128SD32—1732no—
129SD32—1732no—
130SD32—1732no—
131SD32C2×D17312yescomplemented
132D16—1732no—
133D16—1731no—
134D16C2×D17311yescharacteristic, complemented
135M32—1732no—
136M32—1732no—
137C2×C16—1731no—
138C2×C16C2×D17311yescharacteristic, complemented
139#41—22no—
140#41—3462no—
141#41—3462no—
142#41—3462no—
143#41—6922no—
144#41—6922no—
145C2×Q8—3462no—
146C2×D4—3462no—
147C2×D4—6922no—
148C22×C4—1731no—
149C22×C4—6922no—
150Q16—3462no—
151Q16—1732no—
152Q16C22×D17312yes—
153SD16—22no—
154SD16—3462no—
155SD16—3462no—
156SD16—3462no—
157D8—3462no—
158D8—1732no—
159D8C22×D17312yes—
160C2×C8—1731no—
161C2×C8—1732no—
162C2×C8—1732no—
163C2×C8C22×D17311yescharacteristic
164C16—1732no—
165C16C22×D17311yescharacteristic
166C16C22×D17311yescharacteristic
167C23—1731no—
168C23—6922no—
169Q8—22no—
170Q8—3462no—
171Q8—6922no—
172D4—22no—
173D4—42no—
174D4—3462no—
175D4—6922no—
176D4—6922no—
177D4—6922no—
178D4—6922no—
179C2×C4—42no—
180C2×C4—1731no—
181C2×C4—1732no—
182C2×C4—1732no—
183C2×C4—6922no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4—6922no—
187C2×C4D4×D17311yescharacteristic
188C8—1732no—
189C8D4×D17311yescharacteristic
190C8C23×D17311yescharacteristic
191C22—42no—
192C22—13844no—
193C22—1732no—
194C22—6921no—
195C22—6922no—
196C22D8×D17311yescharacteristic
197C4—42no—
198C4—1732no—
199C4—6922no—
200C4D8×D17311yescharacteristic
201C4C2×D4×D17311yescharacteristic
202C2—21no—
203C2—82no—
204C2—3462no—
205C2—13842no—
206C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC4D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#508×C173C2
socleC2×C173C22×D8
hypercenter#508D173
hypocenterC173#3414
LCSG ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D173C2semidirect
2. #26011933C2semidirect
3. #26011930C2semidirect
4. #26011928C2semidirect
5. #26011925C2semidirect
6. #26011923C2semidirect
7. #26011921C2semidirect
8. C2×#19149618C2semidirect
9. #508×C173C2semidirect
12. #19149701C22semidirect
16. Q16×D173C22semidirect
18. D8×D173C22semidirect
20. Dic1384C22semidirect
21. #19149625C22semidirect
22. #19149624C22semidirect
23. #19149623C22semidirect
24. #19149622C22semidirect
25. D2768C22semidirect
26. #19149620C22semidirect
27. #19149618C22semidirect
28. #19149618C22semidirect
29. C2×#14583801C22semidirect
30. #135×C173C22semidirect
31. Q32×C173C22semidirect
32. SD32×C173C22semidirect
33. D16×C173C22semidirect
34. C2×C16×C173C22semidirect
103. C173#3414semidirect
112. #508D173semidirect
120. #135C2×D173semidirect
127. Q32C2×D173semidirect
131. SD32C2×D173semidirect
134. D16C2×D173semidirect
138. C2×C16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#135×D1732
#260119332
#260119302
#260119282
#260119252
#260119232
#260119212
C2×#191496182
#508×C1732
#3414173