Finite Group Databasebeta

← #33244132

Subgroups — #33244132

Order 18688.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#22623069C212yesmaximal, complemented
2#22612245C214yesmaximal, complemented
3#22612243C212yesmaximal, complemented
4#22612241C212yesmaximal, complemented
5C2×D2336C211yesmaximal, characteristic, complemented
6D32×D73C212yesmaximal, complemented
7#22612224C211yesmaximal, characteristic, complemented
8C73×#2262C211yesmaximal, characteristic, complemented
9#17989389—22no—
10#17989024—24no—
11#17989022—22no—
12#17989020C2214yescomplemented
13C2×D1168C2212yescomplemented
14D16×D73C2214yescomplemented
15#17989003C2211yescharacteristic
16#17988424C2214yescomplemented
17D2336C2212yescomplemented
18D2336C2212yescomplemented
19#17988422C2214yescomplemented
20#17988421C2211yescharacteristic, complemented
21#17988420C2212yescomplemented
22C32×D73C2212yescomplemented
23#508×C73C2212yescomplemented
24Q64×C73C2211yescharacteristic, complemented
25SD64×C73C2212yescomplemented
26D32×C73C2211yescharacteristic, complemented
27C2×C32×C73C2211yescharacteristic, complemented
28#14556637—42no—
29#14556618—44no—
30#14556616—42no—
31#14556614—24no—
32C2×D584—22no—
33D8×D73—24no—
34#14556597D411yescharacteristic
35#14556534—24no—
36D1168—22no—
37D1168C2314yes—
38#14556532—24no—
39#14556531D411yescharacteristic
40#14556530D412yes—
41C16×D73C2312yes—
42#14556497C2312yes—
43#135×C73—22no—
44Q32×C73C2312yes—
45SD32×C73—22no—
46D16×C73C2312yes—
47C2×C16×C73C2311yescharacteristic
48C32×C73C2311yescharacteristic
49C32×C73C2311yescharacteristic
50#11924029—44no—
51#11924028—82no—
52#11924028D811yescharacteristic
53C2×D292—42no—
54C2×D292—82no—
55D4×D73—44no—
56D4×D73—84no—
57#11924015—44no—
58D584—24no—
59D584—42no—
60#11924013—44no—
61#11924012D811yescharacteristic
62#11924011D812yes—
63C8×D73C2×D412yes—
64#11924003C2×D412yes—
65#41×C73—42no—
66Q16×C73—22no—
67SD16×C73—42no—
68D8×C73—22no—
69C2×C8×C73C2×D411yescharacteristic
70C16×C73C2×D411yescharacteristic
71C16×C73C2411yescharacteristic
72C22×D73—82no—
73C22×D73—84no—
74Q8×C73—42no—
75D4×C73—42no—
76D4×C73—82no—
77C2×C4×C73—82no—
78C2×C4×C73C2×D811yescharacteristic
79C22⋊D73—84no—
80C22⋊D73D1612yes—
81D292—44no—
82D292—84no—
83D292—82no—
84D292D1611yescharacteristic
85C4×D73—84no—
86C4×D73C2×D812yes—
87Dic146D1611yescharacteristic
88C8×C73C2×D811yescharacteristic
89C8×C73C22×D411yescharacteristic
90#10596831C2×D812yes—
91C22×C73—82no—
92C22×C73C2×D1611yescharacteristic
93C2×D73—84no—
94C2×D73—324no—
95C2×D73—322no—
96C2×D73C2×D1612yes—
97C4×C73—82no—
98C4×C73C2×D1611yescharacteristic
99C4×C73C22×D811yescharacteristic
100Dic73C2×D1612yes—
101#33982—731nomaximal, Sylow
102C2×C73—21no—
103C2×C73—162no—
104C2×C73C22×D1611yescharacteristic
105D73—22no—
106D73—164no—
107#3415—732no—
108#2263—732no—
109#2263—734no—
110#2262—732no—
111#2262D7311yescharacteristic, complemented
112C2×D32—732no—
113C2×D32—731no—
114#2258—731no—
115C73#3398211yescharacteristic, complemented, Sylow
116#576—1462no—
117#509—1464no—
118#509—1462no—
119#508—734no—
120#508C2×D7312yescomplemented
121C2×D16—734no—
122C2×D16—732no—
123#504—731no—
124Q64C2×D7311yescharacteristic, complemented
125SD64—734no—
126SD64C2×D7312yescomplemented
127D32—734no—
128D32—732no—
129D32—732no—
130D32C2×D7311yescharacteristic, complemented
131M64—731no—
132M64—732no—
133C2×C32—732no—
134C2×C32C2×D7311yescharacteristic, complemented
135H(5,𝔽2)—2922no—
136AGL(1,ℤ/8)—2924no—
137AGL(1,ℤ/8)—2922no—
138#135—22no—
139#135—1464no—
140C2×D8—1464no—
141C2×D8—1462no—
142#131—731no—
143Q32C22×D7312yes—
144SD32—22no—
145SD32—1464no—
146D16—734no—
147D16—1464no—
148D16—1462no—
149D16C22×D7312yes—
150M32—731no—
151M32—732no—
152C2×C16—732no—
153C2×C16C22×D7311yescharacteristic
154C32—732no—
155C32C22×D7311yescharacteristic
156C32C22×D7311yescharacteristic
157#41—42no—
158#41—2924no—
159#41—5842no—
160#41—731no—
161C2×D4—2924no—
162C2×D4—2922no—
163C2×D4—5842no—
164C2×D4—5844no—
165Q16—22no—
166SD16—42no—
167SD16—2924no—
168D8—22no—
169D8—2924no—
170D8—2922no—
171D8—1464no—
172M16—731no—
173M16—732no—
174C2×C8—732no—
175C2×C8D4×D7311yescharacteristic
176C16—732no—
177C16D4×D7311yescharacteristic
178C16C23×D7311yescharacteristic
179C23—5844no—
180C23—5842no—
181Q8—42no—
182Q8—731no—
183D4—42no—
184D4—82no—
185D4—2924no—
186D4—5844no—
187D4—5844no—
188D4—5842no—
189D4—732no—
190D4—731no—
191C2×C4—82no—
192C2×C4—5844no—
193C2×C4—732no—
194C2×C4D8×D7311yescharacteristic
195C8—732no—
196C8D8×D7311yescharacteristic
197C8C2×D4×D7311yescharacteristic
198C22—82no—
199C22—23362no—
200C22—23364no—
201C22—5844no—
202C22—732no—
203C22D16×D7311yescharacteristic
204C4—82no—
205C4—732no—
206C4D16×D7311yescharacteristic
207C4C2×D8×D7311yescharacteristic
208C2—21no—
209C2—162no—
210C2—11684no—
211C2—1462no—
212C2C2×D16×D7311yescharacteristic
subgroupG/H
centerC2C2×D16×D73
commutator, cosocleC16×C73C24
FrattiniC16C23×D73
FittingC73×#2262C2
socleC2×C73C22×D16
hypercenter#2262D73
hypocenterC73#33982
LCSG ▷ C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #22623069C2semidirect
2. #22612245C2semidirect
3. #22612243C2semidirect
4. #22612241C2semidirect
5. C2×D2336C2semidirect
6. D32×D73C2semidirect
7. #22612224C2semidirect
8. C73×#2262C2semidirect
12. #17989020C22semidirect
13. C2×D1168C22semidirect
14. D16×D73C22semidirect
16. #17988424C22semidirect
17. D2336C22semidirect
18. D2336C22semidirect
19. #17988422C22semidirect
20. #17988421C22semidirect
21. #17988420C22semidirect
22. C32×D73C22semidirect
23. #508×C73C22semidirect
24. Q64×C73C22semidirect
25. SD64×C73C22semidirect
26. D32×C73C22semidirect
27. C2×C32×C73C22semidirect
111. #2262D73semidirect
115. C73#33982semidirect
120. #508C2×D73semidirect
124. Q64C2×D73semidirect
126. SD64C2×D73semidirect
130. D32C2×D73semidirect
134. C2×C32C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230692
#226122452
#226122432
#226122412
C2×D23362
D32×D732
#226122242
C73×#22622
#3398273