Finite Group Databasebeta

← #46846737

Subgroups — #46846737

Order 34688.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#31774453C211yesmaximal, characteristic, complemented
2#127×D271C211yesmaximal, characteristic
3#31774263C211yesmaximal, characteristic, complemented
4#31774262C211yesmaximal, characteristic, complemented
5#31774261C211yesmaximal, characteristic, complemented
6#31774242C214yesmaximal, complemented
7#31773900C214yesmaximal, complemented
8#31773849C211yesmaximal, characteristic, complemented
9#490×C271C211yesmaximal, characteristic, complemented
10C2×D4×D271—22no—
11C2×D4×D271C2212yes—
12C2×#16864637C2212yes—
13#22576159C2212yescomplemented
14#22576157C2211yescharacteristic
15#22576156—22no—
16#22576156C2212yes—
17#22576152C2211yescharacteristic, complemented
18#22576138—22no—
19#22576125—22no—
20#22576118C2212yes—
21C42×D271C2211yescharacteristic
22#22576082C2214yescomplemented
23#22576081C2214yescomplemented
24#22576080C2214yescomplemented
25#22576071C2211yescharacteristic, complemented
26C8×Dic271C2212yescomplemented
27#127×C271C2211yescharacteristic
28#125×C271C2211yescharacteristic, complemented
29#102×C271C2214yescomplemented
30#97×C271C2211yescharacteristic, complemented
31C23×D271—22no—
32C2×Dic542—22no—
33C2×D1084—22no—
34C2×D1084C2312yes—
35C2×C22⋊D271—22no—
36C2×C22⋊D271—24no—
37D4×D271—28no—
38D4×D271—48no—
39C2×C4×D271D412yes—
40C2×C4×D271C2311yescharacteristic
41#16864637—24no—
42C2×C271⋊C8C2312yes—
43C4⋊Dic271C2312yes—
44Dic271⋊C4—22no—
45Dic271⋊C4—22no—
46(C2×D271)⋊C4—22no—
47C4×Dic271—22no—
48C4×Dic271D412yes—
49C4×Dic271C2311yescharacteristic
50C2×D4×C271—22no—
51C2×D4×C271C2312yes—
52C2×C8×C271C2312yes—
53C4⋊C4×C271—22no—
54C4⋊C4×C271C2312yes—
55C42×C271C2311yescharacteristic
56C22×D271—22no—
57C22×D271—48no—
58C22×D271—44no—
59C22×D271C2×D411yescharacteristic
60Dic542—44no—
61D1084—24no—
62D1084—44no—
63C22⋊D271—44no—
64C22⋊D271—48no—
65C2×Dic271—22no—
66C2×Dic271C2×D411yescharacteristic
67C2×Dic271C2×D412yes—
68C4×D271—24no—
69C4×D271#4114yes—
70C271⋊C8—22no—
71C23×C271—22no—
72D4×C271—24no—
73D4×C271—44no—
74C2×C4×C271—22no—
75C2×C4×C271C2×D412yes—
76C2×C4×C271C2411yescharacteristic
77C8×C271—22no—
78C22×C271—42no—
79C22×C271—44no—
80C22×C271C22×D411yescharacteristic
81C2×D271—22no—
82C2×D271—42no—
83C2×D271—44no—
84C2×D271—88no—
85C2×D271#12412yes—
86C4×C271—22no—
87C4×C271—42no—
88C4×C271C2×#4112yes—
89Dic271—42no—
90Dic271#12412yes—
91Dic271AGL(1,ℤ/8)14yes—
92C2×C271—82no—
93C2×C271C2×#12411yescharacteristic
94C2×C271C2×AGL(1,ℤ/8)12yes—
95D271—22no—
96D271—82no—
97C271#313811yescharacteristic, complemented, Sylow
98#3138—2711nomaximal, Sylow
99#533—2711no—
100C2×#127—2711no—
101#490—2711no—
102#490—2711no—
103#490—2711no—
104#490D27111yescharacteristic, complemented
105#486—2714no—
106#418—2714no—
107C4×M16—2711no—
108C22×D4—2712no—
109C22×D4—5422no—
110C2×M16—2712no—
111C4⋊Q8—2712no—
112#127—2712no—
113#127—2711no—
114#127—5422no—
115#127C2×D27111yescharacteristic
116#125—2711no—
117#125C2×D27111yescharacteristic, complemented
118C22⋊Q8—5422no—
119C4×Q8—5422no—
120#117—2712no—
121C2×C42—2711no—
122#102—2714no—
123#102—2714no—
124#102—2714no—
125#102C2×D27114yescomplemented
126#97—2711no—
127#97C2×D27111yescharacteristic, complemented
128C4×C8—2712no—
129C24—5422no—
130C2×Q8—5422no—
131C2×D4—22no—
132C2×D4—2712no—
133C2×D4—5428no—
134C2×D4—5424no—
135C2×D4—5422no—
136C2×D4—5422no—
137C2×D4—10848no—
138C2×D4C22×D27112yes—
139C22×C4—2711no—
140C22×C4—2712no—
141M16—5424no—
142C2×C8—2712no—
143C2×C8C22×D27112yes—
144C4⋊C4—22no—
145C4⋊C4—2712no—
146C4⋊C4—5422no—
147C4⋊C4—5422no—
148C4⋊C4C22×D27112yes—
149C22⋊C4—5422no—
150C42—2712no—
151C42—2711no—
152C42—5422no—
153C42C22×D27111yescharacteristic
154C23—22no—
155C23—2711no—
156C23—5422no—
157C23—10844no—
158C23—10848no—
159Q8—10844no—
160D4—24no—
161D4—44no—
162D4—5424no—
163D4—10844no—
164D4—10844no—
165D4—10848no—
166C2×C4—22no—
167C2×C4—2711no—
168C2×C4—2712no—
169C2×C4—2714no—
170C2×C4—5424no—
171C2×C4—5422no—
172C2×C4D4×D27112yes—
173C2×C4C23×D27111yescharacteristic
174C8—22no—
175C8—5422no—
176C22—42no—
177C22—44no—
178C22—2712no—
179C22—5422no—
180C22—10842no—
181C22—10844no—
182C22—21688no—
183C22C2×D4×D27111yescharacteristic
184C4—22no—
185C4—42no—
186C4—2712no—
187C4—2714no—
188C4—10842no—
189C4#41×D27112yes—
190C2—82no—
191C2—5422no—
192C2—21682no—
193C2#124×D27111yescharacteristic
194C2AGL(1,ℤ/8)×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#490×C271C2
socleC22×C271C22×D4
prime radical#127×D271C2
hypercenter#490D271
hypocenterC271#3138
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #31774453C2semidirect
3. #31774263C2semidirect
4. #31774262C2semidirect
5. #31774261C2semidirect
6. #31774242C2semidirect
7. #31773900C2semidirect
8. #31773849C2semidirect
9. #490×C271C2semidirect
13. #22576159C22semidirect
17. #22576152C22semidirect
22. #22576082C22semidirect
23. #22576081C22semidirect
24. #22576080C22semidirect
25. #22576071C22semidirect
26. C8×Dic271C22semidirect
28. #125×C271C22semidirect
29. #102×C271C22semidirect
30. #97×C271C22semidirect
97. C271#3138semidirect
104. #490D271semidirect
117. #125C2×D271semidirect
125. #102C2×D271semidirect
127. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744532
#127×D2712
#317742632
#317742622
#317742612
#317742422
#317739002
#317738492
#490×C2712
#3138271