Finite Group Databasebeta

← #31791162

Subgroups — #31791162

Order 17536.

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#22577959C211yesmaximal, characteristic, complemented
2#127×D137C211yesmaximal, characteristic
3#22577769C211yesmaximal, characteristic, complemented
4#22577768C211yesmaximal, characteristic, complemented
5#22577767C211yesmaximal, characteristic, complemented
6#22577748C214yesmaximal, complemented
7#22577406C214yesmaximal, complemented
8#22577355C211yesmaximal, characteristic, complemented
9#490×C137C211yesmaximal, characteristic, complemented
10C2×D4×D137—22no—
11C2×D4×D137C2212yes—
12C2×#14555493C2212yes—
13#17984087C2212yescomplemented
14#17984085C2211yescharacteristic
15#17984084—22no—
16#17984084C2212yes—
17#17984080C2211yescharacteristic, complemented
18#17984066—22no—
19#17984053—22no—
20#17984046C2212yes—
21C42×D137C2211yescharacteristic
22#17984010C2214yescomplemented
23#17984009C2214yescomplemented
24#17984008C2214yescomplemented
25#17983999C2211yescharacteristic, complemented
26C8×Dic137C2212yescomplemented
27#127×C137C2211yescharacteristic
28#125×C137C2211yescharacteristic, complemented
29#102×C137C2214yescomplemented
30#97×C137C2211yescharacteristic, complemented
31C23×D137—22no—
32C2×Dic274—22no—
33C2×D548—22no—
34C2×D548C2312yes—
35C2×C22⋊D137—22no—
36C2×C22⋊D137—24no—
37D4×D137—28no—
38D4×D137—48no—
39C2×C4×D137D412yes—
40C2×C4×D137C2311yescharacteristic
41#14555493—24no—
42C2×#11764040C2312yes—
43C4⋊Dic137C2312yes—
44#14555489—22no—
45#14555489—22no—
46#14555487—22no—
47C4×Dic137—22no—
48C4×Dic137D412yes—
49C4×Dic137C2311yescharacteristic
50C2×D4×C137—22no—
51C2×D4×C137C2312yes—
52C2×C8×C137C2312yes—
53C4⋊C4×C137—22no—
54C4⋊C4×C137C2312yes—
55C42×C137C2311yescharacteristic
56C22×D137—22no—
57C22×D137—48no—
58C22×D137—44no—
59C22×D137C2×D411yescharacteristic
60Dic274—44no—
61D548—24no—
62D548—44no—
63C22⋊D137—44no—
64C22⋊D137—48no—
65C2×Dic137—22no—
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D412yes—
68C4×D137—24no—
69C4×D137#4114yes—
70#11764040—22no—
71C23×C137—22no—
72D4×C137—24no—
73D4×C137—44no—
74C2×C4×C137—22no—
75C2×C4×C137C2×D412yes—
76C2×C4×C137C2411yescharacteristic
77C8×C137—22no—
78C22×C137—42no—
79C22×C137—44no—
80C22×C137C22×D411yescharacteristic
81C2×D137—22no—
82C2×D137—42no—
83C2×D137—44no—
84C2×D137—88no—
85C2×D137#12412yes—
86C4×C137—22no—
87C4×C137—42no—
88C4×C137C2×#4112yes—
89Dic137—42no—
90Dic137#12412yes—
91Dic137AGL(1,ℤ/8)14yes—
92C2×C137—82no—
93C2×C137C2×#12411yescharacteristic
94C2×C137C2×AGL(1,ℤ/8)12yes—
95D137—22no—
96D137—82no—
97C137#313811yescharacteristic, complemented, Sylow
98#3138—1371nomaximal, Sylow
99#533—1371no—
100C2×#127—1371no—
101#490—1371no—
102#490—1371no—
103#490—1371no—
104#490D13711yescharacteristic, complemented
105#486—1374no—
106#418—1374no—
107C4×M16—1371no—
108C22×D4—2742no—
109C22×D4—1372no—
110C2×M16—1372no—
111C4⋊Q8—1372no—
112#127—2742no—
113#127—1372no—
114#127—1371no—
115#127C2×D13711yescharacteristic
116#125—1371no—
117#125C2×D13711yescharacteristic, complemented
118C22⋊Q8—2742no—
119C4×Q8—2742no—
120#117—1372no—
121C2×C42—1371no—
122#102—1374no—
123#102—1374no—
124#102—1374no—
125#102C2×D13714yescomplemented
126#97—1371no—
127#97C2×D13711yescharacteristic, complemented
128C4×C8—1372no—
129C24—2742no—
130C2×Q8—2742no—
131C2×D4—22no—
132C2×D4—2748no—
133C2×D4—2742no—
134C2×D4—2744no—
135C2×D4—2742no—
136C2×D4—5488no—
137C2×D4—1372no—
138C2×D4C22×D13712yes—
139C22×C4—1372no—
140C22×C4—1371no—
141M16—2744no—
142C2×C8—1372no—
143C2×C8C22×D13712yes—
144C4⋊C4—22no—
145C4⋊C4—2742no—
146C4⋊C4—2742no—
147C4⋊C4—1372no—
148C4⋊C4C22×D13712yes—
149C22⋊C4—2742no—
150C42—2742no—
151C42—1372no—
152C42—1371no—
153C42C22×D13711yescharacteristic
154C23—22no—
155C23—2742no—
156C23—5488no—
157C23—5484no—
158C23—1371no—
159Q8—5484no—
160D4—24no—
161D4—44no—
162D4—2744no—
163D4—5484no—
164D4—5484no—
165D4—5488no—
166C2×C4—22no—
167C2×C4—2744no—
168C2×C4—2742no—
169C2×C4—1371no—
170C2×C4—1372no—
171C2×C4—1374no—
172C2×C4D4×D13712yes—
173C2×C4C23×D13711yescharacteristic
174C8—22no—
175C8—2742no—
176C22—44no—
177C22—42no—
178C22—2742no—
179C22—5484no—
180C22—5482no—
181C22—10968no—
182C22—1372no—
183C22C2×D4×D13711yescharacteristic
184C4—22no—
185C4—42no—
186C4—5482no—
187C4—1374no—
188C4—1372no—
189C4#41×D13712yes—
190C2—82no—
191C2—2742no—
192C2—10962no—
193C2#124×D13711yescharacteristic
194C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#490×C137C2
socleC22×C137C22×D4
prime radical#127×D137C2
hypercenter#490D137
hypocenterC137#3138
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #22577959C2semidirect
3. #22577769C2semidirect
4. #22577768C2semidirect
5. #22577767C2semidirect
6. #22577748C2semidirect
7. #22577406C2semidirect
8. #22577355C2semidirect
9. #490×C137C2semidirect
13. #17984087C22semidirect
17. #17984080C22semidirect
22. #17984010C22semidirect
23. #17984009C22semidirect
24. #17984008C22semidirect
25. #17983999C22semidirect
26. C8×Dic137C22semidirect
28. #125×C137C22semidirect
29. #102×C137C22semidirect
30. #97×C137C22semidirect
97. C137#3138semidirect
104. #490D137semidirect
117. #125C2×D137semidirect
125. #102C2×D137semidirect
127. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779592
#127×D1372
#225777692
#225777682
#225777672
#225777482
#225774062
#225773552
#490×C1372
#3138137