Finite Group Databasebeta

← #31790265

Subgroups — #31790265

Order 17536.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#17984114C212yesmaximal, complemented
2C4×#14555510C211yesmaximal, characteristic, complemented
3#22577600C214yesmaximal, complemented
4#22577595C214yesmaximal, complemented
5#22577427C211yesmaximal, characteristic, complemented
6#22577408C212yesmaximal, complemented
7C2×#127×C137C211yesmaximal, characteristic, complemented
8C2×#14555510C2211yescharacteristic
9#17984114—28no—
10C2×#14555498C2214yescomplemented
11C2×#14555495C2214yescomplemented
12C2×#14555494C2212yes—
13C4×Dic274C2212yescomplemented
14C4×D548C2212yescomplemented
15C4×C22⋊D137—22no—
16#17984047C2211yescharacteristic
17#17984045—21no—
18C42×D137—21no—
19C4⋊#11764040C2214yescomplemented
20#17984009C2218yescomplemented
21C22×D4×C137—22no—
22C22×D4×C137C2212yes—
23#127×C137—21no—
24#127×C137—21no—
25#127×C137C2214yes—
26C2×C42×C137C2211yescharacteristic
27#14555510—22no—
28#14555510D414yescomplemented
29C2×Dic274C2311yescharacteristic
30C2×D548C2311yescharacteristic
31C2×C22⋊D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137—22no—
34#14555498—28no—
35#14555495—28no—
36#14555494—24no—
37C2×#11764040C2314yes—
38C4⋊Dic137C2312yes—
39#14555489—22no—
40C22⋊Dic137—21no—
41#14555487—22no—
42C4×Dic137—21no—
43C4×Dic137—22no—
44C24×C137—22no—
45C2×D4×C137—24no—
46C2×D4×C137—22no—
47C2×D4×C137—24no—
48C2×D4×C137—24no—
49C2×D4×C137—24no—
50C2×D4×C137—48no—
51C2×D4×C137C2314yes—
52C22×C4×C137D412yes—
53C22×C4×C137C2311yescharacteristic
54C42×C137D411yescharacteristic
55C42×C137D411yescharacteristic
56C42×C137C2312yes—
57C22×D137—21no—
58Dic274—21no—
59Dic274C2×D412yescomplemented
60D548—21no—
61D548C2×D412yescomplemented
62C22⋊D137—24no—
63C2×Dic137—21no—
64C2×Dic137—22no—
65C4×D137—24no—
66C4×D137—44no—
67#11764040—24no—
68C23×C137—24no—
69C23×C137—48no—
70C23×C137—44no—
71C23×C137C2×D411yescharacteristic
72D4×C137—24no—
73D4×C137—24no—
74D4×C137—48no—
75D4×C137—44no—
76D4×C137—48no—
77C2×C4×C137—24no—
78C2×C4×C137C2×D412yes—
79C2×C4×C137C2×D412yes—
80C2×C4×C137C2×D411yescharacteristic
81C2×C4×C137C2×D412yes—
82C2×C4×C137#4112yes—
83C2×C4×C137C2411yescharacteristic
84C22×C137—22no—
85C22×C137—44no—
86C22×C137—48no—
87C22×C137—88no—
88C22×C137#12112yes—
89C22×C137C22×D411yescharacteristic
90C2×D137—22no—
91C2×D137—42no—
92C4×C137—22no—
93C4×C137#12112yes—
94C4×C137AGL(1,ℤ/8)14yescomplemented
95C4×C137C22×D411yescharacteristic
96C4×C137C2×#4111yescharacteristic
97Dic137—22no—
98Dic137—42no—
99C2×C137—22no—
100C2×C137—84no—
101C2×C137C2×#12111yescharacteristic
102C2×C137C2×AGL(1,ℤ/8)12yes—
103D137—42no—
104C137#303811yescharacteristic, complemented, Sylow
105#3038—1371nomaximal, Sylow
106C2×AGL(1,ℤ/8)—1372no—
107C2×#127D13711yescharacteristic, complemented
108C4×#41—1371no—
109#460—1374no—
110#459—1374no—
111#423—1371no—
112#418—1372no—
113C2×#41—1371no—
114C22×D4—22no—
115C22×D4C2×D13712yes—
116AGL(1,ℤ/8)—2748no—
117C2×SD16—1374no—
118C2×D8—1374no—
119C2×M16—1372no—
120#127—21no—
121#127—21no—
122#127C2×D13714yes—
123C4×Q8—1372no—
124C4×D4—2742no—
125C4×D4—1372no—
126#117—2741no—
127#117—1371no—
128C2×C42—2741no—
129C2×C42C2×D13711yescharacteristic
130C4⋊C8—1374no—
131#102—1378no—
132C24—22no—
133#41—2742no—
134#41—1374no—
135C2×Q8—1371no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—48no—
142C2×D4—2741no—
143C2×D4—1371no—
144C2×D4C22×D13714yes—
145C22×C4—2741no—
146C22×C4—2742no—
147C22×C4C22⋊D13712yes—
148C22×C4C22×D13711yescharacteristic
149SD16—2748no—
150D8—2748no—
151M16—2744no—
152C2×C8—1374no—
153C4⋊C4—2742no—
154C4⋊C4—1372no—
155C22⋊C4—2741no—
156C22⋊C4—2742no—
157C42—2741no—
158C42—2742no—
159C42C22⋊D13711yescharacteristic
160C42C22⋊D13711yescharacteristic
161C42C22×D13712yes—
162C23—24no—
163C23—48no—
164C23—44no—
165C23—2741no—
166C23C2×C22⋊D13711yescharacteristic
167Q8—2741no—
168Q8—1372no—
169D4—24no—
170D4—24no—
171D4—48no—
172D4—44no—
173D4—48no—
174D4—2744no—
175D4—2741no—
176D4—1372no—
177C2×C4—24no—
178C2×C4—2741no—
179C2×C4—2744no—
180C2×C4—2742no—
181C2×C4—5484no—
182C2×C4D4×D13712yes—
183C2×C4C2×C22⋊D13712yes—
184C2×C4C2×C22⋊D13711yescharacteristic
185C2×C4C2×C22⋊D13712yes—
186C2×C4#1455550912yes—
187C2×C4C23×D13711yescharacteristic
188C8—2744no—
189C22—22no—
190C22—48no—
191C22—44no—
192C22—88no—
193C22—2742no—
194C22—5482no—
195C22#1798405912yes—
196C22C22×C22⋊D13711yescharacteristic
197C4—22no—
198C4—2742no—
199C4—5482no—
200C4#1798405912yes—
201C4#1798411414yescomplemented
202C4C2×D4×D13711yescharacteristic
203C4C2×#1455550911yescharacteristic
204C2—22no—
205C2—84no—
206C2—5482no—
207C2C2×#1798405911yescharacteristic
208C2C2×#1798411412yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#127×C137C2
socleC22×C137C22×D4
hypercenterC2×#127D137
hypocenterC137#3038
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C4×#14555510C2semidirect
3. #22577600C2semidirect
4. #22577595C2semidirect
5. #22577427C2semidirect
6. #22577408C2semidirect
7. C2×#127×C137C2semidirect
10. C2×#14555498C22semidirect
11. C2×#14555495C22semidirect
13. C4×Dic274C22semidirect
14. C4×D548C22semidirect
19. C4⋊#11764040C22semidirect
20. #17984009C22semidirect
28. #14555510D4semidirect
59. Dic274C2×D4semidirect
61. D548C2×D4semidirect
94. C4×C137AGL(1,ℤ/8)semidirect
104. C137#3038semidirect
107. C2×#127D137semidirect
201. C4#17984114semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C4×#145555102
#225776002
#225775952
#225774272
#225774082
C2×#127×C1372
#3038137