Finite Group Databasebeta

← #31796036

Subgroups — #31796036

Order 17536.

440 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578373C212yesmaximal, complemented
2#22578247C214yesmaximal, complemented
3C22×#14555487C211yesmaximal, characteristic
4C2×#14555519—22no—
5C22⋊#11764047—22no—
6C22⋊#11764047C2214yescomplemented
7C24×D137C411yescharacteristic
8C23×Dic137C2211yescharacteristic
9C2×#14555487—22no—
10C2×#14555487—22no—
11C2×#14555487C412yes—
12C2×#14555487C2212yes—
13C23×C4×C137C411yescharacteristic
14#14555519—44no—
15C2×#11764047—24no—
16C23×D137—24no—
17C23×D137—21no—
18C23×D137—21no—
19C23×D137—42no—
20C23×D137—42no—
21C23×D137C2×C412yes—
22C22×Dic137—42no—
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137D412yes—
25C22×Dic137D412yes—
26C22×Dic137C2311yescharacteristic
27#14555487—24no—
28#14555487—24no—
29C24×C137C2×C411yescharacteristic
30C22×C4×C137—21no—
31C22×C4×C137—21no—
32C22×C4×C137—42no—
33C22×C4×C137C2×C412yes—
34#11764047—44no—
35C22×D137—24no—
36C22×D137—24no—
37C22×D137—22no—
38C22×D137—22no—
39C22×D137—416no—
40C22×D137—42no—
41C22×D137—42no—
42C22×D137—42no—
43C22×D137—42no—
44C22×D137—42no—
45C22×D137—42no—
46C22×D137M1614yescomplemented
47C2×Dic137—24no—
48C2×Dic137—41no—
49C2×Dic137—42no—
50C2×Dic137—41no—
51C2×Dic137C22⋊C412yes—
52C2×Dic137C2×D412yes—
53C23×C137—22no—
54C23×C137—22no—
55C23×C137—22no—
56C23×C137C22⋊C411yescharacteristic
57C23×C137C22⋊C411yescharacteristic
58C23×C137C22×C411yescharacteristic
59C2×C4×C137—22no—
60C2×C4×C137—42no—
61C2×C4×C137—42no—
62C2×C4×C137—42no—
63C22×C137—28no—
64C22×C137—21no—
65C22×C137—21no—
66C22×C137—21no—
67C22×C137—21no—
68C22×C137—21no—
69C22×C137—21no—
70C22×C137#9912yes—
71C22×C137#10012yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C22×C137C2×M1612yes—
74C2×D137—24no—
75C2×D137—48no—
76C2×D137—48no—
77C2×D137—44no—
78C2×D137—42no—
79C2×D137—42no—
80C2×D137—44no—
81C4×C137—42no—
82Dic137—22no—
83Dic137—22no—
84C2×C137—22no—
85C2×C137—22no—
86C2×C137—22no—
87C2×C137#40711yescharacteristic
88C2×C137C2×#9911yescharacteristic
89C2×C137C2×#10011yescharacteristic
90D137—44no—
91C137#146211yescharacteristic, complemented, Sylow
92#1462—1371nomaximal, Sylow
93C22×C22⋊C4—1371no—
94#407—1372no—
95#323—1374no—
96C25—1371no—
97C23×C4—1371no—
98C23×C4#1058773911yescharacteristic
99C2×M16—2742no—
100C2×C22⋊C4—2742no—
101C2×C22⋊C4—2742no—
102C2×C22⋊C4—1372no—
103C2×C22⋊C4—1372no—
104C22⋊C8—2742no—
105C22⋊C8—1374no—
106C24—2744no—
107C24—2741no—
108C24—2741no—
109C24—5482no—
110C24—5482no—
111C24—1372no—
112C24C2×#1058773911yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—5482no—
117C22×C4—1372no—
118C22×C4—1372no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4C2×#1058773912yes—
122M16—5484no—
123C2×C8—2744no—
124C22⋊C4—2744no—
125C22⋊C4—2744no—
126C23—22no—
127C23—22no—
128C23—22no—
129C23—2744no—
130C23—2744no—
131C23—2742no—
132C23—2742no—
133C23—5482no—
134C23—5482no—
135C23—54816no—
136C23—5482no—
137C23—5482no—
138C23—5482no—
139C23—5482no—
140C23—1374no—
141C23C22⋊#1058773911yescharacteristic
142C23C22⋊#1058773911yescharacteristic
143C23C22×#1058773911yescharacteristic
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—2744no—
149C2×C4—5481no—
150C2×C4—5482no—
151C2×C4—5481no—
152C2×C4—1372no—
153C2×C4—1372no—
154C8—5484no—
155C22—28no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—2744no—
163C22—5484no—
164C22—5488no—
165C22—5488no—
166C22—5484no—
167C22—5482no—
168C22—5482no—
169C22#1798414712yes—
170C22#1798414812yes—
171C22C2×C22⋊#1058773911yescharacteristic
172C22D137⋊M1612yes—
173C4—42no—
174C4—2742no—
175C4—2742no—
176C2—22no—
177C2—22no—
178C2—22no—
179C2—5484no—
180C2#2257837511yescharacteristic
181C2C2×#1798414711yescharacteristic
182C2C2×#1798414811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×D137C4
cosocleC22×Dic137C23
hypercenterC23×C4#10587739
hypocenterC137#1462
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #22578373C2semidirect
2. #22578247C2semidirect
6. C22⋊#11764047C22semidirect
46. C22×D137M16semidirect
91. C137#1462semidirect

Maximal subgroups

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

subgroupindex
#225783732
#225782472
C22×#145554872
#1462137