Finite Group Databasebeta

← #31796116

Subgroups — #31796116

Order 17536.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#22578420C211yesmaximal, characteristic, complemented
2#22578406C211yesmaximal, characteristic, complemented
3#22578251C214yesmaximal, complemented
4C2×#17984050C211yesmaximal, characteristic
5C2×#14555519—21no—
6#17984155—21no—
7#17984155—21no—
8C4⋊#11764047C2212yescomplemented
9C22⋊#11764047—21no—
10C4×#11764047C2212yescomplemented
11C22×D548C411yescharacteristic
12C22×C4×D137—21no—
13#17984050—21no—
14#17984050—21no—
15#17984050C412yes—
16#17984050C2212yes—
17C2×#14555487—21no—
18C2×C4×Dic137C2211yescharacteristic
19C2×C4⋊C4×C137C411yescharacteristic
20#14555519—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C23×D137—21no—
24C2×D548—24no—
25C2×D548—21no—
26C2×D548C2×C412yes—
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29C2×C4×D137—22no—
30C2×C4×D137—22no—
31C2×C4×D137—42no—
32C2×C4×D137—42no—
33#14555487—22no—
34#14555487—22no—
35C4×Dic137C2×C411yescharacteristic
36C4×Dic137D411yescharacteristic
37C4×Dic137D411yescharacteristic
38C4×Dic137C2311yescharacteristic
39C22×C4×C137—21no—
40C22×C4×C137C2×C411yescharacteristic
41C4⋊C4×C137—21no—
42C4⋊C4×C137C2×C412yes—
43#11764047—42no—
44#11764047—42no—
45C22×D137—24no—
46C22×D137—22no—
47C22×D137—42no—
48C22×D137—42no—
49D548—24no—
50D548—22no—
51D548M1614yescomplemented
52C2×Dic137—41no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137D812yes—
56C2×Dic137SD1612yes—
57C2×Dic137C2×D411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C4×D137—44no—
60C4×D137—42no—
61C4×D137—42no—
62C23×C137C22⋊C411yescharacteristic
63C2×C4×C137—22no—
64C2×C4×C137—22no—
65C2×C4×C137—42no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22×C411yescharacteristic
68C22×C137—21no—
69C22×C137—21no—
70C22×C137#10212yes—
71C22×C137C2×C22⋊C411yescharacteristic
72C2×D137—24no—
73C2×D137—48no—
74C2×D137—42no—
75C2×D137—42no—
76C4×C137—21no—
77C4×C137—42no—
78C4×C137C2×M1612yes—
79Dic137—21no—
80Dic137—21no—
81Dic137#10212yes—
82Dic137C2×D811yescharacteristic
83Dic137C2×SD1611yescharacteristic
84C2×C137—22no—
85C2×C137#40711yescharacteristic
86C2×C137C2×#10211yescharacteristic
87C2×C137#42111yescharacteristic
88D137—44no—
89C137#148611yescharacteristic, complemented, Sylow
90#1486—1371nomaximal, Sylow
91C2×C4×D4—1371no—
92#431—1371no—
93#423—1371no—
94D4⋊C8—1374no—
95C22×D4—1371no—
96C23×C4—2741no—
97C2×M16—2741no—
98C4×D4—2741no—
99C4×D4—2741no—
100C4×D4—1372no—
101C4×D4—1372no—
102C2×C4⋊C4#1058773911yescharacteristic
103C2×C22⋊C4—2741no—
104C2×C42—1371no—
105C4⋊C8—2741no—
106C4⋊C8—2741no—
107C4⋊C8—1372no—
108C22⋊C8—2741no—
109C4×C8—1372no—
110C24—2741no—
111C2×D4—2744no—
112C2×D4—2741no—
113C2×D4—1372no—
114C22×C4—21no—
115C22×C4—2742no—
116C22×C4—2742no—
117C22×C4—5482no—
118C22×C4—5482no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4C2×#1058773911yescharacteristic
122M16—5482no—
123C2×C8—2742no—
124C2×C8—2742no—
125C4⋊C4—21no—
126C4⋊C4C2×#1058773912yes—
127C22⋊C4—2742no—
128C22⋊C4—2742no—
129C42—1371no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C23—2744no—
134C23—2742no—
135C23—5482no—
136C23—5482no—
137C23C22⋊#1058773911yescharacteristic
138D4—2744no—
139D4—2742no—
140D4—1374no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—5482no—
145C2×C4—5484no—
146C2×C4—5482no—
147C2×C4—5481no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1372no—
151C2×C4—1372no—
152C2×C4—1371no—
153C2×C4—1371no—
154C2×C4C22⋊#1058773911yescharacteristic
155C2×C4C22×#1058773911yescharacteristic
156C8—5482no—
157C8—5482no—
158C22—21no—
159C22—21no—
160C22—2744no—
161C22—5488no—
162C22—5482no—
163C22—5482no—
164C22#1798415012yes—
165C22C2×C22⋊#1058773911yescharacteristic
166C4—21no—
167C4—42no—
168C4—2741no—
169C4—2741no—
170C4—1371no—
171C4—1372no—
172C4—1371no—
173C4D137⋊M1612yes—
174C2—22no—
175C2—5484no—
176C2#2257837511yescharacteristic
177C2C2×#1798415011yescharacteristic
178C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C4
cosocleC4×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1486
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578420C2semidirect
2. #22578406C2semidirect
3. #22578251C2semidirect
8. C4⋊#11764047C22semidirect
10. C4×#11764047C22semidirect
51. D548M16semidirect
89. C137#1486semidirect

Maximal subgroups

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

subgroupindex
#225784202
#225784062
#225782512
C2×#179840502
#1486137