Finite Group Databasebeta

← #31796194

Subgroups — #31796194

Order 17536.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578375C211yesmaximal, characteristic, complemented
2#22578346C211yesmaximal, characteristic, complemented
3#22578254C214yesmaximal, complemented
4C2×#17984059C211yesmaximal, characteristic
5C2×#14555519—21no—
6C2×C22⋊#10587739—21no—
7#17984145—21no—
8#17984145C2212yescomplemented
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yescomplemented
11C22×C22⋊D137—21no—
12C22×C4×D137C2211yescharacteristic
13#17984059—21no—
14#17984059—21no—
15#17984059C412yescomplemented
16#17984059C2212yes—
17C2×C4⋊Dic137C411yescharacteristic, complemented
18C2×C22⋊Dic137—21no—
19C22×D4×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555519—42no—
22C2×#11764047—22no—
23C22⋊#10587739—42no—
24C23×D137D411yescharacteristic
25C2×C22⋊D137—22no—
26C2×C22⋊D137—22no—
27C2×C22⋊D137—44no—
28C22×Dic137—21no—
29C22×Dic137D411yescharacteristic
30C2×C4×D137—42no—
31C2×C4×D137C2×C411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137C2311yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137C2×C412yes—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39C24×C137—21no—
40C2×D4×C137—21no—
41C2×D4×C137—42no—
42C2×D4×C137C2×C412yes—
43C22×C4×C137C2×C411yescharacteristic
44C2×#10587739—42no—
45C2×#10587739—42no—
46C2×#10587739—42no—
47#11764047—42no—
48C22×D137—22no—
49C22×D137—22no—
50C22×D137—41no—
51C22×D137C22⋊C411yescharacteristic
52C22×D137C2×D411yescharacteristic
53C22⋊D137—44no—
54C22⋊D137—44no—
55C2×Dic137—22no—
56C2×Dic137—22no—
57C2×Dic137—42no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C4×D137—41no—
61C4×D137—41no—
62C4×D137—41no—
63C4×D137—41no—
64C23×C137—22no—
65C23×C137—42no—
66C23×C137—42no—
67C23×C137—42no—
68C23×C137C22⋊C411yescharacteristic
69D4×C137—42no—
70D4×C137—42no—
71C2×C4×C137—22no—
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22×C411yescharacteristic
74C22×C137—21no—
75C22×C137—21no—
76C22×C137—42no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—84no—
80C22×C137#9912yes—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—21no—
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—44no—
88C2×D137AGL(1,ℤ/8)12yes—
89#10587739—82no—
90C4×C137—21no—
91C4×C137—21no—
92Dic137—21no—
93Dic137—21no—
94Dic137—42no—
95C2×C137—22no—
96C2×C137—82no—
97C2×C137C2×#9911yescharacteristic
98C2×C137#41811yescharacteristic
99C2×C137#42111yescharacteristic
100D137—22no—
101D137—41no—
102C137#151011yescharacteristic, complemented, Sylow
103#1510—1371nomaximal, Sylow
104C2×#121—1371no—
105#407—1371no—
106#381—1371no—
107#327—1374no—
108C22×D4—2741no—
109C22×D4#1058773911yescharacteristic, complemented
110C23×C4—1371no—
111C2×M16—2741no—
112#121—2741no—
113#121—2741no—
114#121—1372no—
115#121—1372no—
116C2×C4⋊C4—1371no—
117C2×C22⋊C4—2741no—
118C2×C22⋊C4—2741no—
119C22⋊C8—2741no—
120C22⋊C8—1372no—
121#95—2741no—
122#95—1372no—
123C24—21no—
124C24—1371no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—2742no—
128C2×D4—2742no—
129C2×D4—5484no—
130C2×D4C2×#1058773912yes—
131C22×C4—2741no—
132C22×C4—2742no—
133C22×C4—5482no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139C22×C4C2×#1058773911yescharacteristic
140M16—5482no—
141C2×C8—2742no—
142C4⋊C4—2741no—
143C4⋊C4—1372no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C22⋊C4—5482no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—2742no—
152C23—2742no—
153C23—5481no—
154C23—1371no—
155C23—1371no—
156C23C22⋊#1058773911yescharacteristic
157D4—42no—
158D4—42no—
159D4—5484no—
160D4—5484no—
161C2×C4—22no—
162C2×C4—2742no—
163C2×C4—2742no—
164C2×C4—5481no—
165C2×C4—5481no—
166C2×C4—5481no—
167C2×C4—5481no—
168C2×C4—5482no—
169C2×C4—5482no—
170C2×C4—5482no—
171C2×C4—5482no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4C22⋊#1058773911yescharacteristic
175C2×C4C22×#1058773911yescharacteristic
176C8—5482no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—84no—
183C22—2741no—
184C22—2741no—
185C22—2741no—
186C22—2741no—
187C22—2741no—
188C22—5484no—
189C22—1372no—
190C22#1798414612yes—
191C22C2×C22⋊#1058773911yescharacteristic
192C4—21no—
193C4—21no—
194C4—2741no—
195C4—2741no—
196C4—5482no—
197C4—10962no—
198C2—22no—
199C2—82no—
200C2—2742no—
201C2—5481no—
202C2C2×#1798414611yescharacteristic
203C2#2257839511yescharacteristic
204C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984059C2
cosocleC2×C4×D137C23
hypercenterC22×D4#10587739
hypocenterC137#1510
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578375C2semidirect
2. #22578346C2semidirect
3. #22578254C2semidirect
8. #17984145C22semidirect
10. (C2×#10587739)⋊C4C22semidirect
15. #17984059C4semidirect
17. C2×C4⋊Dic137C4semidirect
19. C22×D4×C137C4semidirect
102. C137#1510semidirect
109. C22×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783752
#225783462
#225782542
C2×#179840592
#1510137