Finite Group Databasebeta

← #31796297

Subgroups — #31796297

Order 17536.

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

normal subgroups only

This group has 344 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#22578421C211yesmaximal, characteristic, complemented
2#22578408C211yesmaximal, characteristic, complemented
3#22578261C214yesmaximal, complemented
4#127×D137C211yesmaximal, characteristic
5C2×#14555520—21no—
6#17984155—21no—
7#17984155C2212yescomplemented
8#17984155C2212yescomplemented
9#17984145—21no—
10C4×#11764047—21no—
11C2×D4×D137—21no—
12C2×D4×D137—21no—
13C2×D4×D137C412yes—
14#17984085C411yescharacteristic
15#17984084—21no—
16#17984084—21no—
17#17984084C2212yes—
18C42×D137C2211yescharacteristic
19#127×C137C411yescharacteristic
20#14555520—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C23×D137—22no—
24C2×D548—21no—
25C2×D548—21no—
26C2×D548C2×C412yes—
27C2×C22⋊D137—22no—
28C2×C22⋊D137—22no—
29C2×C22⋊D137—22no—
30D4×D137—44no—
31D4×D137—44no—
32D4×D137—42no—
33D4×D137—42no—
34C2×C4×D137C2×C411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C4×Dic137D411yescharacteristic
38C4×Dic137D411yescharacteristic
39C4×Dic137C2311yescharacteristic
40C2×D4×C137—21no—
41C2×D4×C137—21no—
42C2×D4×C137C2×C412yes—
43C42×C137C2×C411yescharacteristic
44#11764047—42no—
45#11764047—42no—
46C22×D137—22no—
47C22×D137—44no—
48C22×D137—44no—
49C22×D137—44no—
50C22×D137C22⋊C411yescharacteristic
51D548—42no—
52D548—42no—
53D548—42no—
54C22⋊D137—44no—
55C22⋊D137—42no—
56C22⋊D137—42no—
57C22⋊D137—44no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—22no—
62C4×D137—22no—
63C4×D137D812yes—
64C4×D137SD1612yes—
65C23×C137—22no—
66D4×C137—42no—
67D4×C137—42no—
68D4×C137—42no—
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22×C411yescharacteristic
72C22×C137—42no—
73C22×C137—42no—
74C22×C137—42no—
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—42no—
79C2×D137—42no—
80C2×D137—42no—
81C2×D137—88no—
82C2×D137#10212yes—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137#10212yes—
86Dic137#10012yes—
87Dic137C2×D811yescharacteristic
88Dic137C2×SD1611yescharacteristic
89Dic137AGL(1,ℤ/8)12yes—
90C2×C137—82no—
91C2×C137C2×#10011yescharacteristic
92C2×C137C2×#10211yescharacteristic
93C2×C137#41811yescharacteristic
94D137—22no—
95D137—82no—
96C137#154511yescharacteristic, complemented, Sylow
97#1545—1371nomaximal, Sylow
98C2×#127—1371no—
99#431—1371no—
100#423—1371no—
101#331—1374no—
102C22×D4—2741no—
103C22×D4—2741no—
104C22×D4—1372no—
105C2×M16—2741no—
106#127—2741no—
107#127—2741no—
108#127—1372no—
109#127—1371no—
110#127#1058773911yescharacteristic
111C2×C42—1371no—
112C4⋊C8—2741no—
113C4⋊C8—1372no—
114C4⋊C8—1372no—
115C22⋊C8—2741no—
116C4×C8—2741no—
117C24—2742no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—2742no—
121C2×D4—2742no—
122C2×D4—2742no—
123C2×D4—2741no—
124C2×D4—2741no—
125C2×D4—5484no—
126C2×D4—5484no—
127C2×D4—5482no—
128C2×D4—5482no—
129C2×D4—1372no—
130C2×D4C2×#1058773912yes—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134M16—5482no—
135C2×C8—2742no—
136C2×C8—2742no—
137C42—1371no—
138C42—1371no—
139C42—1371no—
140C42C2×#1058773911yescharacteristic
141C23—22no—
142C23—2742no—
143C23—5484no—
144C23—5484no—
145C23—5484no—
146C23—1371no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—5482no—
151D4—5482no—
152D4—5482no—
153D4—5482no—
154D4—5484no—
155D4—5484no—
156D4—5482no—
157C2×C4—2742no—
158C2×C4—2742no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1372no—
163C2×C4—1372no—
164C2×C4C22⋊#1058773911yescharacteristic
165C2×C4C22⋊#1058773911yescharacteristic
166C2×C4C22×#1058773911yescharacteristic
167C8—5482no—
168C8—5482no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—2741no—
173C22—2741no—
174C22—5482no—
175C22—5482no—
176C22—5482no—
177C22—10968no—
178C22—1372no—
179C22C2×C22⋊#1058773911yescharacteristic
180C4—21no—
181C4—21no—
182C4—1372no—
183C4—1371no—
184C4—1371no—
185C4—1372no—
186C4#1798415012yes—
187C2—82no—
188C2—2742no—
189C2—10962no—
190C2D137⋊#10011yescharacteristic
191C2C2×#1798415011yescharacteristic
192C2#2257839511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC22×C137C2×C22⋊C4
prime radical#127×D137C2
cosocleC4×Dic137C23
hypercenter#127#10587739
hypocenterC137#1545
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578421C2semidirect
2. #22578408C2semidirect
3. #22578261C2semidirect
7. #17984155C22semidirect
8. #17984155C22semidirect
96. C137#1545semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225784082
#225782612
#127×D1372
#1545137