Finite Group Databasebeta

← #31796287

Subgroups — #31796287

Order 17536.

256 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C2×#17984155C211yesmaximal, characteristic, complemented
3#22578262C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C22×#11764047—21no—
7#17984155—21no—
8#17984155C2212yes—
9#17984155C2212yes—
10C22⋊#11764047—21no—
11C4×#11764047—21no—
12C2×#14555509—21no—
13#17984087C2211yescharacteristic
14#17984084C2211yescharacteristic
15#17984074—21no—
16#17984062C411yescharacteristic
17#17984060—21no—
18D4×Dic137C411yescharacteristic
19#17984050C411yescharacteristic
20#17984048—21no—
21C2×C4×Dic137C2211yescharacteristic
22#121×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26#14555509—42no—
27#14555509—42no—
28C2×Dic274—21no—
29C2×D548C2×C411yescharacteristic
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C22⋊D137—21no—
33C22×Dic137—21no—
34C22×Dic137D411yescharacteristic
35C22×Dic137D411yescharacteristic
36C2×C4×D137—21no—
37C4⋊Dic137C2×C411yescharacteristic
38#14555489—21no—
39C22⋊Dic137—21no—
40#14555487—21no—
41#14555487—21no—
42C4×Dic137C2×C411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×D4×C137—21no—
47C2×D4×C137C2×C411yescharacteristic
48C22×C4×C137C2×C411yescharacteristic
49C4⋊C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51#11764047—42no—
52#11764047—42no—
53C22×D137—21no—
54Dic274—41no—
55Dic274—41no—
56D548—41no—
57C22⋊D137—42no—
58C22⋊D137—41no—
59C22⋊D137—41no—
60C22⋊D137—42no—
61C2×Dic137—22no—
62C2×Dic137—21no—
63C2×Dic137—42no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137D812yes—
67C2×Dic137SD1612yes—
68C2×Dic137C2×D411yescharacteristic
69C2×Dic137C2×D411yescharacteristic
70C4×D137—42no—
71C23×C137—21no—
72C23×C137C22⋊C411yescharacteristic
73D4×C137—41no—
74D4×C137—42no—
75C2×C4×C137—21no—
76C2×C4×C137—41no—
77C2×C4×C137C22⋊C411yescharacteristic
78C2×C4×C137C22×C411yescharacteristic
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137#10212yes—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—41no—
87C2×D137—41no—
88C2×D137—41no—
89C4×C137—21no—
90C4×C137—21no—
91C4×C137—41no—
92Dic137—21no—
93Dic137—41no—
94Dic137#10212yes—
95Dic137C2×D811yescharacteristic
96Dic137C2×SD1611yescharacteristic
97Dic137#13512yes—
98C2×C137—22no—
99C2×C137—81no—
100C2×C137#41311yescharacteristic
101C2×C137C2×#10211yescharacteristic
102C2×C137#41611yescharacteristic
103D137—81no—
104C137#154011yescharacteristic, complemented, Sylow
105#1540—1371nomaximal, Sylow
106#532—1371no—
107#431—1371no—
108C2×C4⋊C8—1371no—
109#332—1372no—
110#331—1372no—
111C2×#41—2741no—
112C22×C8—2741no—
113C4⋊Q8—1371no—
114#127—1371no—
115#124—2741no—
116#121—2741no—
117#121—1371no—
118#121#1058773911yescharacteristic
119C4×D4—2741no—
120C4×D4—1371no—
121C4×D4—1371no—
122C2×C42—1371no—
123C4⋊C8—2741no—
124C4⋊C8—1372no—
125C4⋊C8—1372no—
126C22⋊C8—2741no—
127C4×C8—2741no—
128#41—5482no—
129#41—5482no—
130C2×Q8—2741no—
131C2×D4—21no—
132C2×D4—2741no—
133C2×D4—2741no—
134C2×D4—2741no—
135C2×D4—1371no—
136C2×D4C2×#1058773911yescharacteristic
137C22×C4—2741no—
138C22×C4—2741no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4C2×#1058773911yescharacteristic
142C2×C8—2742no—
143C2×C8—2742no—
144C2×C8—5482no—
145C4⋊C4—2741no—
146C4⋊C4—1371no—
147C4⋊C4C2×#1058773911yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—2741no—
150C22⋊C4—2741no—
151C22⋊C4—2741no—
152C42—1371no—
153C42—1371no—
154C42—1371no—
155C42—1371no—
156C23—21no—
157C23—2741no—
158C23C22⋊#1058773911yescharacteristic
159Q8—5481no—
160Q8—5481no—
161D4—41no—
162D4—42no—
163D4—5481no—
164D4—5482no—
165D4—5481no—
166D4—5481no—
167D4—5482no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—2742no—
171C2×C4—2741no—
172C2×C4—5482no—
173C2×C4—5482no—
174C2×C4—1371no—
175C2×C4—1372no—
176C2×C4—1372no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4—1371no—
180C2×C4C22⋊#1058773911yescharacteristic
181C2×C4C22×#1058773911yescharacteristic
182C8—5482no—
183C8—5482no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—5481no—
190C22—5481no—
191C22—5481no—
192C22#1798415012yes—
193C22C2×C22⋊#1058773911yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—2741no—
198C4—5481no—
199C4—1371no—
200C4—1372no—
201C4—1371no—
202C4—1372no—
203C2—22no—
204C2—81no—
205C2—10961no—
206C2#2257839011yescharacteristic
207C2C2×#1798415011yescharacteristic
208C2#2257839311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577948C2
cosocleC4×Dic137C23
hypercenter#121#10587739
hypocenterC137#1540
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578420C2semidirect
2. C2×#17984155C2semidirect
3. #22578262C2semidirect
4. #22578261C2semidirect
104. C137#1540semidirect

Maximal subgroups

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

subgroupindex
#225784202
C2×#179841552
#225782622
#225782612
#225779482
#1540137