Finite Group Databasebeta

← #31796321

Subgroups — #31796321

Order 17536.

246 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578423C211yesmaximal, characteristic, complemented
2#22578406C211yesmaximal, characteristic, complemented
3#22578262C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C2×#14555519—21no—
7#17984155C2212yes—
8#17984155C2212yes—
9C4⋊#11764047—21no—
10C22⋊#11764047—21no—
11#17984143—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
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26#14555509—44no—
27C2×Dic274—21no—
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C22×Dic137D411yescharacteristic
34C22×Dic137D411yescharacteristic
35C2×C4×D137—21no—
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×D4×C137—21no—
46C2×D4×C137C2×C411yescharacteristic
47C22×C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50#11764047—42no—
51#11764047—42no—
52C22×D137—21no—
53Dic274—42no—
54D548—41no—
55C22⋊D137—42no—
56C22⋊D137—41no—
57C22⋊D137—42no—
58C22⋊D137—41no—
59C2×Dic137—22no—
60C2×Dic137—21no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—42no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—42no—
69C23×C137—21no—
70C23×C137C22⋊C411yescharacteristic
71D4×C137—41no—
72D4×C137—42no—
73C2×C4×C137—21no—
74C2×C4×C137—41no—
75C2×C4×C137C22⋊C411yescharacteristic
76C2×C4×C137C22×C411yescharacteristic
77C22×C137—21no—
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—41no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137C2×C22⋊C411yescharacteristic
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—41no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—41no—
90Dic137—21no—
91Dic137—21no—
92Dic137—41no—
93Dic137AGL(1,ℤ/8)11yescharacteristic
94Dic137AGL(1,ℤ/8)12yes—
95Dic137#13711yescharacteristic
96C2×C137—41no—
97C2×C137—81no—
98C2×C137#41311yescharacteristic
99C2×C137#41711yescharacteristic
100C2×C137#41811yescharacteristic
101D137—81no—
102C137#155711yescharacteristic, complemented, Sylow
103#1557—1371nomaximal, Sylow
104#532—1371no—
105#432—1371no—
106#423—1371no—
107#332—1372no—
108#331—1372no—
109C2×#41—2741no—
110C2×M16—2741no—
111C4⋊Q8—1371no—
112#127—1371no—
113#124—2741no—
114#121—2741no—
115#121—1371no—
116#121#1058773911yescharacteristic
117C4×D4—2741no—
118C4×D4—1371no—
119C4×D4—1371no—
120C2×C42—1371no—
121C4⋊C8—2741no—
122C4⋊C8—1372no—
123C4⋊C8—1372no—
124C22⋊C8—2741no—
125#97—2741no—
126#41—5484no—
127C2×Q8—2741no—
128C2×D4—21no—
129C2×D4—2741no—
130C2×D4—2741no—
131C2×D4—2741no—
132C2×D4—1371no—
133C2×D4C2×#1058773911yescharacteristic
134C22×C4—2741no—
135C22×C4—2741no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4C2×#1058773911yescharacteristic
139M16—5482no—
140C2×C8—2742no—
141C2×C8—2742no—
142C4⋊C4—2741no—
143C4⋊C4—1371no—
144C4⋊C4C2×#1058773911yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2741no—
149C42—1371no—
150C42—1371no—
151C42—1371no—
152C42—1371no—
153C23—21no—
154C23—2741no—
155C23C22⋊#1058773911yescharacteristic
156Q8—5482no—
157D4—41no—
158D4—42no—
159D4—5481no—
160D4—5482no—
161D4—5482no—
162D4—5481no—
163D4—5481no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—2741no—
167C2×C4—2742no—
168C2×C4—2741no—
169C2×C4—2741no—
170C2×C4—5482no—
171C2×C4—5482no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4C22⋊#1058773911yescharacteristic
177C2×C4C22×#1058773911yescharacteristic
178C8—5482no—
179C8—5482no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—5481no—
187C22—5481no—
188C22—5481no—
189C22C2×C22⋊#1058773911yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—2741no—
194C4—2741no—
195C4—5481no—
196C4—1371no—
197C4—1371no—
198C4—1372no—
199C2—41no—
200C2—81no—
201C2—10961no—
202C2#2257839011yescharacteristic
203C2#2257839411yescharacteristic
204C2#2257839511yescharacteristic
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#1557
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. #22578423C2semidirect
2. #22578406C2semidirect
3. #22578262C2semidirect
4. #22578261C2semidirect
102. C137#1557semidirect

Maximal subgroups

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

subgroupindex
#225784232
#225784062
#225782622
#225782612
#225779482
#1557137