Finite Group Databasebeta

← #31796126

Subgroups — #31796126

Order 17536.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578406C211yesmaximal, characteristic, complemented
2C4×#14555519C211yesmaximal, characteristic, complemented
3#22578253C212yesmaximal, complemented
4#22578251C212yesmaximal, complemented
5#22577871C211yesmaximal, characteristic
6C2×#14555519—21no—
7C2×#14555519—21no—
8#17984155—21no—
9C4⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047C2212yes—
12C2×#14555510C411yescharacteristic
13#17984053C411yescharacteristic
14#17984053C2211yescharacteristic
15#17984050C411yescharacteristic
16#17984050C2211yescharacteristic
17#17984048—21no—
18#17984048—21no—
19#17984045—21no—
20C2×C4×Dic137C2211yescharacteristic
21C42×D137—21no—
22#117×C137C411yescharacteristic
23#14555519—24no—
24#14555519—42no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27#14555510—22no—
28#14555510—22no—
29C2×Dic274C2×C411yescharacteristic
30C2×D548C2×C411yescharacteristic
31C2×C22⋊D137—21no—
32C22×Dic137D411yescharacteristic
33C22×Dic137D411yescharacteristic
34C2×C4×D137—21no—
35C2×C4×D137—21no—
36C2×C4×D137—21no—
37#14555489—21no—
38#14555489—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137—21no—
42C4×Dic137—21no—
43C4×Dic137C2×C411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137D411yescharacteristic
46C4×Dic137C2311yescharacteristic
47C22×C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C4⋊C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51C42×C137—21no—
52#11764047—24no—
53#11764047—42no—
54C22×D137—21no—
55Dic274—21no—
56Dic274M1612yescomplemented
57D548—21no—
58D548M1612yescomplemented
59C22⋊D137—24no—
60C2×Dic137—21no—
61C2×Dic137—22no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—24no—
69C4×D137—42no—
70C4×D137—41no—
71C4×D137—41no—
72C23×C137C22⋊C411yescharacteristic
73C2×C4×C137—21no—
74C2×C4×C137—21no—
75C2×C4×C137—21no—
76C2×C4×C137—21no—
77C2×C4×C137C22⋊C411yescharacteristic
78C2×C4×C137C22×C411yescharacteristic
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137C2×C22⋊C411yescharacteristic
83C2×D137—22no—
84C2×D137—41no—
85C2×D137—41no—
86C4×C137—21no—
87C4×C137—41no—
88C4×C137—41no—
89C4×C137C2×M1611yescharacteristic
90C4×C137C2×M1611yescharacteristic
91Dic137—22no—
92Dic137—21no—
93Dic137—21no—
94Dic137#10412yes—
95Dic137AGL(1,ℤ/8)11yescharacteristic
96Dic137#13711yescharacteristic
97C2×C137—41no—
98C2×C137#40711yescharacteristic
99C2×C137#41711yescharacteristic
100C2×C137C2×#10411yescharacteristic
101D137—42no—
102C137#148911yescharacteristic, complemented, Sylow
103#1489—1371nomaximal, Sylow
104C4×#41—1371no—
105#423—1371no—
106C4×M16—1371no—
107Q8⋊C8—1372no—
108D4⋊C8—1372no—
109C2×#41—1371no—
110C2×M16—2741no—
111C2×M16—2741no—
112C4×Q8—1371no—
113C4×Q8—1371no—
114C4×D4—2741no—
115C4×D4—2741no—
116C4×D4—1371no—
117C4×D4—1371no—
118#117—2741no—
119#117#1058773911yescharacteristic
120C2×C42—2741no—
121C2×C42—1371no—
122C4⋊C8—2741no—
123C4⋊C8—1372no—
124#97—2741no—
125C4×C8—1372no—
126#41—2742no—
127#41—2742no—
128C2×Q8—1371no—
129C2×D4—2741no—
130C2×D4—1371no—
131C22×C4—2741no—
132C22×C4—2741no—
133C22×C4—2741no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4C2×#1058773911yescharacteristic
137M16—2744no—
138M16—5482no—
139C2×C8—2742no—
140C2×C8—2742no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4C2×#1058773911yescharacteristic
144C4⋊C4C2×#1058773911yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C42—21no—
149C42—2741no—
150C42—2741no—
151C42—1371no—
152C42—1371no—
153C42—1371no—
154C42—1371no—
155C23—2741no—
156C23C22⋊#1058773911yescharacteristic
157Q8—2741no—
158Q8—1372no—
159D4—2744no—
160D4—2741no—
161D4—1372no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—2744no—
167C2×C4—2741no—
168C2×C4—2742no—
169C2×C4—2741no—
170C2×C4—2741no—
171C2×C4—5481no—
172C2×C4—5481no—
173C2×C4—5482no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4C22⋊#1058773911yescharacteristic
179C2×C4C22×#1058773911yescharacteristic
180C8—2744no—
181C8—5482no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—2742no—
186C22—5481no—
187C22—5481no—
188C22C2×C22⋊#1058773911yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—2742no—
193C4—2741no—
194C4—2741no—
195C4—1371no—
196C4—1371no—
197C4—1372no—
198C4D137⋊M1611yescharacteristic
199C4D137⋊M1611yescharacteristic
200C2—41no—
201C2—5482no—
202C2#2257837511yescharacteristic
203C2#2257839411yescharacteristic
204C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#14555510C4
cosocleC4×Dic137C23
hypercenter#117#10587739
hypocenterC137#1489
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #22578406C2semidirect
2. C4×#14555519C2semidirect
3. #22578253C2semidirect
4. #22578251C2semidirect
56. Dic274M16semidirect
58. D548M16semidirect
102. C137#1489semidirect

Maximal subgroups

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

subgroupindex
#225784062
C4×#145555192
#225782532
#225782512
#225778712
#1489137