Finite Group Databasebeta

← #31796045

Subgroups — #31796045

Order 17536.

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

normal subgroups only

This group has 316 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#22578373C212yesmaximal, complemented
2#22578248C212yesmaximal, complemented
3#22578246C212yesmaximal, complemented
4C2×D4×Dic137C211yesmaximal, characteristic
5C2×#14555519—22no—
6C22⋊#11764047—22no—
7C22⋊#11764047C2214yes—
8C23×Dic137C411yescharacteristic
9C23×Dic137C2211yescharacteristic
10D4×Dic137—22no—
11D4×Dic137—22no—
12C2×C4⋊Dic137C2211yescharacteristic
13C2×C22⋊Dic137C411yescharacteristic
14C2×C22⋊Dic137C2211yescharacteristic
15C2×C4×Dic137C411yescharacteristic
16C22×D4×C137C411yescharacteristic
17#14555519—44no—
18C2×#11764047—24no—
19C22×Dic137—21no—
20C22×Dic137—21no—
21C22×Dic137—42no—
22C22×Dic137—42no—
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137C2×C411yescharacteristic
25C22×Dic137C2×C411yescharacteristic
26C22×Dic137D412yes—
27C22×Dic137D412yes—
28C22×Dic137C2311yescharacteristic
29C4⋊Dic137—22no—
30C22⋊Dic137—22no—
31C22⋊Dic137—22no—
32C4×Dic137—21no—
33C4×Dic137—21no—
34C24×C137C2×C411yescharacteristic
35C24×C137C2×C411yescharacteristic
36C2×D4×C137—22no—
37C2×D4×C137—22no—
38C2×D4×C137—22no—
39C22×C4×C137C2×C411yescharacteristic
40#11764047—44no—
41C2×Dic137—22no—
42C2×Dic137—24no—
43C2×Dic137—21no—
44C2×Dic137—21no—
45C2×Dic137—42no—
46C2×Dic137—41no—
47C2×Dic137—42no—
48C2×Dic137—42no—
49C2×Dic137—41no—
50C2×Dic137—41no—
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C2×D412yes—
54C23×C137—21no—
55C23×C137—21no—
56C23×C137—21no—
57C23×C137—21no—
58C23×C137—41no—
59C23×C137—41no—
60C23×C137—41no—
61C23×C137—41no—
62C23×C137C22⋊C412yes—
63C23×C137M1612yescomplemented
64C23×C137C22×C411yescharacteristic
65D4×C137—28no—
66C2×C4×C137—21no—
67C2×C4×C137—21no—
68C2×C4×C137M1612yescomplemented
69C22×C137—21no—
70C22×C137—21no—
71C22×C137—22no—
72C22×C137—22no—
73C22×C137—22no—
74C22×C137—22no—
75C22×C137—42no—
76C22×C137—42no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137C2×C22⋊C411yescharacteristic
82C22×C137C2×M1611yescharacteristic
83C22×C137C2×M1611yescharacteristic
84C4×C137—22no—
85Dic137—21no—
86Dic137—22no—
87Dic137—41no—
88Dic137—41no—
89Dic137#9912yes—
90C2×C137—21no—
91C2×C137—21no—
92C2×C137—42no—
93C2×C137—42no—
94C2×C137#40711yescharacteristic
95C2×C137C2×#9911yescharacteristic
96C2×C137#41311yescharacteristic
97C137#146511yescharacteristic, complemented, Sylow
98#1465—1371nomaximal, Sylow
99C2×C4×D4—1371no—
100#407—1372no—
101#324—1372no—
102#323—1372no—
103C22×D4#1058773911yescharacteristic
104C23×C4—1371no—
105C23×C4—1371no—
106C2×M16—2742no—
107C4×D4—2742no—
108C4×D4—2742no—
109C2×C4⋊C4—1371no—
110C2×C22⋊C4—1371no—
111C2×C22⋊C4—1371no—
112C2×C42—1371no—
113C22⋊C8—2742no—
114C22⋊C8—1374no—
115C24C2×#1058773911yescharacteristic
116C24C2×#1058773911yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C22×C4—2741no—
121C22×C4—2741no—
122C22×C4—5482no—
123C22×C4—5482no—
124C22×C4—1372no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4—1371no—
128C22×C4—1372no—
129C22×C4—1371no—
130C22×C4C2×#1058773911yescharacteristic
131M16—5484no—
132C2×C8—2744no—
133C4⋊C4—2742no—
134C22⋊C4—2742no—
135C22⋊C4—2742no—
136C42—2741no—
137C42—2741no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23C22⋊#1058773912yes—
147C23#1455551912yescomplemented
148C23C22×#1058773911yescharacteristic
149D4—28no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—2744no—
153C2×C4—2742no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—5481no—
157C2×C4—5482no—
158C2×C4—5482no—
159C2×C4—5482no—
160C2×C4—5481no—
161C2×C4—5481no—
162C2×C4—1371no—
163C2×C4—1372no—
164C2×C4—1371no—
165C2×C4#1455551912yescomplemented
166C8—5484no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊#1058773911yescharacteristic
180C22C2×#1455551911yescharacteristic
181C22C2×#1455551911yescharacteristic
182C4—22no—
183C4—2742no—
184C4—2741no—
185C4—5481no—
186C4—5481no—
187C4—1372no—
188C2—21no—
189C2—21no—
190C2—42no—
191C2—42no—
192C2#2257837311yescharacteristic
193C2D137⋊#9911yescharacteristic
194C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting, prime radicalC22×D4×C137C4
socleC22×C137C2×C22⋊C4
cosocleC22×Dic137C23
hypercenterC22×D4#10587739
hypocenterC137#1465
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578373C2semidirect
2. #22578248C2semidirect
3. #22578246C2semidirect
63. C23×C137M16semidirect
68. C2×C4×C137M16semidirect
97. C137#1465semidirect
147. C23#14555519semidirect
165. C2×C4#14555519semidirect

Maximal subgroups

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

subgroupindex
#225783732
#225782482
#225782462
C2×D4×Dic1372
#1465137