Finite Group Databasebeta

← #31796044

Subgroups — #31796044

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#22578374C212yesmaximal, complemented
2#22578249C212yesmaximal, complemented
3#22578246C212yesmaximal, complemented
4C2×C4×C22⋊D137C211yesmaximal, characteristic
5C2×#14555520—22no—
6#17984145—22no—
7C22⋊#11764047C2214yes—
8C22×C22⋊D137C411yescharacteristic
9C22×C4×D137C2211yescharacteristic
10C4×C22⋊D137—22no—
11C4×C22⋊D137—22no—
12C2×#14555489C2211yescharacteristic
13C2×C22⋊Dic137C2211yescharacteristic
14C2×#14555487C411yescharacteristic
15C2×C4×Dic137C411yescharacteristic
16C23×C4×C137C411yescharacteristic
17#14555520—44no—
18C2×#11764047—24no—
19C23×D137C2×C411yescharacteristic
20C2×C22⋊D137—22no—
21C2×C22⋊D137—22no—
22C2×C22⋊D137—22no—
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137C2×C411yescharacteristic
25C22×Dic137C2311yescharacteristic
26C2×C4×D137—42no—
27C2×C4×D137D412yes—
28C2×C4×D137D412yes—
29#14555489—22no—
30C22⋊Dic137—22no—
31#14555487—22no—
32C4×Dic137—21no—
33C4×Dic137—21no—
34C24×C137C2×C411yescharacteristic
35C22×C4×C137—21no—
36C22×C4×C137—21no—
37C22×C4×C137—42no—
38C22×C4×C137C2×C411yescharacteristic
39C22×C4×C137C2×C411yescharacteristic
40#11764047—44no—
41C22×D137—21no—
42C22×D137—21no—
43C22×D137—41no—
44C22×D137—41no—
45C22×D137C22⋊C412yes—
46C22⋊D137—28no—
47C2×Dic137—21no—
48C2×Dic137—21no—
49C2×Dic137—21no—
50C2×Dic137—41no—
51C2×Dic137—41no—
52C2×Dic137M1612yescomplemented
53C2×Dic137C2×D412yes—
54C4×D137—24no—
55C4×D137—42no—
56C23×C137—21no—
57C23×C137—21no—
58C23×C137—41no—
59C23×C137—41no—
60C23×C137M1612yescomplemented
61C23×C137C22×C411yescharacteristic
62C2×C4×C137—22no—
63C2×C4×C137—21no—
64C2×C4×C137—42no—
65C2×C4×C137—42no—
66C2×C4×C137—41no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22⋊C411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137—22no—
72C22×C137—42no—
73C22×C137—42no—
74C22×C137—41no—
75C22×C137—41no—
76C22×C137C2×C22⋊C411yescharacteristic
77C22×C137C2×M1611yescharacteristic
78C22×C137C2×M1611yescharacteristic
79C2×D137—22no—
80C2×D137—22no—
81C2×D137—22no—
82C2×D137—42no—
83C2×D137—42no—
84C4×C137—21no—
85C4×C137—41no—
86C4×C137#9912yes—
87Dic137—22no—
88Dic137—22no—
89Dic137—41no—
90C2×C137—21no—
91C2×C137—21no—
92C2×C137—42no—
93C2×C137#40711yescharacteristic
94C2×C137C2×#9911yescharacteristic
95C2×C137#41311yescharacteristic
96D137—42no—
97C137#146511yescharacteristic, complemented, Sylow
98#1465—1371nomaximal, Sylow
99C2×C4×D4—1371no—
100#407—1372no—
101#324—1372no—
102#323—1372no—
103C22×D4—1371no—
104C23×C4—1371no—
105C23×C4#1058773911yescharacteristic
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—
115C24—1371no—
116C24C2×#1058773911yescharacteristic
117C2×D4—2742no—
118C2×D4—2742no—
119C2×D4—2742no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—5482no—
124C22×C4—1371no—
125C22×C4—1372no—
126C22×C4—1371no—
127C22×C4—1371no—
128C22×C4—1372no—
129C22×C4C2×#1058773911yescharacteristic
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—41no—
141C23—41no—
142C23—2741no—
143C23—2741no—
144C23—5481no—
145C23—5481no—
146C23—1372no—
147C23#1455552012yescomplemented
148C23C22×#1058773911yescharacteristic
149D4—2748no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2744no—
158C2×C4—2741no—
159C2×C4—5482no—
160C2×C4—5481no—
161C2×C4—5481no—
162C2×C4—1372no—
163C2×C4—1372no—
164C2×C4C22⋊#1058773911yescharacteristic
165C2×C4C22⋊#1058773911yescharacteristic
166C8—5484no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—42no—
171C22—42no—
172C22—41no—
173C22—41no—
174C22—2742no—
175C22—2742no—
176C22—2742no—
177C22—5482no—
178C22—5482no—
179C22C2×C22⋊#1058773911yescharacteristic
180C22D137⋊M1611yescharacteristic
181C22C2×#1455552011yescharacteristic
182C4—21no—
183C4—41no—
184C4—2742no—
185C4—2742no—
186C4—5481no—
187C4#1798414612yes—
188C2—21no—
189C2—21no—
190C2—42no—
191C2—5482no—
192C2#2257837411yescharacteristic
193C2C2×#1798414611yescharacteristic
194C2#2257838911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C4
cosocleC22×Dic137C23
hypercenterC23×C4#10587739
hypocenterC137#1465
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #22578374C2semidirect
2. #22578249C2semidirect
3. #22578246C2semidirect
52. C2×Dic137M16semidirect
60. C23×C137M16semidirect
97. C137#1465semidirect
147. C23#14555520semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225782492
#225782462
C2×C4×C22⋊D1372
#1465137