Finite Group Databasebeta

← #31796028

Subgroups — #31796028

Order 17536.

312 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 80 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#22578420C212yesmaximal, complemented
2#22578247C212yesmaximal, complemented
3#22578246C212yesmaximal, complemented
4C2×C4×C22⋊D137C211yesmaximal, characteristic
5#17984155—22no—
6C22⋊#11764047C2214yes—
7C4×#11764047—22no—
8C22×C22⋊D137C411yescharacteristic
9C22×C4×D137C411yescharacteristic
10C4×C22⋊D137—22no—
11C4×C22⋊D137—22no—
12C2×#14555489C411yescharacteristic
13C2×C22⋊Dic137C2211yescharacteristic
14C2×#14555487C2211yescharacteristic
15C2×C4×Dic137C2211yescharacteristic
16C23×C4×C137C411yescharacteristic
17C2×#11764047—24no—
18C23×D137C2×C411yescharacteristic
19C2×C22⋊D137—22no—
20C2×C22⋊D137—22no—
21C2×C22⋊D137C814yescomplemented
22C22×Dic137C2×C411yescharacteristic
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137C2311yescharacteristic
25C2×C4×D137—21no—
26C2×C4×D137—21no—
27C2×C4×D137—42no—
28#14555489—22no—
29C22⋊Dic137—22no—
30#14555487—22no—
31C4×Dic137D412yes—
32C4×Dic137D412yes—
33C24×C137C2×C411yescharacteristic
34C22×C4×C137—21no—
35C22×C4×C137—21no—
36C22×C4×C137—42no—
37C22×C4×C137C2×C411yescharacteristic
38C22×C4×C137C2×C411yescharacteristic
39#11764047—44no—
40C22×D137—21no—
41C22×D137—21no—
42C22×D137—41no—
43C22×D137—41no—
44C22×D137C2×C812yes—
45C22⋊D137—28no—
46C2×Dic137—21no—
47C2×Dic137—41no—
48C2×Dic137—41no—
49C2×Dic137C22⋊C412yes—
50C2×Dic137C2×C812yes—
51C2×Dic137M1612yes—
52C2×Dic137C2×D412yes—
53C4×D137—42no—
54C4×D137—42no—
55C23×C137—21no—
56C23×C137—21no—
57C23×C137—41no—
58C23×C137—41no—
59C23×C137C2×C812yes—
60C23×C137C22×C411yescharacteristic
61C2×C4×C137—22no—
62C2×C4×C137—21no—
63C2×C4×C137—42no—
64C2×C4×C137—42no—
65C2×C4×C137—41no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22⋊C411yescharacteristic
68C22×C137—21no—
69C22×C137—21no—
70C22×C137—22no—
71C22×C137—42no—
72C22×C137—42no—
73C22×C137—41no—
74C22×C137—41no—
75C22×C137C2×C22⋊C411yescharacteristic
76C22×C137C22×C811yescharacteristic
77C22×C137C2×M1611yescharacteristic
78C2×D137—22no—
79C2×D137—42no—
80C2×D137—41no—
81C2×D137—41no—
82C2×D137—42no—
83C4×C137—21no—
84C4×C137—41no—
85C4×C137#10012yes—
86Dic137—22no—
87Dic137—41no—
88Dic137C22⋊C814yes—
89C2×C137—21no—
90C2×C137—21no—
91C2×C137—42no—
92C2×C137C2×C22⋊C811yescharacteristic
93C2×C137#41011yescharacteristic
94C2×C137C2×#10011yescharacteristic
95D137—42no—
96C137#146011yescharacteristic, complemented, Sylow
97#1460—1371nomaximal, Sylow
98C2×C4×D4—1371no—
99#431—1372no—
100#323—1372no—
101#323—1372no—
102C22×D4—1371no—
103C23×C4—1371no—
104C23×C4#1058773911yescharacteristic
105C4×D4—2742no—
106C4×D4—2742no—
107C2×C4⋊C4—1371no—
108C2×C22⋊C4—1371no—
109C2×C22⋊C4—1371no—
110C2×C42—1371no—
111C4⋊C8—2742no—
112C22⋊C8—1374no—
113C4×C8—2742no—
114C24—1371no—
115C24C2×#1058773911yescharacteristic
116C2×D4—2742no—
117C2×D4—2742no—
118C2×D4—1374no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—2741no—
123C22×C4—2741no—
124C22×C4—5482no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4—1371no—
128C22×C4C2×#1058773911yescharacteristic
129C22×C4C2×#1058773911yescharacteristic
130C2×C8—2744no—
131C4⋊C4—2742no—
132C22⋊C4—2742no—
133C22⋊C4—2742no—
134C42—1372no—
135C42—1372no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—2741no—
141C23—2741no—
142C23—5481no—
143C23—5481no—
144C23—1372no—
145C23#1455551812yes—
146C23C22×#1058773911yescharacteristic
147D4—2748no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—2741no—
154C2×C4—5482no—
155C2×C4—5481no—
156C2×C4—5482no—
157C2×C4—5481no—
158C2×C4—1372no—
159C2×C4—1372no—
160C2×C4—1372no—
161C2×C4—1372no—
162C2×C4C22⋊#1058773911yescharacteristic
163C2×C4C22⋊#1058773911yescharacteristic
164C8—5484no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—2742no—
173C22—5482no—
174C22—5482no—
175C22—5481no—
176C22—5481no—
177C22C2×C22⋊#1058773911yescharacteristic
178C22C2×#1455551811yescharacteristic
179C22D137⋊M1611yescharacteristic
180C4—21no—
181C4—41no—
182C4—2742no—
183C4—5481no—
184C4—1374no—
185C4#1798414812yes—
186C2—21no—
187C2—21no—
188C2—42no—
189C2—5482no—
190C2#2257837111yescharacteristic
191C2#2257838211yescharacteristic
192C2C2×#1798414811yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10587739
commutatorC22×C137C22×C8
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C4
cosocleC22×Dic137C23
hypercenterC23×C4#10587739
hypocenterC137#1460
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #22578420C2semidirect
2. #22578247C2semidirect
3. #22578246C2semidirect
21. C2×C22⋊D137C8semidirect
96. C137#1460semidirect

Maximal subgroups

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

subgroupindex
#225784202
#225782472
#225782462
C2×C4×C22⋊D1372
#1460137