Finite Group Databasebeta

← #31796534

Subgroups — #31796534

Order 17536.

412 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#17984147C212yesmaximal
2C4×C22⋊#10587739C212yesmaximal, complemented
3#22578281C212yesmaximal, complemented
4C2×#17984050C211yesmaximal, characteristic
5C2×C22⋊#10587739C2214yes—
6C2×C4×#10587739—22no—
7#17984147C418yescomplemented
8(C2×#10587739)⋊C4—22no—
9C22×D548C2211yescharacteristic
10C22×C4×D137C411yescharacteristic, complemented
11C22×C4×D137C2211yescharacteristic
12#17984050—22no—
13#17984050—22no—
14C2×#14555487C411yescharacteristic, complemented
15C2×#14555487C2211yescharacteristic
16C2×C4×Dic137C411yescharacteristic, complemented
17C2×C4⋊C4×C137C411yescharacteristic, complemented
18C22×#10587739—24no—
19C22⋊#10587739—24no—
20C22⋊#10587739C2×C418yescomplemented
21C4×#10587739—44no—
22C23×D137C2×C411yescharacteristic
23C23×D137C2311yescharacteristic
24C2×D548—22no—
25C2×D548—22no—
26C2×D548C2×C414yes—
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2×C411yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—42no—
32C2×C4×D137—42no—
33C2×C4×D137D412yes—
34C2×C4×D137D412yes—
35#14555487—22no—
36#14555487—22no—
37C4×Dic137—21no—
38C4×Dic137—21no—
39C22×C4×C137C2×C411yescharacteristic
40C22×C4×C137C2×C411yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C4⋊C4×C137—22no—
43C2×#10587739—28no—
44C2×#10587739—44no—
45C2×#10587739—44no—
46C22×D137—21no—
47C22×D137—21no—
48C22×D137—41no—
49C22×D137—41no—
50C22×D137—41no—
51C22×D137—41no—
52C22×D137C4212yescomplemented
53C22×D137C22×C412yes—
54C22×D137C2×D412yes—
55C22×D137#4112yes—
56D548—28no—
57C2×Dic137—22no—
58C2×Dic137—21no—
59C2×Dic137—41no—
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C22⋊C411yescharacteristic
62C4×D137—24no—
63C4×D137—42no—
64C4×D137—42no—
65C4×D137—42no—
66C23×C137C22×C411yescharacteristic
67C2×C4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C4212yescomplemented
73C2×C4×C137C22⋊C412yes—
74C22×C137—21no—
75C22×C137—21no—
76C22×C137C2×C4211yescharacteristic
77C22×C137C2×C22⋊C411yescharacteristic
78C22×C137#11711yescharacteristic
79C2×D137—22no—
80C2×D137—22no—
81C2×D137—22no—
82C2×D137—42no—
83C2×D137—42no—
84C2×D137—42no—
85C2×D137—41no—
86C2×D137—41no—
87C2×D137—42no—
88C2×D137C4×D414yes—
89#10587739—48no—
90C4×C137—22no—
91C4×C137—22no—
92C4×C137—41no—
93Dic137—21no—
94Dic137—41no—
95Dic137#9912yescomplemented
96C2×C137—21no—
97C2×C137—21no—
98C2×C137C4×C22⋊C411yescharacteristic
99C2×C137C2×#9911yescharacteristic
100C2×C137#41011yescharacteristic
101D137—24no—
102D137—42no—
103C137C4×#9911yescharacteristic, complemented, Sylow
104C4×#99—1371nomaximal, Sylow
105C2×C4×D4—1371no—
106C2×#99—1372no—
107C4×C22⋊C4—1372no—
108#342—1372no—
109C22×D4—1371no—
110C23×C4—1371no—
111C23×C4—1371no—
112C4×D4—2742no—
113C4×D4—2742no—
114C2×C4⋊C4#1058773911yescharacteristic, complemented
115C2×C22⋊C4—1371no—
116C2×C22⋊C4—1371no—
117C2×C22⋊C4—1374no—
118C2×C42—2742no—
119C2×C42—1371no—
120#99—1378no—
121#95—2742no—
122C24—1371no—
123C24—1371no—
124C2×D4—2742no—
125C2×D4—2742no—
126C2×D4—1374no—
127C22×C4—2741no—
128C22×C4—2741no—
129C22×C4—2744no—
130C22×C4—5482no—
131C22×C4—5482no—
132C22×C4—1371no—
133C22×C4—1372no—
134C22×C4—1371no—
135C22×C4—1372no—
136C22×C4C2×#1058773911yescharacteristic
137C22×C4C2×#1058773911yescharacteristic
138C22×C4C2×#1058773911yescharacteristic
139C4⋊C4—22no—
140C22⋊C4—2742no—
141C22⋊C4—2744no—
142C22⋊C4—2742no—
143C22⋊C4—1378no—
144C42—2741no—
145C42—2741no—
146C42—5484no—
147C23—2741no—
148C23—2741no—
149C23—5481no—
150C23—5481no—
151C23—5481no—
152C23—5481no—
153C23—1372no—
154C23—1372no—
155C23—1372no—
156C23—1372no—
157C23C22×#1058773911yescharacteristic
158D4—2748no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—2744no—
165C2×C4—2742no—
166C2×C4—2748no—
167C2×C4—2741no—
168C2×C4—5482no—
169C2×C4—5482no—
170C2×C4—5482no—
171C2×C4—5484no—
172C2×C4—5484no—
173C2×C4—5481no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4C4×#1058773912yescomplemented
177C2×C4C22⋊#1058773912yes—
178C22—21no—
179C22—21no—
180C22—2742no—
181C22—2742no—
182C22—2742no—
183C22—5482no—
184C22—5482no—
185C22—5482no—
186C22—5482no—
187C22—5481no—
188C22—5481no—
189C22—1374no—
190C22C2×C4×#1058773911yescharacteristic
191C22C2×C22⋊#1058773911yescharacteristic
192C22#1798416711yescharacteristic
193C4—22no—
194C4—22no—
195C4—41no—
196C4—2741no—
197C4—5488no—
198C4—5481no—
199C4—1372no—
200C2—21no—
201C2—21no—
202C2—2744no—
203C2—5482no—
204C2C4×C22⋊#1058773911yescharacteristic
205C2D137⋊#9911yescharacteristic
206C2#2257838411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC22×C137C2×C42
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C22
cosocleC23×D137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137C4×#99
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊#10587739C2semidirect
3. #22578281C2semidirect
7. #17984147C4semidirect
10. C22×C4×D137C4semidirect
14. C2×#14555487C4semidirect
16. C2×C4×Dic137C4semidirect
17. C2×C4⋊C4×C137C4semidirect
20. C22⋊#10587739C2×C4semidirect
52. C22×D137C42semidirect
72. C2×C4×C137C42semidirect
95. Dic137#99semidirect
103. C137C4×#99semidirect
114. C2×C4⋊C4#10587739semidirect
176. C2×C4C4×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841472
C4×C22⋊#105877392
#225782812
C2×#179840502
C4×#99137