Finite Group Databasebeta

← #31796651

Subgroups — #31796651

Order 17536.

380 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 78 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×#17984146C212yesmaximal
2#22578345C212yesmaximal, complemented
3#22578281C212yesmaximal, complemented
4C2×#17984048C211yesmaximal, characteristic
5C2×C4⋊#10587739—22no—
6C2×C22⋊#10587739C2214yes—
7#17984146—24no—
8(C2×#10587739)⋊C4—22no—
9C22×C22⋊D137C2211yescharacteristic
10C23×Dic137C411yescharacteristic, complemented
11C22×C4×D137C2211yescharacteristic
12#17984048—22no—
13#17984048—22no—
14C2×#14555489C411yescharacteristic, complemented
15C2×#14555487C2211yescharacteristic
16C2×C4×Dic137C411yescharacteristic, complemented
17C2×C22⋊C4×C137C411yescharacteristic, complemented
18C22×#10587739—24no—
19C4⋊#10587739—44no—
20C22⋊#10587739—24no—
21C22⋊#10587739—24no—
22C23×D137C2311yescharacteristic
23C2×C22⋊D137—22no—
24C2×C22⋊D137—22no—
25C2×C22⋊D137D412yes—
26C2×C22⋊D137Q812yes—
27C22×Dic137—21no—
28C22×Dic137—21no—
29C22×Dic137—42no—
30C22×Dic137C2×C411yescharacteristic
31C22×Dic137C2×C411yescharacteristic
32C22×Dic137C2×C411yescharacteristic
33C2×C4×D137—42no—
34C2×C4×D137D412yes—
35C2×C4×D137D412yes—
36#14555489—22no—
37#14555487—22no—
38C4×Dic137—21no—
39C4×Dic137—21no—
40C24×C137C2×C411yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137C2×C411yescharacteristic
43C22⋊C4×C137—22no—
44C2×#10587739—44no—
45C2×#10587739—44no—
46C2×#10587739—44no—
47C22×D137—41no—
48C22×D137—41no—
49C22×D137C2×D412yes—
50C22×D137C2×D411yescharacteristic
51C22×D137C2×Q811yescharacteristic
52C22×D137#4112yes—
53C22⋊D137—24no—
54C22⋊D137—24no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137—22no—
58C2×Dic137—21no—
59C2×Dic137—42no—
60C2×Dic137—42no—
61C2×Dic137—41no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C4⋊C412yescomplemented
65C4×D137—24no—
66C4×D137—42no—
67C23×C137—21no—
68C23×C137—21no—
69C23×C137—41no—
70C23×C137—41no—
71C23×C137C4⋊C412yescomplemented
72C23×C137C22×C411yescharacteristic
73C2×C4×C137—21no—
74C2×C4×C137—41no—
75C2×C4×C137—41no—
76C2×C4×C137C22⋊C412yes—
77C22×C137—21no—
78C22×C137—21no—
79C22×C137—22no—
80C22×C137—42no—
81C22×C137—42no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137C2×C22⋊C411yescharacteristic
85C22×C137C2×C4⋊C411yescharacteristic
86C22×C137#11711yescharacteristic
87C2×D137—22no—
88C2×D137—22no—
89C2×D137—41no—
90C2×D137—41no—
91C2×D137—42no—
92C2×D137#12112yes—
93C2×D137C22⋊Q812yes—
94#10587739—84no—
95C4×C137—22no—
96C4×C137—41no—
97Dic137—22no—
98Dic137—21no—
99Dic137—41no—
100Dic137#9912yescomplemented
101C2×C137—21no—
102C2×C137—21no—
103C2×C137—42no—
104C2×C137#38011yescharacteristic
105C2×C137C2×#9911yescharacteristic
106C2×C137#41011yescharacteristic
107D137—22no—
108D137—22no—
109C137#179911yescharacteristic, complemented, Sylow
110#1799—1371nomaximal, Sylow
111C2×C4×D4—1371no—
112C2×#99—1372no—
113#380—1372no—
114#342—1372no—
115C22×D4—1371no—
116C23×C4—1371no—
117C23×C4—1371no—
118C4×D4—2742no—
119C4×D4—2742no—
120C2×C4⋊C4—2742no—
121C2×C4⋊C4—1371no—
122C2×C22⋊C4—1374no—
123C2×C22⋊C4—1371no—
124C2×C22⋊C4#1058773911yescharacteristic, complemented
125C2×C42—1371no—
126#99—2744no—
127#95—2742no—
128C24—1371no—
129C24C2×#1058773911yescharacteristic
130C2×D4—2742no—
131C2×D4—2742no—
132C2×D4—1372no—
133C2×D4—1372no—
134C22×C4—2741no—
135C22×C4—2741no—
136C22×C4—2744no—
137C22×C4—5482no—
138C22×C4—5482no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4—1372no—
142C22×C4—1371no—
143C22×C4—1372no—
144C22×C4C2×#1058773911yescharacteristic
145C22×C4C2×#1058773911yescharacteristic
146C4⋊C4—2742no—
147C4⋊C4—5484no—
148C22⋊C4—22no—
149C22⋊C4—2742no—
150C22⋊C4—2744no—
151C22⋊C4—2744no—
152C42—2741no—
153C42—2741no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—5481no—
159C23—5481no—
160C23—1372no—
161C23—1371no—
162C23—1371no—
163C23—1372no—
164C23C4⋊#1058773912yescomplemented
165C23C22×#1058773911yescharacteristic
166D4—2744no—
167D4—2744no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—2744no—
172C2×C4—2741no—
173C2×C4—2741no—
174C2×C4—2742no—
175C2×C4—2741no—
176C2×C4—5482no—
177C2×C4—5482no—
178C2×C4—5482no—
179C2×C4—5481no—
180C2×C4—5484no—
181C2×C4—5484no—
182C2×C4—5484no—
183C2×C4—1371no—
184C2×C4—1372no—
185C2×C4—1371no—
186C2×C4C22⋊#1058773912yes—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—2742no—
195C22—2742no—
196C22—5481no—
197C22—5481no—
198C22—5482no—
199C22—1372no—
200C22—1372no—
201C22C2×C22⋊#1058773911yescharacteristic
202C22C2×C4⋊#1058773911yescharacteristic
203C22#1798416711yescharacteristic
204C4—22no—
205C4—41no—
206C4—2742no—
207C4—2741no—
208C4—5481no—
209C4—10964no—
210C4—1372no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2—2742no—
215C2—2742no—
216C2#2257834511yescharacteristic
217C2D137⋊#9911yescharacteristic
218C2#2257838311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C22
cosocleC23×D137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1799
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22578345C2semidirect
3. #22578281C2semidirect
10. C23×Dic137C4semidirect
14. C2×#14555489C4semidirect
16. C2×C4×Dic137C4semidirect
17. C2×C22⋊C4×C137C4semidirect
64. C2×Dic137C4⋊C4semidirect
71. C23×C137C4⋊C4semidirect
100. Dic137#99semidirect
109. C137#1799semidirect
124. C2×C22⋊C4#10587739semidirect
164. C23C4⋊#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841462
#225783452
#225782812
C2×#179840482
#1799137