Finite Group Databasebeta

← #31796625

Subgroups — #31796625

Order 17536.

584 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 86 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×#17984149C212yesmaximal, complemented
2C2×#17984148C212yesmaximal, complemented
3#22578375C212yesmaximal, complemented
4C22×#14555510C211yesmaximal, characteristic
5C2×#14555519C2214yes—
6#17984149—24no—
7#17984148—24no—
8#17984145—24no—
9C2×#14555510—24no—
10C2×#14555510—24no—
11C2×#14555510—24no—
12C22×Dic274C2211yescharacteristic
13C22×D548C411yescharacteristic
14C22×C22⋊D137C411yescharacteristic
15C22×C22⋊D137C2211yescharacteristic
16C22×C4×D137C411yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C23×C4×C137C411yescharacteristic
19#14555519—24no—
20#14555519—24no—
21C2×#11764047—24no—
22C23×D137C2×C411yescharacteristic
23C23×D137C2×C411yescharacteristic
24#14555510—232no—
25C2×Dic274—22no—
26C2×Dic274—22no—
27C2×Dic274D414yes—
28C2×D548—22no—
29C2×D548—22no—
30C2×D548—22no—
31C2×C22⋊D137—22no—
32C2×C22⋊D137—22no—
33C2×C22⋊D137—22no—
34C2×C22⋊D137—22no—
35C2×C22⋊D137—22no—
36C2×C22⋊D137D414yes—
37C22×Dic137C2×C411yescharacteristic
38C22×Dic137C2311yescharacteristic
39C2×C4×D137—22no—
40C2×C4×D137—22no—
41C2×C4×D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137—22no—
44C2×C4×D137D414yes—
45C24×C137C2×C411yescharacteristic
46C22×C4×C137—22no—
47C22×C4×C137—22no—
48C22×C4×C137—22no—
49C22×C4×C137C2×C411yescharacteristic
50C22×C4×C137C2×C411yescharacteristic
51#11764047—44no—
52C22×D137—21no—
53C22×D137—21no—
54C22×D137—21no—
55C22×D137—21no—
56C22×D137—42no—
57C22×D137—42no—
58C22×D137C22⋊C412yes—
59C22×D137C22⋊C412yes—
60Dic274—28no—
61D548—28no—
62C22⋊D137—28no—
63C22⋊D137—28no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137C22⋊C412yes—
67C2×Dic137C2×D412yes—
68C2×Dic137C2×D412yes—
69C2×Dic137C2×D412yes—
70C4×D137—28no—
71C4×D137—28no—
72C23×C137—21no—
73C23×C137—21no—
74C23×C137—42no—
75C23×C137C22⋊C412yes—
76C23×C137C22×C411yescharacteristic
77C2×C4×C137—28no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137C22⋊C412yes—
83C2×C4×C137C22⋊C412yes—
84C22×C137—22no—
85C22×C137—22no—
86C22×C137—42no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137C2×C22⋊C411yescharacteristic
90C22×C137C2×C22⋊C411yescharacteristic
91C22×C137C2×C22⋊C411yescharacteristic
92C2×D137—22no—
93C2×D137—22no—
94C2×D137—24no—
95C2×D137—42no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137—42no—
99C2×D137—42no—
100C4×C137—22no—
101C4×C137—22no—
102Dic137—22no—
103Dic137#12014yes—
104C2×C137—21no—
105C2×C137—21no—
106C2×C137—42no—
107C2×C137#37911yescharacteristic
108C2×C137#41312yes—
109D137—42no—
110D137—42no—
111C137#178511yescharacteristic, complemented, Sylow
112#1785—1371nomaximal, Sylow
113C22×#41—1371no—
114C2×#101—1372no—
115C2×#100—1372no—
116#407—1372no—
117C2×#41—2744no—
118C2×#41—2744no—
119C2×#41—2744no—
120C22×Q8—1371no—
121C22×D4—1371no—
122C22×D4—1371no—
123C22×D4—1371no—
124C23×C4—1371no—
125C23×C4—1371no—
126C23×C4#1058773911yescharacteristic
127C2×M16—1374no—
128#101—2744no—
129#100—2744no—
130C22⋊C8—2744no—
131C24—1371no—
132C24—1371no—
133C24C2×#1058773911yescharacteristic
134#41—27432no—
135C2×Q8—2742no—
136C2×Q8—2742no—
137C2×Q8—1374no—
138C2×D4—2742no—
139C2×D4—2742no—
140C2×D4—2742no—
141C2×D4—2742no—
142C2×D4—2742no—
143C2×D4—2742no—
144C2×D4—2742no—
145C2×D4—2742no—
146C2×D4—1374no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—2742no—
151C22×C4—2742no—
152C22×C4—2742no—
153C22×C4—2742no—
154C22×C4—2742no—
155C22×C4—1371no—
156C22×C4—1374no—
157C22×C4—1371no—
158C22×C4C2×#1058773911yescharacteristic
159C22×C4C2×#1058773911yescharacteristic
160M16—2744no—
161M16—2744no—
162C2×C8—2744no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23—2741no—
167C23—2741no—
168C23—2741no—
169C23—2741no—
170C23—5482no—
171C23—5482no—
172C23—1372no—
173C23—1372no—
174C23C22⋊#1058773912yes—
175C23C22×#1058773911yescharacteristic
176Q8—2748no—
177D4—2748no—
178D4—2748no—
179D4—2748no—
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—2748no—
186C2×C4—2748no—
187C2×C4—2741no—
188C2×C4—2741no—
189C2×C4—1372no—
190C2×C4—1372no—
191C2×C4—1372no—
192C2×C4—1372no—
193C2×C4C22⋊#1058773912yes—
194C2×C4C22⋊#1058773912yes—
195C8—5484no—
196C22—22no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—2742no—
202C22—2742no—
203C22—2744no—
204C22—5482no—
205C22—5482no—
206C22—5482no—
207C22—5482no—
208C22—5482no—
209C22C2×C22⋊#1058773911yescharacteristic
210C22C2×C22⋊#1058773911yescharacteristic
211C22C2×C22⋊#1058773911yescharacteristic
212C4—22no—
213C4—22no—
214C4—2742no—
215C4—1374no—
216C2—21no—
217C2—21no—
218C2—42no—
219C2—5482no—
220C2—5482no—
221C2#2257834411yescharacteristic
222C2#2257838912yes—
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×#14555510C2
cosocleC22×Dic137C23
hypercenterC23×C4#10587739
hypocenterC137#1785
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. C2×#17984149C2semidirect
2. C2×#17984148C2semidirect
3. #22578375C2semidirect
111. C137#1785semidirect

Maximal subgroups

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

subgroupindex
C2×#179841492
C2×#179841482
#225783752
C22×#145555102
#1785137