Finite Group Databasebeta

← #31796618

Subgroups — #31796618

Order 17536.

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

normal subgroups only

This group has 612 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×#17984147C212yesmaximal, complemented
2C2×#17984146C212yesmaximal, complemented
3#22578346C212yesmaximal, complemented
4C22×#14555510C211yesmaximal, characteristic
5C2×C22⋊#10587739C2214yes—
6#17984147—24no—
7#17984146—24no—
8(C2×#10587739)⋊C4—24no—
9C2×#14555510—24no—
10C2×#14555510—24no—
11C2×#14555510—24no—
12C22×Dic274C411yescharacteristic, complemented
13C22×D548C2211yescharacteristic
14C22×C22⋊D137C411yescharacteristic, complemented
15C22×C22⋊D137C2211yescharacteristic
16C22×C4×D137C411yescharacteristic, complemented
17C22×C4×D137C2211yescharacteristic
18C23×C4×C137C411yescharacteristic, complemented
19C22×#10587739—24no—
20C22⋊#10587739—24no—
21C22⋊#10587739—24no—
22C23×D137C2×C411yescharacteristic
23C23×D137C2311yescharacteristic
24#14555510—232no—
25C2×Dic274—22no—
26C2×Dic274—22no—
27C2×Dic274—22no—
28C2×D548—22no—
29C2×D548—22no—
30C2×D548D414yes—
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×Dic137C2×C411yescharacteristic
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
51C2×#10587739—48no—
52C2×#10587739—44no—
53C22×D137—21no—
54C22×D137—21no—
55C22×D137—42no—
56C22×D137—42no—
57C22×D137C22⋊C412yes—
58C22×D137C2×D412yes—
59C22×D137C2×D412yes—
60C22×D137C2×D412yes—
61Dic274—28no—
62D548—28no—
63C22⋊D137—28no—
64C22⋊D137—28no—
65C2×Dic137—21no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137—21no—
69C2×Dic137C22⋊C412yes—
70C2×Dic137C22⋊C412yes—
71C4×D137—28no—
72C4×D137—28no—
73C23×C137—21no—
74C23×C137—21no—
75C23×C137—42no—
76C23×C137C22⋊C412yes—
77C23×C137C22×C411yescharacteristic
78C2×C4×C137—28no—
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137C22⋊C412yes—
84C2×C4×C137C22⋊C412yes—
85C22×C137—22no—
86C22×C137—22no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137—42no—
90C22×C137C2×C22⋊C411yescharacteristic
91C22×C137C2×C22⋊C411yescharacteristic
92C22×C137C2×C22⋊C411yescharacteristic
93C2×D137—24no—
94C2×D137—22no—
95C2×D137—42no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137—42no—
99C2×D137—42no—
100C2×D137#12014yes—
101#10587739—84no—
102C4×C137—22no—
103C4×C137—22no—
104Dic137—22no—
105Dic137—22no—
106C2×C137—21no—
107C2×C137—21no—
108C2×C137—42no—
109C2×C137#37911yescharacteristic
110C2×C137#41012yes—
111D137—24no—
112D137—42no—
113C137#178211yescharacteristic, complemented, Sylow
114#1782—1371nomaximal, Sylow
115C22×#41—1371no—
116C2×#99—1372no—
117C2×#99—1372no—
118#381—1372no—
119C2×#41—2744no—
120C2×#41—2744no—
121C2×#41—2744no—
122C22×Q8—1371no—
123C22×D4—1371no—
124C22×D4—1371no—
125C22×D4—1371no—
126C23×C4—1371no—
127C23×C4—1371no—
128C23×C4#1058773911yescharacteristic, complemented
129C2×C22⋊C4—1374no—
130#99—2744no—
131#99—2744no—
132#95—2744no—
133C24—1371no—
134C24—1371no—
135C24C2×#1058773911yescharacteristic
136#41—27432no—
137C2×Q8—2742no—
138C2×Q8—2742no—
139C2×Q8—2742no—
140C2×D4—2742no—
141C2×D4—2742no—
142C2×D4—2742no—
143C2×D4—2742no—
144C2×D4—2742no—
145C2×D4—2742no—
146C2×D4—2742no—
147C2×D4—1374no—
148C2×D4—1374no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—2742no—
153C22×C4—2742no—
154C22×C4—2742no—
155C22×C4—2742no—
156C22×C4—2742no—
157C22×C4—2744no—
158C22×C4—1371no—
159C22×C4—1371no—
160C22×C4—1374no—
161C22×C4C2×#1058773911yescharacteristic
162C22×C4C2×#1058773911yescharacteristic
163C22⋊C4—2744no—
164C22⋊C4—2744no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—2741no—
169C23—2741no—
170C23—5482no—
171C23—5482no—
172C23—1372no—
173C23—1372no—
174C23—1372no—
175C23—1372no—
176C23C22⋊#1058773912yes—
177C23C22×#1058773911yescharacteristic
178Q8—2748no—
179D4—2748no—
180D4—2748no—
181D4—2748no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—2748no—
188C2×C4—2748no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2741no—
192C2×C4—2741no—
193C2×C4—5488no—
194C2×C4—5484no—
195C2×C4—1372no—
196C2×C4—1372no—
197C2×C4C22⋊#1058773912yes—
198C2×C4C22⋊#1058773912yes—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—2744no—
205C22—2742no—
206C22—5482no—
207C22—5482no—
208C22—5482no—
209C22—5482no—
210C22—5482no—
211C22—1374no—
212C22C2×C22⋊#1058773911yescharacteristic
213C22C2×C22⋊#1058773911yescharacteristic
214C22C2×C22⋊#1058773911yescharacteristic
215C4—22no—
216C4—22no—
217C4—2742no—
218C4—2742no—
219C4—10964no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—2744no—
224C2—5482no—
225C2#2257834411yescharacteristic
226C2#2257838212yes—
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×#14555510C2
cosocleC23×D137C23
hypercenterC23×C4#10587739
hypocenterC137#1782
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×#17984147C2semidirect
2. C2×#17984146C2semidirect
3. #22578346C2semidirect
12. C22×Dic274C4semidirect
14. C22×C22⋊D137C4semidirect
16. C22×C4×D137C4semidirect
18. C23×C4×C137C4semidirect
113. C137#1782semidirect
128. C23×C4#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841472
C2×#179841462
#225783462
C22×#145555102
#1782137