Finite Group Databasebeta

← C2×#22578303 #31797034

Subgroups — C2×#22578303 #31797034

Order 17536.

438 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 58 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×#17984148C211yesmaximal, characteristic, complemented
2C2×#17984146C211yesmaximal, characteristic, complemented
3#22578303C214yesmaximal, direct product factor
4C2×#17984057C211yesmaximal, characteristic
5C2×#14555519—21no—
6C2×C22⋊#10587739—21no—
7#17984148—21no—
8#17984148C2212yescomplemented
9#17984146—21no—
10#17984146C2212yescomplemented
11C22×C22⋊D137—21no—
12C22×C22⋊D137C2211yescharacteristic
13#17984057—21no—
14#17984057—21no—
15#17984057C412yescomplemented
16#17984057C2212yes—
17C2×C22⋊Dic137—21no—
18C2×C22⋊Dic137C411yescharacteristic, complemented
19C25×C137C411yescharacteristic, complemented
20C22×#10587739—41no—
21#14555519—22no—
22#14555519—41no—
23C2×#11764047—41no—
24C22⋊#10587739—22no—
25C22⋊#10587739—41no—
26C23×D137D411yescharacteristic
27C2×C22⋊D137—22no—
28C2×C22⋊D137—22no—
29C2×C22⋊D137—41no—
30C2×C22⋊D137—41no—
31C2×C22⋊D137—44no—
32C2×C22⋊D137C2×C411yescharacteristic
33C2×C22⋊D137D411yescharacteristic
34C2×C22⋊D137D411yescharacteristic
35C2×C22⋊D137C2311yescharacteristic
36C22×Dic137—21no—
37C22×Dic137D411yescharacteristic
38C22⋊Dic137—22no—
39C22⋊Dic137—22no—
40C22⋊Dic137—21no—
41C22⋊Dic137C2×C412yescomplemented
42C24×C137—21no—
43C24×C137—21no—
44C24×C137—44no—
45C24×C137—41no—
46C24×C137—41no—
47C24×C137C2×C411yescharacteristic
48C24×C137C2×C412yescomplemented
49C2×#10587739—42no—
50C2×#10587739—82no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137—21no—
54C22×D137—42no—
55C22×D137C22⋊C411yescharacteristic
56C22×D137C2×D411yescharacteristic
57C22⋊D137—41no—
58C22⋊D137—41no—
59C22⋊D137—41no—
60C22⋊D137—41no—
61C22⋊D137—44no—
62C22⋊D137—44no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—22no—
66C2×Dic137—42no—
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C23×C137—24no—
70C23×C137—21no—
71C23×C137—21no—
72C23×C137—22no—
73C23×C137—42no—
74C23×C137—44no—
75C23×C137—42no—
76C23×C137—42no—
77C23×C137—44no—
78C23×C137—42no—
79C23×C137—44no—
80C23×C137—48no—
81C23×C137—44no—
82C23×C137—41no—
83C23×C137—41no—
84C23×C137C22⋊C411yescharacteristic
85C23×C137C22⋊C411yescharacteristic
86C23×C137C22×C411yescharacteristic
87C22×C137—22no—
88C22×C137—24no—
89C22×C137—21no—
90C22×C137—21no—
91C22×C137—41no—
92C22×C137—41no—
93C22×C137—42no—
94C22×C137—44no—
95C22×C137—42no—
96C22×C137—44no—
97C22×C137—44no—
98C22×C137—48no—
99C22×C137—44no—
100C22×C137—42no—
101C22×C137—42no—
102C22×C137#9911yescharacteristic
103C22×C137#9911yescharacteristic
104C22×C137C2×C22⋊C411yescharacteristic
105C2×D137—21no—
106C2×D137—21no—
107C2×D137—42no—
108C2×D137—41no—
109C2×D137—41no—
110C2×D137—81no—
111#10587739—82no—
112Dic137—21no—
113Dic137—21no—
114Dic137—42no—
115C2×C137—21no—
116C2×C137—21no—
117C2×C137—44no—
118C2×C137—41no—
119C2×C137—41no—
120C2×C137AΣL(1,𝔽16)12yescomplemented
121C2×C137C2×#9911yescharacteristic
122D137—41no—
123D137—41no—
124C137C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
125C2×AΣL(1,𝔽16)—1371nomaximal, Sylow
126C2×#120—1371no—
127C2×#100—1371no—
128C2×#99—1371no—
129AΣL(1,𝔽16)—1374no—
130C25#1058773911yescharacteristic, complemented
131C22×D4—2741no—
132C22×D4—1371no—
133C2×M16—2741no—
134#120—2741no—
135#120—2741no—
136#120—1372no—
137#120—1372no—
138C2×C22⋊C4—2741no—
139C2×C22⋊C4—2741no—
140C2×C22⋊C4—1371no—
141#100—2741no—
142#100—1372no—
143#99—2741no—
144#99—1372no—
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24—1371no—
151C24C2×#1058773911yescharacteristic
152C24C2×#1058773912yescomplemented
153C2×D4—2742no—
154C2×D4—2742no—
155C2×D4—5481no—
156C2×D4—5481no—
157C2×D4—5484no—
158C2×D4—1371no—
159C2×D4—1371no—
160C2×D4—1371no—
161C2×D4—1371no—
162C22×C4—2741no—
163C22×C4—5481no—
164C22×C4—1371no—
165M16—2742no—
166M16—5481no—
167C2×C8—5481no—
168C22⋊C4—2741no—
169C22⋊C4—2742no—
170C22⋊C4—2742no—
171C22⋊C4—2742no—
172C22⋊C4—5481no—
173C22⋊C4—1372no—
174C23—24no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—42no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23—44no—
183C23—42no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—41no—
188C23—41no—
189C23—2741no—
190C23—2741no—
191C23—5482no—
192C23—1371no—
193C23—1371no—
194C23C22⋊#1058773911yescharacteristic
195C23C22⋊#1058773911yescharacteristic
196C23C22×#1058773911yescharacteristic
197D4—5481no—
198D4—5481no—
199D4—5481no—
200D4—5481no—
201D4—5484no—
202D4—5484no—
203C2×C4—2741no—
204C2×C4—2741no—
205C2×C4—2742no—
206C2×C4—5482no—
207C2×C4—5482no—
208C2×C4—10962no—
209C2×C4—1371no—
210C2×C4—1371no—
211C8—5482no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—44no—
222C22—44no—
223C22—48no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—2741no—
228C22—2741no—
229C22—5481no—
230C22—5481no—
231C22—5482no—
232C22—10961no—
233C22#1798414611yescharacteristic
234C22#1798414611yescharacteristic
235C22C2×C22⋊#1058773911yescharacteristic
236C4—2741no—
237C4—2741no—
238C4—5482no—
239C4—10962no—
240C2—21no—
241C2—21no—
242C2—44no—
243C2—41no—
244C2—41no—
245C2—5481no—
246C2—5481no—
247C2#2257830312yesdirect product factor
248C2C2×#1798414611yescharacteristic
subgroupG/H
centerC22#17984146
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC25×C137C4
socleC22×C137#99
prime radicalC2×#17984057C2
cosocleC2×C22⋊D137C23
hypercenterC25#10587739
hypocenterC137C2×AΣL(1,𝔽16)
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C2×#17984148C2semidirect
2. C2×#17984146C2semidirect
3. #22578303C2direct (N × H)
8. #17984148C22semidirect
10. #17984146C22semidirect
15. #17984057C4semidirect
18. C2×C22⋊Dic137C4semidirect
19. C25×C137C4semidirect
41. C22⋊Dic137C2×C4semidirect
48. C24×C137C2×C4semidirect
120. C2×C137AΣL(1,𝔽16)semidirect
124. C137C2×AΣL(1,𝔽16)semidirect
130. C25#10587739semidirect
152. C24C2×#10587739semidirect
247. C2#22578303direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841482
C2×#179841462
#225783032
C2×#179840572
C2×AΣL(1,𝔽16)137