Finite Group Databasebeta

← C2×#11763710 #14545031

Subgroups — C2×#11763710 #14545031

Order 2176.

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×#10587553C211yesmaximal, characteristic, complemented
2C2×#10587551C211yesmaximal, characteristic, complemented
3#11763710C214yesmaximal, direct product factor
4C2×#10587568C211yesmaximal, characteristic
5C24×D17C411yescharacteristic, complemented
6C22×C22⋊D17C2211yescharacteristic
7C22×D68—21no—
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C22⋊C4×C17C411yescharacteristic, complemented
11C2×#63227—21no—
12#10587568—21no—
13#10587568—21no—
14#10587568C412yescomplemented
15#10587568C2212yes—
16#10587553—21no—
17#10587553C2212yescomplemented
18#10587551—21no—
19#10587551C2212yescomplemented
20C24×C17C2×C411yescharacteristic
21C23×D17—21no—
22C23×D17—21no—
23C23×D17—44no—
24C23×D17—41no—
25C23×D17—41no—
26C23×D17C2×C412yescomplemented
27C23×D17D411yescharacteristic
28C22×#595—41no—
29C22×C4×C17—21no—
30C2×C22⋊D17—41no—
31C2×C22⋊D17—41no—
32C2×C22⋊D17C2×C411yescharacteristic
33C2×C22⋊D17D411yescharacteristic
34C2×C22⋊D17D411yescharacteristic
35C2×C22⋊D17C2311yescharacteristic
36C22×Dic17D411yescharacteristic
37C2×D68—22no—
38C2×D68—22no—
39C2×D68—44no—
40C22⋊#595—22no—
41C22⋊#595—41no—
42#63247—22no—
43#63247—41no—
44C2×#3624—41no—
45C22⋊C4×C17—21no—
46C22⋊C4×C17C2×C412yescomplemented
47#63227—22no—
48#63227—22no—
49C23×C17—21no—
50C23×C17—21no—
51C23×C17—42no—
52C23×C17C22⋊C411yescharacteristic
53C23×C17C22⋊C411yescharacteristic
54C23×C17C22×C411yescharacteristic
55C22×D17—24no—
56C22×D17—21no—
57C22×D17—22no—
58C22×D17—21no—
59C22×D17—44no—
60C22×D17—44no—
61C22×D17—42no—
62C22×D17—42no—
63C22×D17—48no—
64C22×D17—44no—
65C22×D17—41no—
66C22×D17—41no—
67C22×D17—42no—
68C22×D17—44no—
69C22×D17—42no—
70C22×D17C22⋊C411yescharacteristic
71C22×D17C2×D411yescharacteristic
72C2×#595—42no—
73C2×#595—82no—
74C2×C4×C17—22no—
75C2×C4×C17—42no—
76C22⋊D17—41no—
77C22⋊D17—41no—
78C22⋊D17—41no—
79C22⋊D17—41no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17C22⋊C411yescharacteristic
83C2×Dic17C2×D411yescharacteristic
84D68—44no—
85D68—44no—
86#3624—42no—
87C2×AΣL(1,𝔽16)—171nomaximal, Sylow
88C22×C17—22no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17#9911yescharacteristic
96C22×C17#9911yescharacteristic
97C22×C17C2×C22⋊C411yescharacteristic
98C2×D17—24no—
99C2×D17—21no—
100C2×D17—21no—
101C2×D17—42no—
102C2×D17—44no—
103C2×D17—48no—
104C2×D17—44no—
105C2×D17—42no—
106C2×D17—44no—
107C2×D17—42no—
108C2×D17—41no—
109C2×D17—41no—
110C2×D17—44no—
111C2×D17—81no—
112#595—82no—
113C4×C17—42no—
114Dic17—21no—
115Dic17—21no—
116C2×#120—171no—
117C2×#100—171no—
118C2×#99—171no—
119AΣL(1,𝔽16)—174no—
120C2×C17—21no—
121C2×C17—21no—
122C2×C17—41no—
123C2×C17—41no—
124C2×C17AΣL(1,𝔽16)12yescomplemented
125C2×C17C2×#9911yescharacteristic
126D17—41no—
127D17—44no—
128D17—41no—
129C25—171no—
130C22×D4—171no—
131C22×D4—341no—
132C2×M16—341no—
133#120—172no—
134#120—172no—
135#120—341no—
136#120—341no—
137C2×C22⋊C4—341no—
138C2×C22⋊C4—341no—
139C2×C22⋊C4#59511yescharacteristic, complemented
140#100—172no—
141#100—341no—
142#99—172no—
143#99—341no—
144C17C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
145C24—171no—
146C24—172no—
147C24—341no—
148C24—341no—
149C24—681no—
150C24—681no—
151C24—684no—
152C24C2×#59511yescharacteristic
153C2×D4—171no—
154C2×D4—171no—
155C2×D4—171no—
156C2×D4—171no—
157C2×D4—342no—
158C2×D4—342no—
159C2×D4—684no—
160C2×D4—681no—
161C2×D4—681no—
162C22×C4—21no—
163C22×C4—171no—
164C22×C4—681no—
165M16—342no—
166M16—681no—
167C2×C8—681no—
168C22⋊C4—21no—
169C22⋊C4—342no—
170C22⋊C4—342no—
171C22⋊C4—342no—
172C22⋊C4—681no—
173C22⋊C4C2×#59512yescomplemented
174C23—21no—
175C23—21no—
176C23—42no—
177C23—171no—
178C23—171no—
179C23—344no—
180C23—342no—
181C23—341no—
182C23—341no—
183C23—682no—
184C23—682no—
185C23—682no—
186C23—682no—
187C23—684no—
188C23—684no—
189C23—684no—
190C23—688no—
191C23—684no—
192C23—681no—
193C23—681no—
194C23C22⋊#59511yescharacteristic
195C23C22⋊#59511yescharacteristic
196C23C22×#59511yescharacteristic
197D4—684no—
198D4—684no—
199D4—681no—
200D4—681no—
201D4—681no—
202D4—681no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—171no—
206C2×C4—171no—
207C2×C4—341no—
208C2×C4—341no—
209C2×C4—682no—
210C2×C4—1362no—
211C8—682no—
212C22—22no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—42no—
218C22—42no—
219C22—344no—
220C22—341no—
221C22—341no—
222C22—681no—
223C22—681no—
224C22—682no—
225C22—684no—
226C22—684no—
227C22—684no—
228C22—688no—
229C22—682no—
230C22—682no—
231C22—684no—
232C22—1361no—
233C22#1058753911yescharacteristic
234C22#1058753911yescharacteristic
235C22C2×C22⋊#59511yescharacteristic
236C4—42no—
237C4—341no—
238C4—341no—
239C4—1362no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—681no—
245C2—684no—
246C2—681no—
247C2#1176371012yesdirect product factor
248C2C2×#1058753911yescharacteristic
subgroupG/H
centerC22#10587539
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17#99
prime radicalC2×#10587568C2
cosocleC2×C22⋊D17C23
hypercenterC2×C22⋊C4#595
hypocenterC17C2×AΣL(1,𝔽16)
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587553C2semidirect
2. C2×#10587551C2semidirect
3. #11763710C2direct (N × H)
5. C24×D17C4semidirect
10. C2×C22⋊C4×C17C4semidirect
14. #10587568C4semidirect
17. #10587553C22semidirect
19. #10587551C22semidirect
26. C23×D17C2×C4semidirect
46. C22⋊C4×C17C2×C4semidirect
124. C2×C17AΣL(1,𝔽16)semidirect
139. C2×C22⋊C4#595semidirect
144. C17C2×AΣL(1,𝔽16)semidirect
173. C22⋊C4C2×#595semidirect
247. C2#11763710direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#105875532
C2×#105875512
#117637102
C2×#105875682
C2×AΣL(1,𝔽16)17