Finite Group Databasebeta

← #14545041

Subgroups — #14545041

Order 2176.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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#11763712C212yesmaximal, complemented
4#11763710C212yesmaximal, complemented
5#11763371C211yesmaximal, characteristic
6C22×C22⋊D17C411yescharacteristic, complemented
7C22×C22⋊D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C22⋊C4×C17C411yescharacteristic, complemented
11#10587616—21no—
12#10587616—21no—
13#10587572—21no—
14#10587572C411yescharacteristic, complemented
15#10587572C2211yescharacteristic
16#10587568—21no—
17#10587568C411yescharacteristic, complemented
18#10587568C2211yescharacteristic
19#10587553—21no—
20#10587553C2212yes—
21#10587551—21no—
22#10587551C2212yes—
23C24×C17C2×C411yescharacteristic
24C23×D17C2×C411yescharacteristic, complemented
25C23×D17D411yescharacteristic
26C22×#595—41no—
27C22×C4×C17—21no—
28C2×C22⋊D17—21no—
29C2×C22⋊D17—21no—
30C2×C22⋊D17—41no—
31C2×C22⋊D17—41no—
32C2×C22⋊D17—41no—
33C2×C22⋊D17—41no—
34C2×C22⋊D17C2×C411yescharacteristic
35C2×C22⋊D17D411yescharacteristic
36C2×C22⋊D17D411yescharacteristic
37C2×C22⋊D17C2311yescharacteristic
38C22×Dic17C2×C411yescharacteristic, complemented
39C22×Dic17D411yescharacteristic
40C2×D68—21no—
41C2×D68—21no—
42C22⋊#595—22no—
43C22⋊#595—41no—
44#63247—22no—
45#63247—41no—
46C2×#3624—41no—
47C22⋊C4×C17—21no—
48C22⋊C4×C17C2×C411yescharacteristic, complemented
49C22⋊C4×C17C2×C411yescharacteristic, complemented
50#63227—21no—
51#63227—21no—
52#63227—21no—
53#63227—21no—
54C4⋊Dic17—21no—
55C4⋊Dic17—21no—
56C23×C17—21no—
57C23×C17—21no—
58C23×C17—81no—
59C23×C17C22⋊C411yescharacteristic
60C23×C17C22⋊C411yescharacteristic
61C23×C17C22×C411yescharacteristic
62C22×D17—21no—
63C22×D17—21no—
64C22×D17—21no—
65C22×D17—41no—
66C22×D17—81no—
67C22×D17—81no—
68C22×D17C22⋊C411yescharacteristic
69C22×D17C2×D411yescharacteristic
70C2×#595—42no—
71C2×#595—82no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—81no—
75C22⋊D17—41no—
76C22⋊D17—41no—
77C22⋊D17—41no—
78C22⋊D17—41no—
79C22⋊D17—42no—
80C22⋊D17—41no—
81C22⋊D17—41no—
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—21no—
85C2×Dic17—41no—
86C2×Dic17C22⋊C411yescharacteristic
87C2×Dic17C2×D411yescharacteristic
88D68—81no—
89D68—81no—
90#3624—42no—
91#2121—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17—41no—
95C22×C17—41no—
96C22×C17—41no—
97C22×C17—81no—
98C22×C17—81no—
99C22×C17#9911yescharacteristic
100C22×C17#9911yescharacteristic
101C22×C17C2×C22⋊C411yescharacteristic
102C2×D17—21no—
103C2×D17—21no—
104C2×D17—42no—
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—41no—
108C2×D17—41no—
109C2×D17—41no—
110C2×D17—81no—
111C2×D17—82no—
112#595—82no—
113C4×C17—41no—
114C4×C17—41no—
115Dic17—21no—
116Dic17—21no—
117Dic17—41no—
118#534—171no—
119C2×#100—171no—
120C2×#99—171no—
121#352—172no—
122AΣL(1,𝔽16)—172no—
123C2×C17—21no—
124C2×C17—21no—
125C2×C17—21no—
126C2×C17—41no—
127C2×C17—41no—
128C2×C17C2×#9911yescharacteristic
129D17—41no—
130D17—41no—
131D17—81no—
132C22×D4—171no—
133C22×D4—171no—
134C2×M16—341no—
135#123—171no—
136#123—171no—
137#123—341no—
138#121—341no—
139#121—341no—
140#120—171no—
141#120—171no—
142#120—341no—
143C2×C22⋊C4—341no—
144C2×C22⋊C4#59511yescharacteristic, complemented
145#100—172no—
146#100—341no—
147#99—172no—
148#99—341no—
149C17#212111yescharacteristic, complemented, Sylow
150C24—171no—
151C24—171no—
152C24C2×#59511yescharacteristic
153C2×D4—171no—
154C2×D4—171no—
155C2×D4—171no—
156C2×D4—171no—
157C2×D4—341no—
158C2×D4—341no—
159C2×D4—341no—
160C2×D4—341no—
161C2×D4—681no—
162C2×D4—681no—
163C2×D4—681no—
164C2×D4—681no—
165C22×C4—21no—
166C22×C4—171no—
167C22×C4—171no—
168C22×C4—681no—
169M16—342no—
170M16—681no—
171C2×C8—681no—
172C4⋊C4—341no—
173C4⋊C4—341no—
174C22⋊C4—21no—
175C22⋊C4—341no—
176C22⋊C4—341no—
177C22⋊C4—341no—
178C22⋊C4—341no—
179C22⋊C4—342no—
180C22⋊C4—681no—
181C22⋊C4C2×#59511yescharacteristic, complemented
182C22⋊C4C2×#59511yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—81no—
186C23—171no—
187C23—171no—
188C23—341no—
189C23—341no—
190C23—341no—
191C23—681no—
192C23—1361no—
193C23—1361no—
194C23C22⋊#59511yescharacteristic
195C23C22⋊#59511yescharacteristic
196C23C22×#59511yescharacteristic
197D4—681no—
198D4—681no—
199D4—681no—
200D4—681no—
201D4—682no—
202D4—681no—
203D4—681no—
204D4—1361no—
205D4—1361no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—81no—
209C2×C4—171no—
210C2×C4—171no—
211C2×C4—341no—
212C2×C4—341no—
213C2×C4—341no—
214C2×C4—681no—
215C2×C4—682no—
216C2×C4—1362no—
217C8—682no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—341no—
226C22—341no—
227C22—681no—
228C22—681no—
229C22—682no—
230C22—681no—
231C22—681no—
232C22—681no—
233C22—1361no—
234C22—1362no—
235C22#1058753911yescharacteristic
236C22#1058753911yescharacteristic
237C22C2×C22⋊#59511yescharacteristic
238C4—41no—
239C4—41no—
240C4—341no—
241C4—341no—
242C4—681no—
243C4—1362no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—681no—
250C2—681no—
251C2—1361no—
252C2C2×#1058753911yescharacteristic
subgroupG/H
centerC2C2×#10587539
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC2×C17C2×#99
prime radical#11763371C2
cosocleC2×C22⋊D17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#2121
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ 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. #11763712C2semidirect
4. #11763710C2semidirect
6. C22×C22⋊D17C4semidirect
10. C2×C22⋊C4×C17C4semidirect
14. #10587572C4semidirect
17. #10587568C4semidirect
24. C23×D17C2×C4semidirect
38. C22×Dic17C2×C4semidirect
48. C22⋊C4×C17C2×C4semidirect
49. C22⋊C4×C17C2×C4semidirect
144. C2×C22⋊C4#595semidirect
149. C17#2121semidirect
181. C22⋊C4C2×#595semidirect
182. C22⋊C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#105875532
C2×#105875512
#117637122
#117637102
#117633712
#212117