Finite Group Databasebeta

← #14545040

Subgroups — #14545040

Order 2176.

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D17⋊#100C211yesmaximal, characteristic, complemented
2D17⋊#99C211yesmaximal, characteristic, complemented
3#11763712C212yesmaximal, complemented
4#11763709C212yesmaximal, complemented
5#11763369C211yesmaximal, characteristic
6C22×C22⋊D17C411yescharacteristic, complemented
7C2×D4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9D17⋊M16—21no—
10C17×#120C411yescharacteristic, complemented
11#10587639C2211yescharacteristic
12#10587624—21no—
13#10587623—21no—
14#10587621C411yescharacteristic, complemented
15#10587572C2211yescharacteristic
16#10587570—21no—
17#10587569—21no—
18C22⋊C4×D17C411yescharacteristic, complemented
19#10587553—21no—
20#10587553C2212yes—
21#10587551C2212yes—
22#10587539—21no—
23C24×C17C2×C411yescharacteristic, complemented
24C23×D17C2×C411yescharacteristic
25C23×D17D411yescharacteristic
26C22×#595—41no—
27C2×D4×C17—21no—
28C2×D4×C17C2×C411yescharacteristic
29C2×C22⋊D17—21no—
30C2×C22⋊D17—21no—
31C2×C22⋊D17—21no—
32C2×C22⋊D17—41no—
33C2×C22⋊D17—41no—
34C2×C22⋊D17D411yescharacteristic
35C2×C22⋊D17C2311yescharacteristic
36C22×Dic17C2×C411yescharacteristic, complemented
37D4×D17—41no—
38D4×D17—41no—
39C2×D68D411yescharacteristic
40C2×C4×D17—21no—
41C2×C4×D17D411yescharacteristic
42C22⋊#595—22no—
43C22⋊#595—41no—
44#63247—22no—
45#63243—41no—
46#63242—41no—
47C22⋊C4×C17—21no—
48C22⋊C4×C17C2×C411yescharacteristic, complemented
49C22⋊Dic17—21no—
50C22⋊Dic17—21no—
51C22⋊Dic17C2×C411yescharacteristic, complemented
52#63227—21no—
53#63227—21no—
54C4⋊Dic17—21no—
55#63225—21no—
56C23×C17—21no—
57C23×C17—41no—
58C23×C17—81no—
59C23×C17C22⋊C411yescharacteristic
60C23×C17C22×C411yescharacteristic
61C22×D17—21no—
62C22×D17—21no—
63C22×D17—21no—
64C22×D17—21no—
65C22×D17—81no—
66C22×D17—81no—
67C22×D17C22⋊C411yescharacteristic
68C22×D17C22⋊C411yescharacteristic
69C22×D17C2×D411yescharacteristic
70C2×#595—42no—
71C2×#595—82no—
72D4×C17—41no—
73D4×C17—81no—
74C2×C4×C17—21no—
75C2×C4×C17C22⋊C411yescharacteristic
76C22⋊D17—41no—
77C22⋊D17—41no—
78C22⋊D17—42no—
79C22⋊D17—41no—
80C22⋊D17—41no—
81C22⋊D17—81no—
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—41no—
85C2×Dic17C2×D411yescharacteristic
86D68—41no—
87C4×D17—21no—
88C4×D17—21no—
89C4×D17—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—41no—
98C22×C17—41no—
99C22×C17—82no—
100C22×C17C2×C22⋊C411yescharacteristic
101C2×D17—21no—
102C2×D17—21no—
103C2×D17—41no—
104C2×D17—42no—
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—81no—
108C2×D17—81no—
109C2×D17—81no—
110C2×D17#9911yescharacteristic
111C2×D17#9911yescharacteristic
112#595—82no—
113C4×C17—21no—
114C4×C17—41no—
115Dic17—21no—
116Dic17—41no—
117Dic17—41no—
118#534—171no—
119C2×#100—171no—
120C2×#99—171no—
121#352—172no—
122AΣL(1,𝔽16)—172no—
123C2×C17—21no—
124C2×C17—41no—
125C2×C17—41no—
126C2×C17—81no—
127C2×C17C2×#9911yescharacteristic
128D17—21no—
129D17—21no—
130D17—41no—
131D17—41no—
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—341no—
142#120#59511yescharacteristic, complemented
143C2×C22⋊C4—171no—
144C2×C22⋊C4—341no—
145#100—172no—
146#100—341no—
147#99—172no—
148#99—341no—
149C17#212111yescharacteristic, complemented, Sylow
150C24—171no—
151C24—171no—
152C24C2×#59511yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—171no—
155C2×D4—171no—
156C2×D4—171no—
157C2×D4—341no—
158C2×D4—341no—
159C2×D4—341no—
160C2×D4—681no—
161C2×D4—681no—
162C2×D4—681no—
163C2×D4—681no—
164C2×D4C2×#59511yescharacteristic
165C22×C4—171no—
166C22×C4—171no—
167C22×C4—341no—
168C22×C4—681no—
169M16—342no—
170M16—681no—
171C2×C8—681no—
172C4⋊C4—341no—
173C4⋊C4—341no—
174C22⋊C4—21no—
175C22⋊C4—171no—
176C22⋊C4—341no—
177C22⋊C4—341no—
178C22⋊C4—341no—
179C22⋊C4—341no—
180C22⋊C4—342no—
181C22⋊C4—681no—
182C22⋊C4C2×#59511yescharacteristic, complemented
183C23—21no—
184C23—41no—
185C23—81no—
186C23—171no—
187C23—171no—
188C23—171no—
189C23—341no—
190C23—341no—
191C23—341no—
192C23—341no—
193C23—1361no—
194C23—1361no—
195C23C22⋊#59511yescharacteristic
196C23C22×#59511yescharacteristic
197D4—41no—
198D4—81no—
199D4—681no—
200D4—681no—
201D4—681no—
202D4—681no—
203D4—681no—
204D4—682no—
205D4—1361no—
206C2×C4—21no—
207C2×C4—171no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—341no—
211C2×C4—341no—
212C2×C4—681no—
213C2×C4—682no—
214C2×C4—1361no—
215C2×C4—1362no—
216C2×C4C22⋊#59511yescharacteristic
217C8—682no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—82no—
226C22—171no—
227C22—171no—
228C22—341no—
229C22—341no—
230C22—681no—
231C22—681no—
232C22—681no—
233C22—682no—
234C22—1361no—
235C22—1361no—
236C22—1361no—
237C22C2×C22⋊#59511yescharacteristic
238C4—21no—
239C4—41no—
240C4—341no—
241C4—681no—
242C4—681no—
243C4—1362no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—81no—
248C2—341no—
249C2—341no—
250C2—681no—
251C2—681no—
252C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#120C4
socleC2×C17C2×#99
prime radical#11763369C2
cosocleC2×C22⋊D17C23
hypercenter#120#595
hypocenterC17#2121
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. D17⋊#100C2semidirect
2. D17⋊#99C2semidirect
3. #11763712C2semidirect
4. #11763709C2semidirect
6. C22×C22⋊D17C4semidirect
10. C17×#120C4semidirect
14. #10587621C4semidirect
18. C22⋊C4×D17C4semidirect
23. C24×C17C2×C4semidirect
36. C22×Dic17C2×C4semidirect
48. C22⋊C4×C17C2×C4semidirect
51. C22⋊Dic17C2×C4semidirect
142. #120#595semidirect
149. C17#2121semidirect
152. C24C2×#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
D17⋊#1002
D17⋊#992
#117637122
#117637092
#117633692
#212117