Finite Group Databasebeta

← #14529241

Subgroups — #14529241

Order 2176.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#10587572C211yesmaximal, characteristic
2C2×#10587563C211yesmaximal, characteristic, complemented
3C2×#10587477C211yesmaximal, characteristic
4C2×#10587477C211yesmaximal, characteristic
5#11762670C211yesmaximal, characteristic, complemented
6#11762576C211yesmaximal, characteristic, complemented
7C17×#342C211yesmaximal, characteristic, complemented
8C22×C22⋊D17C2211yescharacteristic
9C23×Dic17C2211yescharacteristic
10C2×C22⋊C4×C17C2211yescharacteristic
11C2×C22⋊C4×C17C2211yescharacteristic
12C2×C22⋊C4×C17C2211yescharacteristic
13C2×C22⋊Dic17C2211yescharacteristic
14C2×C22⋊Dic17C2211yescharacteristic
15C2×#63227C412yescomplemented
16C2×C4⋊Dic17C412yescomplemented
17C2×#63225—21no—
18C2×C4×Dic17—21no—
19#10587572—22no—
20#10587572—21no—
21#10587572—21no—
22#10587563—22no—
23#10587563—22no—
24#10587502—22no—
25#10587477—22no—
26#10587477—22no—
27C24×C17C2311yescharacteristic
28C23×D17C2×C411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C22×C4×C17C2×C412yes—
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C2×C22⋊D17—21no—
35C2×C22⋊D17—21no—
36C2×C22⋊D17—21no—
37C2×C22⋊D17—21no—
38C22×Dic17—21no—
39C22×Dic17—21no—
40C22×Dic17—21no—
41C22×Dic17—21no—
42C22×Dic17—21no—
43C22×Dic17—21no—
44C22×Dic17C2×C411yescharacteristic
45C22×Dic17C2×C412yes—
46C22×Dic17D412yes—
47C22×Dic17D411yescharacteristic
48C22×Dic17D411yescharacteristic
49C22⋊C4×C17—21no—
50C22⋊C4×C17—21no—
51C22⋊C4×C17—21no—
52C22⋊C4×C17—21no—
53C22⋊C4×C17—21no—
54C22⋊C4×C17D412yes—
55C22⋊Dic17—21no—
56C22⋊Dic17—21no—
57C22⋊Dic17—21no—
58C22⋊Dic17D412yes—
59#63227—22no—
60#63227—22no—
61C4⋊Dic17—22no—
62C4⋊Dic17—22no—
63#63225—24no—
64C4×Dic17—24no—
65C23×C17—42no—
66C23×C17C22×C411yescharacteristic
67C23×C17C2×D411yescharacteristic
68C23×C17C2×D411yescharacteristic
69C23×C17C2×D411yescharacteristic
70C23×C17C2×D411yescharacteristic
71C23×C17#4111yescharacteristic
72C23×C17#4111yescharacteristic
73C22×D17—21no—
74C22×D17—21no—
75C22×D17—21no—
76C22×D17—42no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—41no—
83C22⋊D17—41no—
84C22⋊D17—41no—
85C22⋊D17—41no—
86C22⋊D17—41no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—22no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—41no—
102C2×Dic17C22⋊C412yes—
103C2×Dic17C22⋊C412yes—
104#1899—171nomaximal, Sylow
105C22×C17—22no—
106C22×C17—22no—
107C22×C17—22no—
108C22×C17—41no—
109C22×C17—41no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17C2×C22⋊C411yescharacteristic
113C22×C17C4×D411yescharacteristic
114C22×C17C4×D411yescharacteristic
115C22×C17#12011yescharacteristic
116C22×C17#12111yescharacteristic
117C22×C17#12111yescharacteristic
118C22×C17#12311yescharacteristic
119C2×D17—41no—
120C2×D17—42no—
121C2×D17—81no—
122C2×D17—81no—
123C4×C17—42no—
124C4×C17—42no—
125C4×C17—81no—
126Dic17—22no—
127Dic17—22no—
128Dic17—42no—
129Dic17—41no—
130Dic17—81no—
131C2×#123—171no—
132C2×#117—171no—
133C2×#99—171no—
134C2×#99—171no—
135#381—171no—
136#342—171no—
137#342D1711yescharacteristic, complemented
138C2×C17—21no—
139C2×C17—21no—
140C2×C17—21no—
141C2×C17—21no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17#38611yescharacteristic
145C2×C17#41012yes—
146D17—81no—
147C22×D4—171no—
148C23×C4—171no—
149#123—342no—
150#123—341no—
151#123—341no—
152#117—342no—
153#117—342no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—341no—
156C2×C22⋊C4—171no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—171no—
159C2×C22⋊C4C2×D1711yescharacteristic
160C2×C22⋊C4C2×D1711yescharacteristic
161C2×C22⋊C4C2×D1711yescharacteristic
162C2×C42—341no—
163#99—342no—
164#99—342no—
165#95—342no—
166C17#189911yescharacteristic, complemented, Sylow
167C24—171no—
168C24C22×D1711yescharacteristic
169C2×D4—341no—
170C2×D4—341no—
171C2×D4—341no—
172C2×D4—341no—
173C2×D4—341no—
174C2×D4—341no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—172no—
178C22×C4—172no—
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—341no—
183C22×C4—341no—
184C22×C4—341no—
185C22×C4—341no—
186C22×C4—341no—
187C22×C4—341no—
188C22×C4C4×D1712yes—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—344no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—172no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C22⋊C4—341no—
203C22⋊C4D6812yes—
204C42—344no—
205C23—42no—
206C23—341no—
207C23—341no—
208C23—341no—
209C23—682no—
210C23C2×C4×D1711yescharacteristic
211C23C2×D6811yescharacteristic
212C23#6325211yescharacteristic
213C23D4×D1711yescharacteristic
214C23D4×D1711yescharacteristic
215C23D4×D1711yescharacteristic
216C23#6325611yescharacteristic
217D4—681no—
218D4—681no—
219D4—681no—
220D4—681no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—172no—
228C2×C4—172no—
229C2×C4—342no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—341no—
235C2×C4—341no—
236C2×C4—342no—
237C2×C4—341no—
238C2×C4—341no—
239C2×C4—342no—
240C2×C4—341no—
241C2×C4—682no—
242C2×C4—682no—
243C2×C4—681no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—682no—
252C22—681no—
253C22—1361no—
254C22—1361no—
255C22C4×D6811yescharacteristic
256C22C22⋊C4×D1711yescharacteristic
257C22#1058756811yescharacteristic
258C22#1058757011yescharacteristic
259C22#1058757911yescharacteristic
260C22#1058758011yescharacteristic
261C22#1058758111yescharacteristic
262C4—42no—
263C4—42no—
264C4—81no—
265C4—342no—
266C4—342no—
267C4—682no—
268C4—681no—
269C4—1361no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—1361no—
277C2#1176268711yescharacteristic
278C2#1176278112yes—
subgroupG/H
centerC22C22⋊C4×D17
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC17×#342C2
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C22
cosocleC24×C17C23
hypercenter#342D17
hypocenterC17#1899
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#10587563C2semidirect
5. #11762670C2semidirect
6. #11762576C2semidirect
7. C17×#342C2semidirect
15. C2×#63227C4semidirect
16. C2×C4⋊Dic17C4semidirect
137. #342D17semidirect
166. C17#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#105875722
C2×#105875632
C2×#105874772
#117626702
#117625762
C17×#3422
#189917