Finite Group Databasebeta

← #16949512

Subgroups — #16949512

Order 4352.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#14531504C211yesmaximal, characteristic
2C2×#11762608C211yesmaximal, characteristic
3#101×Dic17C212yesmaximal
4#14526910C212yesmaximal
5C2×C17×#356C211yesmaximal, characteristic
6C2×Q8×Dic17C411yescharacteristic
7C2×Q8×Dic17C2211yescharacteristic
8C2×#10587507C2212yes—
9M16×Dic17—22no—
10#11762716C411yescharacteristic
11#11762716C2211yescharacteristic
12#11762695—22no—
13#11762695—22no—
14#11762675C411yescharacteristic
15#11762655C411yescharacteristic
16#11762608C414yescomplemented
17#11762577—22no—
18C2×C17×C4⋊Q8C2211yescharacteristic
19C2×C17×#101C2212yes—
20C17×#356C414yescomplemented
21C22×Q8×C17C2×C411yescharacteristic
22C22×Q8×C17C2311yescharacteristic
23C2×M16×C17—22no—
24C17×C4⋊Q8—22no—
25C17×C4⋊Q8C2×C412yes—
26C17×C4⋊Q8C2×C412yes—
27C2×C4⋊C4×C17—22no—
28C2×C42×C17C2×C411yescharacteristic
29Q8×Dic17—42no—
30Q8×Dic17—42no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4⋊Dic17D412yes—
35C2×C4×Dic17—21no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D412yes—
39C2×#63223—22no—
40C17×#101—22no—
41C17×#101C2×C414yes—
42#10587507—22no—
43#10587507C2×C414yes—
44#10587502—22no—
45#10587502—22no—
46#10587502—21no—
47#10587502—21no—
48#10587502—22no—
49#10587486—42no—
50C8×Dic17—42no—
51C2×Q8×C17—21no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17C4212yes—
57C2×Q8×C17C22×C411yescharacteristic
58C2×Q8×C17C22×C411yescharacteristic
59C2×Q8×C17#4112yes—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17C22⋊C412yes—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69M16×C17—24no—
70M16×C17—42no—
71C2×C8×C17—42no—
72C4⋊C4×C17—24no—
73C4⋊C4×C17—24no—
74C42×C17—21no—
75C42×C17C4212yes—
76C4⋊Dic17—41no—
77C4⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4⋊Dic17—41no—
80C4×Dic17—22no—
81C4×Dic17—22no—
82C4×Dic17—22no—
83C4×Dic17—42no—
84C4×Dic17—41no—
85#63223—24no—
86#63223—42no—
87C2×#3622—42no—
88#12631—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90Q8×C17—41no—
91Q8×C17—41no—
92Q8×C17—42no—
93Q8×C17—42no—
94Q8×C17—42no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17C2×C4211yescharacteristic
106C2×C4×C17#11711yescharacteristic
107C2×C4×C17C4×D412yes—
108C2×C4×C17C4×D412yes—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—41no—
112C2×Dic17—42no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17#9912yes—
120C8×C17—44no—
121#3622—44no—
122#2327—171no—
123C2×#356—171no—
124C2×#356D1711yescharacteristic
125C4×#101—172no—
126#1394—172no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17C4×C22⋊C411yescharacteristic
130C22×C17C2×#9911yescharacteristic
131C22×C17#41011yescharacteristic
132C4×C17—22no—
133C4×C17—22no—
134C4×C17—21no—
135C4×C17—21no—
136C4×C17—42no—
137C4×C17—42no—
138Dic17—22no—
139Dic17—41no—
140Dic17—42no—
141Dic17—81no—
142Dic17—81no—
143C2×C4⋊Q8C2×D1711yescharacteristic
144C2×C4×Q8—171no—
145C2×C4×Q8—171no—
146C2×#101—172no—
147C2×#101C2×D1712yes—
148C4×M16—342no—
149#391—171no—
150#391—171no—
151#387—342no—
152#387—342no—
153#382—171no—
154#376—171no—
155#356—174no—
156#356Dic1714yescomplemented
157#343—342no—
158C2×C17—21no—
159C2×C17—21no—
160C2×C17C4×#9911yescharacteristic
161C2×C17#213311yescharacteristic
162C2×C17#213511yescharacteristic
163C22×Q8C4×D1711yescharacteristic
164C22×Q8C22×D1711yescharacteristic
165C2×M16—22no—
166C2×M16—342no—
167C4⋊Q8—22no—
168C4⋊Q8C4×D1712yes—
169C4⋊Q8C2×Dic1712yes—
170C4×Q8—682no—
171C4×Q8—682no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—341no—
177C2×C42—171no—
178C2×C42—172no—
179C2×C42—171no—
180C2×C42—341no—
181C2×C42C4×D1711yescharacteristic
182#101—22no—
183#101—174no—
184#101—342no—
185#101C2×Dic1714yes—
186#97—682no—
187C4×C8—682no—
188#95—341no—
189#95—341no—
190#95—342no—
191#95—342no—
192#95—342no—
193C17#1263111yescharacteristic, complemented, Sylow
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C4×Dic1712yes—
200C2×Q8C2×C4×D1711yescharacteristic
201C2×Q8#6325412yes—
202C2×Q8C22×Dic1711yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—171no—
206C22×C4—172no—
207C22×C4—171no—
208C22×C4—341no—
209C22×C4—341no—
210C22×C4C2×C4×D1711yescharacteristic
211C22×C4D4×D1712yes—
212M16—24no—
213M16—42no—
214M16—344no—
215M16—682no—
216C2×C8—42no—
217C2×C8—682no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—681no—
221C4⋊C4—682no—
222C4⋊C4—681no—
223C4⋊C4—682no—
224C42—21no—
225C42—342no—
226C42—342no—
227C42—342no—
228C42—681no—
229C42—682no—
230C42C4×Dic1712yes—
231C23C22⋊C4×D1711yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235Q8—42no—
236Q8—42no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—172no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—681no—
251C2×C4—682no—
252C2×C4—681no—
253C2×C4—681no—
254C2×C4—681no—
255C2×C4—682no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4#1058756311yescharacteristic
259C2×C4#1058756712yes—
260C2×C4C2×C4×Dic1711yescharacteristic
261C2×C4D4×Dic1712yes—
262C8—44no—
263C8—684no—
264C22—21no—
265C22—21no—
266C22C22⋊C4×Dic1711yescharacteristic
267C22#99×D1711yescharacteristic
268C22#1176278111yescharacteristic
269C4—22no—
270C4—22no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—342no—
276C4—681no—
277C4—682no—
278C4—1361no—
279C4—1361no—
280C2—21no—
281C2—21no—
282C2#99×Dic1711yescharacteristic
283C2#1453047211yescharacteristic
284C2#1453047711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC2×C4×C17C2×C42
FrattiniC22×Q8C22×D17
FittingC2×C17×#356C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenterC2×#356D17
hypocenterC17#12631
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
16. #11762608C4semidirect
20. C17×#356C4semidirect
156. #356Dic17semidirect
193. C17#12631semidirect

Maximal subgroups

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

subgroupindex
#145315042
C2×#117626082
#101×Dic172
#145269102
C2×C17×#3562
#1263117