Finite Group Databasebeta

← #356×Dic17 #16949508

Subgroups — #356×Dic17 #16949508

Order 4352.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4⋊Q8×Dic17C211yesmaximal, 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#11762708—22no—
13#11762679C411yescharacteristic
14C4⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic
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—41no—
30Q8×Dic17—41no—
31Q8×Dic17—41no—
32Q8×Dic17—41no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17D412yes—
38C2×C4×Dic17—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17D412yes—
44C2×#63223—22no—
45C17×#101—22no—
46C17×#101C2×C414yes—
47#10587507—22no—
48#10587507C2×C414yes—
49#10587502—22no—
50#10587502—22no—
51#10587486—42no—
52C8×Dic17—42no—
53C2×Q8×C17—21no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C2×Q8×C17—41no—
58C2×Q8×C17C4212yes—
59C2×Q8×C17C22×C411yescharacteristic
60C2×Q8×C17C22×C411yescharacteristic
61C2×Q8×C17#4112yes—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C412yes—
70C22×Dic17C22⋊C411yescharacteristic
71M16×C17—24no—
72M16×C17—42no—
73C2×C8×C17—42no—
74C4⋊C4×C17—24no—
75C4⋊C4×C17—24no—
76C42×C17—21no—
77C42×C17C4212yes—
78C4⋊Dic17—41no—
79C4⋊Dic17—42no—
80C4⋊Dic17—42no—
81C4⋊Dic17—44no—
82C4⋊Dic17—41no—
83C4×Dic17—24no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—21no—
87C4×Dic17—21no—
88C4×Dic17—42no—
89C4×Dic17—41no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92#63223—24no—
93#63223—42no—
94C2×#3622—42no—
95C4×#356—171nomaximal, Sylow
96C23×C17C2×C22⋊C411yescharacteristic
97Q8×C17—41no—
98Q8×C17—41no—
99Q8×C17—42no—
100Q8×C17—42no—
101Q8×C17—42no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17C2×C4211yescharacteristic
113C2×C4×C17#11711yescharacteristic
114C2×C4×C17C4×D412yes—
115C2×C4×C17C4×D412yes—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—21no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C2×Dic17#9911yescharacteristic
127C2×Dic17#9911yescharacteristic
128C8×C17—44no—
129#3622—44no—
130C4×C4⋊Q8—171no—
131C2×#356—171no—
132C2×#356D1711yescharacteristic
133C4×#101—172no—
134#1394—172no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17C4×C22⋊C411yescharacteristic
138C22×C17C2×#9911yescharacteristic
139C22×C17#41011yescharacteristic
140C4×C17—24no—
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—21no—
144C4×C17—21no—
145C4×C17—42no—
146Dic17—21no—
147Dic17—42no—
148Dic17—42no—
149Dic17—41no—
150Dic17—81no—
151Dic17#35612yesdirect product factor
152C2×C4⋊Q8C2×D1711yescharacteristic
153C2×C4×Q8—171no—
154C2×C4×Q8—171no—
155C2×#101—172no—
156C2×#101C2×D1712yes—
157C4×M16—342no—
158#391—171no—
159#391—171no—
160#389—342no—
161#384—171no—
162C4×C4⋊C4—342no—
163C43—171no—
164#356—174no—
165#356Dic1714yesdirect product factor
166#343—342no—
167C2×C17—21no—
168C2×C17—21no—
169C2×C17C4×#9911yescharacteristic
170C2×C17C2×#35611yescharacteristic
171C2×C17#213211yescharacteristic
172C22×Q8C4×D1711yescharacteristic
173C22×Q8C22×D1711yescharacteristic
174C2×M16—22no—
175C2×M16—342no—
176C4⋊Q8—22no—
177C4⋊Q8C4×D1712yes—
178C4⋊Q8C2×Dic1712yes—
179C4×Q8—681no—
180C4×Q8—681no—
181C4×Q8—681no—
182C4×Q8—681no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—172no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—341no—
188C2×C4⋊C4—342no—
189C2×C42—171no—
190C2×C42—172no—
191C2×C42—171no—
192C2×C42—341no—
193C2×C42—341no—
194C2×C42—341no—
195C2×C42C4×D1711yescharacteristic
196#101—22no—
197#101—174no—
198#101—342no—
199#101C2×Dic1714yes—
200#97—682no—
201C4×C8—682no—
202#95—342no—
203#95—342no—
204C17C4×#35611yescharacteristic, complemented, Sylow
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8C4×Dic1712yes—
211C2×Q8C2×C4×D1711yescharacteristic
212C2×Q8#6325412yes—
213C2×Q8C22×Dic1711yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—171no—
217C22×C4—172no—
218C22×C4—171no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4C2×C4×D1711yescharacteristic
222C22×C4D4×D1712yes—
223M16—24no—
224M16—42no—
225M16—344no—
226M16—682no—
227C2×C8—42no—
228C2×C8—682no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—681no—
232C4⋊C4—682no—
233C4⋊C4—681no—
234C4⋊C4—684no—
235C4⋊C4—682no—
236C42—21no—
237C42—342no—
238C42—342no—
239C42—341no—
240C42—341no—
241C42—344no—
242C42—681no—
243C42—682no—
244C42—681no—
245C42—681no—
246C42C4×Dic1712yes—
247C23C22⋊C4×D1711yescharacteristic
248Q8—41no—
249Q8—42no—
250Q8—41no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—171no—
264C2×C4—171no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—681no—
270C2×C4—681no—
271C2×C4—681no—
272C2×C4—682no—
273C2×C4—682no—
274C2×C4—681no—
275C2×C4#1058756311yescharacteristic
276C2×C4#1058756712yes—
277C2×C4C2×C4×Dic1711yescharacteristic
278C2×C4D4×Dic1712yes—
279C8—44no—
280C8—684no—
281C22—21no—
282C22—21no—
283C22C22⋊C4×Dic1711yescharacteristic
284C22#99×D1711yescharacteristic
285C22#1176278111yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—172no—
293C4—341no—
294C4—682no—
295C4—681no—
296C4—682no—
297C4—1361no—
298C2—21no—
299C2—21no—
300C2#99×Dic1711yescharacteristic
301C2D17×#35611yescharacteristic
302C2#1453046811yescharacteristic
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
hypocenterC17C4×#356
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
151. Dic17#356direct (N × H)
165. #356Dic17direct (N × H)
204. C17C4×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic172
C2×#117626082
#101×Dic172
#145269102
C2×C17×#3562
C4×#35617