Finite Group Databasebeta

← #16950595

Subgroups — #16950595

Order 4352.

566 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 87 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#14531367C211yesmaximal, characteristic
2#14528706C212yesmaximal
3#14526911C211yesmaximal, characteristic, complemented
4#14526910C212yesmaximal, complemented
5C17×#1394C211yesmaximal, characteristic, complemented
6C22×Q8×D17C2211yescharacteristic
7C2×#10587578C411yescharacteristic
8C2×#10587558C411yescharacteristic
9C2×#10587507C2212yes—
10C2×#10587495—22no—
11#11762772—22no—
12#11762716C2211yescharacteristic
13#11762715C411yescharacteristic
14#11762714C411yescharacteristic
15#11762713—22no—
16#11762671—22no—
17C2×C17×#101C2212yes—
18C17×#391C2211yescharacteristic
19C22×Q8×C17C2311yescharacteristic
20C2×Q8×D17—22no—
21C2×Q8×D17—21no—
22C2×Q8×D17—21no—
23C2×Q8×D17—42no—
24C2×Q8×D17—42no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17D412yes—
27C22×Dic34C2×C411yescharacteristic
28C22×Dic34D412yes—
29C2×M16×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×#63223—22no—
39#10587578—42no—
40#10587558—42no—
41C17×#101—22no—
42C17×#101—22no—
43C17×#95—22no—
44#10587507—22no—
45#10587507—22no—
46#10587502—22no—
47#10587502—22no—
48#10587502—22no—
49#10587495—44no—
50#10587491—44no—
51C23×D17C22⋊C411yescharacteristic
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17#4112yes—
55C2×Q8×C17#4111yescharacteristic
56C2×Q8×C17#4111yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D412yes—
61C22×Dic17—21no—
62C22×Dic17—21no—
63C22×Dic17C22⋊C412yes—
64C22×Dic17C22⋊C411yescharacteristic
65Q8×D17—48no—
66Q8×D17—44no—
67Q8×D17—44no—
68C2×C4×D17—22no—
69C2×C4×D17—21no—
70C2×C4×D17—21no—
71C2×C4×D17—22no—
72C2×C4×D17—24no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—42no—
76C2×Dic34—21no—
77C2×Dic34—21no—
78C2×Dic34—22no—
79C2×Dic34—22no—
80C2×Dic34—24no—
81C2×Dic34—41no—
82C2×Dic34—41no—
83C2×Dic34—42no—
84M16×C17—42no—
85M16×C17—42no—
86C2×C8×C17—42no—
87C4⋊C4×C17—41no—
88C42×C17—41no—
89#63227—42no—
90#63227—42no—
91C4⋊Dic17—41no—
92#63225—42no—
93C4×Dic17—42no—
94C4×Dic17—41no—
95#63223—42no—
96#63223—42no—
97C2×#3622—42no—
98#12862—171nomaximal, Sylow
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17#9912yes—
105Q8×C17—42no—
106Q8×C17—41no—
107Q8×C17—41no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17#11711yescharacteristic
118C2×C4×C17#11711yescharacteristic
119C2×C4×C17#12312yes—
120C2×C4×C17#12312yes—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17#9912yes—
132C4×D17—24no—
133C4×D17—24no—
134C4×D17—44no—
135C4×D17—44no—
136Dic34—44no—
137Dic34—44no—
138Dic34—42no—
139Dic34—42no—
140C8×C17—82no—
141#3622—82no—
142#2310—171no—
143#1784—172no—
144#1394—172no—
145#1394—171no—
146#1394D1711yescharacteristic, complemented
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#38111yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#9911yescharacteristic
152C2×D17—22no—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—42no—
156C2×D17—42no—
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—81no—
162C4×C17—81no—
163Dic17—22no—
164Dic17—42no—
165Dic17—81no—
166Dic17—81no—
167C23×Q8—171no—
168C2×#117—171no—
169C2×#117—171no—
170C2×#101—172no—
171C2×#101—342no—
172C2×#101C2×D1712yes—
173#407—342no—
174#391—171no—
175#391—171no—
176#391—171no—
177#391—342no—
178#391C2×D1711yescharacteristic
179#381—342no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17#181911yescharacteristic
183C2×C17#213311yescharacteristic
184C2×C17#213311yescharacteristic
185D17—42no—
186C22×Q8—171no—
187C22×Q8—172no—
188C22×Q8—342no—
189C22×Q8—341no—
190C22×Q8—341no—
191C22×Q8—682no—
192C22×Q8—682no—
193C22×Q8C22×D1711yescharacteristic
194C23×C4—171no—
195C23×C4—172no—
196C2×M16—22no—
197C2×M16—342no—
198#117—682no—
199#117—682no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4C4×D1711yescharacteristic
203C2×C22⋊C4—341no—
204C2×C22⋊C4—341no—
205C2×C42—171no—
206C2×C42—341no—
207C2×C42C4×D1711yescharacteristic
208#101—22no—
209#101—22no—
210#101—342no—
211#101—342no—
212#101—684no—
213C22⋊C8—684no—
214#95—22no—
215#95—342no—
216#95—342no—
217#95—342no—
218C17#1286211yescharacteristic, complemented, Sylow
219C24—171no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—344no—
223C2×Q8—341no—
224C2×Q8—342no—
225C2×Q8—341no—
226C2×Q8—342no—
227C2×Q8—681no—
228C2×Q8—681no—
229C2×Q8—682no—
230C2×Q8—688no—
231C2×Q8—684no—
232C2×Q8—684no—
233C2×Q8#6325212yes—
234C2×Q8#6325411yescharacteristic
235C2×Q8#6325611yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—171no—
239C22×C4—172no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—344no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—682no—
250C22×C4C2×C4×D1711yescharacteristic
251C22×C4D4×D1712yes—
252M16—42no—
253M16—42no—
254M16—682no—
255M16—682no—
256C2×C8—42no—
257C2×C8—682no—
258C4⋊C4—41no—
259C4⋊C4—681no—
260C4⋊C4—682no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C42—41no—
264C42—681no—
265C42—682no—
266C23—172no—
267C23—341no—
268C23—341no—
269C23—681no—
270C23—681no—
271C23C22⋊C4×D1711yescharacteristic
272Q8—42no—
273Q8—41no—
274Q8—41no—
275Q8—684no—
276Q8—684no—
277Q8—682no—
278Q8—682no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—172no—
289C2×C4—344no—
290C2×C4—344no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—342no—
296C2×C4—684no—
297C2×C4—684no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—682no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4#1058755811yescharacteristic
304C2×C4#1058756912yes—
305C2×C4#1058757811yescharacteristic
306C2×C4#1058758012yes—
307C8—82no—
308C8—1362no—
309C22—21no—
310C22—21no—
311C22—342no—
312C22—342no—
313C22—342no—
314C22—682no—
315C22—682no—
316C22#1176267111yescharacteristic
317C22#99×D1711yescharacteristic
318C22#99×D1711yescharacteristic
319C4—22no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—342no—
326C4—682no—
327C4—1361no—
328C4—1361no—
329C2—21no—
330C2—21no—
331C2—682no—
332C2#1452885711yescharacteristic
333C2#1453047011yescharacteristic
334C2#1453047211yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#1394D17
hypocenterC17#12862
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #14526911C2semidirect
4. #14526910C2semidirect
5. C17×#1394C2semidirect
146. #1394D17semidirect
218. C17#12862semidirect

Maximal subgroups

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

subgroupindex
#145313672
#145287062
#145269112
#145269102
C17×#13942
#1286217