Finite Group Databasebeta

← #16949955

Subgroups — #16949955

Order 4352.

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

normal subgroups only

This group has 714 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
1#14530453C212yesmaximal
2C2×#11762604C212yesmaximal, complemented
3#14528892C212yesmaximal, complemented
4C17×#3510C211yesmaximal, characteristic
5C2×#10587505C2214yes—
6#11762604—22no—
7#11762604—22no—
8#11762604—22no—
9#11762602—22no—
10#11762575—24no—
11C17×#559—22no—
12C17×#556—22no—
13C17×#551—24no—
14C17×#542—22no—
15C17×#536C411yescharacteristic, complemented
16C17×#535C2211yescharacteristic
17C17×#533C411yescharacteristic, complemented
18C17×#529—22no—
19C2×C17×#124C2211yescharacteristic
20C17×#519C411yescharacteristic, complemented
21C17×#518C2211yescharacteristic
22C4×#41×C17C411yescharacteristic, complemented
23C2×#41×C17C2×C411yescharacteristic
24C22×Q8×C17C2×C411yescharacteristic
25C22×D4×C17C2311yescharacteristic
26C17×C4⋊Q8—21no—
27C17×C4⋊Q8—21no—
28C17×#127D412yes—
29C17×#126—24no—
30C17×#126—24no—
31C17×#125—21no—
32C17×#124—24no—
33C17×#124—21no—
34C17×#124—22no—
35C17×#124D412yes—
36C17×#124D412yes—
37C17×#124D412yes—
38C17×#123—24no—
39C17×#123—24no—
40C17×#123—22no—
41C17×C22⋊Q8—24no—
42C17×C22⋊Q8—22no—
43C17×C22⋊Q8—22no—
44C17×#121—22no—
45C17×#120—22no—
46C4×Q8×C17—21no—
47C4×Q8×C17—22no—
48C4×Q8×C17—22no—
49C4×Q8×C17—22no—
50C4×D4×C17—24no—
51C4×D4×C17—21no—
52C4×D4×C17—22no—
53C4×D4×C17—22no—
54C4×D4×C17—22no—
55C17×#117—24no—
56C17×#117—22no—
57C17×#117—21no—
58C17×#117C2×C411yescharacteristic
59C17×#117C2×C411yescharacteristic
60C17×#117C2×C411yescharacteristic
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17D414yes—
63C2×C42×C17—21no—
64C2×C42×C17C2×C411yescharacteristic
65C2×C22⋊Dic17—24no—
66#10587505—24no—
67#10587505—24no—
68C24×C17C2×D412yes—
69#41×C17—42no—
70C2×Q8×C17—22no—
71C2×Q8×C17—21no—
72C2×Q8×C17—42no—
73C2×Q8×C17C22⋊C412yes—
74C2×Q8×C17C22⋊C411yescharacteristic
75C2×D4×C17—21no—
76C2×D4×C17—24no—
77C2×D4×C17—42no—
78C2×D4×C17C22⋊C411yescharacteristic
79C2×D4×C17C2×D412yes—
80C2×D4×C17C2×D412yes—
81C22×C4×C17—21no—
82C22×C4×C17—22no—
83C22×C4×C17—21no—
84C22×C4×C17—24no—
85C22×C4×C17—21no—
86C22×C4×C17C22×C411yescharacteristic
87C22×Dic17—44no—
88C4⋊C4×C17—24no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—22no—
91C4⋊C4×C17—22no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—21no—
97C22⋊C4×C17—24no—
98C22⋊C4×C17—22no—
99C22⋊C4×C17—21no—
100C22⋊C4×C17—24no—
101C22⋊C4×C17—24no—
102C22⋊C4×C17—24no—
103C22⋊C4×C17—24no—
104C22⋊C4×C17C22⋊C412yes—
105C22⋊C4×C17C22⋊C412yes—
106C42×C17—24no—
107C42×C17—22no—
108C42×C17—21no—
109C42×C17—21no—
110C42×C17—21no—
111C42×C17C22⋊C411yescharacteristic
112C42×C17C22⋊C411yescharacteristic
113C42×C17C22⋊C412yes—
114C22⋊Dic17—44no—
115C22⋊Dic17—44no—
116#12728—171nomaximal, Sylow
117C23×C17—24no—
118C23×C17—21no—
119C23×C17—82no—
120C23×C17C2×C22⋊C411yescharacteristic
121C23×C17#12014yes—
122Q8×C17—42no—
123Q8×C17—41no—
124Q8×C17—42no—
125Q8×C17—82no—
126D4×C17—42no—
127D4×C17—42no—
128D4×C17—41no—
129D4×C17—42no—
130D4×C17—84no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—21no—
134C2×C4×C17—22no—
135C2×C4×C17—22no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—22no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—82no—
142C2×C4×C17—84no—
143C2×C4×C17—81no—
144C2×C4×C17C2×C22⋊C411yescharacteristic
145C2×C4×C17C2×C22⋊C411yescharacteristic
146C2×Dic17—84no—
147C2×Dic17—164no—
148#3510D1711yescharacteristic
149#2127—172no—
150C2×#354—172no—
151#1828—172no—
152C22×C17—24no—
153C22×C17—41no—
154C22×C17—44no—
155C22×C17—42no—
156C22×C17—41no—
157C22×C17—84no—
158C22×C17#37911yescharacteristic
159C22×C17#41012yes—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—42no—
163C4×C17—41no—
164C4×C17—42no—
165C4×C17—41no—
166C4×C17—42no—
167C4×C17—41no—
168C4×C17—42no—
169Dic17—164no—
170#559—22no—
171#556—22no—
172#551—24no—
173#542—22no—
174#536Dic1711yescharacteristic, complemented
175#535C2×D1711yescharacteristic
176#533Dic1711yescharacteristic, complemented
177#529—22no—
178C2×#124C2×D1711yescharacteristic
179#519Dic1711yescharacteristic, complemented
180#518C2×D1711yescharacteristic
181C4×#41Dic1711yescharacteristic, complemented
182C2×#99—174no—
183#354—342no—
184#354—342no—
185#354—342no—
186#353—342no—
187#342—344no—
188C2×C17—21no—
189C2×C17—22no—
190C2×C17—44no—
191C2×C17—81no—
192C2×C17#178211yescharacteristic
193C2×#41C2×Dic1711yescharacteristic
194C22×Q8C2×Dic1711yescharacteristic
195C22×D4C22×D1711yescharacteristic
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198#127C22⋊D1712yes—
199#126—24no—
200#126—24no—
201#125—21no—
202#124—24no—
203#124—21no—
204#124—22no—
205#124C22⋊D1712yes—
206#124C22⋊D1712yes—
207#124C22⋊D1712yes—
208#123—24no—
209#123—24no—
210#123—22no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#121—22no—
215#120—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—24no—
226#117—22no—
227#117—21no—
228#117C2×Dic1711yescharacteristic
229#117C2×Dic1711yescharacteristic
230#117C2×Dic1711yescharacteristic
231C2×C22⋊C4—22no—
232C2×C22⋊C4—344no—
233C2×C22⋊C4C22⋊D1714yes—
234C2×C42—21no—
235C2×C42C2×Dic1711yescharacteristic
236#99—344no—
237#99—344no—
238C17#1272811yescharacteristic, complemented, Sylow
239C24C2×C22⋊D1712yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic1712yes—
245C2×Q8C22⋊Dic1711yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊Dic1711yescharacteristic
250C2×D4C2×C22⋊D1712yes—
251C2×D4C2×C22⋊D1712yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—684no—
258C22×C4C22×Dic1711yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—684no—
276C22⋊C4—684no—
277C22⋊C4C22⋊Dic1712yes—
278C22⋊C4C22⋊Dic1712yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊Dic1711yescharacteristic
285C42C22⋊Dic1711yescharacteristic
286C42C22⋊Dic1712yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊Dic1711yescharacteristic
291C23#1058763914yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—2724no—
315C2×C4—1364no—
316C2×C4C2×C22⋊Dic1711yescharacteristic
317C2×C4C2×C22⋊Dic1711yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22#1176266511yescharacteristic
325C22#1176278312yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—2724no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2#1452870211yescharacteristic
subgroupG/H
centerC2#14528702
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#3510C2
socleC2×C17#1782
prime radicalC17×#535C22
cosocleC22×D4×C17C23
hypercenter#3510D17
hypocenterC17#12728
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. C2×#11762604C2semidirect
3. #14528892C2semidirect
15. C17×#536C4semidirect
17. C17×#533C4semidirect
20. C17×#519C4semidirect
22. C4×#41×C17C4semidirect
174. #536Dic17semidirect
176. #533Dic17semidirect
179. #519Dic17semidirect
181. C4×#41Dic17semidirect
238. C17#12728semidirect

Maximal subgroups

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

subgroupindex
#145304532
C2×#117626042
#145288922
C17×#35102
#1272817