Finite Group Databasebeta

← #16945451

Subgroups — #16945451

Order 4352.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533101C211yesmaximal, characteristic
2#14527105C211yesmaximal, characteristic
3#14526807C214yesmaximal, complemented
4C17×#1799C211yesmaximal, characteristic
5C22×C4⋊Dic17C411yescharacteristic, complemented
6#11762862—21no—
7#11762748C412yescomplemented
8#11762745C2212yes—
9#11762707—21no—
10#11762689—21no—
11#11762681—21no—
12#11762669—21no—
13#11762669C411yescharacteristic, complemented
14#11762533—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#99—21no—
18C17×#380—21no—
19C17×#342C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17—41no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C22⋊C4×C17C2311yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17C2×C412yes—
39C2×#63225—22no—
40C17×#99—42no—
41C17×#95—41no—
42#10587502—22no—
43#10587502—22no—
44#10587502C2×C412yes—
45C22⋊#3622—22no—
46C4⋊#3622—41no—
47C4×#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C2×D411yescharacteristic
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17SD1612yes—
53C2×D4×C17Q1612yes—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—42no—
69C22×Dic17—82no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—82no—
72C22⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—42no—
75C22⋊C4×C17—42no—
76C42×C17—21no—
77C42×C17—21no—
78C22⋊Dic17—42no—
79C22⋊Dic17—82no—
80C4⋊Dic17—42no—
81C4⋊Dic17—41no—
82C4⋊Dic17—41no—
83C4⋊Dic17—42no—
84#63225—82no—
85C2×#3622—42no—
86#11899—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10312yes—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17C2×SD1611yescharacteristic
98C23×C17C2×Q1611yescharacteristic
99D4×C17—42no—
100D4×C17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—81no—
114C2×C4×C17—82no—
115C2×C4×C17—82no—
116C2×C4×C17—82no—
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#9911yescharacteristic
119C2×C4×C17#10312yes—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—84no—
127#3622—82no—
128#2487—171no—
129#1799D1711yescharacteristic
130#1460—171no—
131#1352—174no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—22no—
136C22×C17—21no—
137C22×C17—42no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10311yescharacteristic
148C22×C17#42111yescharacteristic
149C4×C17—21no—
150C4×C17—22no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—162no—
155C4×C17#35212yes—
156Dic17—82no—
157Dic17—82no—
158C2×C4×D4C2×D1711yescharacteristic
159C22×C4⋊C4—171no—
160#431—341no—
161C2×#99—21no—
162#400—172no—
163#399—172no—
164#389—341no—
165#386—341no—
166#385—341no—
167#380—21no—
168#380—171no—
169#380—341no—
170#342C2×D1712yes—
171#323—172no—
172#323—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—41no—
176C2×C17—42no—
177C2×C17—41no—
178C2×C17#152011yescharacteristic
179C2×C17C2×#35211yescharacteristic
180C2×C17#212811yescharacteristic
181C22×D4C22⋊D1711yescharacteristic
182C23×C4—341no—
183C23×C4C4×D1711yescharacteristic
184C23×C4C22⋊D1711yescharacteristic
185C4×D4—41no—
186C4×D4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—172no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—681no—
193C2×C4⋊C4—681no—
194C2×C4⋊C4C4×D1711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—341no—
197C2×C22⋊C4—342no—
198C2×C22⋊C4D6811yescharacteristic
199C2×C22⋊C4C22×D1711yescharacteristic
200C2×C42D6811yescharacteristic
201C4⋊C8—681no—
202#99—42no—
203C22⋊C8—342no—
204C4×C8—681no—
205#95—41no—
206#95—172no—
207#95—342no—
208#95—342no—
209C17#1189911yescharacteristic, complemented, Sylow
210C24#6322711yescharacteristic
211C24C2×C22⋊D1711yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#6322912yes—
215C2×D4#6323112yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—682no—
226C22×C4—1362no—
227C22×C4#6322711yescharacteristic
228C22×C4#6322711yescharacteristic
229C22×C4#6322711yescharacteristic
230C22×C4C2×C4×D1711yescharacteristic
231C22×C4C2×D6811yescharacteristic
232C2×C8—682no—
233C4⋊C4—41no—
234C4⋊C4—82no—
235C4⋊C4—682no—
236C4⋊C4—682no—
237C4⋊C4—681no—
238C4⋊C4—681no—
239C4⋊C4—1362no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—682no—
245C22⋊C4—1362no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1058748112yes—
257C23C2×#6322711yescharacteristic
258C23C2×#6322911yescharacteristic
259C23C2×#6323111yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—1364no—
285C2×C4#1058747711yescharacteristic
286C2×C4#1058747711yescharacteristic
287C2×C4#1058748112yes—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1058747711yescharacteristic
304C22C2×#1058748111yescharacteristic
305C22#1176282711yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—162no—
312C4—1362no—
313C4—1362no—
314C4#1176259812yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#1452735511yescharacteristic
321C2C2×#1176259811yescharacteristic
322C2#1453045511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1799D17
hypocenterC17#11899
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14526807C2semidirect
5. C22×C4⋊Dic17C4semidirect
7. #11762748C4semidirect
13. #11762669C4semidirect
209. C17#11899semidirect

Maximal subgroups

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

subgroupindex
#145331012
#145271052
#145268072
C17×#17992
#1189917