Finite Group Databasebeta

← #16945342

Subgroups — #16945342

Order 4352.

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

normal subgroups only

This group has 472 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#14533092C211yesmaximal, characteristic
2#14527105C211yesmaximal, characteristic
3#14526783C214yesmaximal, complemented
4C17×#1799C211yesmaximal, characteristic
5C2×#10587621—21no—
6C22×C4⋊Dic17C411yescharacteristic, complemented
7#11762862—21no—
8#11762739C412yescomplemented
9#11762734C2212yes—
10#11762707—21no—
11#11762670—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—
34#10587621—42no—
35#10587621—42no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—41no—
39C2×C4⋊Dic17—41no—
40C2×C4⋊Dic17C2×C412yes—
41C2×#63225—22no—
42C17×#99—42no—
43C17×#95—41no—
44#10587502—22no—
45#10587502—22no—
46#10587502C2×C412yes—
47C22⋊#3622—22no—
48C4⋊#3622—41no—
49C4×#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17D812yes—
55C2×D4×C17SD1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—42no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—82no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—21no—
80C42×C17—21no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—42no—
88#63225—44no—
89C2×#3622—42no—
90#11874—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17#10212yes—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17C2×D811yescharacteristic
102C23×C17C2×SD1611yescharacteristic
103D4×C17—42no—
104D4×C17—42no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—81no—
118C2×C4×C17—82no—
119C2×C4×C17—82no—
120C2×C4×C17—82no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#10212yes—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—84no—
131#3622—82no—
132#2486—171no—
133#1799D1711yescharacteristic
134#1460—171no—
135#1344—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—81no—
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10211yescharacteristic
152C22×C17#42111yescharacteristic
153C4×C17—21no—
154C4×C17—22no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—162no—
159C4×C17#35212yes—
160Dic17—82no—
161Dic17—82no—
162C2×#123—341no—
163C2×C4×D4C2×D1711yescharacteristic
164C22×C4⋊C4—171no—
165#431—341no—
166C2×#99—21no—
167#398—172no—
168#397—172no—
169#389—341no—
170#381—341no—
171#380—21no—
172#380—171no—
173#380—341no—
174#342C2×D1712yes—
175#323—172no—
176#323—341no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—41no—
180C2×C17—42no—
181C2×C17—41no—
182C2×C17#151911yescharacteristic
183C2×C17C2×#35211yescharacteristic
184C2×C17#212911yescharacteristic
185C22×D4C22⋊D1711yescharacteristic
186C23×C4—341no—
187C23×C4C4×D1711yescharacteristic
188C23×C4C22⋊D1711yescharacteristic
189#123—682no—
190#123—682no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—172no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—681no—
199C2×C4⋊C4—681no—
200C2×C4⋊C4C4×D1711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—341no—
204C2×C22⋊C4D6811yescharacteristic
205C2×C22⋊C4C22×D1711yescharacteristic
206C2×C42D6811yescharacteristic
207C4⋊C8—681no—
208#99—42no—
209C22⋊C8—342no—
210C4×C8—681no—
211#95—41no—
212#95—172no—
213#95—342no—
214#95—342no—
215C17#1187411yescharacteristic, complemented, Sylow
216C24#6322711yescharacteristic
217C24C2×C22⋊D1711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#6322812yes—
221C2×D4#6323012yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—682no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4#6322711yescharacteristic
235C22×C4#6322711yescharacteristic
236C22×C4#6322711yescharacteristic
237C22×C4C2×C4×D1711yescharacteristic
238C22×C4C2×D6811yescharacteristic
239C2×C8—682no—
240C4⋊C4—41no—
241C4⋊C4—82no—
242C4⋊C4—682no—
243C4⋊C4—682no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—684no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—682no—
252C22⋊C4—682no—
253C22⋊C4—682no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1058748012yes—
265C23C2×#6322711yescharacteristic
266C23C2×#6322811yescharacteristic
267C23C2×#6323011yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—1364no—
293C2×C4#1058747711yescharacteristic
294C2×C4#1058747711yescharacteristic
295C2×C4#1058748012yes—
296C8—1362no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1058747711yescharacteristic
312C22C2×#1058748011yescharacteristic
313C22#1176282711yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—162no—
320C4—1362no—
321C4—1362no—
322C4#1176259812yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#1452735111yescharacteristic
329C2C2×#1176259811yescharacteristic
330C2#1453045811yescharacteristic
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#11874
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. #14526783C2semidirect
6. C22×C4⋊Dic17C4semidirect
8. #11762739C4semidirect
13. #11762669C4semidirect
215. C17#11874semidirect

Maximal subgroups

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

subgroupindex
#145330922
#145271052
#145267832
C17×#17992
#1187417