Finite Group Databasebeta

← #16945375

Subgroups — #16945375

Order 4352.

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

normal subgroups only

This group has 460 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#14528776C211yesmaximal, characteristic
3#14526791C214yesmaximal, complemented
4C17×#1459C211yesmaximal, characteristic
5C2×#10587621—21no—
6C22×C4⋊Dic17C411yescharacteristic
7#11762833—21no—
8C2×#10587504—21no—
9#11762739C2212yes—
10#11762734C412yes—
11#11762707—21no—
12#11762670—21no—
13#11762669—21no—
14#11762669C411yescharacteristic
15#11762578C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C22×C8×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—21no—
33#10587621—42no—
34#10587621—42no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17—41no—
39C2×C4⋊Dic17C2×C412yes—
40C2×#63225—22no—
41C2×#63223—22no—
42C17×C22⋊C8—22no—
43C17×C22⋊C8—41no—
44#10587504—42no—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48C4⋊#3622—41no—
49C4⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
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×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—42no—
70C22×Dic17—42no—
71C22×Dic17—42no—
72C2×C8×C17—42no—
73C2×C8×C17—82no—
74C4⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C42×C17—21no—
78C42×C17—21no—
79C22⋊Dic17—42no—
80C22⋊Dic17—44no—
81C4⋊Dic17—42no—
82C4⋊Dic17—41no—
83C4⋊Dic17—41no—
84C4⋊Dic17—42no—
85#63225—42no—
86#63225—42no—
87#63223—42no—
88#63223—42no—
89C2×#3622—42no—
90#11881—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×C17#10212yes—
101C23×C17C2×C22⋊C411yescharacteristic
102D4×C17—42no—
103D4×C17—42no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—81no—
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#9911yescharacteristic
119C2×C4×C17C2×D811yescharacteristic
120C2×C4×C17C2×SD1611yescharacteristic
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—84no—
128C8×C17—82no—
129#3622—82no—
130#2486—171no—
131#1800—171no—
132#1459D1711yescharacteristic
133#1346—174no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10211yescharacteristic
148C22×C17#42111yescharacteristic
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17#35212yes—
156Dic17—82no—
157Dic17—82no—
158C2×#123—341no—
159C2×C4×D4C2×D1711yescharacteristic
160C22×C4⋊C4—171no—
161#423—341no—
162C2×#100—341no—
163C2×C22⋊C8—21no—
164#398—172no—
165#397—172no—
166#389—341no—
167#381—341no—
168#380—171no—
169#380—341no—
170#343—172no—
171#324C2×D1712yes—
172#323—21no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—42no—
176C2×C17—42no—
177C2×C17#151911yescharacteristic
178C2×C17C2×#35211yescharacteristic
179C2×C17#213511yescharacteristic
180C22×D4D6811yescharacteristic
181C23×C4—341no—
182C23×C4C4×D1711yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184C2×M16—342no—
185C22×C8—41no—
186#123—682no—
187#123—682no—
188C4×D4—41no—
189C4×D4—41no—
190C2×C4⋊C4—172no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—681no—
195C2×C4⋊C4—681no—
196C2×C4⋊C4C22×D1711yescharacteristic
197C2×C22⋊C4—341no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C4×D1711yescharacteristic
200C2×C22⋊C4C22⋊D1711yescharacteristic
201C2×C42D6811yescharacteristic
202C4⋊C8—681no—
203C4⋊C8—681no—
204#100—682no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207#95—172no—
208#95—342no—
209#95—342no—
210C17#1188111yescharacteristic, complemented, Sylow
211C24#6322711yescharacteristic
212C24#6322711yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#6321912yes—
216C2×D4D13612yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—682no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4#6322711yescharacteristic
229C22×C4#6322711yescharacteristic
230C22×C4C2×C4×D1711yescharacteristic
231C22×C4C2×D6811yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—682no—
234M16—682no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—682no—
238C4⋊C4—41no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C4⋊C4—681no—
242C4⋊C4—681no—
243C4⋊C4—682no—
244C4⋊C4—682no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—682no—
248C22⋊C4—684no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1058749212yes—
260C23#1058749212yes—
261C23C2×#6322711yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—682no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—1364no—
284C2×C4#1058747711yescharacteristic
285C2×C4#1058747711yescharacteristic
286C2×C4C2×#6321911yescharacteristic
287C2×C4C2×D13611yescharacteristic
288C8—82no—
289C8—1362no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1058747711yescharacteristic
303C22C2×#1058749211yescharacteristic
304C22#1176282511yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—1362no—
312C4—1362no—
313C4#1176259812yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#1452735311yescharacteristic
319C2C2×#1176259811yescharacteristic
320C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11881
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526791C2semidirect
210. C17#11881semidirect

Maximal subgroups

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

subgroupindex
#145330922
#145287762
#145267912
C17×#14592
#1188117