Finite Group Databasebeta

← #16945472

Subgroups — #16945472

Order 4352.

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

normal subgroups only

This group has 568 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#14532149C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526813C214yesmaximal, complemented
4C17×#1800C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic
6C2×#10587610—21no—
7C2×C22⋊#3622—21no—
8#11762729C2212yes—
9#11762721C412yes—
10#11762715—21no—
11#11762689—21no—
12#11762681—21no—
13#11762669C411yescharacteristic
14#11762535C2212yes—
15#11762533—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#423—21no—
18C2×C17×#100—21no—
19C17×#343C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C22×Dic34—21no—
25C22×Dic34—44no—
26C22×Dic34—41no—
27C22×Dic34—41no—
28C22×Dic34C2×C412yes—
29C2×M16×C17—22no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39#10587610—44no—
40C2×#63225—22no—
41C22×#3622—41no—
42C17×C4⋊C8—41no—
43C17×C4⋊C8—41no—
44C17×#100—42no—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17SD1612yes—
55C2×D4×C17Q1612yes—
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—44no—
70C22×Dic17—42no—
71C2×Dic34—24no—
72C2×Dic34—42no—
73C2×Dic34—48no—
74C2×Dic34—44no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—42no—
78C2×Dic34—44no—
79C2×Dic34—44no—
80M16×C17—42no—
81M16×C17—42no—
82C2×C8×C17—42no—
83C4⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C42×C17—21no—
87C42×C17—21no—
88C22⋊Dic17—44no—
89C22⋊Dic17—42no—
90C4⋊Dic17—42no—
91#63225—44no—
92C2×#3622—42no—
93C2×#3622—82no—
94#11904—171nomaximal, Sylow
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17#10312yes—
104C23×C17#10312yes—
105C23×C17C2×C22⋊C411yescharacteristic
106D4×C17—42no—
107D4×C17—42no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—81no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17C2×SD1611yescharacteristic
124C2×C4×C17C2×Q1611yescharacteristic
125C2×Dic17—24no—
126C2×Dic17—42no—
127C2×Dic17—48no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132Dic34—44no—
133Dic34—44no—
134Dic34—44no—
135Dic34—44no—
136C8×C17—82no—
137#3622—82no—
138#2409—171no—
139#1800D1711yescharacteristic
140#1459—171no—
141#1354—174no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17C2×#9911yescharacteristic
155C22×C17C2×#10311yescharacteristic
156C22×C17#42111yescharacteristic
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17AΣL(1,𝔽16)12yes—
164Dic17—44no—
165Dic17—82no—
166C23×Q8—171no—
167C2×C22⋊Q8—341no—
168C2×C4×D4C2×D1711yescharacteristic
169#423—21no—
170C2×#100—21no—
171C2×C22⋊C8—341no—
172#395—172no—
173#393—172no—
174#391—341no—
175#386—341no—
176#385—341no—
177#380—171no—
178#343C2×D1712yes—
179#324—172no—
180#323—341no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—42no—
184C2×C17—42no—
185C2×C17#152011yescharacteristic
186C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
187C2×C17#213511yescharacteristic
188C22×Q8—172no—
189C22×Q8—341no—
190C22×Q8—681no—
191C22×Q8—681no—
192C22×Q8—684no—
193C22×D4C22⋊D1711yescharacteristic
194C23×C4—341no—
195C23×C4C4×D1711yescharacteristic
196C23×C4D6811yescharacteristic
197C2×M16—22no—
198C22×C8—681no—
199C22⋊Q8—684no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4C22×D1711yescharacteristic
205C2×C22⋊C4—342no—
206C2×C22⋊C4—341no—
207C2×C22⋊C4C4×D1711yescharacteristic
208C2×C22⋊C4D6811yescharacteristic
209C2×C42C22⋊D1711yescharacteristic
210C4⋊C8—41no—
211C4⋊C8—41no—
212#100—42no—
213C22⋊C8—342no—
214C22⋊C8—681no—
215#95—172no—
216#95—342no—
217#95—342no—
218C17#1190411yescharacteristic, complemented, Sylow
219C24#6322711yescharacteristic
220C24#6322711yescharacteristic
221C2×Q8—344no—
222C2×Q8—682no—
223C2×Q8—682no—
224C2×Q8—684no—
225C2×Q8—684no—
226C2×Q8—688no—
227C2×Q8—684no—
228C2×Q8—681no—
229C2×Q8—681no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#6322912yes—
233C2×D4#6323112yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—684no—
243C22×C4—682no—
244C22×C4#6322711yescharacteristic
245C22×C4#6322711yescharacteristic
246C22×C4C2×C4×D1711yescharacteristic
247C22×C4C2×D6811yescharacteristic
248C22×C4C2×C22⋊D1711yescharacteristic
249M16—42no—
250M16—42no—
251C2×C8—42no—
252C2×C8—682no—
253C2×C8—1362no—
254C4⋊C4—41no—
255C4⋊C4—682no—
256C4⋊C4—684no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—684no—
260C22⋊C4—682no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1058748112yes—
272C23#1058748112yes—
273C23C2×#6322711yescharacteristic
274Q8—684no—
275Q8—684no—
276Q8—684no—
277Q8—684no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—344no—
294C2×C4—682no—
295C2×C4—688no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4#1058747711yescharacteristic
301C2×C4#1058747711yescharacteristic
302C2×C4C2×#6322911yescharacteristic
303C2×C4C2×#6323111yescharacteristic
304C8—82no—
305C8—1362no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1058747711yescharacteristic
319C22C2×#1058748111yescharacteristic
320C22#1176282711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—684no—
328C4—1362no—
329C4#1176259712yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#1452735511yescharacteristic
335C2C2×#1176259711yescharacteristic
336C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1800C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1800D17
hypocenterC17#11904
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #14526813C2semidirect
218. C17#11904semidirect

Maximal subgroups

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

subgroupindex
#145321492
#145271022
#145268132
C17×#18002
#1190417