Finite Group Databasebeta

← #16926058

Subgroups — #16926058

Order 4352.

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

normal subgroups only

This group has 738 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
1C2×#11763529C211yesmaximal, characteristic
2#14527457C212yesmaximal, complemented
3#14527212C212yesmaximal, complemented
4#14527105C211yesmaximal, characteristic
5C17×#1800C211yesmaximal, characteristic
6#11763529—24no—
7#11763529—24no—
8C2×#10587555C2211yescharacteristic
9C2×#10587565—22no—
10C2×#10587622—21no—
11C2×#10587610C412yes—
12C2×C4×Dic34C2211yescharacteristic
13C2×#10587573—21no—
14C2×#10587612C412yes—
15#11762862C2212yes—
16#11762556—24no—
17#11762539—24no—
18#11762533—22no—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#423C2212yes—
21C2×C17×#100—21no—
22C17×#343—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C412yes—
25C22×Dic34—21no—
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—22no—
28C4×D4×C17—22no—
29C4×D4×C17—22no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17D412yes—
32C2×C42×C17C2311yescharacteristic
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35#10587622—28no—
36#10587612—24no—
37#10587612—24no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C412yes—
41#10587610—24no—
42#10587610—24no—
43C2×#63225—22no—
44C2×#63225—21no—
45C2×C4×Dic17—22no—
46C2×C4×Dic17—21no—
47#10587573—28no—
48#10587565—216no—
49#10587555—22no—
50#10587555D414yes—
51C4×Dic34—22no—
52C4×Dic34D414yes—
53C17×C4⋊C8—22no—
54C17×C4⋊C8—22no—
55C17×#100—42no—
56C22⋊#3622—22no—
57C4⋊#3622—22no—
58C4×#3622—22no—
59C24×C17C22⋊C412yes—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—21no—
63C2×D4×C17—21no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—44no—
67C22×C4×C17C22⋊C412yes—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—22no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×Dic34—21no—
75C2×Dic34—24no—
76C2×Dic34—42no—
77C2×Dic34—42no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80C2×Dic34C22⋊C412yes—
81M16×C17—42no—
82M16×C17—42no—
83C2×C8×C17—42no—
84C4⋊C4×C17—22no—
85C22⋊C4×C17—24no—
86C42×C17C2×D412yes—
87C42×C17C2×D412yes—
88C22⋊Dic17—28no—
89C22⋊Dic17—28no—
90C4⋊Dic17—21no—
91C4⋊Dic17—22no—
92C4⋊Dic17—24no—
93C4⋊Dic17C22⋊C412yes—
94C4⋊Dic17C22⋊C414yes—
95#63225—28no—
96#63225—24no—
97C4×Dic17—28no—
98C4×Dic17—24no—
99C2×#3622—42no—
100#8389—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—22no—
103C23×C17—22no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17C2×C22⋊C411yescharacteristic
107D4×C17—42no—
108D4×C17—42no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—41no—
112C2×C4×C17—44no—
113C2×C4×C17—44no—
114C2×C4×C17—42no—
115C2×C4×C17C2×C22⋊C411yescharacteristic
116C2×C4×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17#12012yes—
118C2×C4×C17#12012yes—
119C2×C4×C17#13512yes—
120C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C17#13711yescharacteristic
122C2×Dic17—24no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129Dic34—24no—
130Dic34—42no—
131Dic34—44no—
132Dic34—44no—
133C8×C17—82no—
134#3622—82no—
135C2×#556—171no—
136#1800D1711yescharacteristic
137#1546—172no—
138#1487—172no—
139#1460—171no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—82no—
146C22×C17—82no—
147C22×C17#37911yescharacteristic
148C22×C17#41611yescharacteristic
149C22×C17#41711yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—42no—
153C4×C17#41312yes—
154C4×C17#44912yes—
155C4×C17#45212yes—
156Dic17—42no—
157Dic17—42no—
158Dic17—44no—
159#556—344no—
160#556—344no—
161C2×C4⋊Q8—171no—
162C2×#126—342no—
163C2×#124—341no—
164C2×C22⋊Q8—172no—
165C2×C4×Q8—171no—
166C2×C4×Q8—341no—
167C2×C4×D4C2×D1711yescharacteristic
168C2×#117—172no—
169#431—172no—
170#423C2×D1712yes—
171C2×#100—21no—
172#343—21no—
173#332—344no—
174Q8⋊C8—344no—
175#323—342no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—82no—
179C2×C17#178511yescharacteristic
180C2×C17#178811yescharacteristic
181C2×C17#179611yescharacteristic
182C22×Q8—171no—
183C22×Q8—341no—
184C22×D4C22⋊D1711yescharacteristic
185C23×C4C4×D1712yes—
186C2×M16—22no—
187C4⋊Q8—174no—
188C4⋊Q8—342no—
189#126—3416no—
190#124—348no—
191C22⋊Q8—344no—
192C22⋊Q8—344no—
193C4×Q8—174no—
194C4×Q8—348no—
195C4×Q8—342no—
196C4×D4—22no—
197C4×D4—22no—
198#117—344no—
199#117—344no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—172no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—342no—
205C2×C4⋊C4C22⋊D1711yescharacteristic
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4D6812yes—
209C2×C42—341no—
210C2×C42—342no—
211C2×C42C22×D1711yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—342no—
215#100—42no—
216C22⋊C8—342no—
217C4×C8—342no—
218C17#838911yescharacteristic, complemented, Sylow
219C24#6322712yes—
220C2×Q8—172no—
221C2×Q8—341no—
222C2×Q8—344no—
223C2×Q8—682no—
224C2×Q8—681no—
225C2×Q8—681no—
226C2×Q8—682no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—341no—
235C22×C4—342no—
236C22×C4—341no—
237C22×C4#6322712yes—
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×D6811yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—682no—
245C4⋊C4—22no—
246C4⋊C4—172no—
247C4⋊C4—174no—
248C4⋊C4—348no—
249C4⋊C4—341no—
250C4⋊C4—342no—
251C4⋊C4—344no—
252C4⋊C4—344no—
253C22⋊C4—24no—
254C22⋊C4—348no—
255C22⋊C4—348no—
256C42—348no—
257C42—344no—
258C42D4×D1712yes—
259C42D4×D1712yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×#6322711yescharacteristic
266Q8—344no—
267Q8—682no—
268Q8—684no—
269Q8—684no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—342no—
279C2×C4—344no—
280C2×C4—342no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—682no—
285C2×C4C22⋊C4×D1711yescharacteristic
286C2×C4C22⋊C4×D1711yescharacteristic
287C2×C4#1058756812yes—
288C2×C4#1058761811yescharacteristic
289C2×C4#1058762312yes—
290C2×C4#1058762811yescharacteristic
291C2×C4#1058763612yes—
292C8—82no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1176266411yescharacteristic
302C22#1176280311yescharacteristic
303C22#1176280811yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—682no—
308C4—684no—
309C4—682no—
310C4#1176279012yes—
311C4#1176295212yes—
312C4#1176296912yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#1452870911yescharacteristic
317C2#1452872211yescharacteristic
318C2#1452876011yescharacteristic
subgroupG/H
centerC22#11762664
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1800C2
socleC22×C17#379
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#1800D17
hypocenterC17#8389
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #14527457C2semidirect
3. #14527212C2semidirect
218. C17#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#117635292
#145274572
#145272122
#145271052
C17×#18002
#838917