Finite Group Databasebeta

← #17950020

Subgroups — #17950020

Order 4352.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544639C211yesmaximal, characteristic
2#14544445C211yesmaximal, characteristic
3#14543605C214yesmaximal, complemented
4#14537137C211yesmaximal, characteristic
5C22×#63242—21no—
6C2×C4⋊#3624—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C2×#10587555C411yescharacteristic
12C2×#10587583C412yes—
13C2×C42×D17C2211yescharacteristic
14#11762714C2212yes—
15#11762708—21no—
16#11762708—21no—
17#11762666—21no—
18#11762666—21no—
19C17×#384C411yescharacteristic
20C22×C4×D17C2×C411yescharacteristic
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×Dic34C2×C412yes—
24C22×#3624—22no—
25C22×#3624—22no—
26C2×#63242—24no—
27C2×#63242—42no—
28C2×#63242—42no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17C2×C411yescharacteristic
33C2×#63227—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17C2311yescharacteristic
40#10587583—44no—
41C42×D17—42no—
42#10587555—42no—
43#10587543—42no—
44C4⋊#3624—42no—
45#10587502—22no—
46#10587502—22no—
47C23×D17C22⋊C411yescharacteristic
48C22×C4×C17—22no—
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C22⋊C411yescharacteristic
51C22×C4×C17C22×C411yescharacteristic
52C22×Dic17—22no—
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C2×D411yescharacteristic
55C22×Dic17C2×D411yescharacteristic
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—41no—
61C2×C4×D17—41no—
62C2×C4×D17—41no—
63C2×C4×D17SD1612yes—
64C2×C4×D17SD1612yes—
65C2×C4×D17Q1612yes—
66C2×C4×D17Q1612yes—
67C2×Dic34—24no—
68C2×Dic34—42no—
69C2×Dic34—42no—
70C2×#3624—24no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76#63242—48no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C42×C17—22no—
81#63227—44no—
82C4⋊Dic17—44no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4×Dic17—22no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88#7667—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—21no—
94C22×D17#10012yes—
95C22×D17#10312yes—
96C22×D17#10312yes—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17#9912yes—
105C2×C4×C17#10312yes—
106C2×C4×C17#10312yes—
107C2×C4×C17#10412yes—
108C2×C4×C17#13112yes—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17C2×SD1611yescharacteristic
120C2×Dic17C2×SD1611yescharacteristic
121C2×Dic17C2×Q1611yescharacteristic
122C2×Dic17C2×Q1611yescharacteristic
123C4×D17—24no—
124C4×D17—24no—
125C4×D17—42no—
126C4×D17—42no—
127C4×D17—44no—
128Dic34—84no—
129#3624—44no—
130#3624—82no—
131#2857—171no—
132#1797—171no—
133C2×#336—171no—
134#1272—174no—
135C22×C17#40811yescharacteristic
136C22×C17C2×#9911yescharacteristic
137C22×C17C2×#10011yescharacteristic
138C22×C17C2×#10311yescharacteristic
139C22×C17C2×#10311yescharacteristic
140C22×C17C2×#10411yescharacteristic
141C22×C17#42111yescharacteristic
142C2×D17—22no—
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—22no—
148C2×D17#33314yes—
149C4×C17—22no—
150C4×C17—22no—
151C4×C17—82no—
152C4×C17#32814yes—
153Dic17—21no—
154Dic17—21no—
155Dic17—21no—
156Dic17—21no—
157Dic17—41no—
158Dic17—82no—
159C23×C8—341no—
160C2×C4⋊Q8—171no—
161C2×C22⋊Q8—172no—
162C22×C42—171no—
163C2×C4⋊C8—341no—
164C2×C22⋊C8—341no—
165#391—172no—
166#389—341no—
167#389—341no—
168#384#59511yescharacteristic
169#380—341no—
170#380—341no—
171#336—172no—
172#336—172no—
173#336—341no—
174C2×C17#147111yescharacteristic
175C2×C17#148111yescharacteristic
176C2×C17#149311yescharacteristic
177C2×C17C2×#32811yescharacteristic
178C2×C17#152011yescharacteristic
179C2×C17#153011yescharacteristic
180C2×C17C2×#33311yescharacteristic
181D17—24no—
182C22×Q8—172no—
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—171no—
186C22×C8—342no—
187C22×C8—344no—
188C22×C8—342no—
189C22×C8—682no—
190C22×C8—682no—
191C4⋊Q8—682no—
192C22⋊Q8—684no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4C2×#59512yes—
199C2×C22⋊C4—342no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—682no—
204C2×C42C2×#59511yescharacteristic
205C4⋊C8—682no—
206C22⋊C8—682no—
207#95—342no—
208#95—342no—
209C17#766711yescharacteristic, complemented, Sylow
210C24—171no—
211C2×Q8—344no—
212C2×Q8—682no—
213C2×Q8—682no—
214C22×C4—22no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—172no—
218C22×C4—172no—
219C22×C4—172no—
220C22×C4—172no—
221C22×C4—171no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—681no—
228C22×C4—681no—
229C22×C4—681no—
230C22×C4C22⋊#59511yescharacteristic
231C22×C4C22⋊#59511yescharacteristic
232C22×C4C22×#59511yescharacteristic
233C2×C8—344no—
234C2×C8—682no—
235C2×C8—688no—
236C2×C8—682no—
237C2×C8—682no—
238C2×C8—682no—
239C2×C8—682no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—682no—
244C4⋊C4—684no—
245C4⋊C4—682no—
246C22⋊C4—684no—
247C42—22no—
248C42—342no—
249C42—681no—
250C42—681no—
251C23—172no—
252C23—172no—
253C23—172no—
254C23—341no—
255C23—341no—
256C23—341no—
257C23—341no—
258C23C2×C22⋊#59511yescharacteristic
259Q8—1364no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—171no—
268C2×C4—171no—
269C2×C4—171no—
270C2×C4—171no—
271C2×C4—341no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—344no—
276C2×C4—344no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—684no—
283C2×C4—682no—
284C2×C4—682no—
285C2×C4—682no—
286C2×C4#1058753912yes—
287C2×C4#1058754812yes—
288C2×C4#1058754912yes—
289C2×C4#1058754912yes—
290C2×C4#1058768012yes—
291C8—684no—
292C8—1362no—
293C22—174no—
294C22—342no—
295C22—342no—
296C22—342no—
297C22—342no—
298C22—342no—
299C22—342no—
300C22#1176378411yescharacteristic
301C22C2×#1058753911yescharacteristic
302C22D17⋊#10011yescharacteristic
303C22C2×#1058754911yescharacteristic
304C22C2×#1058754911yescharacteristic
305C22C2×#1058754811yescharacteristic
306C22#1176380711yescharacteristic
307C4—22no—
308C4—22no—
309C4—82no—
310C4—341no—
311C4—341no—
312C4—341no—
313C4—341no—
314C4—681no—
315C4—1362no—
316C4#1176366214yes—
317C2—344no—
318C2#1454406111yescharacteristic
319C2#1454409411yescharacteristic
320C2#1454413411yescharacteristic
321C2C2×#1176366211yescharacteristic
322C2#1454422311yescharacteristic
323C2#1454425411yescharacteristic
324C2D17⋊#33311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#7667
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #14543605C2semidirect
209. C17#7667semidirect

Maximal subgroups

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

subgroupindex
#145446392
#145444452
#145436052
#145371372
#766717