Finite Group Databasebeta

← #16922995

Subgroups — #16922995

Order 4352.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#14537136C211yesmaximal, characteristic
2#14528764C211yesmaximal, characteristic
3#14526606C214yesmaximal, complemented
4C17×#1817C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587610—21no—
7C2×C4⋊#3622—21no—
8C2×C22⋊#3622—21no—
9#11762715—21no—
10#11762707—21no—
11#11762707C2212yes—
12#11762679C411yescharacteristic
13#11762669—21no—
14#11762669C412yes—
15#11762564C2212yes—
16C22×C42×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#336—21no—
19C17×#336C2212yes—
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C22×C8×C17—22no—
25C2×C42×C17—42no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C42×C17C2311yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C412yes—
35C2×C4⋊Dic17C2×C412yes—
36#10587610—42no—
37#10587610—42no—
38C2×#63225—22no—
39C22×#3622—22no—
40#10587555—42no—
41C17×C22⋊C8—42no—
42#10587502—22no—
43#10587502—22no—
44C22⋊#3622—42no—
45C4⋊#3622—42no—
46C24×C17C22⋊C411yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—41no—
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17D812yes—
59C22×C4×C17SD1612yes—
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C2×Dic34—42no—
66C2×Dic34—42no—
67C2×Dic34—42no—
68C2×C8×C17—42no—
69C2×C8×C17—42no—
70C2×C8×C17—42no—
71C42×C17—22no—
72C42×C17—22no—
73C42×C17—41no—
74C42×C17—41no—
75C22⋊Dic17—44no—
76C4⋊Dic17—24no—
77C4⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—42no—
81#63225—42no—
82#63225—42no—
83C2×#3622—42no—
84C2×#3622—42no—
85C2×#3622—42no—
86#7728—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17#10212yes—
94C23×C17C2×C22⋊C411yescharacteristic
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—44no—
110C2×C4×C17—44no—
111C2×C4×C17#9912yes—
112C2×C4×C17#10112yes—
113C2×C4×C17#10212yes—
114C2×C4×C17#13112yes—
115C2×C4×C17C2×D811yescharacteristic
116C2×C4×C17C2×SD1611yescharacteristic
117C2×C4×C17#13712yes—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124Dic34—84no—
125C8×C17—82no—
126#3622—82no—
127#2857—171no—
128#1817D1711yescharacteristic
129#1797—171no—
130#1271—174no—
131C22×C17—22no—
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17#40811yescharacteristic
139C22×C17C2×#9911yescharacteristic
140C22×C17C2×#10111yescharacteristic
141C22×C17C2×#10211yescharacteristic
142C22×C17#41911yescharacteristic
143C22×C17#42111yescharacteristic
144C22×C17#42111yescharacteristic
145C4×C17—22no—
146C4×C17—22no—
147C4×C17—22no—
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—22no—
151C4×C17—21no—
152Dic17—82no—
153Dic17—82no—
154C2×C4⋊Q8—171no—
155C2×C22⋊Q8—341no—
156C22×C42C2×D1711yescharacteristic
157C2×C4⋊C8—341no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8—341no—
160#391—341no—
161#389—172no—
162#389—341no—
163#384—171no—
164#380—172no—
165#380—341no—
166#336—21no—
167#336—172no—
168#336C2×D1712yes—
169C2×C17—42no—
170C2×C17#147211yescharacteristic
171C2×C17#148011yescharacteristic
172C2×C17#149711yescharacteristic
173C2×C17#151111yescharacteristic
174C2×C17#151911yescharacteristic
175C2×C17#153711yescharacteristic
176C2×C17#154611yescharacteristic
177C22×Q8—341no—
178C23×C4C4×D1711yescharacteristic
179C23×C4D6811yescharacteristic
180C23×C4C22⋊D1711yescharacteristic
181C22×C8—22no—
182C22×C8—342no—
183C4⋊Q8—682no—
184C22⋊Q8—682no—
185C22⋊Q8—682no—
186C2×C4⋊C4—172no—
187C2×C4⋊C4—172no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C22⋊C4—342no—
193C2×C42—42no—
194C2×C42C4×D1711yescharacteristic
195C2×C42D6811yescharacteristic
196C2×C42C22⋊D1711yescharacteristic
197C2×C42C22×D1711yescharacteristic
198C4⋊C8—682no—
199C22⋊C8—42no—
200C22⋊C8—682no—
201#95—342no—
202#95—342no—
203C17#772811yescharacteristic, complemented, Sylow
204C24#6322711yescharacteristic
205C2×Q8—682no—
206C2×Q8—682no—
207C2×Q8—682no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—342no—
217C22×C4—342no—
218C22×C4#6321912yes—
219C22×C4D13612yes—
220C22×C4#6322711yescharacteristic
221C22×C4#6322711yescharacteristic
222C22×C4#6322711yescharacteristic
223C22×C4C2×C4×D1711yescharacteristic
224C22×C4C2×D6811yescharacteristic
225C22×C4C2×C22⋊D1711yescharacteristic
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—682no—
230C2×C8—682no—
231C2×C8—682no—
232C4⋊C4—344no—
233C4⋊C4—682no—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C4⋊C4—682no—
237C4⋊C4—682no—
238C4⋊C4—682no—
239C22⋊C4—684no—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1058749212yes—
251C23C2×#6322711yescharacteristic
252Q8—1364no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4—682no—
272C2×C4—682no—
273C2×C4—682no—
274C2×C4—682no—
275C2×C4#1058747712yes—
276C2×C4#1058749212yes—
277C2×C4#1058749512yes—
278C2×C4C2×#6321911yescharacteristic
279C2×C4C2×D13611yescharacteristic
280C2×C4#1058759312yes—
281C2×C4#1058763712yes—
282C8—82no—
283C8—1362no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#1176277711yescharacteristic
292C22C2×#1058747711yescharacteristic
293C22C2×#1058749511yescharacteristic
294C22C2×#1058749211yescharacteristic
295C22#1176281711yescharacteristic
296C22#1176282511yescharacteristic
297C22#1176282711yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—22no—
304C4—22no—
305C4—1362no—
306C4—1362no—
307C2—42no—
308C2#1452714911yescharacteristic
309C2#1452718411yescharacteristic
310C2#1452725611yescharacteristic
311C2#1452731611yescharacteristic
312C2#1452735311yescharacteristic
313C2#1452742111yescharacteristic
314C2#1452745511yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1817C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1817D17
hypocenterC17#7728
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #14526606C2semidirect
203. C17#7728semidirect

Maximal subgroups

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

subgroupindex
#145371362
#145287642
#145266062
C17×#18172
#772817