Finite Group Databasebeta

← #16931991

Subgroups — #16931991

Order 4352.

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

normal subgroups only

This group has 490 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#14537813C211yesmaximal, characteristic
2#14533980C211yesmaximal, characteristic
3#14527410C212yesmaximal, complemented
4#14526641C212yesmaximal, complemented
5C17×#1466C211yesmaximal, characteristic
6C2×#10587556C2211yescharacteristic
7#11762881—24no—
8#11762866C2212yes—
9#11762773—21no—
10C2×C22⋊#3622—21no—
11#11762732—21no—
12#11762732—21no—
13#11762712C411yescharacteristic
14#11762712C411yescharacteristic
15#11762682—21no—
16#11762682—21no—
17#11762678C411yescharacteristic
18#11762674C411yescharacteristic
19C4×C4⋊Dic17C2211yescharacteristic
20#11762550—24no—
21C2×C4×D4×C17C2211yescharacteristic
22C17×#432C2212yes—
23C17×#323—21no—
24C17×#323—21no—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C4×D4×C17—22no—
29C4×D4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C2×#63223—22no—
48C22×#3622—22no—
49#10587556—22no—
50#10587556—22no—
51C17×C4⋊C8—22no—
52C17×C22⋊C8—22no—
53C17×#97—22no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C22⋊#3622—22no—
58C22⋊#3622—22no—
59C22⋊#3622—22no—
60C4⋊#3622—22no—
61#10587474D414yes—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22⋊C411yescharacteristic
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C2×C8×C17—42no—
83C4⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—22no—
86C42×C17C2×D412yes—
87C42×C17#4112yes—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C22⋊Dic17—81no—
91C22⋊Dic17—81no—
92C4⋊Dic17—22no—
93C4⋊Dic17—22no—
94C4⋊Dic17—42no—
95#63225—41no—
96#63225—42no—
97#63225—41no—
98#63225—81no—
99#63225—81no—
100C4×Dic17—41no—
101C4×Dic17—41no—
102#63223—82no—
103C2×#3622—24no—
104C2×#3622—82no—
105#9449—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17#10112yes—
116C23×C17C2×C22⋊C411yescharacteristic
117D4×C17—44no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17C4×D412yes—
131C2×C4×C17#12011yescharacteristic
132C2×C4×C17#12112yes—
133C2×C4×C17#12311yescharacteristic
134C2×Dic17—22no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C8×C17—82no—
147#3622—44no—
148#2973—171no—
149#2566—171no—
150#1534—172no—
151#1466D1711yescharacteristic
152#1285—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—22no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17#38611yescharacteristic
166C22×C17C2×#10111yescharacteristic
167C22×C17#41311yescharacteristic
168C4×C17—22no—
169C4×C17—22no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—41no—
173C4×C17—41no—
174Dic17—42no—
175Dic17—81no—
176Dic17—81no—
177Dic17—81no—
178C2×#125—171no—
179C2×C4×D4C2×D1711yescharacteristic
180#435—344no—
181#432—172no—
182#432C2×D1712yes—
183#407—341no—
184C2×C22⋊C8—341no—
185#396—341no—
186#396—341no—
187#390—171no—
188#390—171no—
189#385—341no—
190#385—341no—
191#384—171no—
192#382—171no—
193C4×C4⋊C4—171no—
194#330—344no—
195#323—21no—
196#323—21no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17#188411yescharacteristic
202C2×C17#190111yescharacteristic
203C2×C17#191111yescharacteristic
204C22×D4C4×D1711yescharacteristic
205C23×C4D6811yescharacteristic
206C23×C4D6811yescharacteristic
207C2×M16—342no—
208C22×C8—342no—
209#125—342no—
210#125—342no—
211C4×D4—22no—
212C4×D4—22no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4C4×D1711yescharacteristic
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4C22⋊D1711yescharacteristic
226C2×C22⋊C4C22⋊D1711yescharacteristic
227C2×C42—171no—
228C2×C42—171no—
229C2×C42C22×D1711yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—342no—
232C22⋊C8—22no—
233C22⋊C8—342no—
234C22⋊C8—342no—
235C22⋊C8—342no—
236#97—22no—
237#97—174no—
238#95—341no—
239#95—341no—
240#95—341no—
241C17#944911yescharacteristic, complemented, Sylow
242C24#6322711yescharacteristic
243C24#6322711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4#6322711yescharacteristic
258C22×C4#6322711yescharacteristic
259C22×C4C2×C4×D1711yescharacteristic
260C22×C4C2×D6811yescharacteristic
261C22×C4C2×C22⋊D1711yescharacteristic
262M16—1362no—
263C2×C8—42no—
264C2×C8—344no—
265C2×C8—1362no—
266C4⋊C4—22no—
267C4⋊C4—342no—
268C4⋊C4—342no—
269C4⋊C4—682no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—682no—
273C4⋊C4—1361no—
274C4⋊C4—1361no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—682no—
278C22⋊C4—682no—
279C22⋊C4—1361no—
280C22⋊C4—1361no—
281C42—681no—
282C42—681no—
283C42D4×D1712yes—
284C42#6325412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1058749512yes—
295C23C2×#6322711yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—342no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4#1058756712yes—
322C2×C4#1058756811yescharacteristic
323C2×C4#1058757211yescharacteristic
324C2×C4#1058762512yes—
325C8—82no—
326C8—684no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1176268611yescharacteristic
340C22C2×#1058749511yescharacteristic
341C22#1176279111yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—682no—
349C4—1361no—
350C4—1361no—
351C4—1361no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#1452916511yescharacteristic
357C2#1452925011yescharacteristic
358C2#1452931411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1466C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#1466D17
hypocenterC17#9449
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #14527410C2semidirect
4. #14526641C2semidirect
241. C17#9449semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145339802
#145274102
#145266412
C17×#14662
#944917