Finite Group Databasebeta

← #16945196

Subgroups — #16945196

Order 4352.

390 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 69 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#14534354C211yesmaximal, characteristic
2#14528860C211yesmaximal, characteristic
3#14527095C211yesmaximal, characteristic
4#14526805C211yesmaximal, characteristic
5#14526785C211yesmaximal, characteristic
6#14526783C211yesmaximal, characteristic
7C17×#1351C211yesmaximal, characteristic
8C2×Q8×Dic17C411yescharacteristic, complemented
9C2×#10587563C2211yescharacteristic
10#11762859—21no—
11#11762741C411yescharacteristic, complemented
12#11762734C2211yescharacteristic
13#11762728C411yescharacteristic, complemented
14#11762703—21no—
15#11762696—21no—
16#11762692—21no—
17#11762684—21no—
18#11762676C411yescharacteristic, complemented
19#11762670—21no—
20#11762576C2211yescharacteristic
21#11762575—21no—
22#11762534—21no—
23#11762533C2211yescharacteristic
24C17×#393C2211yescharacteristic
25C17×#342C2211yescharacteristic
26C17×#323C2211yescharacteristic
27C22×Q8×C17C2×C411yescharacteristic
28C23×Dic17D411yescharacteristic
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17—21no—
31C2×C22⋊C4×C17C2311yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17D411yescharacteristic
35Q8×Dic17—41no—
36Q8×Dic17—41no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225C2×C411yescharacteristic
42C2×#63225D411yescharacteristic
43C2×C4×Dic17—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46#10587563—41no—
47#10587563—41no—
48C17×C22⋊C8—21no—
49C17×#95—21no—
50C17×#95C2×C411yescharacteristic
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—41no—
57#10587502—41no—
58#10587502C2×C411yescharacteristic
59C22⋊#3622—21no—
60#10587485—41no—
61C8×Dic17—41no—
62C24×C17C2×D411yescharacteristic
63C2×Q8×C17—42no—
64C2×Q8×C17—41no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—41no—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17C22⋊C411yescharacteristic
80C22×Dic17C22⋊C411yescharacteristic
81C22×Dic17C22⋊C411yescharacteristic
82C2×C8×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90C22⋊Dic17—41no—
91C22⋊Dic17—41no—
92C4⋊Dic17—42no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96#63225—41no—
97#63225—41no—
98#63225—42no—
99C4×Dic17—21no—
100C4×Dic17—21no—
101C4×Dic17—21no—
102C4×Dic17—21no—
103C4×Dic17—42no—
104C2×#3622—41no—
105#11842—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17C2×C22⋊C411yescharacteristic
111C23×C17#13511yescharacteristic
112C23×C17#13511yescharacteristic
113Q8×C17—42no—
114Q8×C17—42no—
115C2×C4×C17—22no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
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—
146C2×Dic17—41no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17#9911yescharacteristic
151C2×Dic17#9911yescharacteristic
152C8×C17—81no—
153#3622—81no—
154#2599—171no—
155#1820—171no—
156#1458—171no—
157#1351—171no—
158#1351D1711yescharacteristic
159#1344—171no—
160#1344—171no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—81no—
170C22×C17C2×#9911yescharacteristic
171C22×C17#41611yescharacteristic
172C22×C17#42111yescharacteristic
173C4×C17—42no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—81no—
177C4×C17—161no—
178Dic17—21no—
179Dic17—21no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—81no—
184Dic17—81no—
185Dic17—161no—
186C2×C4×Q8—171no—
187C2×#117—171no—
188#431—341no—
189#398—171no—
190#397—171no—
191#395—171no—
192#393C2×D1711yescharacteristic
193#388—341no—
194#387—341no—
195#387—341no—
196#385—341no—
197#383—171no—
198#381—341no—
199#342—171no—
200#342—341no—
201#342C2×D1711yescharacteristic
202#324—341no—
203#323—171no—
204#323C2×D1711yescharacteristic
205C2×C17—21no—
206C2×C17—21no—
207C2×C17—41no—
208C2×C17—41no—
209C2×C17#151611yescharacteristic
210C2×C17#212011yescharacteristic
211C2×C17#212911yescharacteristic
212C22×Q8C4×D1711yescharacteristic
213C23×C4—171no—
214C4×Q8—681no—
215C4×Q8—681no—
216#117—681no—
217#117—681no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—171no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4C22×D1711yescharacteristic
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—341no—
233C4⋊C8—681no—
234C22⋊C8—21no—
235C22⋊C8—341no—
236C4×C8—681no—
237#95—21no—
238#95—171no—
239#95—341no—
240#95—341no—
241#95—341no—
242#95—341no—
243#95—341no—
244#95—681no—
245#95—681no—
246#95C4×D1711yescharacteristic
247C17#1184211yescharacteristic, complemented, Sylow
248C24D4×D1711yescharacteristic
249C2×Q8—42no—
250C2×Q8—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4C2×C4×D1711yescharacteristic
267C22×C4D4×D1711yescharacteristic
268C2×C8—41no—
269C2×C8—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—682no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—682no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—681no—
282C22⋊C4—681no—
283C22⋊C4—681no—
284C22⋊C4—681no—
285C22⋊C4—681no—
286C42—341no—
287C42—341no—
288C42—341no—
289C42—341no—
290C42—682no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23C22⋊C4×D1711yescharacteristic
296C23#1058760211yescharacteristic
297C23#1058760511yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—171no—
314C2×C4—171no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4—1361no—
337C8—81no—
338C8—1361no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22#99×D1711yescharacteristic
349C22#1176280411yescharacteristic
350C22#1176282411yescharacteristic
351C4—42no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—161no—
356C4—2721no—
357C4—341no—
358C4—341no—
359C4—681no—
360C4—681no—
361C4—681no—
362C4—1361no—
363C4—1361no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#1452733911yescharacteristic
369C2#1453042211yescharacteristic
370C2#1453045711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1351C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1351D17
hypocenterC17#11842
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
8. C2×Q8×Dic17C4semidirect
11. #11762741C4semidirect
13. #11762728C4semidirect
18. #11762676C4semidirect
247. C17#11842semidirect

Maximal subgroups

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

subgroupindex
#145343542
#145288602
#145270952
#145268052
#145267852
#145267832
C17×#13512
#1184217