Finite Group Databasebeta

← #16927641

Subgroups — #16927641

Order 4352.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537760C211yesmaximal, characteristic
2C4×#10587478C211yesmaximal, characteristic, complemented
3#14527545C211yesmaximal, characteristic, complemented
4#14527309C211yesmaximal, characteristic
5#14526640C211yesmaximal, characteristic, complemented
6#14526639C211yesmaximal, characteristic, complemented
7C17×#1509C211yesmaximal, characteristic
8#11762940C411yescharacteristic, complemented
9Q8×#3622C411yescharacteristic, complemented
10#11762887C411yescharacteristic, complemented
11#11762881—21no—
12D4×#3622—21no—
13D4×#3622C411yescharacteristic, complemented
14#11762870—21no—
15#11762866C2211yescharacteristic
16#11762866C2211yescharacteristic
17C2×#10587478—21no—
18#11762765—21no—
19C2×C4×#3622C2211yescharacteristic
20C4×C4⋊Dic17C2211yescharacteristic
21#11762571—21no—
22#11762561—21no—
23#11762559—21no—
24#11762546—21no—
25#11762543—21no—
26C17×#529C2211yescharacteristic
27C17×#432C2211yescharacteristic
28C4×C4⋊C4×C17C2211yescharacteristic
29C17×#328—21no—
30C17×#327—21no—
31C17×#126—21no—
32C17×#125Q811yescharacteristic
33C17×#124Q811yescharacteristic
34C17×#123—21no—
35C17×C22⋊Q8D411yescharacteristic
36C17×#121D411yescharacteristic
37C4×Q8×C17C2×C411yescharacteristic
38C4×D4×C17—21no—
39C4×D4×C17C2×C411yescharacteristic
40C17×#117—21no—
41C2×C4⋊C4×C17—21no—
42C2×C42×C17—21no—
43C2×C42×C17C2311yescharacteristic
44C2×C4⋊Dic17—21no—
45C2×C4×Dic17—21no—
46C2×#63223—21no—
47C22×#3622—21no—
48C22×#3622D411yescharacteristic
49C22×#3622D411yescharacteristic
50C17×C4⋊C8—21no—
51C17×C22⋊C8—21no—
52C17×#97—21no—
53C17×#95—21no—
54#10587502—21no—
55C22⋊#3622—21no—
56C22⋊#3622—21no—
57C22⋊#3622—21no—
58C22⋊#3622—21no—
59#10587478—24no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4⋊#3622C2×C411yescharacteristic
64C4⋊#3622C2×C411yescharacteristic
65#10587474—21no—
66#10587474—21no—
67#10587474—21no—
68C4×#3622—21no—
69C4×#3622C2×C411yescharacteristic
70C4×#3622C2×C411yescharacteristic
71C4×#3622D411yescharacteristic
72C4×#3622D411yescharacteristic
73C2×Q8×C17C4⋊C411yescharacteristic
74C2×D4×C17—21no—
75C2×D4×C17C4⋊C411yescharacteristic
76C22×C4×C17—21no—
77C22×C4×C17—41no—
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17#4111yescharacteristic
81C22×Dic17—41no—
82C2×C8×C17—41no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—21no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17C4⋊C411yescharacteristic
90C4⋊C4×C17C4⋊C411yescharacteristic
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C22⋊C4×C17—21no—
95C22⋊C4×C17—21no—
96C42×C17—21no—
97C42×C17—81no—
98C42×C17C22×C411yescharacteristic
99C42×C17C2×D411yescharacteristic
100C42×C17C2×Q811yescharacteristic
101C42×C17#4111yescharacteristic
102C4⋊Dic17—22no—
103C4⋊Dic17—41no—
104C4×Dic17—81no—
105#63223—42no—
106C2×#3622—22no—
107C2×#3622—21no—
108C2×#3622—21no—
109C2×#3622—41no—
110C2×#3622—42no—
111C2×#3622—41no—
112C2×#3622C22⋊C411yescharacteristic
113C2×#3622C22⋊C411yescharacteristic
114C2×#3622C22⋊C411yescharacteristic
115C2×#3622C22⋊C411yescharacteristic
116#8697—171nomaximal, Sylow
117C23×C17—21no—
118C23×C17#12111yescharacteristic
119Q8×C17—41no—
120D4×C17—41no—
121D4×C17—41no—
122D4×C17—41no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—42no—
134C2×C4×C17—81no—
135C2×C4×C17—81no—
136C2×C4×C17C2×C22⋊C411yescharacteristic
137C2×C4×C17C2×C4⋊C411yescharacteristic
138C2×C4×C17#11711yescharacteristic
139C2×C4×C17#12111yescharacteristic
140C2×C4×C17C22⋊Q811yescharacteristic
141C2×C4×C17C22⋊Q811yescharacteristic
142C2×Dic17—42no—
143C2×Dic17—81no—
144C2×Dic17—81no—
145C8×C17—81no—
146#3622—21no—
147#3622—21no—
148#3622—21no—
149#3622—41no—
150#3622—41no—
151#3622—81no—
152#3622#10412yes—
153#2966—171no—
154C4×#107—171no—
155#1568—171no—
156#1509—171no—
157#1509D1711yescharacteristic
158#1285—171no—
159#1285—171no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17#38011yescharacteristic
166C22×C17#47211yescharacteristic
167C22×C17#47311yescharacteristic
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—82no—
177C4×C17C2×#10411yescharacteristic
178C4×C17#42111yescharacteristic
179Dic17—82no—
180#529C2×D1711yescharacteristic
181#446—171no—
182C8×Q8—171no—
183#436—171no—
184#435—341no—
185C8×D4—171no—
186C8×D4—341no—
187#433—341no—
188#432—171no—
189#432—171no—
190#432C2×D1711yescharacteristic
191C2×#107—341no—
192#405—341no—
193C2×C4×C8—171no—
194C4×C4⋊C4—171no—
195C4×C4⋊C4C2×D1711yescharacteristic
196#340—341no—
197#335—341no—
198#334—341no—
199#328—21no—
200#328—341no—
201#327—21no—
202#327—341no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—81no—
206C2×C17#180611yescharacteristic
207C2×C17#181211yescharacteristic
208C2×C17#181611yescharacteristic
209C2×M16—341no—
210C22×C8—171no—
211C22×C8—171no—
212C22×C8—341no—
213#126—21no—
214#125Dic3411yescharacteristic
215#124Dic3411yescharacteristic
216#123—21no—
217C22⋊Q8C22⋊D1711yescharacteristic
218#121C22⋊D1711yescharacteristic
219C4×Q8C4×D1711yescharacteristic
220C4×D4—21no—
221C4×D4C4×D1711yescharacteristic
222#117—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—341no—
225C2×C42—21no—
226C2×C42—341no—
227C2×C42C22×D1711yescharacteristic
228#107—344no—
229C4⋊C8—21no—
230C4⋊C8—171no—
231C4⋊C8—171no—
232C4⋊C8—341no—
233C4⋊C8—341no—
234C4⋊C8—341no—
235C22⋊C8—21no—
236C22⋊C8—341no—
237C22⋊C8—341no—
238C22⋊C8—341no—
239C22⋊C8—341no—
240#97—21no—
241#97—341no—
242#97—341no—
243#97—341no—
244C4×C8—171no—
245C4×C8—171no—
246C4×C8—171no—
247C4×C8—171no—
248C4×C8—341no—
249#95—21no—
250#95—341no—
251C17#869711yescharacteristic, complemented, Sylow
252C2×Q8#6322511yescharacteristic
253C2×D4—21no—
254C2×D4#6322511yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—681no—
258C22×C4D4×D1711yescharacteristic
259C22×C4#6325411yescharacteristic
260C22×C4C2×C22⋊D1711yescharacteristic
261M16—682no—
262C2×C8—41no—
263C2×C8—171no—
264C2×C8—171no—
265C2×C8—171no—
266C2×C8—171no—
267C2×C8—342no—
268C2×C8—341no—
269C2×C8—341no—
270C2×C8—681no—
271C2×C8—682no—
272C2×C8—681no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—342no—
280C4⋊C4—681no—
281C4⋊C4#6322511yescharacteristic
282C4⋊C4#6322511yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—1361no—
291C42C2×Dic3411yescharacteristic
292C42C2×C4×D1711yescharacteristic
293C42D4×D1711yescharacteristic
294C42#6325411yescharacteristic
295C23—21no—
296C23#1058762511yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—682no—
315C2×C4—1361no—
316C2×C4—1361no—
317C2×C4#1058756311yescharacteristic
318C2×C4#1058756411yescharacteristic
319C2×C4#1058756411yescharacteristic
320C2×C4C22⋊C4×D1711yescharacteristic
321C2×C4C2×#6322511yescharacteristic
322C2×C4#1058762511yescharacteristic
323C8—81no—
324C8—172no—
325C8—341no—
326C8—341no—
327C8—341no—
328C8—681no—
329C8—681no—
330C8—1361no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1176266711yescharacteristic
337C22#1176307711yescharacteristic
338C22#1176308211yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—1362no—
349C4#104×D1711yescharacteristic
350C4#1176282411yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#1452880211yescharacteristic
355C2#1452882811yescharacteristic
356C2#1452884611yescharacteristic
subgroupG/H
centerC22#11762667
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1509C2
socleC22×C17#380
prime radicalC17×#121D4
cosocleC2×C42×C17C23
hypercenter#1509D17
hypocenterC17#8697
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#10587478C2semidirect
3. #14527545C2semidirect
5. #14526640C2semidirect
6. #14526639C2semidirect
8. #11762940C4semidirect
9. Q8×#3622C4semidirect
10. #11762887C4semidirect
13. D4×#3622C4semidirect
251. C17#8697semidirect

Maximal subgroups

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

subgroupindex
#145377602
C4×#105874782
#145275452
#145273092
#145266402
#145266392
C17×#15092
#869717