Finite Group Databasebeta

← #16927567

Subgroups — #16927567

Order 4352.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537495C211yesmaximal, characteristic
2#14528955C211yesmaximal, characteristic, complemented
3C2×#11762559C211yesmaximal, characteristic, complemented
4#14527462C211yesmaximal, characteristic
5#14527459C211yesmaximal, characteristic
6#14526658C211yesmaximal, characteristic, complemented
7C17×#1292C211yesmaximal, characteristic, complemented
8C2×#10587593—21no—
9#11763359C2211yescharacteristic
10#11762886—21no—
11#11762886C412yes—
12#11762874—21no—
13#11762874C412yes—
14C2×#10587493—21no—
15C2×#10587493—21no—
16#11762833C2211yescharacteristic
17C2×C4⋊#3622C2211yescharacteristic
18#11762777—21no—
19C2×C4×#3622C2211yescharacteristic
20#11762559—21no—
21#11762559—21no—
22#11762556—21no—
23#11762554—21no—
24#11762552—21no—
25#11762552—21no—
26C17×#423C2211yescharacteristic
27C17×#423C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C2×#63252—21no—
30C2×M16×C17—21no—
31C2×M16×C17—21no—
32C22×C8×C17—21no—
33C2×C42×C17C2311yescharacteristic
34#10587616—21no—
35#10587616Q812yes—
36C2×#63223—21no—
37C22×#3622—21no—
38C22×#3622D411yescharacteristic
39C22×#3622D411yescharacteristic
40#10587593—42no—
41#10587593—42no—
42C2×#63218—22no—
43C2×C8×D17—22no—
44#10587561—21no—
45#10587560D411yescharacteristic
46C4×D68—21no—
47C4×D68C2×C412yes—
48#10587555D411yescharacteristic
49C17×C4⋊C8—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C4⋊C8C2×C412yes—
55#10587493—42no—
56#10587493—42no—
57#10587491—22no—
58#10587491—22no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4×#3622C2×C411yescharacteristic
64C4×#3622C2×C411yescharacteristic
65C4×#3622D411yescharacteristic
66C4×#3622D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×Q811yescharacteristic
69C22×C4×C17#4111yescharacteristic
70C2×C22⋊D17—22no—
71#63252—41no—
72#63252—41no—
73#63252—41no—
74#63252—41no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68C4⋊C412yes—
78C2×C4×D17—22no—
79C2×Dic34—21no—
80M16×C17—41no—
81M16×C17—41no—
82M16×C17—41no—
83M16×C17—41no—
84C2×C8×C17—22no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17—81no—
88C42×C17C22×C411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17C2×D411yescharacteristic
91C42×C17#4111yescharacteristic
92#63227—22no—
93#63227—22no—
94C4⋊Dic17—21no—
95C4⋊Dic17C4⋊C412yes—
96#63223—41no—
97#63223—41no—
98C2×#3622—21no—
99C2×#3622—21no—
100C2×#3622—41no—
101C2×#3622—41no—
102C2×#3622—81no—
103C2×#3622C22⋊C411yescharacteristic
104C2×#3622C22⋊C411yescharacteristic
105C2×#3622C22⋊C411yescharacteristic
106C2×#3622C22⋊C411yescharacteristic
107C2×#3622D812yes—
108C2×#3622SD1612yes—
109#63218—44no—
110C8×D17—44no—
111#8682—171nomaximal, Sylow
112C23×C17C22⋊Q811yescharacteristic
113C22×D17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17C2×C22⋊C411yescharacteristic
118C2×C4×C17C2×C4⋊C411yescharacteristic
119C2×C4×C17#11711yescharacteristic
120C2×C4×C17#12111yescharacteristic
121C2×C4×C17#12111yescharacteristic
122C2×C4×C17C22⋊Q811yescharacteristic
123C2×C4×C17C2×D811yescharacteristic
124C2×C4×C17C2×SD1611yescharacteristic
125C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C17#13711yescharacteristic
127C22⋊D17—42no—
128C22⋊D17—42no—
129C2×Dic17—22no—
130D68—42no—
131D68—41no—
132D68—41no—
133D68—41no—
134D68—41no—
135C4×D17—42no—
136C4×D17—42no—
137Dic34—41no—
138Dic34—41no—
139C8×C17—42no—
140C8×C17—81no—
141C8×C17—81no—
142#3622—21no—
143#3622—21no—
144#3622—81no—
145#3622—81no—
146#3622#10212yes—
147#3622#10212yes—
148#2931—171no—
149#1848—171no—
150C2×#334—171no—
151#1547—171no—
152#1547—171no—
153#1292—171no—
154#1292D1711yescharacteristic, complemented
155C22×C17—21no—
156C22×C17—21no—
157C22×C17#38011yescharacteristic
158C22×C17#47411yescharacteristic
159C22×C17#47611yescharacteristic
160C2×D17—42no—
161C2×D17—42no—
162C2×D17—42no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17C2×#10211yescharacteristic
167C4×C17#41711yescharacteristic
168C4×C17#45911yescharacteristic
169C4×C17#46111yescharacteristic
170C4×C17#47111yescharacteristic
171C4×C17#47111yescharacteristic
172Dic17—42no—
173C2×#131—341no—
174#532—171no—
175#436—172no—
176#436—341no—
177C8×D4—172no—
178C8×D4—341no—
179C2×#108—341no—
180C2×#108—341no—
181#423—171no—
182#423C2×D1711yescharacteristic
183#423C2×D1711yescharacteristic
184C2×C4⋊C8—171no—
185C2×C4⋊C8C2×D1711yescharacteristic
186#408—341no—
187C2×C4×C8—171no—
188#334—341no—
189#334—341no—
190#332—341no—
191#332—341no—
192#331—341no—
193#331—341no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17#180211yescharacteristic
197C2×C17#181311yescharacteristic
198C2×C17#181511yescharacteristic
199D17—82no—
200C2×#41—341no—
201#131—682no—
202#131—682no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—342no—
206C2×M16—341no—
207C22×C8—21no—
208C22×C8—171no—
209C22×C8—171no—
210C22×C8—342no—
211C22×C8—341no—
212C4⋊Q8—171no—
213#127—171no—
214#124—341no—
215#121—172no—
216#121—341no—
217C4×D4—172no—
218C4×D4—341no—
219C2×C42C22×D1711yescharacteristic
220#108—682no—
221#108—682no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—341no—
228C4⋊C8—341no—
229C4⋊C8—341no—
230C4⋊C8—341no—
231C4⋊C8C4×D1712yes—
232C22⋊C8—342no—
233C22⋊C8—342no—
234C4×C8—171no—
235C4×C8—171no—
236C4×C8—171no—
237C4×C8—171no—
238C17#868211yescharacteristic, complemented, Sylow
239#41—681no—
240#41—681no—
241#41—681no—
242#41—681no—
243C2×Q8—341no—
244C2×D4—172no—
245C2×D4—342no—
246C2×D4—341no—
247C2×D4—341no—
248C22×C4—342no—
249C22×C4#6325211yescharacteristic
250C22×C4D4×D1711yescharacteristic
251C22×C4Q8×D1711yescharacteristic
252M16—41no—
253M16—41no—
254M16—41no—
255M16—41no—
256M16—684no—
257M16—681no—
258M16—681no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C2×C8—171no—
264C2×C8—171no—
265C2×C8—172no—
266C2×C8—172no—
267C2×C8—171no—
268C2×C8—171no—
269C2×C8—341no—
270C2×C8—341no—
271C2×C8—684no—
272C2×C8—681no—
273C2×C8—681no—
274C2×C8—1361no—
275C4⋊C4—172no—
276C4⋊C4—341no—
277C22⋊C4—342no—
278C22⋊C4—342no—
279C42C2×C4×D1711yescharacteristic
280C42#6325211yescharacteristic
281C42D4×D1711yescharacteristic
282C42D4×D1711yescharacteristic
283C23—342no—
284C23#1058758211yescharacteristic
285Q8—681no—
286Q8—681no—
287D4—682no—
288D4—682no—
289D4—682no—
290D4—681no—
291D4—681no—
292D4—681no—
293D4—681no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—342no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4#1058755811yescharacteristic
301C2×C4C22⋊C4×D1711yescharacteristic
302C2×C4#1058757011yescharacteristic
303C2×C4#1058757011yescharacteristic
304C2×C4C4⋊C4×D1711yescharacteristic
305C2×C4#1058758211yescharacteristic
306C2×C4D8×D1711yescharacteristic
307C2×C4SD16×D1711yescharacteristic
308C2×C4#1058760011yescharacteristic
309C2×C4#1058760411yescharacteristic
310C8—42no—
311C8—81no—
312C8—81no—
313C8—172no—
314C8—172no—
315C8—341no—
316C8—341no—
317C8—1361no—
318C8—1361no—
319C22—21no—
320C22—21no—
321C22—682no—
322C22—682no—
323C22—682no—
324C22#1176266811yescharacteristic
325C22#1176308811yescharacteristic
326C22#1176309811yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—682no—
331C4#102×D1711yescharacteristic
332C4#1176280911yescharacteristic
333C4#1176300311yescharacteristic
334C4#1176301311yescharacteristic
335C4#1176307111yescharacteristic
336C4#1176307511yescharacteristic
337C2—21no—
338C2—21no—
339C2—1362no—
340C2#1452878311yescharacteristic
341C2#1452883311yescharacteristic
342C2#1452884211yescharacteristic
subgroupG/H
centerC22#11762668
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1292C2
socleC22×C17#380
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#1292D17
hypocenterC17#8682
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #14528955C2semidirect
3. C2×#11762559C2semidirect
6. #14526658C2semidirect
7. C17×#1292C2semidirect
154. #1292D17semidirect
238. C17#8682semidirect

Maximal subgroups

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

subgroupindex
#145374952
#145289552
C2×#117625592
#145274622
#145274592
#145266582
C17×#12922
#868217