Finite Group Databasebeta

← #16927536

Subgroups — #16927536

Order 4352.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537496C211yesmaximal, characteristic
2#14528946C211yesmaximal, characteristic, complemented
3#14527530C211yesmaximal, characteristic, complemented
4#14527344C211yesmaximal, characteristic
5#14526643C211yesmaximal, characteristic, complemented
6#14526642C211yesmaximal, characteristic, complemented
7C17×#1517C211yesmaximal, characteristic
8C2×#10587634—21no—
9#11762887—21no—
10#11762887C412yescomplemented
11D4×#3622—21no—
12D4×#3622C412yescomplemented
13#11762862C2211yescharacteristic
14C2×#10587478—21no—
15C2×#10587479—21no—
16#11762833C2211yescharacteristic
17#11762776—21no—
18C2×C4×#3622C2211yescharacteristic
19#11762675C2211yescharacteristic
20#11762561—21no—
21#11762559—21no—
22#11762543—22no—
23C17×#532C2211yescharacteristic
24C17×#431C2211yescharacteristic
25C17×#382C2211yescharacteristic
26C17×#327—22no—
27C2×#41×C17—21no—
28C17×C4⋊Q8Q811yescharacteristic
29C17×#127Q811yescharacteristic
30C17×#124—21no—
31C17×#121—21no—
32C17×#121D412yes—
33C4×D4×C17—21no—
34C4×D4×C17C2×C412yes—
35C2×C4⋊C4×C17—21no—
36C2×C42×C17C2311yescharacteristic
37#10587634—42no—
38#10587634—42no—
39C2×C4⋊Dic17—21no—
40C2×#63223—22no—
41C2×#63223—21no—
42C22×#3622—22no—
43C22×#3622D411yescharacteristic
44C22×#3622D411yescharacteristic
45C17×C4⋊C8—21no—
46C17×C22⋊C8—21no—
47C4×C8×C17—21no—
48C17×#95—21no—
49C17×#95—21no—
50#10587502—21no—
51#10587502—21no—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54C22⋊#3622—21no—
55#10587479—42no—
56#10587478—42no—
57C4⋊#3622—21no—
58C4⋊#3622—21no—
59C4⋊#3622C2×C412yes—
60C4×#3622—21no—
61C4×#3622C2×C411yescharacteristic
62C4×#3622C2×C411yescharacteristic
63C4×#3622D411yescharacteristic
64C4×#3622D411yescharacteristic
65#41×C17—41no—
66#41×C17—41no—
67#41×C17—41no—
68#41×C17—41no—
69C2×Q8×C17—21no—
70C2×D4×C17—22no—
71C2×D4×C17—21no—
72C2×D4×C17—21no—
73C2×D4×C17C4⋊C412yes—
74C22×C4×C17—22no—
75C22×C4×C17—41no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C22×Dic17—41no—
80C2×C8×C17—41no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17C4⋊C412yes—
85C22⋊C4×C17—22no—
86C22⋊C4×C17—22no—
87C42×C17C22×C411yescharacteristic
88C42×C17C2×D411yescharacteristic
89C42×C17C2×Q811yescharacteristic
90C42×C17#4111yescharacteristic
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63223—41no—
94#63223—41no—
95#63223—44no—
96C2×#3622—21no—
97C2×#3622—21no—
98C2×#3622—22no—
99C2×#3622—44no—
100C2×#3622—41no—
101C2×#3622C22⋊C411yescharacteristic
102C2×#3622C22⋊C411yescharacteristic
103C2×#3622C22⋊C411yescharacteristic
104C2×#3622C22⋊C411yescharacteristic
105C2×#3622D812yes—
106C2×#3622SD1612yes—
107#8676—171nomaximal, Sylow
108C23×C17—22no—
109C23×C17#12111yescharacteristic
110Q8×C17—41no—
111Q8×C17—41no—
112D4×C17—42no—
113D4×C17—42no—
114D4×C17—42no—
115D4×C17—42no—
116D4×C17—41no—
117D4×C17—41no—
118C2×C4×C17—22no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17C2×C22⋊C411yescharacteristic
127C2×C4×C17C2×C4⋊C411yescharacteristic
128C2×C4×C17#11711yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17C22⋊Q811yescharacteristic
131C2×C4×C17C22⋊Q811yescharacteristic
132C2×C4×C17C2×D811yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
134C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C17#13711yescharacteristic
136C2×Dic17—81no—
137C2×Dic17—81no—
138C2×Dic17—81no—
139C8×C17—81no—
140#3622—21no—
141#3622—21no—
142#3622—42no—
143#3622—81no—
144#3622#10212yes—
145#3622#10212yes—
146#2931—171no—
147#1846—171no—
148#1564—171no—
149#1517—171no—
150#1517D1711yescharacteristic
151#1286—171no—
152#1286—171no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17#38011yescharacteristic
159C22×C17#45911yescharacteristic
160C22×C17#46111yescharacteristic
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—42no—
166C4×C17—161no—
167C4×C17C2×#10211yescharacteristic
168C4×C17#41711yescharacteristic
169C4×C17#47411yescharacteristic
170C4×C17#47611yescharacteristic
171Dic17—161no—
172C2×#131—341no—
173#532C2×D1711yescharacteristic
174#436—172no—
175#436—341no—
176C8×D4—172no—
177C8×D4—341no—
178#431—171no—
179#431C2×D1711yescharacteristic
180C2×#107—341no—
181C2×#106—341no—
182#423—171no—
183#408—341no—
184C2×C4×C8—171no—
185#382—171no—
186#382C2×D1711yescharacteristic
187#335—341no—
188#334—341no—
189#327—22no—
190#327—342no—
191C2×C17—21no—
192C2×C17—21no—
193C2×C17—82no—
194C2×C17#180211yescharacteristic
195C2×C17#181211yescharacteristic
196C2×C17#181411yescharacteristic
197C2×#41—21no—
198#131—682no—
199#131—682no—
200C2×M16—342no—
201C2×M16—341no—
202C22×C8—171no—
203C22×C8—171no—
204C22×C8—342no—
205C4⋊Q8Dic3411yescharacteristic
206#127Dic3411yescharacteristic
207#124—21no—
208#121—21no—
209#121C22⋊D1712yes—
210C4×D4—21no—
211C4×D4C4×D1712yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—341no—
214C2×C42C22×D1711yescharacteristic
215#107—682no—
216#106—682no—
217C4⋊C8—21no—
218C4⋊C8—172no—
219C4⋊C8—341no—
220C4⋊C8—341no—
221C22⋊C8—21no—
222C22⋊C8—342no—
223C22⋊C8—342no—
224C22⋊C8—341no—
225C4×C8—21no—
226C4×C8—171no—
227C4×C8—171no—
228C4×C8—171no—
229C4×C8—171no—
230C4×C8—341no—
231#95—21no—
232#95—21no—
233#95—341no—
234#95—341no—
235C17#867611yescharacteristic, complemented, Sylow
236#41—41no—
237#41—41no—
238#41—41no—
239#41—41no—
240C2×Q8—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4#6322512yes—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—681no—
248C22×C4D4×D1711yescharacteristic
249C22×C4#6325411yescharacteristic
250C22×C4C2×C22⋊D1711yescharacteristic
251M16—684no—
252M16—681no—
253M16—681no—
254C2×C8—41no—
255C2×C8—171no—
256C2×C8—171no—
257C2×C8—172no—
258C2×C8—172no—
259C2×C8—171no—
260C2×C8—171no—
261C2×C8—341no—
262C2×C8—341no—
263C2×C8—342no—
264C2×C8—684no—
265C2×C8—681no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4#6322512yes—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C42C2×Dic3411yescharacteristic
275C42C2×C4×D1711yescharacteristic
276C42D4×D1711yescharacteristic
277C42#6325411yescharacteristic
278C23—22no—
279C23#1058762511yescharacteristic
280Q8—41no—
281Q8—41no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—41no—
287D4—41no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—1361no—
297C2×C4—1361no—
298C2×C4—1361no—
299C2×C4#1058756311yescharacteristic
300C2×C4#1058756411yescharacteristic
301C2×C4#1058756411yescharacteristic
302C2×C4C22⋊C4×D1711yescharacteristic
303C2×C4D8×D1711yescharacteristic
304C2×C4#1058759711yescharacteristic
305C2×C4SD16×D1711yescharacteristic
306C2×C4#1058760111yescharacteristic
307C2×C4C2×#6322511yescharacteristic
308C2×C4#1058762511yescharacteristic
309C8—81no—
310C8—172no—
311C8—172no—
312C8—341no—
313C8—341no—
314C8—682no—
315C8—1361no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22#1176266711yescharacteristic
322C22#1176300211yescharacteristic
323C22#1176301211yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—161no—
330C4—2721no—
331C4#102×D1711yescharacteristic
332C4#1176280711yescharacteristic
333C4#1176308711yescharacteristic
334C4#1176309711yescharacteristic
335C2—21no—
336C2—21no—
337C2—82no—
338C2#1452878211yescharacteristic
339C2#1452882811yescharacteristic
340C2#1452883611yescharacteristic
subgroupG/H
centerC22#11762667
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1517C2
socleC22×C17#380
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#1517D17
hypocenterC17#8676
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #14528946C2semidirect
3. #14527530C2semidirect
5. #14526643C2semidirect
6. #14526642C2semidirect
10. #11762887C4semidirect
12. D4×#3622C4semidirect
235. C17#8676semidirect

Maximal subgroups

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

subgroupindex
#145374962
#145289462
#145275302
#145273442
#145266432
#145266422
C17×#15172
#867617