Finite Group Databasebeta

← #16927258

Subgroups — #16927258

Order 4352.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762887C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic
3#14527542C211yesmaximal, characteristic, complemented
4#14527539C211yesmaximal, characteristic, complemented
5#14526658C211yesmaximal, characteristic, complemented
6#14526658C211yesmaximal, characteristic, complemented
7C17×#1800C211yesmaximal, characteristic
8C22×#63223C412yes—
9#11762887—21no—
10#11762887—21no—
11#11762887—21no—
12#11762887—21no—
13#11762833C2211yescharacteristic
14#11762833C2211yescharacteristic
15C2×C4⋊#3622C2211yescharacteristic
16C2×#10587504—21no—
17C2×C22⋊#3622C412yes—
18C2×C4×#3622C2211yescharacteristic
19#11762578—21no—
20#11762561—21no—
21#11762561—21no—
22#11762559—21no—
23#11762559—21no—
24C2×C4×D4×C17C2211yescharacteristic
25C17×#423C2211yescharacteristic
26C17×#423C2211yescharacteristic
27C2×C17×#100—21no—
28C17×#343—21no—
29C22×D4×C17D411yescharacteristic
30C23×C4×C17C2×C412yes—
31C2×M16×C17—21no—
32C2×M16×C17—21no—
33C4×D4×C17—21no—
34C4×D4×C17—21no—
35C4×D4×C17—21no—
36C4×D4×C17—21no—
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17Q812yes—
39C2×C42×C17C2311yescharacteristic
40C2×#63223—22no—
41C2×#63223—22no—
42C2×#63223—21no—
43C2×#63223—21no—
44C2×#63223—44no—
45C22×#3622C2×C411yescharacteristic
46C22×#3622C2×C412yes—
47C22×#3622C2×C411yescharacteristic
48C17×C4⋊C8—21no—
49C17×C4⋊C8—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8—21no—
52C17×#100—42no—
53#10587504—42no—
54C22⋊#3622—22no—
55C22⋊#3622—22no—
56C4⋊#3622—21no—
57C4⋊#3622—21no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4×#3622D411yescharacteristic
63C4×#3622D411yescharacteristic
64C4×#3622D411yescharacteristic
65C4×#3622D411yescharacteristic
66C24×C17C4⋊C412yes—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17—21no—
71C2×D4×C17—21no—
72C2×D4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17C4⋊C412yes—
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×Q811yescharacteristic
81M16×C17—41no—
82M16×C17—41no—
83M16×C17—41no—
84M16×C17—41no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17—21no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C42×C17C2×D411yescharacteristic
92C42×C17C2×D411yescharacteristic
93C42×C17#4111yescharacteristic
94C42×C17#4111yescharacteristic
95#63223—44no—
96#63223—44no—
97#63223—41no—
98#63223—41no—
99#63223—41no—
100#63223—41no—
101C2×#3622—22no—
102C2×#3622—22no—
103C2×#3622—41no—
104C2×#3622—42no—
105C2×#3622—41no—
106C2×#3622—41no—
107C2×#3622C22⋊C411yescharacteristic
108C2×#3622C22⋊C411yescharacteristic
109C2×#3622C22⋊C411yescharacteristic
110C2×#3622C22⋊C411yescharacteristic
111#8618—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17—22no—
114C23×C17—22no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17C2×C4⋊C411yescharacteristic
118D4×C17—41no—
119D4×C17—41no—
120D4×C17—41no—
121D4×C17—41no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—22no—
125C2×C4×C17—41no—
126C2×C4×C17—44no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17C2×C22⋊C411yescharacteristic
131C2×C4×C17#11711yescharacteristic
132C2×C4×C17#12111yescharacteristic
133C2×C4×C17#12111yescharacteristic
134C2×C4×C17C22⋊Q811yescharacteristic
135C2×C4×C17C22⋊Q811yescharacteristic
136C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C17#13711yescharacteristic
139C2×C4×C17#13711yescharacteristic
140C8×C17—81no—
141C8×C17—81no—
142#3622—21no—
143#3622—21no—
144#3622—21no—
145#3622—42no—
146#3622—81no—
147#3622—81no—
148#3622#10012yescomplemented
149C2×#436—171no—
150#1800—171no—
151#1800D1711yescharacteristic
152#1567—171no—
153#1566—171no—
154#1292—171no—
155#1292—171no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—42no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—82no—
162C22×C17—82no—
163C22×C17#38011yescharacteristic
164C22×C17#42811yescharacteristic
165C22×C17#42811yescharacteristic
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—42no—
170C4×C17C2×#10011yescharacteristic
171C4×C17#41311yescharacteristic
172C4×C17#46811yescharacteristic
173C4×C17#46811yescharacteristic
174C4×C17#46911yescharacteristic
175C4×C17#47011yescharacteristic
176C22×M16—172no—
177C2×C4×D4C2×D1711yescharacteristic
178#436—341no—
179#436—341no—
180#436—341no—
181#436—341no—
182#423—171no—
183#423—171no—
184#423C2×D1711yescharacteristic
185#423C2×D1711yescharacteristic
186C2×C4⋊C8—171no—
187C2×#100—21no—
188C2×#100—341no—
189C2×C22⋊C8—172no—
190C2×C4×C8—171no—
191#343—21no—
192#343—341no—
193#335—341no—
194#335—341no—
195#334—341no—
196#334—341no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—82no—
200C2×C17#180011yescharacteristic
201C2×C17#180911yescharacteristic
202C2×C17#181311yescharacteristic
203C22×D4C22⋊D1711yescharacteristic
204C23×C4C4×D1712yes—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—342no—
208C2×M16—342no—
209C2×M16—341no—
210C2×M16—341no—
211C2×M16—684no—
212C22×C8—171no—
213C22×C8—171no—
214C22×C8—172no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C2×C4⋊C4C22⋊D1711yescharacteristic
220C2×C22⋊C4Dic3412yes—
221C2×C42C22×D1711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—341no—
227C4⋊C8—341no—
228C4⋊C8—341no—
229C4⋊C8—341no—
230C4⋊C8—341no—
231C4⋊C8—341no—
232#100—42no—
233#100—682no—
234C22⋊C8—342no—
235C22⋊C8—342no—
236C4×C8—171no—
237C4×C8—171no—
238C4×C8—171no—
239C4×C8—171no—
240C17#861811yescharacteristic, complemented, Sylow
241C24#6322512yes—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4#6322512yes—
253C22×C4C2×Dic3411yescharacteristic
254C22×C4C2×C4×D1711yescharacteristic
255C22×C4C2×C22⋊D1711yescharacteristic
256M16—41no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—684no—
261M16—684no—
262M16—681no—
263M16—681no—
264M16—681no—
265M16—681no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—171no—
271C2×C8—171no—
272C2×C8—342no—
273C2×C8—342no—
274C2×C8—681no—
275C2×C8—682no—
276C2×C8—681no—
277C2×C8—681no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C42D4×D1711yescharacteristic
283C42D4×D1711yescharacteristic
284C42#6325411yescharacteristic
285C42#6325411yescharacteristic
286C23—22no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23C2×#6322511yescharacteristic
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4#1058756311yescharacteristic
305C2×C4#1058756411yescharacteristic
306C2×C4#1058756411yescharacteristic
307C2×C4C22⋊C4×D1711yescharacteristic
308C2×C4#1058761811yescharacteristic
309C2×C4#1058762511yescharacteristic
310C2×C4#1058762511yescharacteristic
311C2×C4#1058762811yescharacteristic
312C2×C4#1058763511yescharacteristic
313C2×C4#1058763711yescharacteristic
314C8—81no—
315C8—81no—
316C8—172no—
317C8—341no—
318C8—341no—
319C8—341no—
320C8—682no—
321C8—1361no—
322C8—1361no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22#1176266711yescharacteristic
331C22#1176284911yescharacteristic
332C22#1176285211yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4#100×D1711yescharacteristic
338C4#1176279011yescharacteristic
339C4#1176305311yescharacteristic
340C4#1176305711yescharacteristic
341C4#1176306011yescharacteristic
342C4#1176306511yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2#1452877411yescharacteristic
347C2#1452881511yescharacteristic
348C2#1452883211yescharacteristic
subgroupG/H
centerC22#11762667
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1800C2
socleC22×C17#380
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#1800D17
hypocenterC17#8618
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #14527542C2semidirect
4. #14527539C2semidirect
5. #14526658C2semidirect
6. #14526658C2semidirect
148. #3622#100semidirect
240. C17#8618semidirect

Maximal subgroups

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

subgroupindex
C2×#117628872
#145287762
#145275422
#145275392
#145266582
C17×#18002
#861817