Finite Group Databasebeta

← #16951192

Subgroups — #16951192

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541887C211yesmaximal, characteristic, complemented
2#14537685C211yesmaximal, characteristic
3#14530411C211yesmaximal, characteristic, complemented
4#14530397C211yesmaximal, characteristic, complemented
5#14526872C211yesmaximal, characteristic, complemented
6#14526871C211yesmaximal, characteristic, complemented
7C2×C17×#350C211yesmaximal, characteristic, complemented
8#11763607C411yescharacteristic
9C2×#10587587C2211yescharacteristic
10#11763226—22no—
11#11763226—22no—
12C2×#10587483—22no—
13C2×#10587483C2211yescharacteristic
14C2×#10587483C2211yescharacteristic
15#11762929—22no—
16#11762924—22no—
17#11762857C411yescharacteristic
18#11762826C411yescharacteristic
19#11762825C411yescharacteristic
20#11762764C2211yescharacteristic
21#11762592—22no—
22#11762588—22no—
23C2×C17×#131C2211yescharacteristic
24C2×C17×M32C2211yescharacteristic
25C22×C16×C17C2211yescharacteristic
26C17×#350—22no—
27C2×#41×C17C2×C411yescharacteristic
28C2×#63252C2×C411yescharacteristic
29C17×#131—21no—
30C17×#131—21no—
31C17×#131D412yes—
32C17×#131D412yes—
33C2×M16×C17—21no—
34C2×M16×C17C2×C411yescharacteristic
35C22×C8×C17—21no—
36C22×C8×C17C2311yescharacteristic
37#10587637C812yes—
38#10587636—22no—
39#10587635C812yes—
40#10587628—21no—
41#10587618—21no—
42#10587612C2×C411yescharacteristic
43C2×#63223C2×C411yescharacteristic
44C2×#63223C2×C411yescharacteristic
45#10587587—21no—
46#10587587—21no—
47#10587587—21no—
48#10587587—21no—
49C2×#63218—21no—
50C2×C8×D17—21no—
51C17×M32—21no—
52C17×M32—21no—
53C2×C16×C17—21no—
54C2×C16×C17—21no—
55C2×C16×C17—21no—
56C2×C16×C17—21no—
57#10587508—22no—
58#10587496—22no—
59#10587493—22no—
60#10587486—21no—
61C8×Dic17—21no—
62#10587483—24no—
63#10587483—21no—
64#10587483—21no—
65#10587483—21no—
66#10587483D412yes—
67C2×#63214—24no—
68C2×#63214—21no—
69C2×#63214D412yes—
70#41×C17—21no—
71#41×C17—21no—
72#41×C17C22⋊C412yes—
73#41×C17C2×C812yes—
74C2×Q8×C17M1611yescharacteristic
75C2×D4×C17—21no—
76C2×D4×C17M1611yescharacteristic
77C22×C4×C17—21no—
78C22×C4×C17C22×C411yescharacteristic
79C2×C22⋊D17—21no—
80#63252—21no—
81#63252—21no—
82#63252—21no—
83#63252—21no—
84C2×D68C2×C811yescharacteristic
85C2×C4×D17—21no—
86C2×Dic34C2×C811yescharacteristic
87M16×C17—21no—
88M16×C17—22no—
89M16×C17—22no—
90M16×C17C22⋊C412yes—
91C2×C8×C17—21no—
92C2×C8×C17—22no—
93C2×C8×C17—22no—
94C2×C8×C17C2×D411yescharacteristic
95C2×C8×C17C2×D411yescharacteristic
96C2×C8×C17C2×D411yescharacteristic
97C2×C8×C17C2×D411yescharacteristic
98C2×C8×C17#4111yescharacteristic
99C2×C8×C17#4111yescharacteristic
100C22⋊Dic17—21no—
101#63231—22no—
102#63230—22no—
103#63229—22no—
104#63228—22no—
105C4⋊Dic17—21no—
106C4×Dic17—21no—
107#63223—21no—
108#63223—21no—
109#63223—21no—
110#63223C2×C812yes—
111C2×#3622—21no—
112C2×#3622—21no—
113#63218—41no—
114#63218—41no—
115C8×D17—41no—
116C8×D17—41no—
117C16×C17—41no—
118C16×C17—41no—
119#63214—22no—
120#63214—22no—
121#12979—171nomaximal, Sylow
122C23×C17—21no—
123C23×C17#13111yescharacteristic
124C22×D17—21no—
125Q8×C17—21no—
126Q8×C17C22⋊C812yes—
127D4×C17—22no—
128D4×C17—22no—
129D4×C17—21no—
130D4×C17C22⋊C812yes—
131C2×C4×C17—21no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17C2×C22⋊C411yescharacteristic
135C2×C4×C17C4×D411yescharacteristic
136C2×C4×C17C4×D411yescharacteristic
137C2×C4×C17C22×C811yescharacteristic
138C2×C4×C17C2×M1611yescharacteristic
139C2×C4×C17#13111yescharacteristic
140C22⋊D17—41no—
141C22⋊D17—41no—
142C2×Dic17—21no—
143C2×Dic17—41no—
144D68—21no—
145D68—21no—
146C4×D17—41no—
147C4×D17—41no—
148Dic34—21no—
149Dic34—21no—
150C8×C17—22no—
151C8×C17#12011yescharacteristic
152C8×C17#12111yescharacteristic
153C8×C17#12111yescharacteristic
154C8×C17#12311yescharacteristic
155#3622—22no—
156#3622—41no—
157#3407—171no—
158#2955—171no—
159#2117—171no—
160C2×#350D1711yescharacteristic, complemented
161#2113—171no—
162#1379—171no—
163#1378—171no—
164C22×C17—22no—
165C22×C17—41no—
166C22×C17—81no—
167C22×C17#40811yescharacteristic
168C22×C17C8×D411yescharacteristic
169C22×C17#43511yescharacteristic
170C2×D17—42no—
171C2×D17—41no—
172C4×C17—22no—
173C4×C17#38611yescharacteristic
174C4×C17C2×C22⋊C811yescharacteristic
175C4×C17C8×D411yescharacteristic
176C4×C17#43511yescharacteristic
177Dic17—41no—
178Dic17—81no—
179#575—171no—
180C2×#131—171no—
181C2×#131C2×D1711yescharacteristic
182#504—342no—
183#504—342no—
184C2×M32—171no—
185C2×M32—171no—
186C2×M32—342no—
187C2×M32C2×D1711yescharacteristic
188C22×C16C2×D1711yescharacteristic
189#444—342no—
190#443—342no—
191#430—171no—
192#421—171no—
193#421—171no—
194#405—171no—
195#350—22no—
196#350—342no—
197C22⋊C16—342no—
198C2×C17—21no—
199C2×C17—21no—
200C2×C17—21no—
201C2×C17—42no—
202C2×C17#186911yescharacteristic
203D17—81no—
204C2×#41—171no—
205C2×#41C4×D1711yescharacteristic
206#137—172no—
207#137—341no—
208AGL(1,ℤ/8)—172no—
209AGL(1,ℤ/8)—341no—
210#135—342no—
211#131—21no—
212#131—21no—
213#131—341no—
214#131—341no—
215#131—341no—
216#131—341no—
217#131D6812yes—
218#131C22⋊D1712yes—
219C2×M16—21no—
220C2×M16—171no—
221C2×M16—171no—
222C2×M16—341no—
223C2×M16C4×D1711yescharacteristic
224C22×C8—21no—
225C22×C8—341no—
226C22×C8C22×D1711yescharacteristic
227#117—171no—
228M32—21no—
229M32—21no—
230M32—172no—
231M32—344no—
232M32—341no—
233M32—341no—
234M32—341no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—172no—
240C2×C16—344no—
241C2×C16—341no—
242#108—342no—
243#104—342no—
244#104—342no—
245#97—341no—
246C4×C8—341no—
247C17#1297911yescharacteristic, complemented, Sylow
248#41—21no—
249#41—21no—
250#41—341no—
251#41—341no—
252#41—341no—
253#41—341no—
254#41C8×D1712yes—
255#41#6322712yes—
256C2×Q8—171no—
257C2×Q8#6321811yescharacteristic
258C2×D4—21no—
259C2×D4—171no—
260C2×D4—341no—
261C2×D4#6321811yescharacteristic
262C22×C4—21no—
263C22×C4—341no—
264C22×C4C2×C4×D1711yescharacteristic
265Q16—342no—
266SD16—342no—
267SD16—342no—
268D8—342no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—172no—
273M16—341no—
274M16—341no—
275M16—341no—
276M16—681no—
277M16—681no—
278M16#6322712yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—341no—
283C2×C8—341no—
284C2×C8—681no—
285C2×C8—681no—
286C2×C8C2×D6811yescharacteristic
287C2×C8D4×D1711yescharacteristic
288C2×C8D4×D1711yescharacteristic
289C2×C8#6325411yescharacteristic
290C2×C8#6325411yescharacteristic
291C2×C8C2×C22⋊D1711yescharacteristic
292C4⋊C4—341no—
293C22⋊C4—341no—
294C42—341no—
295C16—41no—
296C16—41no—
297C16—342no—
298C16—342no—
299C23—21no—
300C23—341no—
301C23#1058759311yescharacteristic
302Q8—21no—
303Q8—341no—
304Q8—341no—
305Q8#1058749112yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—341no—
310D4—341no—
311D4—681no—
312D4—681no—
313D4#1058749112yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—341no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4#1058756711yescharacteristic
322C2×C4#1058756711yescharacteristic
323C2×C4C2×C8×D1711yescharacteristic
324C2×C4C2×#6321811yescharacteristic
325C2×C4#1058759311yescharacteristic
326C2×C4C2×#6322711yescharacteristic
327C8—22no—
328C8—342no—
329C8—681no—
330C8#1058756811yescharacteristic
331C8#1058757211yescharacteristic
332C8#1058762511yescharacteristic
333C8#1058762511yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—682no—
338C22—681no—
339C22#1176277711yescharacteristic
340C22#1176287211yescharacteristic
341C22#1176287711yescharacteristic
342C4—22no—
343C4—681no—
344C4—1361no—
345C4#1176268611yescharacteristic
346C4C2×#1058749111yescharacteristic
347C4#1176287211yescharacteristic
348C4#1176287711yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—1361no—
354C2#1452907411yescharacteristic
subgroupG/H
centerC8#10587568
commutatorC2×C4×C17C22×C8
FrattiniC22×C8C22×D17
FittingC2×C17×#350C2
socleC2×C17#1869
prime radical#11763607C4
cosocleC22×C8×C17C23
hypercenterC2×#350D17
hypocenterC17#12979
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. #14541887C2semidirect
3. #14530411C2semidirect
4. #14530397C2semidirect
5. #14526872C2semidirect
6. #14526871C2semidirect
7. C2×C17×#350C2semidirect
160. C2×#350D17semidirect
247. C17#12979semidirect

Maximal subgroups

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

subgroupindex
#145418872
#145376852
#145304112
#145303972
#145268722
#145268712
C2×C17×#3502
#1297917