Finite Group Databasebeta

← #16933652

Subgroups — #16933652

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763155C211yesmaximal, characteristic
2#14529731C211yesmaximal, characteristic, complemented
3#14529712C211yesmaximal, characteristic
4#14528654C211yesmaximal, characteristic, complemented
5C2×#11762561C211yesmaximal, characteristic, complemented
6C2×#11762557C211yesmaximal, characteristic, complemented
7C2×C17×#331C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9#11763155—22no—
10#11763155C414yescomplemented
11C2×C4⋊#3622C2211yescharacteristic
12C2×#10587503—21no—
13C2×#10587503—21no—
14C2×#10587503—22no—
15C2×#10587486—21no—
16C2×C4×#3622C2211yescharacteristic
17C4×C4⋊Dic17C2211yescharacteristic
18#11762572—22no—
19#11762563—21no—
20#11762561—22no—
21#11762557—22no—
22C2×C17×#127C2211yescharacteristic
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C17×#331—22no—
26C22×D4×C17—21no—
27C22×D4×C17—21no—
28C22×D4×C17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C17×#127—21no—
33C17×#127—21no—
34C17×#127—21no—
35C17×#127C2×C412yes—
36C2×C42×C17C2311yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D412yes—
40C2×C4×Dic17—21no—
41C22×#3622—21no—
42C22×#3622D411yescharacteristic
43C22×#3622D411yescharacteristic
44#10587555—21no—
45#10587555—21no—
46#10587555—21no—
47#10587555C2×C412yes—
48C17×C4⋊C8—21no—
49C17×C4⋊C8—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8D412yes—
52#10587503—24no—
53#10587503—24no—
54#10587503—44no—
55#10587502—21no—
56#10587486—42no—
57C4⋊#3622—21no—
58C4⋊#3622—21no—
59C4×#3622C2×C412yes—
60C4×#3622D412yes—
61C24×C17—41no—
62C2×D4×C17—22no—
63C2×D4×C17—22no—
64C2×D4×C17—22no—
65C2×D4×C17—41no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C2×D4×C17—42no—
72C2×D4×C17—41no—
73C2×D4×C17—42no—
74C2×D4×C17—82no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17#4111yescharacteristic
78C22×Dic17—22no—
79C22×Dic17—41no—
80C2×Dic34—22no—
81C2×Dic34—41no—
82C2×Dic34—82no—
83C2×C8×C17—22no—
84C2×C8×C17—41no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17—42no—
88C42×C17C22×C411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17C2×D411yescharacteristic
91C42×C17#4111yescharacteristic
92C4⋊Dic17—22no—
93C4⋊Dic17—22no—
94C4⋊Dic17—22no—
95C4⋊Dic17—22no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4×Dic17—41no—
99C4×Dic17—41no—
100C2×#3622—21no—
101C2×#3622—21no—
102C2×#3622—21no—
103C2×#3622—21no—
104C2×#3622—41no—
105C2×#3622—42no—
106C2×#3622C22⋊C412yes—
107C2×#3622C22⋊C412yes—
108#9747—171nomaximal, Sylow
109C23×C17—42no—
110C23×C17—81no—
111C23×C17—81no—
112C23×C17—82no—
113C23×C17—82no—
114C23×C17#12111yescharacteristic
115D4×C17—42no—
116D4×C17—42no—
117D4×C17—44no—
118D4×C17—82no—
119D4×C17—82no—
120D4×C17—81no—
121D4×C17—81no—
122D4×C17—81no—
123D4×C17—81no—
124C2×C4×C17—22no—
125C2×C4×C17C2×C22⋊C411yescharacteristic
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17C4×D411yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12411yescharacteristic
130C2×C4×C17#12711yescharacteristic
131C2×C4×C17#13511yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17AGL(1,ℤ/8)12yes—
135C2×C4×C17#13711yescharacteristic
136C2×C4×C17#13712yes—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—82no—
142Dic34—82no—
143Dic34—82no—
144C8×C17—42no—
145C8×C17—81no—
146#3622—22no—
147#3622—22no—
148#3622—81no—
149C2×#486—171no—
150#1987—171no—
151#1982—171no—
152#1770—171no—
153C2×#335—171no—
154C2×#333—171no—
155C2×#331D1711yescharacteristic
156C22×C17—82no—
157C22×C17—82no—
158C22×C17—82no—
159C22×C17—162no—
160C22×C17—162no—
161C22×C17#39011yescharacteristic
162C22×C17#46311yescharacteristic
163C22×C17#46411yescharacteristic
164C22×C17#46812yes—
165C22×C17#47212yes—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17#41611yescharacteristic
169C4×C17#41711yescharacteristic
170C4×C17#44012yes—
171C4×C17#46311yescharacteristic
172C4×C17#46411yescharacteristic
173C4×C17#49611yescharacteristic
174C4×C17#49711yescharacteristic
175Dic17—82no—
176Dic17—82no—
177C2×C4⋊Q8—171no—
178C2×#127C2×D1711yescharacteristic
179#486—174no—
180#486—342no—
181C2×C4⋊C8—171no—
182C2×C4⋊C8C2×D1711yescharacteristic
183C2×C4⋊C8C2×D1711yescharacteristic
184C2×#102—341no—
185C2×#102—341no—
186C2×#102—342no—
187C2×#97—341no—
188C2×C4×C8—171no—
189C4×C4⋊C4—171no—
190#340—342no—
191#336—341no—
192#335—342no—
193#333—342no—
194#331—22no—
195C2×C17—162no—
196C2×C17#169712yes—
197C2×C17#169812yes—
198C2×C17#197311yescharacteristic
199C2×C17#197811yescharacteristic
200C2×C17#198311yescharacteristic
201C22×Q8—341no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—171no—
208C22×C8—171no—
209C22×C8—341no—
210C4⋊Q8—172no—
211C4⋊Q8—341no—
212C4⋊Q8—341no—
213C4⋊Q8—341no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127C4×D1712yes—
218C2×C4⋊C4—172no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C42—341no—
222C2×C42C22×D1711yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—341no—
227C4⋊C8—341no—
228C4⋊C8C22⋊D1712yes—
229#102—344no—
230#102—344no—
231#102—684no—
232#97—682no—
233C4×C8—172no—
234C4×C8—172no—
235#95—341no—
236C17#974711yescharacteristic, complemented, Sylow
237C24—41no—
238C2×Q8—342no—
239C2×Q8—681no—
240C2×Q8—1362no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C22×C4—342no—
255C22×C4—681no—
256C22×C4#6325211yescharacteristic
257C22×C4D4×D1711yescharacteristic
258C22×C4D4×D1711yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—172no—
265C2×C8—172no—
266C2×C8—341no—
267C2×C8—341no—
268C2×C8—341no—
269C2×C8—341no—
270C2×C8—682no—
271C2×C8—681no—
272C4⋊C4—342no—
273C4⋊C4—342no—
274C4⋊C4—342no—
275C4⋊C4—342no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C42—681no—
279C42—681no—
280C42C2×C4×D1711yescharacteristic
281C42D4×D1711yescharacteristic
282C42#6325411yescharacteristic
283C42C2×C22⋊D1711yescharacteristic
284C23—42no—
285C23—81no—
286C23—81no—
287C23—82no—
288C23—82no—
289C23#1058757011yescharacteristic
290Q8—1362no—
291Q8—1362no—
292D4—42no—
293D4—42no—
294D4—44no—
295D4—82no—
296D4—82no—
297D4—81no—
298D4—81no—
299D4—81no—
300D4—81no—
301C2×C4—22no—
302C2×C4—682no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—1362no—
307C2×C4C22⋊C4×D1711yescharacteristic
308C2×C4#1058756711yescharacteristic
309C2×C4#1058757111yescharacteristic
310C2×C4#1058759711yescharacteristic
311C2×C4#1058759712yes—
312C2×C4#1058759811yescharacteristic
313C2×C4#1058760112yes—
314C2×C4#1058760111yescharacteristic
315C2×C4#1058760211yescharacteristic
316C2×C4C4×C22⋊D1711yescharacteristic
317C2×C4#1058762511yescharacteristic
318C2×C4#1058762611yescharacteristic
319C8—42no—
320C8—81no—
321C8—342no—
322C8—342no—
323C8—1361no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—162no—
328C22—162no—
329C22#1176270911yescharacteristic
330C22#1176302311yescharacteristic
331C22#1176303011yescharacteristic
332C22#1176305412yes—
333C22#1176307812yes—
334C4—21no—
335C4—21no—
336C4—1362no—
337C4—1362no—
338C4#1176280211yescharacteristic
339C4#1176280711yescharacteristic
340C4#1176290612yes—
341C4#1176302211yescharacteristic
342C4#1176302911yescharacteristic
343C4#1176319711yescharacteristic
344C4#1176320011yescharacteristic
345C2—162no—
346C2#1452831612yes—
347C2#1452832312yes—
348C2#1452966611yescharacteristic
349C2#1452969111yescharacteristic
350C2#1452971511yescharacteristic
subgroupG/H
centerC23#10587570
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#331C2
socleC23×C17#121
prime radicalC2×C17×#127C22
cosocleC2×C42×C17C23
hypercenterC2×#331D17
hypocenterC17#9747
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. #14529731C2semidirect
4. #14528654C2semidirect
5. C2×#11762561C2semidirect
6. C2×#11762557C2semidirect
10. #11763155C4semidirect
236. C17#9747semidirect

Maximal subgroups

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

subgroupindex
C2×#117631552
#145297312
#145297122
#145286542
C2×#117625612
C2×#117625572
C2×C17×#3312
#974717