Finite Group Databasebeta

← #16933595

Subgroups — #16933595

Order 4352.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763155C211yesmaximal, characteristic
2#14529707C211yesmaximal, characteristic, complemented
3C4⋊C4×#3622C211yesmaximal, characteristic, complemented
4C2×#11762561C211yesmaximal, characteristic, complemented
5C2×#11762556C211yesmaximal, characteristic, complemented
6C2×#11762553C211yesmaximal, characteristic
7C17×#1982C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9#11763155—22no—
10#11763155C414yescomplemented
11C2×C4⋊#3622C2211yescharacteristic
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587481—22no—
14C2×#10587503—21no—
15C2×#10587503—21no—
16C2×C4×#3622—21no—
17C2×C4×#3622C2211yescharacteristic
18#11762564—21no—
19#11762561—22no—
20#11762556—22no—
21#11762553—22no—
22C2×C17×#127C2211yescharacteristic
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×#102—22no—
25C4×C4⋊C4×C17C2211yescharacteristic
26C22×D4×C17—21no—
27C22×D4×C17—21no—
28C22×D4×C17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C17×#127—21no—
32C17×#127—21no—
33C17×#127—21no—
34C17×#127C2×C412yes—
35C2×C4⋊C4×C17D412yes—
36C2×C42×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C22×#3622—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×#102—44no—
51C17×#95—21no—
52#10587503—24no—
53#10587503—24no—
54#10587481—44no—
55C4⋊#3622—21no—
56C4⋊#3622—21no—
57C4⋊#3622—21no—
58C4⋊#3622D412yes—
59C4×#3622—42no—
60C4×#3622C2×C412yes—
61C4×#3622D412yes—
62C24×C17—41no—
63C2×D4×C17—22no—
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17—42no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17—42no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17—82no—
76C22×C4×C17—22no—
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C22×Dic17—41no—
81C2×Dic34—22no—
82C2×Dic34—42no—
83C2×Dic34—42no—
84C2×Dic34—41no—
85C2×C8×C17—41no—
86C2×C8×C17—42no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17D814yes—
89C42×C17—41no—
90C42×C17—41no—
91C42×C17C22×C411yescharacteristic
92C42×C17C2×D411yescharacteristic
93C42×C17C2×D411yescharacteristic
94C42×C17#4111yescharacteristic
95C4⋊Dic17—22no—
96C4⋊Dic17—22no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C2×#3622—21no—
100C2×#3622—21no—
101C2×#3622—21no—
102C2×#3622—21no—
103C2×#3622—22no—
104C2×#3622—41no—
105C2×#3622—41no—
106C2×#3622—41no—
107C2×#3622—42no—
108C2×#3622C22⋊C412yes—
109C2×#3622C22⋊C412yes—
110#9738—171nomaximal, Sylow
111C23×C17—42no—
112C23×C17—81no—
113C23×C17—81no—
114C23×C17—82no—
115C23×C17—82no—
116C23×C17#12111yescharacteristic
117D4×C17—42no—
118D4×C17—42no—
119D4×C17—44no—
120D4×C17—82no—
121D4×C17—82no—
122D4×C17—81no—
123D4×C17—81no—
124D4×C17—81no—
125D4×C17—81no—
126C2×C4×C17—22no—
127C2×C4×C17—24no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17C2×C22⋊C411yescharacteristic
131C2×C4×C17C4×D411yescharacteristic
132C2×C4×C17C4×D411yescharacteristic
133C2×C4×C17#12111yescharacteristic
134C2×C4×C17#12411yescharacteristic
135C2×C4×C17#12711yescharacteristic
136C2×C4×C17C2×D812yes—
137C2×C4×C17#13511yescharacteristic
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17#13512yes—
140C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C17#13711yescharacteristic
142C2×Dic17—42no—
143C2×Dic17—82no—
144Dic34—44no—
145Dic34—81no—
146Dic34—81no—
147C8×C17—81no—
148#3622—22no—
149#3622—22no—
150#3622—42no—
151#3622—81no—
152C2×#486—171no—
153#1982D1711yescharacteristic
154#1981—171no—
155C8×C4⋊C4—171no—
156C2×#335—171no—
157C2×#332—171no—
158C2×#331—171no—
159C22×C17—82no—
160C22×C17—82no—
161C22×C17—82no—
162C22×C17—162no—
163C22×C17—162no—
164C22×C17#39011yescharacteristic
165C22×C17#46311yescharacteristic
166C22×C17#46411yescharacteristic
167C22×C17#46612yes—
168C22×C17#47112yes—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—44no—
172C4×C17#41611yescharacteristic
173C4×C17#41711yescharacteristic
174C4×C17C4×D812yes—
175C4×C17#46311yescharacteristic
176C4×C17#46411yescharacteristic
177C4×C17#49311yescharacteristic
178C4×C17#49511yescharacteristic
179Dic17—82no—
180C2×C4⋊Q8—171no—
181C2×#127C2×D1711yescharacteristic
182#486—174no—
183#486—342no—
184C2×C4⋊C8—171no—
185C2×C4⋊C8—171no—
186C2×C4⋊C8C2×D1711yescharacteristic
187C2×#103—342no—
188C2×#102—22no—
189C2×#102—341no—
190C2×#102—341no—
191C2×C4×C8—171no—
192C2×C4×C8—341no—
193C4×C4⋊C4C2×D1711yescharacteristic
194#336—341no—
195#335—342no—
196#332—342no—
197#331—342no—
198C2×C17—162no—
199C2×C17#168112yes—
200C2×C17#168212yes—
201C2×C17#197011yescharacteristic
202C2×C17#197711yescharacteristic
203C2×C17#198311yescharacteristic
204C22×Q8—341no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×C8—21no—
209C22×C8—171no—
210C22×C8—171no—
211C22×C8—341no—
212C22×C8—341no—
213C4⋊Q8—172no—
214C4⋊Q8—341no—
215C4⋊Q8—341no—
216C4⋊Q8—341no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127C4×D1712yes—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4C22⋊D1712yes—
224C2×C42—21no—
225C2×C42C22×D1711yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—172no—
229C4⋊C8—341no—
230C4⋊C8—341no—
231C4⋊C8—341no—
232#103—684no—
233#102—44no—
234#102—344no—
235#102—344no—
236C4×C8—172no—
237C4×C8—172no—
238C4×C8—682no—
239#95—21no—
240C17#973811yescharacteristic, complemented, Sylow
241C24—41no—
242C2×Q8—342no—
243C2×Q8—681no—
244C2×Q8—682no—
245C2×Q8—682no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—82no—
259C22×C4—22no—
260C22×C4—681no—
261C22×C4D4×D1711yescharacteristic
262C22×C4#6325411yescharacteristic
263C22×C4C2×C22⋊D1711yescharacteristic
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—172no—
267C2×C8—172no—
268C2×C8—341no—
269C2×C8—341no—
270C2×C8—341no—
271C2×C8—341no—
272C2×C8—342no—
273C2×C8—681no—
274C2×C8—681no—
275C2×C8—682no—
276C2×C8—681no—
277C4⋊C4—22no—
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4#6322814yes—
283C42—41no—
284C42—41no—
285C42C2×C4×D1711yescharacteristic
286C42#6325211yescharacteristic
287C42D4×D1711yescharacteristic
288C42D4×D1711yescharacteristic
289C23—42no—
290C23—81no—
291C23—81no—
292C23—82no—
293C23—82no—
294C23#1058762511yescharacteristic
295Q8—684no—
296Q8—1361no—
297Q8—1361no—
298D4—42no—
299D4—42no—
300D4—44no—
301D4—82no—
302D4—82no—
303D4—81no—
304D4—81no—
305D4—81no—
306D4—81no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—682no—
312C2×C4—1362no—
313C2×C4C22⋊C4×D1711yescharacteristic
314C2×C4#1058756711yescharacteristic
315C2×C4#1058757011yescharacteristic
316C2×C4#1058757111yescharacteristic
317C2×C4#1058759711yescharacteristic
318C2×C4#1058759811yescharacteristic
319C2×C4#1058760111yescharacteristic
320C2×C4#1058760211yescharacteristic
321C2×C4C4×C22⋊D1711yescharacteristic
322C2×C4C2×#6322812yes—
323C2×C4#1058762611yescharacteristic
324C2×C4#1058763612yes—
325C8—81no—
326C8—342no—
327C8—342no—
328C8—682no—
329C8—1361no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—162no—
334C22—162no—
335C22#1176270911yescharacteristic
336C22#1176302211yescharacteristic
337C22#1176302911yescharacteristic
338C22#1176304312yes—
339C22#1176307012yes—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—1362no—
344C4#1176280211yescharacteristic
345C4#1176280711yescharacteristic
346C4C4×#6322812yes—
347C4#1176302311yescharacteristic
348C4#1176303011yescharacteristic
349C4#1176318511yescharacteristic
350C4#1176319111yescharacteristic
351C2—162no—
352C2#1452821912yes—
353C2#1452822612yes—
354C2#1452964911yescharacteristic
355C2#1452969011yescharacteristic
356C2#1452971511yescharacteristic
subgroupG/H
centerC23#10587625
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1982C2
socleC23×C17#121
prime radicalC2×C17×#127C22
cosocleC2×C42×C17C23
hypercenter#1982D17
hypocenterC17#9738
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #14529707C2semidirect
3. C4⋊C4×#3622C2semidirect
4. C2×#11762561C2semidirect
5. C2×#11762556C2semidirect
10. #11763155C4semidirect
240. C17#9738semidirect

Maximal subgroups

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

subgroupindex
C2×#117631552
#145297072
C4⋊C4×#36222
C2×#117625612
C2×#117625562
C2×#117625532
C17×#19822
#973817