Finite Group Databasebeta

← #16923277

Subgroups — #16923277

Order 4352.

446 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14535743C211yesmaximal, characteristic
2#14530114C211yesmaximal, characteristic, complemented
3#14530027C211yesmaximal, characteristic, complemented
4#14526608C211yesmaximal, characteristic, complemented
5#14526606C211yesmaximal, characteristic, complemented
6#14526605C211yesmaximal, characteristic
7C17×#2023C211yesmaximal, characteristic
8C2×#10587565—21no—
9C2×#10587479—21no—
10C2×#10587506—21no—
11C2×#10587506—21no—
12#11762745—21no—
13#11762732—21no—
14#11762723—21no—
15#11762707C2211yescharacteristic
16#11762707C2211yescharacteristic
17#11762704—21no—
18#11762702—21no—
19#11762571—21no—
20#11762564C2211yescharacteristic
21#11762564C2211yescharacteristic
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C17×#102—21no—
24C17×#390C2211yescharacteristic
25C17×#340—21no—
26C17×#336C2211yescharacteristic
27C22×Q8×C17—21no—
28C22×D4×C17D411yescharacteristic
29C22×C8×C17—21no—
30C17×C4⋊Q8—42no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C42×C17C2311yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C2×C4⋊Dic17D411yescharacteristic
43C2×C4⋊Dic17D411yescharacteristic
44C2×C4⋊Dic17D411yescharacteristic
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63225—21no—
49C2×C4×Dic17—21no—
50C22×#3622—21no—
51C22×#3622—21no—
52#10587565—42no—
53#10587565—42no—
54C17×#102—42no—
55#10587506—42no—
56#10587506—42no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587479—42no—
61C24×C17—21no—
62C2×Q8×C17—22no—
63C2×Q8×C17—41no—
64C2×Q8×C17—41no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C2×Q8×C17—41no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C22×C4×C17—21no—
73C22×C4×C17—41no—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C2×C8×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C42×C17—41no—
94C22⋊Dic17—42no—
95C22⋊Dic17—42no—
96C22⋊Dic17—81no—
97C22⋊Dic17—81no—
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103C4⋊Dic17SD1612yes—
104#63225—42no—
105#63225—41no—
106#63225—41no—
107#63225—41no—
108#63225—41no—
109#63225—81no—
110C4×Dic17—42no—
111C2×#3622—22no—
112C2×#3622—41no—
113C2×#3622—41no—
114C2×#3622—41no—
115C2×#3622—41no—
116C2×#3622—41no—
117#7776—171nomaximal, Sylow
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17#12011yescharacteristic
126Q8×C17—42no—
127Q8×C17—42no—
128Q8×C17—81no—
129Q8×C17—81no—
130D4×C17—42no—
131D4×C17—81no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—42no—
142C2×C4×C17—81no—
143C2×C4×C17—81no—
144C2×C4×C17C2×SD1611yescharacteristic
145C2×C4×C17#13511yescharacteristic
146C2×C4×C17#13511yescharacteristic
147C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C17#13711yescharacteristic
150C2×Dic17—22no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C8×C17—81no—
163#3622—42no—
164#3622—81no—
165#2727—171no—
166#2058—171no—
167#2043—171no—
168#2023D1711yescharacteristic
169#1271—171no—
170#1271—171no—
171#1270—171no—
172C22×C17—81no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17—81no—
178C22×C17—81no—
179C22×C17#44911yescharacteristic
180C22×C17#45011yescharacteristic
181C22×C17#45211yescharacteristic
182C22×C17#45411yescharacteristic
183C22×C17#45511yescharacteristic
184C22×C17#45611yescharacteristic
185C22×C17B(4,𝔽2)11yescharacteristic
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—82no—
193C4×C17—81no—
194Dic17—42no—
195Dic17—81no—
196Dic17—81no—
197Dic17—81no—
198C2×C4⋊Q8C2×D1711yescharacteristic
199C2×#126—341no—
200C2×#106—341no—
201C2×#103—341no—
202C2×#103—341no—
203C2×#102—21no—
204#399—341no—
205#396—341no—
206#394—341no—
207#390C2×D1711yescharacteristic
208#389—171no—
209#389—171no—
210#389—341no—
211#388—341no—
212#340—21no—
213#340—341no—
214#336—171no—
215#336—171no—
216#336C2×D1711yescharacteristic
217C2×C17—161no—
218C2×C17#159711yescharacteristic
219C2×C17#160411yescharacteristic
220C2×C17#160611yescharacteristic
221C2×C17#163511yescharacteristic
222C2×C17#164211yescharacteristic
223C2×C17#165211yescharacteristic
224C2×C17#166411yescharacteristic
225C22×Q8—21no—
226C22×D4C22⋊D1711yescharacteristic
227C22×C8—21no—
228C22×C8—341no—
229C22×C8—341no—
230C4⋊Q8—42no—
231#126—682no—
232#126—682no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—171no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—341no—
240C2×C4⋊C4—341no—
241C2×C4⋊C4—341no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4C22⋊D1711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—341no—
247C2×C22⋊C4—341no—
248C2×C22⋊C4—341no—
249C2×C22⋊C4—341no—
250C2×C42—341no—
251C2×C42C22×D1711yescharacteristic
252#106—682no—
253#103—682no—
254#103—682no—
255#102—42no—
256#95—341no—
257#95—341no—
258#95—341no—
259C17#777611yescharacteristic, complemented, Sylow
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4D4×D1711yescharacteristic
278C22×C4D4×D1711yescharacteristic
279C22×C4C2×C22⋊D1711yescharacteristic
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—342no—
284C2×C8—681no—
285C2×C8—681no—
286C2×C8—681no—
287C2×C8—681no—
288C2×C8—681no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—172no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4—341no—
298C4⋊C4—681no—
299C4⋊C4—681no—
300C4⋊C4—682no—
301C4⋊C4—681no—
302C4⋊C4—681no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—1361no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—682no—
311C22⋊C4—682no—
312C22⋊C4—1361no—
313C22⋊C4—1361no—
314C42—41no—
315C42—682no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1058762311yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—342no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4#1058759711yescharacteristic
355C2×C4#1058759811yescharacteristic
356C2×C4SD16×D1711yescharacteristic
357C2×C4#1058760011yescharacteristic
358C2×C4#1058762811yescharacteristic
359C2×C4#1058763611yescharacteristic
360C8—81no—
361C8—682no—
362C8—1361no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1176295111yescharacteristic
371C22#1176295811yescharacteristic
372C22#1176296911yescharacteristic
373C22#1176298011yescharacteristic
374C22#1176298611yescharacteristic
375C22#1176299011yescharacteristic
376C22#1176299511yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—81no—
385C4—682no—
386C4—1361no—
387C4—1361no—
388C4—1361no—
389C2—161no—
390C2#1452771011yescharacteristic
391C2#1452775111yescharacteristic
392C2#1452776211yescharacteristic
393C2#1452793411yescharacteristic
394C2#1452798111yescharacteristic
395C2#1452805311yescharacteristic
396C2#1452811711yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2023C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#2023D17
hypocenterC17#7776
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
2. #14530114C2semidirect
3. #14530027C2semidirect
4. #14526608C2semidirect
5. #14526606C2semidirect
259. C17#7776semidirect

Maximal subgroups

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

subgroupindex
#145357432
#145301142
#145300272
#145266082
#145266062
#145266052
C17×#20232
#777617