Finite Group Databasebeta

← #14536723

Subgroups — #14536723

Order 2176.

470 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762737C211yesmaximal, characteristic, complemented
2#11762723C211yesmaximal, characteristic, complemented
3#11762723C211yesmaximal, characteristic, complemented
4#11762723C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762712C211yesmaximal, characteristic, complemented
7#11762712C211yesmaximal, characteristic, complemented
8#11762706C211yesmaximal, characteristic, complemented
9#11762703C211yesmaximal, characteristic, complemented
10#11762703C211yesmaximal, characteristic, complemented
11#11762702C211yesmaximal, characteristic, complemented
12#11762702C211yesmaximal, characteristic, complemented
13#11762697C211yesmaximal, characteristic, complemented
14#11762696C211yesmaximal, characteristic, complemented
15C17×#392C211yesmaximal, characteristic
16C22×D4×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×C4×Dic17C2211yescharacteristic
41C2×C4×Dic17C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43C17×#95C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C24×C17C2311yescharacteristic
53C24×C17C2311yescharacteristic
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22⋊C4×C17—22no—
76C22⋊C4×C17—22no—
77C22⋊C4×C17—41no—
78C22⋊Dic17—22no—
79C22⋊Dic17—22no—
80C22⋊Dic17—22no—
81C22⋊Dic17—41no—
82C22⋊Dic17—41no—
83C22⋊Dic17—41no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92#63225—22no—
93#63225—22no—
94#63225—41no—
95#63225—41no—
96C4×Dic17—41no—
97C4×Dic17—41no—
98C4×Dic17—41no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17C2411yescharacteristic
121D4×C17—44no—
122D4×C17—44no—
123D4×C17—44no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17#4112yes—
135C2×C4×C17#4112yes—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17#4112yes—
160#2803—171nomaximal, Sylow
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17C2×#4111yescharacteristic
183C22×C17C2×#4111yescharacteristic
184C22×C17C2×#4111yescharacteristic
185C22×C17H(5,𝔽2)11yescharacteristic
186C22×C17H(5,𝔽2)11yescharacteristic
187C22×C17H(5,𝔽2)11yescharacteristic
188C22×C17H(5,𝔽2)11yescharacteristic
189C4×C17—22no—
190C4×C17—22no—
191C4×C17—41no—
192C4×C17—41no—
193Dic17—22no—
194Dic17—41no—
195Dic17—41no—
196Dic17—41no—
197Dic17—41no—
198Dic17—41no—
199Dic17—41no—
200Dic17—41no—
201#397—171no—
202#394—171no—
203#394—171no—
204#394—171no—
205#394—171no—
206#392D1711yescharacteristic
207#390—171no—
208#390—171no—
209#389—171no—
210#388—171no—
211#388—171no—
212#388—171no—
213#388—171no—
214#388—171no—
215#387—171no—
216C2×C17—81no—
217C2×C17—81no—
218C2×C17—81no—
219C2×C17#53811yescharacteristic
220C2×C17#53811yescharacteristic
221C2×C17#54011yescharacteristic
222C2×C17#55311yescharacteristic
223C2×C17#55511yescharacteristic
224C2×C17#55911yescharacteristic
225C2×C17#56011yescharacteristic
226C22×D4C2×D1711yescharacteristic
227C22×D4C2×D1711yescharacteristic
228C22×D4C2×D1711yescharacteristic
229C2×C4⋊C4—171no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—171no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4—171no—
238C2×C22⋊C4—171no—
239C2×C22⋊C4—171no—
240C2×C22⋊C4—171no—
241C2×C22⋊C4—171no—
242C2×C22⋊C4—171no—
243C2×C22⋊C4—171no—
244C2×C22⋊C4—171no—
245C2×C22⋊C4—171no—
246C2×C22⋊C4—171no—
247C2×C22⋊C4C2×D1711yescharacteristic
248C2×C22⋊C4C2×D1711yescharacteristic
249C2×C22⋊C4C2×D1711yescharacteristic
250C2×C42—171no—
251C2×C42—171no—
252C2×C42—171no—
253#95—171no—
254#95—171no—
255#95—171no—
256#95—171no—
257#95—171no—
258#95—171no—
259#95—171no—
260#95C2×D1711yescharacteristic
261C17#280311yescharacteristic, complemented, Sylow
262C24C22×D1711yescharacteristic
263C24C22×D1711yescharacteristic
264C24C22×D1711yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4C22×D1711yescharacteristic
283C22×C4C22×D1711yescharacteristic
284C22×C4C22×D1711yescharacteristic
285C22×C4C22×D1711yescharacteristic
286C4⋊C4—342no—
287C4⋊C4—342no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—342no—
296C22⋊C4—342no—
297C22⋊C4—342no—
298C22⋊C4—681no—
299C22⋊C4—681no—
300C22⋊C4—681no—
301C22⋊C4—681no—
302C22⋊C4—681no—
303C22⋊C4—681no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C42—681no—
308C42—681no—
309C42—681no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C23×D1711yescharacteristic
332D4—44no—
333D4—44no—
334D4—44no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—172no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—341no—
365C2×C4—341no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4#6325412yes—
370C2×C4#6325412yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22C2×#6325411yescharacteristic
393C22C2×#6325411yescharacteristic
394C22#1058769111yescharacteristic
395C22#1058769111yescharacteristic
396C22#1058769111yescharacteristic
397C22#1058769111yescharacteristic
398C22#41×D1711yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—342no—
404C4—681no—
405C4—681no—
406C4—681no—
407C4—681no—
408C4—681no—
409C4—681no—
410C4—681no—
411C2—81no—
412C2—81no—
413C2—81no—
414C2#1176339411yescharacteristic
415C2#1176339411yescharacteristic
416C2#1176340911yescharacteristic
417C2#1176350611yescharacteristic
418C2#1176352311yescharacteristic
419C2#1176354511yescharacteristic
420C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC17×#392C2
hypercenter#392D17
hypocenterC17#2803
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #11762737C2semidirect
2. #11762723C2semidirect
3. #11762723C2semidirect
4. #11762723C2semidirect
5. #11762723C2semidirect
6. #11762712C2semidirect
7. #11762712C2semidirect
8. #11762706C2semidirect
9. #11762703C2semidirect
10. #11762703C2semidirect
11. #11762702C2semidirect
12. #11762702C2semidirect
13. #11762697C2semidirect
14. #11762696C2semidirect
261. C17#2803semidirect

Maximal subgroups

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

subgroupindex
#117627372
#117627232
#117627122
#117627062
#117627032
#117627022
#117626972
#117626962
C17×#3922
#280317