Finite Group Databasebeta

← #14535576

Subgroups — #14535576

Order 2176.

598 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587581C211yesmaximal, characteristic, complemented
2#11762735C211yesmaximal, characteristic, complemented
3#11762726C211yesmaximal, characteristic, complemented
4#11762725C211yesmaximal, characteristic, complemented
5#11762718C211yesmaximal, characteristic, complemented
6#11762718C211yesmaximal, characteristic, complemented
7#11762718C211yesmaximal, characteristic, complemented
8#11762711C211yesmaximal, characteristic, complemented
9#11762710C211yesmaximal, characteristic, complemented
10#11762709C211yesmaximal, characteristic, complemented
11#11762700C211yesmaximal, characteristic, complemented
12#11762695C211yesmaximal, characteristic, complemented
13#11762685C211yesmaximal, characteristic, complemented
14#11762680C211yesmaximal, characteristic, complemented
15C2×C17×#126C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×D68C2211yescharacteristic
20C22×D68C2211yescharacteristic
21C22×D68C2211yescharacteristic
22C22×C4×D17C2211yescharacteristic
23C17×#126—24no—
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C4⋊C4×C17C2211yescharacteristic
26C2×C4⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic
29C2×C22⋊C4×C17C2211yescharacteristic
30C2×C42×C17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×#63227C2211yescharacteristic
43C2×C4⋊Dic17C2211yescharacteristic
44C2×#63225C2211yescharacteristic
45C2×C4×Dic17C2211yescharacteristic
46#10587581—24no—
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C23×D17C2311yescharacteristic
56C23×D17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—22no—
64C2×C22⋊D17—22no—
65C2×C22⋊D17—41no—
66C2×C22⋊D17—41no—
67C2×C22⋊D17—41no—
68C2×C22⋊D17—41no—
69C2×C22⋊D17—41no—
70C2×C22⋊D17—41no—
71C2×C22⋊D17—41no—
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C2×D68—22no—
78C2×D68—22no—
79C2×D68—22no—
80C2×D68—22no—
81C2×D68—22no—
82C2×D68—41no—
83C2×D68—41no—
84C2×D68—41no—
85C2×D68—41no—
86C2×C4×D17—22no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—22no—
91C4⋊C4×C17D414yescomplemented
92C22⋊C4×C17—22no—
93C22⋊C4×C17—22no—
94C22⋊C4×C17—22no—
95C42×C17—22no—
96C22⋊Dic17—41no—
97#63227—22no—
98#63227—22no—
99#63227—22no—
100#63227—22no—
101#63227—41no—
102#63227—41no—
103#63227—41no—
104#63227—41no—
105#63227—41no—
106#63227—41no—
107#63227—41no—
108C4⋊Dic17—41no—
109#63225—41no—
110C4×Dic17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17C2411yescharacteristic
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17—21no—
132C22×D17—21no—
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—21no—
138C22×D17—21no—
139C22×D17#4112yes—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17C2×D412yes—
156C2×C4×C17C2×D412yes—
157C2×C4×C17C2×D412yes—
158C22⋊D17—44no—
159C22⋊D17—44no—
160C22⋊D17—44no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17—21no—
170C2×Dic17—21no—
171C2×Dic17—21no—
172C2×Dic17—21no—
173C2×Dic17—21no—
174C2×Dic17—21no—
175C2×Dic17#4112yes—
176D68—44no—
177D68—44no—
178D68—44no—
179C4×D17—44no—
180#2714—171nomaximal, Sylow
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17C22×D411yescharacteristic
189C22×C17C2×#4111yescharacteristic
190C22×C17C2×#4111yescharacteristic
191C22×C17H(5,𝔽2)11yescharacteristic
192C22×C17H(5,𝔽2)11yescharacteristic
193C22×C17H(5,𝔽2)11yescharacteristic
194C22×C17H(5,𝔽2)11yescharacteristic
195C2×D17—22no—
196C2×D17—22no—
197C2×D17—22no—
198C2×D17—41no—
199C2×D17—41no—
200C2×D17—41no—
201C2×D17—41no—
202C2×D17—41no—
203C2×D17—41no—
204C2×D17—41no—
205C2×D17—41no—
206C2×D17—41no—
207C2×D17—41no—
208C2×D17—41no—
209C2×D17—41no—
210C2×D17—41no—
211C2×D17—41no—
212C2×D17—41no—
213C2×D17—41no—
214C2×D17—41no—
215C2×D17—41no—
216C4×C17—22no—
217C4×C17—22no—
218C4×C17—22no—
219C4×C17—41no—
220C4×C17—41no—
221C4×C17—41no—
222Dic17—22no—
223Dic17—41no—
224Dic17—41no—
225Dic17—41no—
226Dic17—41no—
227C2×#126D1711yescharacteristic, complemented
228C2×#121—171no—
229#397—171no—
230#394—171no—
231#394—171no—
232#392—171no—
233#392—171no—
234#392—171no—
235#390—171no—
236#390—171no—
237#390—171no—
238#388—171no—
239#387—171no—
240#386—171no—
241#385—171no—
242C2×C17—81no—
243C2×C17#53511yescharacteristic
244C2×C17#53811yescharacteristic
245C2×C17#53811yescharacteristic
246C2×C17#54611yescharacteristic
247C2×C17#55011yescharacteristic
248C2×C17#55311yescharacteristic
249C2×C17#56011yescharacteristic
250D17—42no—
251D17—81no—
252D17—81no—
253C22×D4—171no—
254C22×D4—171no—
255C22×D4—171no—
256C22×D4—171no—
257C22×D4—171no—
258C22×D4—171no—
259C23×C4—171no—
260#126—24no—
261#121—344no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—171no—
264C2×C4⋊C4C2×D1711yescharacteristic
265C2×C4⋊C4C2×D1711yescharacteristic
266C2×C4⋊C4C2×D1711yescharacteristic
267C2×C22⋊C4—171no—
268C2×C22⋊C4—171no—
269C2×C22⋊C4—171no—
270C2×C22⋊C4—171no—
271C2×C22⋊C4—171no—
272C2×C22⋊C4—171no—
273C2×C22⋊C4—171no—
274C2×C22⋊C4—171no—
275C2×C22⋊C4—171no—
276C2×C22⋊C4—171no—
277C2×C22⋊C4—171no—
278C2×C22⋊C4—171no—
279C2×C22⋊C4C2×D1711yescharacteristic
280C2×C22⋊C4C2×D1711yescharacteristic
281C2×C22⋊C4C2×D1711yescharacteristic
282C2×C42—171no—
283C2×C42C2×D1711yescharacteristic
284#95—171no—
285#95—171no—
286#95—171no—
287#95—171no—
288#95—171no—
289#95—171no—
290C17#271411yescharacteristic, complemented, Sylow
291C24—171no—
292C24—171no—
293C24—171no—
294C24C22×D1711yescharacteristic
295C2×D4—342no—
296C2×D4—342no—
297C2×D4—342no—
298C2×D4—342no—
299C2×D4—342no—
300C2×D4—342no—
301C2×D4—342no—
302C2×D4—681no—
303C2×D4—681no—
304C2×D4—681no—
305C2×D4—681no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—681no—
313C22×C4—171no—
314C22×C4—171no—
315C22×C4—171no—
316C22×C4—171no—
317C22×C4—171no—
318C22×C4—342no—
319C22×C4—681no—
320C22×C4—681no—
321C22×C4C22×D1711yescharacteristic
322C22×C4C22×D1711yescharacteristic
323C22×C4C22×D1711yescharacteristic
324C22×C4C22×D1711yescharacteristic
325C22×C4C22×D1711yescharacteristic
326C22×C4C22×D1711yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—681no—
330C4⋊C4—681no—
331C4⋊C4C22⋊D1714yescomplemented
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—342no—
336C22⋊C4—342no—
337C22⋊C4—342no—
338C22⋊C4—342no—
339C22⋊C4—681no—
340C22⋊C4—681no—
341C22⋊C4—681no—
342C22⋊C4—681no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C22⋊C4—681no—
347C42—22no—
348C42—681no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—172no—
357C23—341no—
358C23—341no—
359C23—341no—
360C23—341no—
361C23—341no—
362C23—341no—
363C23—341no—
364C23—341no—
365C23—341no—
366C23—341no—
367C23—341no—
368C23—341no—
369C23—341no—
370C23—341no—
371C23—341no—
372C23—341no—
373C23—341no—
374C23—341no—
375C23—341no—
376C23—341no—
377C23C23×D1711yescharacteristic
378D4—684no—
379D4—684no—
380D4—684no—
381D4—684no—
382D4—684no—
383D4—684no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—172no—
400C2×C4—341no—
401C2×C4—341no—
402C2×C4—341no—
403C2×C4—341no—
404C2×C4—341no—
405C2×C4—341no—
406C2×C4—341no—
407C2×C4—341no—
408C2×C4—341no—
409C2×C4—341no—
410C2×C4—341no—
411C2×C4—341no—
412C2×C4—341no—
413C2×C4—341no—
414C2×C4—684no—
415C2×C4C2×C22⋊D1712yes—
416C2×C4C2×C22⋊D1712yes—
417C2×C4C2×C22⋊D1712yes—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—342no—
426C22—342no—
427C22—342no—
428C22—681no—
429C22—681no—
430C22—681no—
431C22—681no—
432C22—681no—
433C22—681no—
434C22—681no—
435C22—681no—
436C22—681no—
437C22—681no—
438C22—681no—
439C22—681no—
440C22—681no—
441C22—681no—
442C22—681no—
443C22—681no—
444C22—681no—
445C22—681no—
446C22#41×D1711yescharacteristic
447C22#41×D1711yescharacteristic
448C22#1058769611yescharacteristic
449C22#1058769611yescharacteristic
450C22#1058769611yescharacteristic
451C22#1058769611yescharacteristic
452C22C22×C22⋊D1711yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—342no—
460C4—681no—
461C4—681no—
462C4—681no—
463C4—681no—
464C2—81no—
465C2—682no—
466C2—1361no—
467C2—1361no—
468C2#1176337311yescharacteristic
469C2#1176339811yescharacteristic
470C2#1176339811yescharacteristic
471C2#1176344711yescharacteristic
472C2#1176347811yescharacteristic
473C2#1176351011yescharacteristic
474C2#1176355011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#126C2
hypercenterC2×#126D17
hypocenterC17#2714
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10587581C2semidirect
2. #11762735C2semidirect
3. #11762726C2semidirect
4. #11762725C2semidirect
5. #11762718C2semidirect
6. #11762718C2semidirect
7. #11762718C2semidirect
8. #11762711C2semidirect
9. #11762710C2semidirect
10. #11762709C2semidirect
11. #11762700C2semidirect
12. #11762695C2semidirect
13. #11762685C2semidirect
14. #11762680C2semidirect
15. C2×C17×#126C2semidirect
91. C4⋊C4×C17D4semidirect
227. C2×#126D17semidirect
290. C17#2714semidirect
331. C4⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875812
#117627352
#117627262
#117627252
#117627182
#117627112
#117627102
#117627092
#117627002
#117626952
#117626852
#117626802
C2×C17×#1262
#271417