Finite Group Databasebeta

← #14535461

Subgroups — #14535461

Order 2176.

540 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587565C211yesmaximal, characteristic, complemented
2C2×#10587615C211yesmaximal, characteristic, complemented
3#11762746C211yesmaximal, characteristic, complemented
4#11762732C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762725C211yesmaximal, characteristic, complemented
7#11762724C211yesmaximal, characteristic, complemented
8#11762717C211yesmaximal, characteristic
9#11762712C211yesmaximal, characteristic, complemented
10#11762709C211yesmaximal, characteristic, complemented
11#11762706C211yesmaximal, characteristic, complemented
12#11762698C211yesmaximal, characteristic, complemented
13#11762692C211yesmaximal, characteristic, complemented
14#11762669C211yesmaximal, characteristic, complemented
15C17×#386C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29#10587615—24no—
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×C4⋊Dic17C2211yescharacteristic
37C2×C4⋊Dic17C2211yescharacteristic
38C2×C4⋊Dic17C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×C4×Dic17C2211yescharacteristic
45C2×C4×Dic17C2211yescharacteristic
46#10587565—24no—
47C17×#95C2211yescharacteristic
48C17×#95C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C24×C17C2311yescharacteristic
55C23×D17C2311yescharacteristic
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—41no—
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—22no—
68C2×C22⋊D17—22no—
69C2×C22⋊D17—41no—
70C2×C22⋊D17—41no—
71C2×C22⋊D17—41no—
72C2×C22⋊D17—41no—
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22⋊C4×C17—22no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊Dic17—22no—
84C22⋊Dic17—22no—
85C22⋊Dic17—22no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89#63227—22no—
90#63227—22no—
91#63227—22no—
92#63227—41no—
93#63227—41no—
94C4⋊Dic17—22no—
95C4⋊Dic17—22no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98#63225—22no—
99#63225—22no—
100#63225—41no—
101#63225—41no—
102#63225D414yescomplemented
103C4×Dic17—22no—
104C4×Dic17—41no—
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#4112yes—
119C23×C17C2411yescharacteristic
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127D4×C17—44no—
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—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—44no—
142C2×C4×C17C2×D412yes—
143C2×C4×C17#4112yes—
144C22⋊D17—44no—
145C22⋊D17—44no—
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—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17C2×D412yes—
166C2×Dic17C2×D412yes—
167#2705—171nomaximal, Sylow
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—22no—
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×C17C22×D411yescharacteristic
183C22×C17C2×#4111yescharacteristic
184C22×C17C2×#4111yescharacteristic
185C22×C17H(5,𝔽2)11yescharacteristic
186C22×C17H(5,𝔽2)11yescharacteristic
187C22×C17H(5,𝔽2)11yescharacteristic
188C22×C172−1+411yescharacteristic
189C2×D17—41no—
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—41no—
196C4×C17—22no—
197C4×C17—22no—
198C4×C17—41no—
199C4×C17—41no—
200C4×C17—41no—
201Dic17—22no—
202Dic17—22no—
203Dic17—41no—
204Dic17—41no—
205Dic17—41no—
206Dic17—41no—
207Dic17—41no—
208C2×#126—171no—
209C2×#123—171no—
210#399—171no—
211#396—171no—
212#396—171no—
213#394—171no—
214#394—171no—
215#392—171no—
216#390—171no—
217#390—171no—
218#389—171no—
219#388—171no—
220#387—171no—
221#386D1711yescharacteristic, complemented
222#380—171no—
223C2×C17—42no—
224C2×C17—81no—
225C2×C17#53711yescharacteristic
226C2×C17#53911yescharacteristic
227C2×C17#54211yescharacteristic
228C2×C17#54611yescharacteristic
229C2×C17#55011yescharacteristic
230C2×C17#55111yescharacteristic
231C2×C17#56011yescharacteristic
232D17—81no—
233C22×D4—171no—
234C22×D4—171no—
235C22×D4C2×D1711yescharacteristic
236C23×C4C2×D1711yescharacteristic
237#126—344no—
238#123—344no—
239C2×C4⋊C4—171no—
240C2×C4⋊C4—171no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—171no—
243C2×C4⋊C4—171no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4—171no—
248C2×C22⋊C4—171no—
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4—171no—
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—171no—
256C2×C22⋊C4—171no—
257C2×C22⋊C4—171no—
258C2×C22⋊C4—171no—
259C2×C22⋊C4C2×D1711yescharacteristic
260C2×C22⋊C4C2×D1711yescharacteristic
261C2×C22⋊C4C2×D1711yescharacteristic
262C2×C42—171no—
263C2×C42—171no—
264#95—171no—
265#95—171no—
266#95—171no—
267#95—171no—
268#95C2×D1711yescharacteristic
269#95C2×D1711yescharacteristic
270C17#270511yescharacteristic, complemented, Sylow
271C24—171no—
272C24C22×D1711yescharacteristic
273C24C22×D1711yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—342no—
278C2×D4—342no—
279C2×D4—681no—
280C2×D4—681no—
281C2×D4—681no—
282C2×D4—681no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—171no—
290C22×C4—171no—
291C22×C4—171no—
292C22×C4—171no—
293C22×C4C22×D1711yescharacteristic
294C22×C4C22×D1711yescharacteristic
295C22×C4C22×D1711yescharacteristic
296C22×C4C22×D1711yescharacteristic
297C22×C4C22×D1711yescharacteristic
298C4⋊C4—174no—
299C4⋊C4—342no—
300C4⋊C4—342no—
301C4⋊C4—342no—
302C4⋊C4—342no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—681no—
306C4⋊C4—681no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—342no—
311C22⋊C4—342no—
312C22⋊C4—342no—
313C22⋊C4—342no—
314C22⋊C4—342no—
315C22⋊C4—342no—
316C22⋊C4—681no—
317C22⋊C4—681no—
318C22⋊C4—681no—
319C22⋊C4—681no—
320C22⋊C4—681no—
321C42—342no—
322C42—681no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—341no—
337C23—341no—
338C23—341no—
339C23—341no—
340C23—341no—
341C23—341no—
342C23—341no—
343C23#6325412yes—
344C23C23×D1711yescharacteristic
345D4—44no—
346D4—684no—
347D4—684no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—172no—
363C2×C4—172no—
364C2×C4—341no—
365C2×C4—341no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—341no—
373C2×C4—341no—
374C2×C4—341no—
375C2×C4—341no—
376C2×C4—341no—
377C2×C4—341no—
378C2×C4—341no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4#6325212yes—
384C2×C4D4×D1712yes—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—681no—
400C22—681no—
401C22—681no—
402C22—681no—
403C22—681no—
404C22—681no—
405C22—681no—
406C22C2×#6325211yescharacteristic
407C22C2×D4×D1711yescharacteristic
408C22C2×#6325411yescharacteristic
409C22#1058769111yescharacteristic
410C22#1058769111yescharacteristic
411C22#1058769611yescharacteristic
412C22#1058769711yescharacteristic
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—342no—
419C4—342no—
420C4—681no—
421C4—681no—
422C4—681no—
423C4—681no—
424C4—681no—
425C2—42no—
426C2—81no—
427C2—1361no—
428C2#1176338811yescharacteristic
429C2#1176340311yescharacteristic
430C2#1176342411yescharacteristic
431C2#1176344611yescharacteristic
432C2#1176348011yescharacteristic
433C2#1176348511yescharacteristic
434C2#1176354711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radical#11762717C2
hypercenter#386D17
hypocenterC17#2705
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#10587565C2semidirect
2. C2×#10587615C2semidirect
3. #11762746C2semidirect
4. #11762732C2semidirect
5. #11762732C2semidirect
6. #11762725C2semidirect
7. #11762724C2semidirect
9. #11762712C2semidirect
10. #11762709C2semidirect
11. #11762706C2semidirect
12. #11762698C2semidirect
13. #11762692C2semidirect
14. #11762669C2semidirect
15. C17×#386C2semidirect
102. #63225D4semidirect
221. #386D17semidirect
270. C17#2705semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
C2×#105876152
#117627462
#117627322
#117627252
#117627242
#117627172
#117627122
#117627092
#117627062
#117626982
#117626922
#117626692
C17×#3862
#270517