Finite Group Databasebeta

← #14535279

Subgroups — #14535279

Order 2176.

536 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587564C211yesmaximal, characteristic, complemented
2C2×#10587632C211yesmaximal, characteristic, complemented
3#11762746C211yesmaximal, characteristic, complemented
4#11762743C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762730C211yesmaximal, characteristic, complemented
7#11762722C211yesmaximal, characteristic, complemented
8#11762716C211yesmaximal, characteristic, complemented
9#11762709C211yesmaximal, characteristic
10#11762691C211yesmaximal, characteristic, complemented
11#11762686C211yesmaximal, characteristic
12#11762685C211yesmaximal, characteristic
13#11762684C211yesmaximal, characteristic, complemented
14#11762668C211yesmaximal, characteristic, complemented
15C17×#393C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C22×C4×D17C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26#10587632—24no—
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×C4×Dic17C2211yescharacteristic
42#10587564—24no—
43C17×#95C2211yescharacteristic
44C17×#95C2211yescharacteristic
45C17×#95C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C2×Q8×C17—22no—
56C2×Q8×C17—22no—
57C2×Q8×C17—41no—
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—41no—
65C2×C22⋊D17—41no—
66C2×C22⋊D17Q814yescomplemented
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—41no—
80C2×Dic34—22no—
81C2×Dic34—41no—
82C2×Dic34—41no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—41no—
86C22⋊Dic17—22no—
87C22⋊Dic17—41no—
88#63227—22no—
89#63227—22no—
90#63227—41no—
91#63227—41no—
92#63227—41no—
93C4⋊Dic17—22no—
94C4⋊Dic17—22no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97#63225—22no—
98#63225—22no—
99#63225—41no—
100#63225—41no—
101#63225D414yescomplemented
102C4×Dic17—41no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17C2×Q812yes—
110C23×C17C2411yescharacteristic
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17C2×Q812yes—
118Q8×C17—44no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
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—21no—
135C2×C4×C17C2×D412yes—
136C2×C4×C17#4112yes—
137C22⋊D17—44no—
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—44no—
157C2×Dic17C2×D412yes—
158C2×Dic17C2×D412yes—
159C2×Dic17C2×Q812yes—
160C4×D17—44no—
161Dic34—44no—
162#2692—171nomaximal, Sylow
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17C22×D411yescharacteristic
171C22×C17C22×Q811yescharacteristic
172C22×C17C2×#4111yescharacteristic
173C22×C17H(5,𝔽2)11yescharacteristic
174C22×C17H(5,𝔽2)11yescharacteristic
175C22×C17H(5,𝔽2)11yescharacteristic
176C22×C172−1+411yescharacteristic
177C2×D17—22no—
178C2×D17—22no—
179C2×D17—22no—
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C4×C17—22no—
185C4×C17—22no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190Dic17—22no—
191Dic17—22no—
192Dic17—22no—
193Dic17—41no—
194Dic17—41no—
195Dic17—41no—
196Dic17—41no—
197C2×C22⋊Q8—171no—
198C2×C22⋊Q8—171no—
199#399—171no—
200#398—171no—
201#396—171no—
202#395—171no—
203#393—171no—
204#393D1711yescharacteristic, complemented
205#391—171no—
206#390—171no—
207#387—171no—
208#386—171no—
209#386—171no—
210#385—171no—
211#380—171no—
212C2×C17—42no—
213C2×C17#53511yescharacteristic
214C2×C17#54211yescharacteristic
215C2×C17#54311yescharacteristic
216C2×C17#54611yescharacteristic
217C2×C17D4×Q811yescharacteristic
218C2×C17#55411yescharacteristic
219C2×C17#56211yescharacteristic
220D17—42no—
221C22×Q8—171no—
222C22×Q8C2×D1711yescharacteristic
223C22×D4—171no—
224C23×C4—171no—
225C23×C4—171no—
226C22⋊Q8—344no—
227C22⋊Q8—344no—
228C2×C4⋊C4—171no—
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×C4⋊C4—171no—
236C2×C4⋊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⋊C4C2×D1711yescharacteristic
245C2×C22⋊C4C2×D1711yescharacteristic
246C2×C22⋊C4C2×D1711yescharacteristic
247C2×C42—171no—
248#95—171no—
249#95—171no—
250#95—171no—
251#95—171no—
252#95—171no—
253#95—171no—
254#95—171no—
255#95C2×D1711yescharacteristic
256#95C2×D1711yescharacteristic
257#95C2×D1711yescharacteristic
258C17#269211yescharacteristic, complemented, Sylow
259C24—171no—
260C24C22×D1711yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—342no—
265C2×Q8—681no—
266C2×Q8—681no—
267C2×D4—174no—
268C2×D4—681no—
269C2×D4—681no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—342no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—681no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4C22×D1711yescharacteristic
284C22×C4C22×D1711yescharacteristic
285C22×C4C22×D1711yescharacteristic
286C22×C4C22×D1711yescharacteristic
287C22×C4C22×D1711yescharacteristic
288C22×C4C22×D1711yescharacteristic
289C4⋊C4—174no—
290C4⋊C4—342no—
291C4⋊C4—342no—
292C4⋊C4—342no—
293C4⋊C4—342no—
294C4⋊C4—681no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—681no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—41no—
301C22⋊C4—342no—
302C22⋊C4—342no—
303C22⋊C4—342no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C42—681no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—172no—
316C23—341no—
317C23—341no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23—341no—
322C23Q8×D1712yes—
323C23C23×D1711yescharacteristic
324Q8—44no—
325Q8—684no—
326D4—684no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
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—172no—
344C2×C4—172no—
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—684no—
365C2×C4—684no—
366C2×C4D4×D1712yes—
367C2×C4#6325412yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—342no—
376C22—342no—
377C22—342no—
378C22—681no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22C2×D4×D1711yescharacteristic
383C22C2×#6325411yescharacteristic
384C22#1058769111yescharacteristic
385C22C2×Q8×D1711yescharacteristic
386C22#1058769611yescharacteristic
387C22#1058769611yescharacteristic
388C22#1058769711yescharacteristic
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—342no—
396C4—342no—
397C4—342no—
398C4—681no—
399C4—681no—
400C4—681no—
401C4—681no—
402C2—42no—
403C2—682no—
404C2#1176337711yescharacteristic
405C2#1176342811yescharacteristic
406C2#1176343411yescharacteristic
407C2#1176344611yescharacteristic
408C2#1176347211yescharacteristic
409C2#1176351711yescharacteristic
410C2#1176355311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2692
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587564C2semidirect
2. C2×#10587632C2semidirect
3. #11762746C2semidirect
4. #11762743C2semidirect
5. #11762732C2semidirect
6. #11762730C2semidirect
7. #11762722C2semidirect
8. #11762716C2semidirect
10. #11762691C2semidirect
13. #11762684C2semidirect
14. #11762668C2semidirect
15. C17×#393C2semidirect
66. C2×C22⋊D17Q8semidirect
101. #63225D4semidirect
204. #393D17semidirect
258. C17#2692semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
C2×#105876322
#117627462
#117627432
#117627322
#117627302
#117627222
#117627162
#117627092
#117626912
#117626862
#117626852
#117626842
#117626682
C17×#3932
#269217