Finite Group Databasebeta

← #14536833

Subgroups — #14536833

Order 2176.

464 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587575C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic
3#11762745C211yesmaximal, characteristic, complemented
4#11762744C211yesmaximal, characteristic, complemented
5#11762738C211yesmaximal, characteristic, complemented
6#11762737C211yesmaximal, characteristic, complemented
7#11762732C211yesmaximal, characteristic, complemented
8#11762731C211yesmaximal, characteristic, complemented
9#11762731C211yesmaximal, characteristic, complemented
10#11762730C211yesmaximal, characteristic, complemented
11#11762729C211yesmaximal, characteristic, complemented
12#11762725C211yesmaximal, characteristic
13#11762724C211yesmaximal, characteristic
14#11762674C211yesmaximal, characteristic, complemented
15C17×#396C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43#10587575—24no—
44#10587571—24no—
45C17×#95C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C2×C22⋊D17—22no—
62C2×C22⋊D17—41no—
63C2×C22⋊D17—41no—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C2×Dic34—22no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C22⋊Dic17—22no—
81C22⋊Dic17—41no—
82C22⋊Dic17—41no—
83#63227—22no—
84#63227—22no—
85#63227—41no—
86#63227—41no—
87#63227—41no—
88#63227—41no—
89C4⋊Dic17—22no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92#63225—22no—
93#63225—22no—
94#63225—22no—
95#63225—22no—
96#63225—41no—
97#63225—41no—
98C4×Dic17D414yes—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17C2411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
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×C17C2×D412yes—
132C22⋊D17—44no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
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×Dic17C2×D412yes—
153C2×Dic17C2×D412yes—
154Dic34—44no—
155#2820—171nomaximal, Sylow
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17C22×D411yescharacteristic
164C22×C17H(5,𝔽2)11yescharacteristic
165C22×C17H(5,𝔽2)11yescharacteristic
166C22×C17H(5,𝔽2)11yescharacteristic
167C22×C172−1+411yescharacteristic
168C22×C172−1+411yescharacteristic
169C22×C172−1+411yescharacteristic
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183Dic17—22no—
184Dic17—22no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190C2×#125—171no—
191C2×#124—171no—
192#399—171no—
193#399—171no—
194#397—171no—
195#397—171no—
196#396—171no—
197#396—171no—
198#396—171no—
199#396D1711yescharacteristic, complemented
200#395—171no—
201#395—171no—
202#394—171no—
203#394—171no—
204#382—171no—
205C2×C17—81no—
206C2×C17#53511yescharacteristic
207C2×C17#53611yescharacteristic
208C2×C17#53711yescharacteristic
209C2×C17#56211yescharacteristic
210C2×C17#56211yescharacteristic
211C2×C17#56311yescharacteristic
212C2×C17#56311yescharacteristic
213D17—81no—
214C22×Q8—171no—
215C22×D4—171no—
216#125—344no—
217#124—344no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4C2×D1711yescharacteristic
228C2×C4⋊C4C2×D1711yescharacteristic
229C2×C4⋊C4C2×D1711yescharacteristic
230C2×C22⋊C4—171no—
231C2×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4—171no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4—171no—
238C2×C22⋊C4—171no—
239C2×C22⋊C4C2×D1711yescharacteristic
240C2×C22⋊C4C2×D1711yescharacteristic
241C2×C22⋊C4C2×D1711yescharacteristic
242C2×C42—171no—
243#95—171no—
244#95—171no—
245#95—171no—
246#95—171no—
247#95—171no—
248#95—171no—
249#95—171no—
250#95C2×D1711yescharacteristic
251C17#282011yescharacteristic, complemented, Sylow
252C24—171no—
253C24C22×D1711yescharacteristic
254C2×Q8—342no—
255C2×Q8—681no—
256C2×Q8—681no—
257C2×D4—342no—
258C2×D4—681no—
259C2×D4—681no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4C22×D1711yescharacteristic
268C22×C4C22×D1711yescharacteristic
269C22×C4C22×D1711yescharacteristic
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—342no—
277C4⋊C4—342no—
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—342no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—342no—
289C22⋊C4—342no—
290C22⋊C4—342no—
291C22⋊C4—681no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—681no—
297C42—174no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—341no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23—341no—
310C23—341no—
311C23—341no—
312C23C23×D1711yescharacteristic
313Q8—684no—
314D4—684no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—172no—
333C2×C4—172no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4D4×D1712yes—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—681no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22C2×D4×D1711yescharacteristic
369C22#1058769111yescharacteristic
370C22#1058769111yescharacteristic
371C22#1058769411yescharacteristic
372C22#1058769611yescharacteristic
373C22#1058769711yescharacteristic
374C22#1058769711yescharacteristic
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—342no—
382C4—342no—
383C4—681no—
384C4—681no—
385C4—681no—
386C4—681no—
387C4—681no—
388C2—81no—
389C2—1361no—
390C2#1176337511yescharacteristic
391C2#1176338211yescharacteristic
392C2#1176338611yescharacteristic
393C2#1176355411yescharacteristic
394C2#1176355511yescharacteristic
395C2#1176356011yescharacteristic
396C2#1176356111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC22×C22⋊D17C22
hypercenter#396D17
hypocenterC17#2820
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#10587575C2semidirect
3. #11762745C2semidirect
4. #11762744C2semidirect
5. #11762738C2semidirect
6. #11762737C2semidirect
7. #11762732C2semidirect
8. #11762731C2semidirect
9. #11762731C2semidirect
10. #11762730C2semidirect
11. #11762729C2semidirect
14. #11762674C2semidirect
15. C17×#396C2semidirect
199. #396D17semidirect
251. C17#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#105875752
C2×#105875712
#117627452
#117627442
#117627382
#117627372
#117627322
#117627312
#117627302
#117627292
#117627252
#117627242
#117626742
C17×#3962
#282017