Finite Group Databasebeta

← #14539893

Subgroups — #14539893

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763430C211yesmaximal, characteristic, complemented
2#11763201C211yesmaximal, characteristic, complemented
3#11763191C211yesmaximal, characteristic, complemented
4#11763168C211yesmaximal, characteristic, complemented
5#11763159C211yesmaximal, characteristic, complemented
6#11763028C211yesmaximal, characteristic, complemented
7#11763021C211yesmaximal, characteristic, complemented
8#11763016C211yesmaximal, characteristic, complemented
9#11763011C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762965C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic
13#11762948C211yesmaximal, characteristic, complemented
14#11762870C211yesmaximal, characteristic, complemented
15C17×#536C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C17×C4⋊Q8C2211yescharacteristic
19C17×C4⋊Q8C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C17×#124C2211yescharacteristic
22C17×#123—21no—
23C17×#123—21no—
24C17×C22⋊Q8—21no—
25C17×C22⋊Q8—21no—
26C17×#117C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28C2×#63231C2211yescharacteristic
29C2×#63231C2211yescharacteristic
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32C2×#63230C2211yescharacteristic
33C2×#63229C2211yescharacteristic
34C2×#63228C2211yescharacteristic
35#10587616C2211yescharacteristic
36#10587610C2211yescharacteristic
37#10587584—21no—
38#10587582—21no—
39#10587571—21no—
40#10587569—21no—
41#10587561C2211yescharacteristic
42C4×D68C2211yescharacteristic
43#10587555C2211yescharacteristic
44C4×Dic34C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587506C2211yescharacteristic
49#10587506C2211yescharacteristic
50#10587506C2211yescharacteristic
51#10587503C2211yescharacteristic
52#10587503C2211yescharacteristic
53C22⋊#3622C2211yescharacteristic
54C22⋊#3622C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56C4⋊#3622C2211yescharacteristic
57#10587474C2211yescharacteristic
58C4×#3622C2211yescharacteristic
59#41×C17—41no—
60#41×C17—41no—
61C2×Q8×C17—21no—
62C2×Q8×C17—41no—
63C2×Q8×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×Q8×C17C2311yescharacteristic
66C2×Q8×C17C2311yescharacteristic
67C2×D4×C17—21no—
68C2×D4×C17C2311yescharacteristic
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17C2311yescharacteristic
72C2×C22⋊D17—21no—
73C2×D68C2311yescharacteristic
74C2×C4×D17—21no—
75C2×Dic34—21no—
76C2×Dic34C2311yescharacteristic
77C4⋊C4×C17—21no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17D411yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17C2311yescharacteristic
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92#63231—41no—
93#63231—41no—
94#63231—41no—
95#63230—41no—
96#63230—41no—
97#63230—41no—
98#63229—41no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17C2311yescharacteristic
106C4⋊Dic17C2311yescharacteristic
107#63225—21no—
108#63225—21no—
109C4×Dic17—21no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117Q8×C17—21no—
118Q8×C17—21no—
119Q8×C17—21no—
120Q8×C17—21no—
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—42no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—42no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2411yescharacteristic
140C22⋊D17—81no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144D68—21no—
145D68—21no—
146C4×D17—81no—
147Dic34—21no—
148Dic34—21no—
149Dic34—81no—
150#3622—21no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3225—171nomaximal, Sylow
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17C22×D411yescharacteristic
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17H(5,𝔽2)11yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—41no—
176Dic17—41no—
177Dic17—41no—
178#542—171no—
179#536D1711yescharacteristic, complemented
180#497—171no—
181#495—171no—
182#489—171no—
183#487—171no—
184#464—171no—
185#463—171no—
186#462—171no—
187#461—171no—
188#452—171no—
189#451—171no—
190#449—171no—
191#448—171no—
192#433—171no—
193C2×C17—41no—
194C2×C17—81no—
195C2×C17#53511yescharacteristic
196C2×C17#57711yescharacteristic
197C2×C17#57811yescharacteristic
198D17—81no—
199C2×#41C2×D1711yescharacteristic
200C22×Q8C2×D1711yescharacteristic
201C2×Q16—171no—
202C2×Q16—171no—
203C2×Q16—171no—
204C2×SD16—171no—
205C2×SD16—171no—
206C2×SD16—171no—
207C2×SD16—171no—
208C2×D8—171no—
209C4⋊Q8—171no—
210C4⋊Q8C2×D1711yescharacteristic
211C4⋊Q8C2×D1711yescharacteristic
212#126—341no—
213#124—171no—
214#124—341no—
215#124C2×D1711yescharacteristic
216#124C2×D1711yescharacteristic
217#123—21no—
218#123—21no—
219#123—341no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—171no—
223C22⋊Q8—341no—
224#121—171no—
225C4×Q8—171no—
226C4×D4—171no—
227#117C2×D1711yescharacteristic
228C4⋊C8—171no—
229C4⋊C8—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#103—171no—
235#103—171no—
236#102—171no—
237#102—171no—
238C22⋊C8—171no—
239C22⋊C8—171no—
240#97—171no—
241C4×C8—171no—
242C17#322511yescharacteristic, complemented, Sylow
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—171no—
249C2×Q8—341no—
250C2×Q8C22×D1711yescharacteristic
251C2×Q8C22×D1711yescharacteristic
252C2×Q8C22×D1711yescharacteristic
253C2×D4—21no—
254C2×D4—171no—
255C2×D4—341no—
256C2×D4C22×D1711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—341no—
260C22×C4C22×D1711yescharacteristic
261Q16—681no—
262Q16—681no—
263Q16—681no—
264SD16—681no—
265SD16—681no—
266SD16—681no—
267SD16—681no—
268D8—681no—
269C2×C8—171no—
270C2×C8—171no—
271C2×C8—171no—
272C2×C8—171no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4C22⋊D1711yescharacteristic
283C4⋊C4C22⋊D1711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4C22⋊D1711yescharacteristic
294C22⋊C4C22⋊D1711yescharacteristic
295C42—341no—
296C42C22×D1711yescharacteristic
297C42C22×D1711yescharacteristic
298C23—21no—
299C23—341no—
300C23C2×C22⋊D1711yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—341no—
309Q8—341no—
310Q8—1361no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—341no—
315D4—341no—
316D4—1361no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—1361no—
328C2×C4C2×C22⋊D1711yescharacteristic
329C2×C4C2×C22⋊D1711yescharacteristic
330C2×C4C2×C22⋊D1711yescharacteristic
331C2×C4C2×C22⋊D1711yescharacteristic
332C2×C4C2×C22⋊D1711yescharacteristic
333C2×C4C23×D1711yescharacteristic
334C8—341no—
335C8—341no—
336C8—341no—
337C8—341no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—681no—
345C22—681no—
346C22—681no—
347C22C22×C22⋊D1711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—681no—
357C4—681no—
358C4—681no—
359C4#1058769111yescharacteristic
360C4#1058769111yescharacteristic
361C2—41no—
362C2—81no—
363C2—1361no—
364C2#1176337611yescharacteristic
365C2#1176361711yescharacteristic
366C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#536C2
socleC22×C17C22×D4
prime radical#11762954C2
hypercenter#536D17
hypocenterC17#3225
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #11763430C2semidirect
2. #11763201C2semidirect
3. #11763191C2semidirect
4. #11763168C2semidirect
5. #11763159C2semidirect
6. #11763028C2semidirect
7. #11763021C2semidirect
8. #11763016C2semidirect
9. #11763011C2semidirect
10. #11762971C2semidirect
11. #11762965C2semidirect
13. #11762948C2semidirect
14. #11762870C2semidirect
15. C17×#536C2semidirect
179. #536D17semidirect
242. C17#3225semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117632012
#117631912
#117631682
#117631592
#117630282
#117630212
#117630162
#117630112
#117629712
#117629652
#117629542
#117629482
#117628702
C17×#5362
#322517