Finite Group Databasebeta

← #14540910

Subgroups — #14540910

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763499C211yesmaximal, characteristic, complemented
2#11763173C211yesmaximal, characteristic, complemented
3#11763130C211yesmaximal, characteristic, complemented
4#11763125C211yesmaximal, characteristic, complemented
5#11763116C211yesmaximal, characteristic, complemented
6#11763053C211yesmaximal, characteristic, complemented
7#11763036C211yesmaximal, characteristic, complemented
8#11763006C211yesmaximal, characteristic, complemented
9#11762971C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762914C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762810C211yesmaximal, characteristic, complemented
14#11762805C211yesmaximal, characteristic, complemented
15C17×#550C211yesmaximal, characteristic
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic
18C17×#127—21no—
19C17×#127C2211yescharacteristic
20C17×#121—21no—
21C17×#121—21no—
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28#10587621—21no—
29C2×#63229C2211yescharacteristic
30C2×#63229C2211yescharacteristic
31#10587612C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33#10587610C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587575—21no—
38#10587565—21no—
39#10587563—21no—
40#10587556C2211yescharacteristic
41C4×Dic34C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587481C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59#41×C17—22no—
60#41×C17—22no—
61#41×C17—41no—
62#41×C17—41no—
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C22×Dic17—21no—
78C2×Dic34C2311yescharacteristic
79C4⋊C4×C17—21no—
80C4⋊C4×C17D411yescharacteristic
81C4⋊C4×C17C2311yescharacteristic
82C22⋊C4×C17—21no—
83C22⋊C4×C17—21no—
84C22⋊C4×C17D411yescharacteristic
85C22⋊C4×C17D411yescharacteristic
86C42×C17—21no—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91#63231—22no—
92#63229—41no—
93#63229—41no—
94C4⋊Dic17—21no—
95C4⋊Dic17C2311yescharacteristic
96C4⋊Dic17C2311yescharacteristic
97C4⋊Dic17C2311yescharacteristic
98#63225—21no—
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225—21no—
103C4×Dic17—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C23×C17C2×D411yescharacteristic
115Q8×C17—21no—
116Q8×C17#4112yes—
117D4×C17—21no—
118D4×C17—21no—
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—41no—
122D4×C17—42no—
123D4×C17—42no—
124D4×C17—42no—
125D4×C17—81no—
126D4×C17—81no—
127D4×C17—81no—
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—42no—
135C2×C4×C17—42no—
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2411yescharacteristic
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—42no—
146Dic34—21no—
147Dic34—21no—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3622—21no—
152#3317—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—22no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17C2×#4111yescharacteristic
173C4×C17H(5,𝔽2)11yescharacteristic
174Dic17—41no—
175Dic17—41no—
176Dic17—41no—
177Dic17—41no—
178#552—171no—
179#550D1711yescharacteristic
180#490—171no—
181#482—171no—
182#481—171no—
183#479—171no—
184#468—171no—
185#465—171no—
186#460—171no—
187#452—171no—
188#452—171no—
189#441—171no—
190#435—171no—
191#417—171no—
192#416—171no—
193C2×C17—41no—
194C2×C17—41no—
195C2×C17—81no—
196C2×C17—81no—
197C2×C17#54611yescharacteristic
198C2×C17#57511yescharacteristic
199C2×C17#57711yescharacteristic
200C2×#41C2×D1711yescharacteristic
201C2×#41C2×D1711yescharacteristic
202C2×Q16—171no—
203C2×SD16—171no—
204C2×SD16—171no—
205C2×M16—171no—
206C22×C8—171no—
207#127—21no—
208#127C2×D1711yescharacteristic
209#126—341no—
210#125—171no—
211#125—341no—
212#123—341no—
213C22⋊Q8—171no—
214C22⋊Q8—171no—
215#121—21no—
216#121—21no—
217#121C2×D1711yescharacteristic
218#121C2×D1711yescharacteristic
219C4×Q8—171no—
220C4×Q8C2×D1711yescharacteristic
221C4×D4—21no—
222C4×D4C2×D1711yescharacteristic
223#117—171no—
224#117—341no—
225C2×C4⋊C4—171no—
226#107—171no—
227#106—171no—
228#106—171no—
229C4⋊C8—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#102—171no—
235#102—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240C22⋊C8—171no—
241C22⋊C8—171no—
242#97—171no—
243C17#331711yescharacteristic, complemented, Sylow
244#41—22no—
245#41—22no—
246#41—41no—
247#41—41no—
248C2×Q8—171no—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C22⋊D1711yescharacteristic
257C2×D4C22×D1711yescharacteristic
258C2×D4C22×D1711yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263C22×C4C22×D1711yescharacteristic
264Q16—342no—
265SD16—681no—
266SD16—681no—
267M16—681no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—171no—
271C2×C8—171no—
272C2×C8—681no—
273C4⋊C4—21no—
274C4⋊C4—171no—
275C4⋊C4—171no—
276C4⋊C4—171no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4C22⋊D1711yescharacteristic
284C4⋊C4C22×D1711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4C22⋊D1711yescharacteristic
291C22⋊C4C22⋊D1711yescharacteristic
292C42—21no—
293C42—341no—
294C42—341no—
295C42C22×D1711yescharacteristic
296C23—21no—
297C23—21no—
298C23C2×C22⋊D1711yescharacteristic
299C23C2×C22⋊D1711yescharacteristic
300Q8—21no—
301Q8—341no—
302Q8—341no—
303Q8#6325412yes—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—41no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—682no—
328C2×C4C2×C22⋊D1711yescharacteristic
329C2×C4C2×C22⋊D1711yescharacteristic
330C2×C4C2×C22⋊D1711yescharacteristic
331C2×C4C2×C22⋊D1711yescharacteristic
332C2×C4C23×D1711yescharacteristic
333C8—341no—
334C8—341no—
335C8—341no—
336C8—341no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C22×C22⋊D1711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—681no—
357C4—681no—
358C4—681no—
359C4—681no—
360C4C2×#6325411yescharacteristic
361C4#1058769111yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2#1176345311yescharacteristic
367C2#1176360711yescharacteristic
368C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#550C2
socleC22×C17C22×D4
hypercenter#550D17
hypocenterC17#3317
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #11763499C2semidirect
2. #11763173C2semidirect
3. #11763130C2semidirect
4. #11763125C2semidirect
5. #11763116C2semidirect
6. #11763053C2semidirect
7. #11763036C2semidirect
8. #11763006C2semidirect
9. #11762971C2semidirect
10. #11762971C2semidirect
11. #11762914C2semidirect
12. #11762881C2semidirect
13. #11762810C2semidirect
14. #11762805C2semidirect
243. C17#3317semidirect

Maximal subgroups

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

subgroupindex
#117634992
#117631732
#117631302
#117631252
#117631162
#117630532
#117630362
#117630062
#117629712
#117629142
#117628812
#117628102
#117628052
C17×#5502
#331717