Finite Group Databasebeta

← #14539660

Subgroups — #14539660

Order 2176.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763380C211yesmaximal, characteristic, complemented
2#11763147C211yesmaximal, characteristic, complemented
3#11763144C211yesmaximal, characteristic, complemented
4#11763139C211yesmaximal, characteristic, complemented
5#11763130C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763053C211yesmaximal, characteristic, complemented
8#11763048C211yesmaximal, characteristic, complemented
9#11763036C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762971C211yesmaximal, characteristic, complemented
12#11762964C211yesmaximal, characteristic, complemented
13#11762947C211yesmaximal, characteristic, complemented
14#11762776C211yesmaximal, characteristic, complemented
15C17×#537C211yesmaximal, characteristic
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C17×C22⋊Q8—21no—
20C17×C22⋊Q8C2211yescharacteristic
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121—21no—
23C17×#121—21no—
24C17×#121—21no—
25C17×#121C2211yescharacteristic
26C17×#121C2211yescharacteristic
27C2×C4⋊C4×C17C2211yescharacteristic
28#10587630—21no—
29C2×#63231C2211yescharacteristic
30C2×#63231C2211yescharacteristic
31#10587622—21no—
32#10587621—21no—
33#10587621—21no—
34C2×#63229C2211yescharacteristic
35C2×#63229C2211yescharacteristic
36#10587612C2211yescharacteristic
37#10587610C2211yescharacteristic
38#10587610C2211yescharacteristic
39#10587610C2211yescharacteristic
40#10587610C2211yescharacteristic
41C2×#63223C2211yescharacteristic
42C22×#3622C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47C22⋊#3622C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic
49C22⋊#3622C2211yescharacteristic
50C22⋊#3622C2211yescharacteristic
51#10587481C2211yescharacteristic
52#10587481C2211yescharacteristic
53#10587481C2211yescharacteristic
54#10587481C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587479C2211yescharacteristic
57#10587478C2211yescharacteristic
58#10587478C2211yescharacteristic
59#41×C17—41no—
60#41×C17—41no—
61#41×C17—41no—
62#41×C17—41no—
63C2×Q8×C17D411yescharacteristic
64C2×Q8×C17C2311yescharacteristic
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17D411yescharacteristic
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C22×Dic17—21no—
78C2×Dic34—21no—
79C2×Dic34—41no—
80C2×Dic34—41no—
81C2×Dic34C2311yescharacteristic
82C2×Dic34C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97#63231—41no—
98#63231—41no—
99#63229—41no—
100#63229—41no—
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108#63223—41no—
109C2×#3622—41no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C23×C17C2×D411yescharacteristic
117C23×C17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120Q8×C17—41no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—41no—
124D4×C17—41no—
125D4×C17—41no—
126D4×C17—42no—
127D4×C17—81no—
128D4×C17—81no—
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—42no—
139C2×C4×C17—81no—
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2×D411yescharacteristic
142C2×C4×C17C2×D411yescharacteristic
143C2×C4×C17C2411yescharacteristic
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—42no—
149Dic34—21no—
150Dic34—21no—
151Dic34—21no—
152Dic34—21no—
153Dic34—42no—
154#3622—21no—
155#3622—21no—
156#3622—21no—
157#3622—21no—
158#3201—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17C22×D411yescharacteristic
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17H(5,𝔽2)11yescharacteristic
179C4×C17H(5,𝔽2)11yescharacteristic
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#537D1711yescharacteristic
185#536—171no—
186#485—171no—
187#484—171no—
188#483—171no—
189#482—171no—
190#470—171no—
191#468—171no—
192#467—171no—
193#465—171no—
194#452—171no—
195#452—171no—
196#451—171no—
197#448—171no—
198#408—171no—
199C2×C17—41no—
200C2×C17—41no—
201C2×C17—41no—
202C2×C17—81no—
203C2×C17#53411yescharacteristic
204C2×C17#57711yescharacteristic
205C2×C17#57811yescharacteristic
206C2×#41C2×D1711yescharacteristic
207C2×#41C2×D1711yescharacteristic
208C22×Q8—171no—
209C2×Q16—171no—
210C2×Q16—171no—
211C2×SD16—171no—
212C2×SD16—171no—
213C2×M16—171no—
214C22×C8—171no—
215C4⋊Q8—341no—
216#124—341no—
217#123—341no—
218#123—341no—
219C22⋊Q8—21no—
220C22⋊Q8—171no—
221C22⋊Q8—171no—
222C22⋊Q8—171no—
223C22⋊Q8—171no—
224C22⋊Q8C2×D1711yescharacteristic
225C22⋊Q8C2×D1711yescharacteristic
226#121—21no—
227#121—21no—
228#121—21no—
229#121C2×D1711yescharacteristic
230#121C2×D1711yescharacteristic
231#117—171no—
232C2×C4⋊C4C2×D1711yescharacteristic
233#107—171no—
234#107—171no—
235#106—171no—
236#106—171no—
237#103—171no—
238#103—171no—
239#103—171no—
240#103—171no—
241#103—171no—
242#103—171no—
243#102—171no—
244#102—171no—
245C22⋊C8—171no—
246C22⋊C8—171no—
247C22⋊C8—171no—
248C22⋊C8—171no—
249C17#320111yescharacteristic, complemented, Sylow
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—171no—
255C2×Q8—171no—
256C2×Q8—341no—
257C2×Q8—681no—
258C2×Q8—681no—
259C2×Q8C22⋊D1711yescharacteristic
260C2×Q8C22×D1711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4C22⋊D1711yescharacteristic
265C2×D4C22⋊D1711yescharacteristic
266C2×D4C22⋊D1711yescharacteristic
267C2×D4C22×D1711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C22×C4C22×D1711yescharacteristic
274Q16—681no—
275Q16—681no—
276SD16—681no—
277SD16—681no—
278M16—681no—
279C2×C8—171no—
280C2×C8—171no—
281C2×C8—171no—
282C2×C8—171no—
283C2×C8—681no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—171no—
288C4⋊C4—171no—
289C4⋊C4—341no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4C22×D1711yescharacteristic
294C4⋊C4C22×D1711yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C42—341no—
305C23—21no—
306C23C2×C22⋊D1711yescharacteristic
307C23C2×C22⋊D1711yescharacteristic
308C23C2×C22⋊D1711yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—41no—
312Q8—341no—
313Q8—341no—
314Q8—341no—
315Q8—341no—
316Q8—682no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—682no—
341C2×C4C2×C22⋊D1711yescharacteristic
342C2×C4C2×C22⋊D1711yescharacteristic
343C2×C4C2×C22⋊D1711yescharacteristic
344C2×C4C23×D1711yescharacteristic
345C8—341no—
346C8—341no—
347C8—341no—
348C8—341no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×C22⋊D1711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—681no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4#1058769111yescharacteristic
373C4#1058769111yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2#1176336811yescharacteristic
379C2#1176362011yescharacteristic
380C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#537C2
socleC22×C17C22×D4
hypercenter#537D17
hypocenterC17#3201
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #11763380C2semidirect
2. #11763147C2semidirect
3. #11763144C2semidirect
4. #11763139C2semidirect
5. #11763130C2semidirect
6. #11763065C2semidirect
7. #11763053C2semidirect
8. #11763048C2semidirect
9. #11763036C2semidirect
10. #11762971C2semidirect
11. #11762971C2semidirect
12. #11762964C2semidirect
13. #11762947C2semidirect
14. #11762776C2semidirect
249. C17#3201semidirect

Maximal subgroups

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

subgroupindex
#117633802
#117631472
#117631442
#117631392
#117631302
#117630652
#117630532
#117630482
#117630362
#117629712
#117629642
#117629472
#117627762
C17×#5372
#320117