Finite Group Databasebeta

← #14540204

Subgroups — #14540204

Order 2176.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763430C211yesmaximal, characteristic
2#11763113C211yesmaximal, characteristic, complemented
3#11763106C211yesmaximal, characteristic, complemented
4#11763104C211yesmaximal, characteristic, complemented
5#11763094C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763057C211yesmaximal, characteristic
8#11763048C211yesmaximal, characteristic, complemented
9#11763040C211yesmaximal, characteristic
10#11763031C211yesmaximal, characteristic, complemented
11#11763031C211yesmaximal, characteristic, complemented
12#11763016C211yesmaximal, characteristic, complemented
13#11763011C211yesmaximal, characteristic, complemented
14#11762836C211yesmaximal, characteristic, complemented
15C17×#541C211yesmaximal, characteristic, complemented
16C17×C4⋊Q8C2211yescharacteristic
17C17×#126—21no—
18C17×#126—21no—
19C17×#125—21no—
20C17×#125—21no—
21C17×#125C2211yescharacteristic
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×Q8×C17C2211yescharacteristic
26C17×#117C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28C2×#63231C2211yescharacteristic
29C2×#63230C2211yescharacteristic
30C2×#63230C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587610C2211yescharacteristic
33C2×#63223C2211yescharacteristic
34C22×#3622C2211yescharacteristic
35#10587584—21no—
36#10587582—21no—
37#10587571—21no—
38#10587569—21no—
39#10587561C2211yescharacteristic
40C4×D68C2211yescharacteristic
41#10587555C2211yescharacteristic
42C4×Dic34C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587481C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587479C2211yescharacteristic
52#10587479C2211yescharacteristic
53#10587478C2211yescharacteristic
54#10587478C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56C4⋊#3622C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic
59C2×Q8×C17C2311yescharacteristic
60C2×Q8×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—21no—
63C2×D68C2311yescharacteristic
64C2×C4×D17—21no—
65C2×Dic34—21no—
66C2×Dic34C2311yescharacteristic
67C4⋊C4×C17—21no—
68C4⋊C4×C17—21no—
69C4⋊C4×C17—21no—
70C4⋊C4×C17—21no—
71C4⋊C4×C17—21no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17—21no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17D411yescharacteristic
76C4⋊C4×C17D411yescharacteristic
77C4⋊C4×C17C2311yescharacteristic
78C4⋊C4×C17C2311yescharacteristic
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C22⋊C4×C17D411yescharacteristic
82C22⋊C4×C17D411yescharacteristic
83C42×C17—21no—
84C42×C17—21no—
85C42×C17C2311yescharacteristic
86C42×C17C2311yescharacteristic
87C22⋊Dic17—21no—
88#63231—41no—
89#63231—41no—
90#63230—41no—
91#63230—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101C4×Dic17—21no—
102#63223—41no—
103C2×#3622—41no—
104C2×#3622C2311yescharacteristic
105C2×#3622C2311yescharacteristic
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C23×C17C2×D411yescharacteristic
109C22×D17—21no—
110Q8×C17—21no—
111Q8×C17—21no—
112Q8×C17—21no—
113Q8×C17—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×C17C2×D411yescharacteristic
121C2×C4×C17C2×D411yescharacteristic
122C2×C4×C17C2×D411yescharacteristic
123C2×C4×C17C2×D411yescharacteristic
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2411yescharacteristic
126C22⋊D17—81no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130D68—21no—
131D68—21no—
132C4×D17—81no—
133Dic34—21no—
134Dic34—21no—
135Dic34—81no—
136#3622—21no—
137#3622—21no—
138#3622—21no—
139#3622—21no—
140#3258—171nomaximal, Sylow
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17C22×D411yescharacteristic
145C2×D17—41no—
146C2×D17—41no—
147C2×D17—41no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17H(5,𝔽2)11yescharacteristic
158C4×C172−1+411yescharacteristic
159Dic17—41no—
160Dic17—41no—
161Dic17—41no—
162#542—171no—
163#541D1711yescharacteristic, complemented
164#479—171no—
165#478—171no—
166#477—171no—
167#475—171no—
168#470—171no—
169#468—171no—
170#467—171no—
171#465—171no—
172#464—171no—
173#464—171no—
174#462—171no—
175#461—171no—
176#424—171no—
177C2×C17—41no—
178C2×C17#53711yescharacteristic
179C2×C17#57711yescharacteristic
180C2×C17#57811yescharacteristic
181D17—81no—
182C2×Q16—171no—
183C2×Q16—171no—
184C2×SD16—171no—
185C2×SD16—171no—
186C2×M16—171no—
187C22×C8—171no—
188C4⋊Q8—171no—
189C4⋊Q8C2×D1711yescharacteristic
190#126—21no—
191#126—21no—
192#126—341no—
193#125—21no—
194#125—21no—
195#125C2×D1711yescharacteristic
196#124—171no—
197#124—341no—
198#123—341no—
199C22⋊Q8—171no—
200C22⋊Q8—341no—
201C22⋊Q8C2×D1711yescharacteristic
202C22⋊Q8C2×D1711yescharacteristic
203#121—171no—
204C4×Q8—171no—
205C4×Q8C2×D1711yescharacteristic
206C4×Q8C2×D1711yescharacteristic
207C4×D4—171no—
208#117C2×D1711yescharacteristic
209#107—171no—
210#107—171no—
211#106—171no—
212#106—171no—
213C4⋊C8—171no—
214C4⋊C8—171no—
215C4⋊C8—171no—
216C4⋊C8—171no—
217#103—171no—
218#103—171no—
219#103—171no—
220#103—171no—
221#103—171no—
222#103—171no—
223#102—171no—
224#102—171no—
225C17#325811yescharacteristic, complemented, Sylow
226C2×Q8—171no—
227C2×Q8—341no—
228C2×Q8C22×D1711yescharacteristic
229C2×Q8C22×D1711yescharacteristic
230C2×D4—171no—
231C2×D4—341no—
232C22×C4—341no—
233C22×C4C22×D1711yescharacteristic
234Q16—681no—
235Q16—681no—
236SD16—681no—
237SD16—681no—
238M16—681no—
239C2×C8—171no—
240C2×C8—171no—
241C2×C8—171no—
242C2×C8—171no—
243C2×C8—681no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—171no—
253C4⋊C4—171no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4C22⋊D1711yescharacteristic
258C4⋊C4C22⋊D1711yescharacteristic
259C4⋊C4C22×D1711yescharacteristic
260C4⋊C4C22×D1711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—341no—
264C22⋊C4—341no—
265C22⋊C4—341no—
266C22⋊C4—341no—
267C22⋊C4—341no—
268C22⋊C4C22⋊D1711yescharacteristic
269C22⋊C4C22⋊D1711yescharacteristic
270C42—21no—
271C42—21no—
272C42—341no—
273C42C22×D1711yescharacteristic
274C42C22×D1711yescharacteristic
275C23—341no—
276C23C2×C22⋊D1711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—341no—
282Q8—341no—
283Q8—1361no—
284D4—341no—
285D4—341no—
286D4—1361no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—1361no—
297C2×C4C2×C22⋊D1711yescharacteristic
298C2×C4C2×C22⋊D1711yescharacteristic
299C2×C4C2×C22⋊D1711yescharacteristic
300C2×C4C2×C22⋊D1711yescharacteristic
301C2×C4C2×C22⋊D1711yescharacteristic
302C2×C4C23×D1711yescharacteristic
303C8—341no—
304C8—341no—
305C8—341no—
306C8—341no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—681no—
311C22—681no—
312C22—681no—
313C22C22×C22⋊D1711yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—681no—
324C4—681no—
325C4—681no—
326C4#1058769611yescharacteristic
327C4#1058769711yescharacteristic
328C2—41no—
329C2—1361no—
330C2#1176338711yescharacteristic
331C2#1176361711yescharacteristic
332C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#541C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#541D17
hypocenterC17#3258
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #11763113C2semidirect
3. #11763106C2semidirect
4. #11763104C2semidirect
5. #11763094C2semidirect
6. #11763065C2semidirect
8. #11763048C2semidirect
10. #11763031C2semidirect
11. #11763031C2semidirect
12. #11763016C2semidirect
13. #11763011C2semidirect
14. #11762836C2semidirect
15. C17×#541C2semidirect
163. #541D17semidirect
225. C17#3258semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117631132
#117631062
#117631042
#117630942
#117630652
#117630572
#117630482
#117630402
#117630312
#117630162
#117630112
#117628362
C17×#5412
#325817