Finite Group Databasebeta

← #17979397

Subgroups — #17979397

Order 4352.

1800 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545330C213yesmaximal
2#14545319C216yesmaximal
3#14545318C216yesmaximal
4#14545284C216yesmaximal
5#14545278C216yesmaximal
6#125×#595C213yesmaximal
7D17×#564C211yesmaximal, characteristic
8#11763760C2213yes—
9#11763760C2213yes—
10#11763760C2216yes—
11#11763760C2216yes—
12#11763760C2216yes—
13#11763760C2216yes—
14#11763758C2216yes—
15#11763758C2216yes—
16#11763758C2216yes—
17#11763758C2216yes—
18#11763758C2216yes—
19#11763758C2216yes—
20#11763758C2216yes—
21#11763758C2216yes—
22#11763758C2216yes—
23#11763758C2216yes—
24#11763755C2212yes—
25#11763754C2212yes—
26#11763753C2213yes—
27C4⋊C4×#595C2216yes—
28C4⋊C4×#595C2213yes—
29C4⋊C4×#595C2216yes—
30C4⋊C4×#595C2213yes—
31C4⋊C4×#595C2216yes—
32C4⋊C4×#595C2216yes—
33C4×C4⋊#595C2216yes—
34#11763745C2216yes—
35C42×#595C2211yescharacteristic
36C17⋊#564C413yescomplemented
37C17⋊#564C416yescomplemented
38C17⋊#564C416yescomplemented
39#125×D17C2213yes—
40#125×D17C2216yes—
41#125×D17C2216yes—
42C17×#564C411yescharacteristic, complemented
43C2×C4⋊#595C2316yes—
44C2×C4⋊#595C2316yes—
45C2×C4⋊#595C2313yes—
46C2×C4⋊#595C2313yes—
47C2×C4⋊#595C2316yes—
48C2×C4⋊#595C2316yes—
49C2×C4×#595C2313yes—
50C2×C4×#595C2316yes—
51C2×C4×#595C2313yes—
52C2×C4×#595C2316yes—
53C2×C4×#595C2316yes—
54C2×C4×#595C2316yes—
55C17×#125C2×C413yes—
56C17×#125C2×C416yes—
57C17×#125C2×C416yes—
58C4⋊C4×D17C2316yes—
59C4⋊C4×D17C2313yes—
60C4⋊C4×D17C2316yes—
61C4⋊C4×D17C2313yes—
62C4⋊C4×D17C2316yes—
63C4⋊C4×D17C2316yes—
64#10587576C2×C413yes—
65#10587576C2×C413yes—
66#10587576C2×C416yes—
67#10587576C2×C416yes—
68#10587576C2×C416yes—
69#10587576C2×C416yes—
70#10587575C2×C416yes—
71#10587575C2×C416yes—
72#10587575C2×C416yes—
73#10587575C2×C416yes—
74#10587575C2×C416yes—
75#10587575C2×C416yes—
76#10587575C2×C416yes—
77#10587575C2×C416yes—
78#10587575C2×C416yes—
79#10587575C2×C416yes—
80C42×D17C2311yescharacteristic
81C42×D17C2312yes—
82C42×D17C2312yes—
83#10587556C2×C413yes—
84#10587556C2×C416yes—
85#10587556C2×C416yes—
86(C2×#595)⋊C4C2316yes—
87(C2×#595)⋊C4C2316yes—
88(C2×#595)⋊C4C2316yes—
89(C2×#595)⋊C4C2316yes—
90(C2×#595)⋊C4C2316yes—
91(C2×#595)⋊C4C2316yes—
92(C2×#595)⋊C4C2316yes—
93(C2×#595)⋊C4C2316yes—
94(C2×#595)⋊C4C2316yes—
95(C2×#595)⋊C4C2316yes—
96C22×#595C2416yes—
97C22×#595C2411yescharacteristic
98C22×#595C2416yes—
99C22×#595C2413yes—
100C2×C4×D17C2413yes—
101C2×C4×D17C2416yes—
102C2×C4×D17C2416yes—
103C4⋊#595—212no—
104C4⋊#595—46no—
105C4⋊#595—43no—
106C4⋊#595—46no—
107C4⋊#595—46no—
108C4⋊#595—43no—
109C4×#595—26no—
110C4×#595—212no—
111C4×#595—46no—
112C4×#595—46no—
113C4×#595—43no—
114C4×#595—46no—
115C4⋊C4×C17C22×C413yes—
116C4⋊C4×C17C22×C416yes—
117C4⋊C4×C17C22×C416yes—
118C4⋊C4×C17C22×C413yes—
119C4⋊C4×C17C22×C416yes—
120C4⋊C4×C17C22×C416yes—
121C42×C17C22×C411yescharacteristic
122C42×C17C22×C412yes—
123C42×C17C22×C412yes—
124C4⋊Dic17C22×C413yes—
125C4⋊Dic17C22×C413yes—
126C4⋊Dic17C22×C416yes—
127C4⋊Dic17C22×C416yes—
128C4⋊Dic17C22×C416yes—
129C4⋊Dic17C22×C416yes—
130#63225C22×C416yes—
131#63225C22×C416yes—
132#63225C22×C416yes—
133#63225C22×C416yes—
134#63225C22×C416yes—
135#63225C22×C416yes—
136#63225C22×C416yes—
137#63225C22×C416yes—
138#63225C22×C416yes—
139#63225C22×C416yes—
140C4×Dic17C22×C413yes—
141C4×Dic17C22×C416yes—
142C4×Dic17C22×C416yes—
143#34530—171nomaximal, Sylow
144C22×D17C2511yescharacteristic
145C2×#595—23no—
146C2×#595—23no—
147C2×#595—26no—
148C2×#595—26no—
149C2×#595—26no—
150C2×#595—26no—
151C2×#595—26no—
152C2×#595—23no—
153C2×#595—26no—
154C2×#5952−1+416yes—
155C2×C4×C17C23×C413yes—
156C2×C4×C17C23×C416yes—
157C2×C4×C17C23×C416yes—
158C2×Dic17C23×C413yes—
159C2×Dic17C23×C416yes—
160C2×Dic17C23×C416yes—
161C4×D17—23no—
162C4×D17—26no—
163C4×D17—26no—
164C4×D17—26no—
165C4×D17—23no—
166C4×D17—26no—
167C2×#564—171no—
168#2382—173no—
169#2371—176no—
170#2370—176no—
171#2336—176no—
172#2330—176no—
173C4×#125—173no—
174C22×C17C24×C411yescharacteristic
175C2×D17C2×2−1+413yes—
176C2×D17#58513yes—
177#595—26no—
178#595—46no—
179#595—46no—
180#595—41no—
181C4×C17—23no—
182C4×C17—26no—
183C4×C17—26no—
184Dic17—23no—
185Dic17—26no—
186Dic17—26no—
187#564—176no—
188#564—176no—
189#564—173no—
190#564#59511yescharacteristic, complemented
191C2×#125—173no—
192C2×#125—176no—
193C2×#125—176no—
194#389—173no—
195#389—173no—
196#389—176no—
197#389—176no—
198#389—176no—
199#389—176no—
200#387—176no—
201#387—176no—
202#387—176no—
203#387—176no—
204#387—176no—
205#387—176no—
206#387—176no—
207#387—176no—
208#387—176no—
209#387—176no—
210#384—172no—
211#383—172no—
212#382—173no—
213C4×C4⋊C4—176no—
214C4×C4⋊C4—173no—
215C4×C4⋊C4—176no—
216C4×C4⋊C4—173no—
217C4×C4⋊C4—176no—
218C4×C4⋊C4—176no—
219C4×C4⋊C4—176no—
220#376—176no—
221C43—171no—
222C2×C17C4×2−1+413yes—
223D17#356313yes—
224D17#356711yescharacteristic
225#125—176no—
226#125—176no—
227#125—176no—
228#125—176no—
229#125—176no—
230#125—176no—
231#125—176no—
232#125—176no—
233#125—176no—
234#125—176no—
235#125—173no—
236#125—173no—
237#125—176no—
238#125—176no—
239#125—176no—
240#125—176no—
241#125—173no—
242#125—176no—
243#125—176no—
244#125C2×#59513yes—
245#125C2×#59516yes—
246#125C2×#59516yes—
247C2×C4⋊C4—173no—
248C2×C4⋊C4—176no—
249C2×C4⋊C4—176no—
250C2×C4⋊C4—173no—
251C2×C4⋊C4—176no—
252C2×C4⋊C4—176no—
253C2×C4⋊C4—176no—
254C2×C4⋊C4—173no—
255C2×C4⋊C4—173no—
256C2×C4⋊C4—176no—
257C2×C4⋊C4—176no—
258C2×C4⋊C4—176no—
259C2×C42—171no—
260C2×C42—172no—
261C2×C42—172no—
262C2×C42—173no—
263C2×C42—176no—
264C2×C42—173no—
265C2×C42—176no—
266C2×C42—176no—
267C2×C42—176no—
268#95—176no—
269#95—176no—
270#95—176no—
271#95—176no—
272#95—176no—
273#95—176no—
274#95—176no—
275#95—176no—
276#95—176no—
277#95—176no—
278C17#3453011yescharacteristic, complemented, Sylow
279C22×C4—173no—
280C22×C4—176no—
281C22×C4—173no—
282C22×C4—176no—
283C22×C4—176no—
284C22×C4—176no—
285C22×C4—171no—
286C4⋊C4—176no—
287C4⋊C4—176no—
288C4⋊C4—176no—
289C4⋊C4—176no—
290C4⋊C4—176no—
291C4⋊C4—176no—
292C4⋊C4—176no—
293C4⋊C4—176no—
294C4⋊C4—176no—
295C4⋊C4—176no—
296C4⋊C4—173no—
297C4⋊C4—176no—
298C4⋊C4—176no—
299C4⋊C4—173no—
300C4⋊C4—176no—
301C4⋊C4—176no—
302C4⋊C4—3412no—
303C4⋊C4—686no—
304C4⋊C4—683no—
305C4⋊C4—686no—
306C4⋊C4—686no—
307C4⋊C4—683no—
308C4⋊C4C22×#59513yes—
309C4⋊C4C22×#59516yes—
310C4⋊C4C22×#59516yes—
311C4⋊C4C22×#59513yes—
312C4⋊C4C22×#59516yes—
313C4⋊C4C22×#59516yes—
314C42—173no—
315C42—176no—
316C42—176no—
317C42—346no—
318C42—3412no—
319C42—686no—
320C42—686no—
321C42—683no—
322C42—686no—
323C42C22×#59511yescharacteristic
324C42C22×#59512yes—
325C42C22×#59512yes—
326C23—171no—
327C2×C4—176no—
328C2×C4—173no—
329C2×C4—176no—
330C2×C4—176no—
331C2×C4—343no—
332C2×C4—346no—
333C2×C4—346no—
334C2×C4—343no—
335C2×C4—346no—
336C2×C4—346no—
337C2×C4—343no—
338C2×C4—343no—
339C2×C4—346no—
340C2×C4—346no—
341C2×C4—346no—
342C2×C4—346no—
343C2×C4—343no—
344C2×C4—346no—
345C2×C4—346no—
346C2×C4C23×#59513yes—
347C2×C4C23×#59516yes—
348C2×C4C23×#59516yes—
349C22—173no—
350C22—173no—
351C22C24×#59511yescharacteristic
352C4—23no—
353C4—26no—
354C4—26no—
355C4—346no—
356C4—343no—
357C4—346no—
358C4—346no—
359C4—686no—
360C4—686no—
361C4—681no—
362C2—173no—
363C2—171no—
364C22−1+4×#59513yes—
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#564C4
prime radical, cosocleC22×D17C25
hypercenter#564#595
hypocenterC17#34530
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
36. C17⋊#564C4semidirect
37. C17⋊#564C4semidirect
38. C17⋊#564C4semidirect
42. C17×#564C4semidirect
190. #564#595semidirect
278. C17#34530semidirect

Maximal subgroups

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

subgroupindex
#145453302
#145453192
#145453182
#145452842
#145452782
#125×#5952
D17×#5642
#3453017