Finite Group Databasebeta

← #17949765

Subgroups — #17949765

Order 4352.

482 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544712C211yesmaximal, characteristic, complemented
2#14544700C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543601C212yesmaximal
5#14531242C211yesmaximal, characteristic
6C2×#10587544—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587579C411yescharacteristic
13C2×#10587578—21no—
14#11762711C411yescharacteristic
15#11762710C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762701—21no—
18#11762699—21no—
19C4⋊C4×Dic17C411yescharacteristic
20#11762656—21no—
21C42×Dic17C2211yescharacteristic
22C17×#389C411yescharacteristic
23C22×D68C2×C411yescharacteristic
24C22×C4×D17—21no—
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×#63227—21no—
32C2×#63227—21no—
33C2×#63227—21no—
34C2×#63227—21no—
35C2×#63227—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587579—22no—
46#10587579C814yescomplemented
47#10587578—44no—
48#10587544—42no—
49C17×#95—21no—
50#10587502—21no—
51#10587502—21no—
52C23×D17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—21no—
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C2×D68—21no—
64C2×D68—41no—
65C2×D68—41no—
66C2×D68C2×C812yes—
67C2×C4×D17—24no—
68C2×C4×D17—42no—
69C2×C4×D17—42no—
70C2×#3624—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C4⋊C4×C17—21no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17C2×C812yes—
80C42×C17—41no—
81#63227—24no—
82#63227—42no—
83#63227—42no—
84#63227—42no—
85#63227—41no—
86#63227—41no—
87C4⋊Dic17—42no—
88C4⋊Dic17—41no—
89C4×Dic17—21no—
90C4×Dic17—22no—
91C4×Dic17—21no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94C4×Dic17—41no—
95C4×Dic17—42no—
96C4×Dic17C2×C812yes—
97C4×Dic17M1612yes—
98#7594—171nomaximal, Sylow
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—22no—
101C22×D17—41no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17—41no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17C22×C811yescharacteristic
122C2×C4×C17C2×M1611yescharacteristic
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17C22⋊C812yes—
135C2×Dic17C22⋊C812yes—
136C2×Dic17AGL(1,ℤ/8)11yescharacteristic
137C2×Dic17AGL(1,ℤ/8)12yes—
138C2×Dic17#13711yescharacteristic
139D68—42no—
140D68—42no—
141C4×D17—48no—
142#3624—82no—
143#3624—82no—
144#2293—171no—
145#1839—171no—
146#1831—171no—
147#1271—172no—
148#1270—172no—
149C22×C17C2×C22⋊C811yescharacteristic
150C22×C17#41011yescharacteristic
151C22×C17#41311yescharacteristic
152C22×C17#41711yescharacteristic
153C22×C17#41811yescharacteristic
154C22×C17#42111yescharacteristic
155C22×C17#42111yescharacteristic
156C2×D17—42no—
157C2×D17—42no—
158C2×D17—42no—
159C2×D17—81no—
160C2×D17—81no—
161C2×D17—81no—
162C2×D17—81no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—41no—
166C4×C17—42no—
167C4×C17—41no—
168C4×C17—81no—
169Dic17—22no—
170Dic17—22no—
171Dic17—21no—
172Dic17—21no—
173Dic17—22no—
174Dic17—81no—
175C2×C4×D4—171no—
176C2×#117—341no—
177C2×C4⋊C8—341no—
178#390—171no—
179#390—171no—
180#389—171no—
181#389#59511yescharacteristic
182#388—341no—
183#388—341no—
184C4×C4⋊C4—171no—
185C4×C22⋊C4—341no—
186C43—171no—
187#336—172no—
188#336—172no—
189#336—341no—
190#336—341no—
191#336—341no—
192C2×C17#145811yescharacteristic
193C2×C17#147711yescharacteristic
194C2×C17#147811yescharacteristic
195C2×C17#151211yescharacteristic
196C2×C17#151411yescharacteristic
197C2×C17#153511yescharacteristic
198C2×C17#155711yescharacteristic
199D17—82no—
200C22×D4—171no—
201C23×C4—341no—
202C22×C8—342no—
203C22×C8—342no—
204C4×D4—174no—
205C4×D4—342no—
206#117—684no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4C2×#59511yescharacteristic
211C2×C4⋊C4C2×#59511yescharacteristic
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4—341no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C42—171no—
218C2×C42—171no—
219C2×C42—171no—
220C2×C42—171no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—341no—
224C2×C42C2×#59511yescharacteristic
225C4⋊C8—682no—
226#95—21no—
227#95—341no—
228#95—341no—
229C17#759411yescharacteristic, complemented, Sylow
230C24—341no—
231C2×D4—172no—
232C2×D4—341no—
233C2×D4—681no—
234C2×D4—681no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—344no—
242C22×C4—341no—
243C22×C4—682no—
244C22×C4—682no—
245C22×C4C22⋊#59511yescharacteristic
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22×#59511yescharacteristic
248C2×C8—682no—
249C2×C8—682no—
250C2×C8—682no—
251C2×C8—682no—
252C2×C8—682no—
253C2×C8—682no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—681no—
258C4⋊C4—682no—
259C4⋊C4#6324212yes—
260C22⋊C4—344no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C22⋊C4—681no—
265C22⋊C4—681no—
266C42—41no—
267C42—172no—
268C42—172no—
269C42—341no—
270C42—342no—
271C42—341no—
272C42—681no—
273C42—681no—
274C42—681no—
275C42—682no—
276C23—342no—
277C23—681no—
278C23—681no—
279C23—681no—
280C23—681no—
281C23—681no—
282C23—681no—
283C23C2×C22⋊#59511yescharacteristic
284D4—682no—
285D4—682no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—172no—
301C2×C4—172no—
302C2×C4—171no—
303C2×C4—171no—
304C2×C4—172no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—688no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4C2×#6324211yescharacteristic
318C2×C4D17⋊M1611yescharacteristic
319C8—1362no—
320C8—1362no—
321C22—682no—
322C22—682no—
323C22—682no—
324C22—1361no—
325C22—1361no—
326C22—1361no—
327C22—1361no—
328C22#1176377711yescharacteristic
329C22#1176379011yescharacteristic
330C22#1176379611yescharacteristic
331C22#1176380011yescharacteristic
332C22#1176380111yescharacteristic
333C22#1176380611yescharacteristic
334C22#1176380811yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—42no—
340C4—81no—
341C4—341no—
342C4—341no—
343C4—342no—
344C4—342no—
345C4—342no—
346C4—1361no—
347C2—1362no—
348C2#1454401811yescharacteristic
349C2#1454408211yescharacteristic
350C2#1454408411yescharacteristic
351C2#1454419811yescharacteristic
352C2#1454420511yescharacteristic
353C2#1454427111yescharacteristic
354C2#1454431611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7594
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #14544712C2semidirect
2. #14544700C2semidirect
3. #14543602C2semidirect
46. #10587579C8semidirect
229. C17#7594semidirect

Maximal subgroups

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

subgroupindex
#145447122
#145447002
#145436022
#145436012
#145312422
#759417