Finite Group Databasebeta

← #17903965

Subgroups — #17903965

Order 4352.

1770 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542189C212yesmaximal
2#14539322C218yesmaximal, complemented
3#14539314C218yesmaximal, complemented
4#14539307C214yesmaximal, complemented
5#14539296C214yesmaximal, complemented
6#14537374C211yesmaximal, characteristic
7C2×#11762845C212yesmaximal
8C2×#11762767C211yesmaximal, characteristic
9C17×#2911C211yesmaximal, characteristic
10C23×#3622C2211yescharacteristic
11#11763440C2218yes—
12#11763437—24no—
13#11763437C2218yes—
14#11763435—24no—
15#11763366—24no—
16#11763362—24no—
17#11763362C2218yes—
18C2×#10587612C2212yes—
19C2×#10587612C2211yescharacteristic
20#11763217C22116yescomplemented
21#11763216C22116yescomplemented
22#11763213C2218yescomplemented
23#11763210C2218yescomplemented
24#11763209C2218yescomplemented
25#11763206C2218yescomplemented
26#11762851C2218yes—
27#11762847C2214yes—
28#11762845C2214yes—
29#11762845C2214yes—
30#11762845C2214yes—
31#11762845C2214yes—
32C2×#10587478C2214yes—
33C2×#10587479C2214yes—
34#11762767C2214yes—
35#11762767C2214yes—
36C2×#10587486C2212yes—
37C2×C8×Dic17C2212yes—
38C22×M16×C17C2211yescharacteristic
39C2×C17×#117C2212yes—
40C17×#428C2218yes—
41C17×#427C2214yes—
42C23×C4×C17C2311yescharacteristic
43C2×M16×C17Q818yescomplemented
44C2×M16×C17C2314yes—
45C22×C8×C17C2312yes—
46C17×#117—22no—
47C17×#117—22no—
48C17×#117C2314yes—
49C17×#117C2314yes—
50C2×C4⋊C4×C17C2314yes—
51C2×C22⋊C4×C17—22no—
52C2×C42×C17—22no—
53C2×C22⋊Dic17—22no—
54C2×C22⋊Dic17D412yes—
55#10587612—22no—
56#10587612—22no—
57#10587612D414yes—
58#10587612C2314yes—
59#10587612C2314yes—
60#10587612C2314yes—
61C2×C4⋊Dic17D412yes—
62C2×C4⋊Dic17C2314yes—
63#10587610—216no—
64C2×C4×Dic17—22no—
65C2×C4×Dic17C2312yes—
66C22×#3622Q818yescomplemented
67C22×#3622C2311yescharacteristic
68C22×#3622C2311yescharacteristic
69C22×#3622C2312yes—
70C22×#3622C2312yes—
71#10587576—28no—
72#10587576C23116yes—
73#10587574—28no—
74#10587574C23116yes—
75#10587564—216no—
76#10587564—216no—
77#10587556—28no—
78#10587555—28no—
79C4×Dic34—216no—
80C17×#107C2318yes—
81C17×#106C2318yes—
82#10587489C2318yes—
83#10587488C2318yes—
84#10587486C2318yes—
85C8×Dic17C2314yes—
86C8×Dic17C2314yes—
87#10587479C2318yes—
88#10587479C2318yes—
89#10587478C2318yes—
90#10587478C2318yes—
91C24×C17C2×D411yescharacteristic
92C22×C4×C17—24no—
93C22×C4×C17C2×D412yes—
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×Q814yes—
96C22×C4×C17C2×Q814yes—
97C22×C4×C17C2411yescharacteristic
98C22×C4×C17C2412yes—
99C22×Dic17—24no—
100C22×Dic17C2×D414yes—
101C2×Dic34—216no—
102M16×C17C2×Q8116yes—
103C2×C8×C17C2×Q818yes—
104C2×C8×C17C2414yes—
105C4⋊C4×C17—24no—
106C4⋊C4×C17C2418yes—
107C22⋊C4×C17—24no—
108C22⋊C4×C17—24no—
109C42×C17—24no—
110C42×C17—24no—
111C22⋊Dic17—24no—
112C22⋊Dic17—24no—
113C22⋊Dic17C2×D418yes—
114C4⋊Dic17—24no—
115C4⋊Dic17—216no—
116C4⋊Dic17C2×D418yes—
117C4⋊Dic17C2418yes—
118#63225—216no—
119#63225—216no—
120C4×Dic17—24no—
121C4×Dic17—24no—
122C4×Dic17C2×D414yes—
123C4×Dic17C2414yes—
124C2×#3622C2×Q814yes—
125C2×#3622C2×Q814yes—
126C2×#3622C2×Q818yes—
127C2×#3622C2×Q818yes—
128C2×#3622C2411yescharacteristic
129C2×#3622C2412yes—
130C2×#3622C2411yescharacteristic
131#60660—171nomaximal, Sylow
132C23×C17—42no—
133C23×C17C4⋊Q814yes—
134C23×C17C22×D411yescharacteristic
135C23×C17C22×D412yes—
136C2×C4×C17—28no—
137C2×C4×C17—84no—
138C2×C4×C17C4⋊Q818yes—
139C2×C4×C17C4⋊Q814yes—
140C2×C4×C17C22×D412yes—
141C2×C4×C17C22×D411yescharacteristic
142C2×C4×C17C22×Q812yes—
143C2×C4×C17C22×Q814yes—
144C2×C4×C17C22×Q812yes—
145C2×C4×C17C22×Q814yes—
146C2×C4×C17C2511yescharacteristic
147C2×Dic17—28no—
148C2×Dic17—44no—
149C2×Dic17—84no—
150C2×Dic17C22×D418yes—
151Dic34—816no—
152C8×C17C22×Q818yes—
153#3622C22×Q812yes—
154#3622C22×Q814yes—
155#3622C22×Q812yes—
156#3458—172no—
157#3164—178no—
158#3163—178no—
159#3162—174no—
160#3160—174no—
161#2911—171no—
162#2911D1711yescharacteristic
163C2×#427—172no—
164C2×#405—171no—
165C22×C17—22no—
166C22×C17—24no—
167C22×C17—44no—
168C22×C17C2×C4⋊Q812yes—
169C22×C17C2×C4⋊Q814yes—
170C22×C17C23×D411yescharacteristic
171C4×C17—48no—
172C4×C17C2×C4⋊Q814yes—
173C4×C17C2×C4⋊Q812yes—
174C4×C17C23×Q811yescharacteristic
175C4×C17C23×Q811yescharacteristic
176Dic17—28no—
177Dic17—48no—
178C22×M16C2×D1711yescharacteristic
179C23×C8—171no—
180#544—178no—
181#544—178no—
182#544—344no—
183#543—344no—
184#533—178no—
185#533—344no—
186#533—344no—
187C2×#117—171no—
188C2×#117—172no—
189C2×#117C2×D1712yes—
190#501—1716no—
191#501—1716no—
192#500—178no—
193#499—178no—
194#499—178no—
195#498—178no—
196#428—178no—
197#428C2×D1718yes—
198#427—174no—
199#427—174no—
200#427—174no—
201#427—174no—
202#427—174no—
203#427C2×D1714yes—
204C2×#107—174no—
205C2×#106—174no—
206#405—174no—
207#405—174no—
208C2×#97—172no—
209C2×C4×C8—172no—
210C2×C17—22no—
211C2×C17—24no—
212C2×C17C22×C4⋊Q811yescharacteristic
213C2×C17#358412yes—
214C23×C4C22×D1711yescharacteristic
215C2×M16Dic3418yescomplemented
216C2×M16C22×D1714yes—
217C22×C8—178no—
218C22×C8—171no—
219C22×C8—171no—
220C22×C8—172no—
221C22×C8—172no—
222C22×C8C22×D1712yes—
223C4⋊Q8—1716no—
224C4⋊Q8—348no—
225C4⋊Q8—348no—
226#125—1716no—
227#125—348no—
228#125—348no—
229C22⋊Q8—3416no—
230C22⋊Q8—3416no—
231C22⋊Q8—3416no—
232C4×Q8—3416no—
233#117—22no—
234#117—22no—
235#117—174no—
236#117—174no—
237#117—174no—
238#117—174no—
239#117—342no—
240#117—342no—
241#117C22×D1714yes—
242#117C22×D1714yes—
243C2×C4⋊C4—172no—
244C2×C4⋊C4—174no—
245C2×C4⋊C4C22×D1714yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—172no—
248C2×C22⋊C4—342no—
249C2×C42—22no—
250C2×C42—172no—
251C2×C42—342no—
252#107—178no—
253#107—178no—
254#107—178no—
255#107C22×D1718yes—
256#106—178no—
257#106—178no—
258#106—178no—
259#106C22×D1718yes—
260#97—178no—
261C4×C8—174no—
262C4×C8—174no—
263C17#6066011yescharacteristic, complemented, Sylow
264C24D4×D1711yescharacteristic
265C2×Q8—3416no—
266C22×C4—24no—
267C22×C4—174no—
268C22×C4—344no—
269C22×C4C2×Dic3414yes—
270C22×C4D4×D1711yescharacteristic
271C22×C4D4×D1712yes—
272C22×C4Q8×D1714yes—
273C22×C4C23×D1712yes—
274C22×C4C23×D1711yescharacteristic
275M16C2×Dic34116yes—
276C2×C8—174no—
277C2×C8—174no—
278C2×C8—178no—
279C2×C8—178no—
280C2×C8—171no—
281C2×C8—171no—
282C2×C8—172no—
283C2×C8C2×Dic3418yes—
284C2×C8C23×D1714yes—
285C4⋊C4—24no—
286C4⋊C4—178no—
287C4⋊C4—178no—
288C4⋊C4—3416no—
289C4⋊C4—3416no—
290C4⋊C4—3416no—
291C4⋊C4—344no—
292C4⋊C4C23×D1718yes—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—178no—
296C22⋊C4—344no—
297C22⋊C4—344no—
298C42—24no—
299C42—24no—
300C42—174no—
301C42—174no—
302C42—344no—
303C42—344no—
304C23—42no—
305C23#1058757414yes—
306C23C2×D4×D1711yescharacteristic
307C23C2×D4×D1712yes—
308Q8—13616no—
309C2×C4—28no—
310C2×C4—84no—
311C2×C4—178no—
312C2×C4—348no—
313C2×C4—684no—
314C2×C4—1364no—
315C2×C4#1058757414yes—
316C2×C4#1058757418yes—
317C2×C4C22×Dic3412yes—
318C2×C4C22×Dic3414yes—
319C2×C4C2×D4×D1712yes—
320C2×C4C2×D4×D1711yescharacteristic
321C2×C4C2×Q8×D1714yes—
322C2×C4C2×Q8×D1712yes—
323C2×C4C24×D1711yescharacteristic
324C8—172no—
325C8—172no—
326C8—174no—
327C8C22×Dic3418yes—
328C22—24no—
329C22—22no—
330C22—44no—
331C22C2×#1058757412yes—
332C22C2×#1058757414yes—
333C22C22×D4×D1711yescharacteristic
334C4—48no—
335C4—348no—
336C4—688no—
337C4C2×#1058757414yes—
338C4C2×#1058757412yes—
339C4C22×Q8×D1711yescharacteristic
340C4C23×Dic3411yescharacteristic
341C2—24no—
342C2—22no—
343C2C22×#1058757411yescharacteristic
344C2#1454355712yes—
subgroupG/H
centerC2×C4C2×D4×D17
commutator, cosocleC2×C4×C17C25
FrattiniC2×C4C24×D17
FittingC17×#2911C2
socleC22×C17C23×D4
prime radicalC24×C17C2×D4
hypercenter#2911D17
hypocenterC17#60660
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
2. #14539322C2semidirect
3. #14539314C2semidirect
4. #14539307C2semidirect
5. #14539296C2semidirect
20. #11763217C22semidirect
21. #11763216C22semidirect
22. #11763213C22semidirect
23. #11763210C22semidirect
24. #11763209C22semidirect
25. #11763206C22semidirect
43. C2×M16×C17Q8semidirect
66. C22×#3622Q8semidirect
215. C2×M16Dic34semidirect
263. C17#60660semidirect

Maximal subgroups

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

subgroupindex
#145421892
#145393222
#145393142
#145393072
#145392962
#145373742
C2×#117628452
C2×#117627672
C17×#29112
#6066017