Finite Group Databasebeta

← #519×#64408 #34030529

Subgroups — #519×#64408 #34030529

Order 18688.

3402 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
1C2×#17989655C211yesmaximal, characteristic
2C22×Q8×#64408C211yesmaximal, characteristic
3#22626371C218yesmaximal, complemented
4#22626368C216yesmaximal
5C4×Q8×#64408C218yesmaximal, complemented
6#117×#64408C216yesmaximal
7#519×D73C211yesmaximal, characteristic
8#17989655C4116yescomplemented
9C2×Q8×#64408C4116yescomplemented
10C2×Q8×#64408C2214yes—
11C2×Q8×#64408C2214yes—
12C2×Q8×#64408C2218yes—
13C2×#14556670C2216yes—
14C22×C4⋊#64408C2213yes—
15C22×C4×#64408C2213yes—
16C4⋊C4×#64408C22112yes—
17C4⋊C4×#64408C22112yes—
18C4⋊C4×#64408C22124yes—
19C4×C4⋊#64408C22124yes—
20C22⋊C4×#64408C2214yes—
21C22⋊C4×#64408C22112yes—
22#17989523C22112yes—
23C42×#64408C22112yes—
24C22×Q8×D73C2211yescharacteristic
25#17989065C411yescharacteristic, complemented
26#17989064C416yescomplemented
27#17989063C418yescomplemented
28C4×Q8×D73C2218yes—
29#117×D73C2216yes—
30#519×C73C411yescharacteristic, complemented
31C23×#64408C2311yescharacteristic
32C23×#64408C2313yes—
33Q8×#64408C2×C4132yes—
34Q8×#64408C2×C4132yes—
35Q8×#64408C2×C4164yescomplemented
36#14556670C2×C4148yes—
37C2×C4⋊#64408C2×C4124yes—
38C2×C4⋊#64408C23112yes—
39C2×C4⋊#64408C23112yes—
40C2×C22⋊#64408C2314yes—
41C2×C4×#64408C2×C4124yes—
42C2×C4×#64408C2318yes—
43C2×C4×#64408C23124yes—
44C2×C4×#64408C2316yes—
45C2×C4×#64408C2316yes—
46C2×C4×#64408C23112yes—
47(C2×#64408)⋊C4C2318yes—
48(C2×#64408)⋊C4C23124yes—
49C22×Dic146C2×C413yes—
50C2×Q8×D73C2×C418yes—
51C2×Q8×D73C2×C418yes—
52C2×Q8×D73C2×C414yes—
53C2×Q8×D73C2314yes—
54C22×C4×D73C2313yes—
55Q8×Dic73C2×C418yescomplemented
56C4×Dic146C2×C4124yescomplemented
57#14556556C2×C4124yescomplemented
58#14556550C2×C416yes—
59#14556549C2×C4112yes—
60#14556548C2×C4112yes—
61#14556547C2×C4112yes—
62C4⋊C4×D73C23112yes—
63C22⋊C4×D73C2314yes—
64C42×D73C23112yes—
65C22×Q8×C73C2×C411yescharacteristic
66C4×Q8×C73C2×C418yescomplemented
67#117×C73C2×C416yes—
68C22×#64408—24no—
69C22×#64408—46no—
70C22×#64408C22×C414yes—
71C22×#64408C22×C4112yes—
72C22×#64408C2411yescharacteristic
73C22×#64408C2416yes—
74C22×#64408C2418yes—
75C22×#64408C2411yescharacteristic
76C4⋊#64408C22×C4148yes—
77C4⋊#64408C22×C4148yes—
78C22⋊#64408C22×C4116yes—
79C4×#64408—216no—
80C4×#64408—424no—
81C4×#64408C22×C4124yes—
82C4×#64408C22×C4124yes—
83C4×#64408C22×C4148yes—
84C23×D73C2411yescharacteristic
85C2×Dic146C42124yescomplemented
86C2×Dic146C22×C4112yes—
87Q8×D73C42132yescomplemented
88Q8×D73C22×C4116yes—
89Q8×D73C22×C4116yes—
90C22×Dic73C22×C413yes—
91C2×C4×D73C22×C416yes—
92C2×C4×D73C22×C4112yes—
93C2×C4×D73C22×C4112yes—
94C2×C4×D73C2418yes—
95C2×C4×D73C2416yes—
96C4⋊Dic73C22×C4112yes—
97#11924007C22×C4124yes—
98C22⋊Dic73C22×C414yes—
99#11924005C22×C418yes—
100C4×Dic73C22×C4124yes—
101C4×Dic73C22×C4112yes—
102C2×Q8×C73C4218yescomplemented
103C2×Q8×C73C22×C414yes—
104C22×C4×C73C22×C413yes—
105C4⋊C4×C73C22×C4112yes—
106C22⋊C4×C73C22×C414yes—
107C42×C73C22×C4112yes—
108C23×C73C23×C411yescharacteristic
109C22×D73—22no—
110C22×D73—22no—
111C22×D73C23×C412yes—
112C22×D73C23×C412yes—
113C22×D73C23×C411yescharacteristic
114C22×D73C2511yescharacteristic
115C2×#64408—28no—
116C2×#64408—22no—
117C2×#64408—28no—
118C2×#64408—212no—
119C2×#64408—28no—
120C2×#64408—412no—
121C2×#64408C23×C412yes—
122C2×#64408C23×C4116yes—
123C2×#64408C23×C4112yes—
124C2×#64408C23×C412yes—
125C2×#644082−1+414yes—
126Q8×C73C2×C42116yes—
127C2×C4×C73C2×C42112yes—
128C2×C4×C73C23×C418yes—
129C2×C4×C73C23×C416yes—
130C2×Dic73C2×C42112yes—
131C2×Dic73C23×C418yes—
132C2×Dic73C23×C416yes—
133C4×D73—28no—
134C4×D73—28no—
135C4×D73C2×C42124yes—
136C4×D73C23×C4112yes—
137C4×D73C23×C4112yes—
138Dic146C2×C42148yes—
139C22×C73—22no—
140C22×C73C22×C4212yes—
141C22×C73C24×C411yescharacteristic
142C2×D73—22no—
143C2×D73—22no—
144C2×D73—22no—
145C2×D73—24no—
146C2×D73C22×C4212yes—
147C2×D73C24×C411yescharacteristic
148C2×D73C24×C411yescharacteristic
149C2×D73C2×2−1+412yes—
150C2×D73#58512yes—
151#64408—22no—
152#64408—212no—
153#64408—216no—
154#64408#51914yesdirect product factor
155C4×C73—28no—
156C4×C73C22×C42112yes—
157Dic73—28no—
158Dic73C22×C42112yes—
159C4×#519—731nomaximal, Sylow
160C2×C73—22no—
161C2×C73C23×C4211yescharacteristic
162C2×C73C4×2−1+412yes—
163D73—22no—
164D73C2×#51911yescharacteristic
165D73#342811yescharacteristic
166D73C4×2−1+412yes—
167C2×#519—731no—
168C2×#519—731no—
169C22×C4×Q8—731no—
170#2276—738no—
171#2274—736no—
172C42×Q8—738no—
173C4×#117—736no—
174C73C4×#51911yescharacteristic, complemented, Sylow
175C23×Q8—731no—
176#519—7316no—
177#519—731no—
178#519—738no—
179#519—736no—
180#519#6440811yescharacteristic, direct product factor
181C2×C4×Q8—7316no—
182C2×C4×Q8—734no—
183C2×C4×Q8—734no—
184C2×C4×Q8—738no—
185C2×C4×Q8—738no—
186C2×#117—736no—
187C2×#117—736no—
188C22×C4⋊C4—733no—
189C22×C42—733no—
190C4×C4⋊C4—7324no—
191C4×C4⋊C4—7312no—
192C4×C4⋊C4—7312no—
193C4×C4⋊C4—7324no—
194C4×C22⋊C4—734no—
195C4×C22⋊C4—7312no—
196#376—7312no—
197C43—7312no—
198C22×Q8—738no—
199C22×Q8—738no—
200C22×Q8—734no—
201C22×Q8—734no—
202C22×Q8—733no—
203C22×Q8C2×#6440811yescharacteristic
204C23×C4—733no—
205C23×C4—733no—
206C23×C4—731no—
207C4×Q8—7332no—
208C4×Q8—7332no—
209C4×Q8—7364no—
210C4×Q8—738no—
211C4×Q8—7324no—
212C4×Q8—7324no—
213C4×Q8C2×#6440818yescomplemented
214#117—7348no—
215#117—736no—
216#117—7312no—
217#117—7312no—
218#117—7312no—
219#117C2×#6440816yes—
220C2×C4⋊C4—7312no—
221C2×C4⋊C4—7324no—
222C2×C4⋊C4—7312no—
223C2×C4⋊C4—7312no—
224C2×C22⋊C4—734no—
225C2×C22⋊C4—734no—
226C2×C42—7324no—
227C2×C42—7312no—
228C2×C42—736no—
229C2×C42—736no—
230C2×C42—7324no—
231C2×C42—7312no—
232C2×C42—738no—
233#95—738no—
234#95—7324no—
235C24—731no—
236C2×Q8—7332no—
237C2×Q8—7316no—
238C2×Q8—7316no—
239C2×Q8—7312no—
240C2×Q8—7324no—
241C2×Q8C4×#6440818yescomplemented
242C2×Q8C22×#6440814yes—
243C22×C4—2926no—
244C22×C4—7312no—
245C22×C4—731no—
246C22×C4—733no—
247C22×C4—736no—
248C22×C4—736no—
249C22×C4—7312no—
250C22×C4—7312no—
251C22×C4—734no—
252C22×C4—731no—
253C22×C4—738no—
254C22×C4—736no—
255C22×C4—738no—
256C22×C4—1464no—
257C22×C4C22×#6440813yes—
258C4⋊C4—7348no—
259C4⋊C4—7348no—
260C4⋊C4—7324no—
261C4⋊C4—7312no—
262C4⋊C4C22×#64408112yes—
263C22⋊C4—7316no—
264C22⋊C4—734no—
265C22⋊C4—738no—
266C22⋊C4C22×#6440814yes—
267C42—29224no—
268C42—7324no—
269C42—7324no—
270C42—7348no—
271C42—7324no—
272C42—7312no—
273C42—14616no—
274C42C22×#64408112yes—
275C23—731no—
276C23—732no—
277C23—732no—
278C23—731no—
279C23—1462no—
280C23—1462no—
281C23C23×#6440811yescharacteristic
282Q8—7348no—
283Q8C2×C4×#64408116yes—
284C2×C4—29212no—
285C2×C4—732no—
286C2×C4—734no—
287C2×C4—7312no—
288C2×C4—736no—
289C2×C4—7312no—
290C2×C4—7312no—
291C2×C4—7324no—
292C2×C4—732no—
293C2×C4—7316no—
294C2×C4—7312no—
295C2×C4—738no—
296C2×C4—1468no—
297C2×C4—1468no—
298C2×C4—1468no—
299C2×C4—1468no—
300C2×C4—14612no—
301C2×C4—1468no—
302C2×C4—1462no—
303C2×C4C2×C4×#64408112yes—
304C2×C4C23×#6440818yes—
305C2×C4C23×#6440816yes—
306C22—22no—
307C22—731no—
308C22—731no—
309C22—732no—
310C22—732no—
311C22—732no—
312C22—1462no—
313C22—1462no—
314C22—1464no—
315C22—1462no—
316C22C22×C4×#6440812yes—
317C22C24×#6440811yescharacteristic
318C4—28no—
319C4—734no—
320C4—7312no—
321C4—14612no—
322C4—1462no—
323C4—14616no—
324C4—1468no—
325C4C22×C4×#64408112yes—
326C2—22no—
327C2—731no—
328C2—731no—
329C2—732no—
330C2—1462no—
331C2C23×C4×#6440811yescharacteristic
332C22−1+4×#6440812yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutatorC2×C73C23×C42
Fitting#519×C73C4
socleC22×C73C24×C4
prime radicalC23×D73C24
cosocleC22×D73C25
hypercenter#519#64408
hypocenterC73C4×#519
LCSG ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #22626371C2semidirect
5. C4×Q8×#64408C2semidirect
8. #17989655C4semidirect
9. C2×Q8×#64408C4semidirect
25. #17989065C4semidirect
26. #17989064C4semidirect
27. #17989063C4semidirect
30. #519×C73C4semidirect
35. Q8×#64408C2×C4semidirect
55. Q8×Dic73C2×C4semidirect
56. C4×Dic146C2×C4semidirect
57. #14556556C2×C4semidirect
66. C4×Q8×C73C2×C4semidirect
85. C2×Dic146C42semidirect
87. Q8×D73C42semidirect
102. C2×Q8×C73C42semidirect
154. #64408#519direct (N × H)
174. C73C4×#519semidirect
180. #519#64408direct (N × H)
213. C4×Q8C2×#64408semidirect
241. C2×Q8C4×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179896552
C22×Q8×#644082
#226263712
#226263682
C4×Q8×#644082
#117×#644082
#519×D732
C4×#51973