Finite Group Databasebeta

← #28587340

Subgroups — #28587340

Order 15104.

938 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837530C212yesmaximal
2C42×Dic118C211yesmaximal, characteristic
3#21423970C212yesmaximal
4#21423968C212yesmaximal
5C8×Dic59⋊C4C212yesmaximal
6#21423950C211yesmaximal, characteristic
7#21423950C211yesmaximal, characteristic
8#21423949C211yesmaximal, characteristic
9#97×Dic59C211yesmaximal, characteristic
10#97×Dic59C211yesmaximal, characteristic
11C4×#97×C59C211yesmaximal, characteristic
12C2×C4×Dic118C412yes—
13C2×C4×Dic118C2211yescharacteristic
14#16837530C4116yescomplemented
15C2×C4⋊(C59⋊C8)C2212yes—
16C2×Dic59⋊C8C2212yes—
17C2×Dic59⋊C8C2212yes—
18C2×#14273638C2211yescharacteristic
19C2×#14273638C2211yescharacteristic
20C2×#14273637C2212yes—
21C2×#14273637C2212yes—
22C2×C8×Dic59C2212yes—
23C2×C8×Dic59C2212yes—
24C4×Dic59⋊C4C412yes—
25C4×Dic59⋊C4C412yes—
26C4×Dic59⋊C4C2211yescharacteristic
27C4×Dic59⋊C4C2211yescharacteristic
28C4×C4⋊Dic59C412yes—
29C4×C4⋊Dic59C2211yescharacteristic
30C42×Dic59C2211yescharacteristic
31C42×Dic59C2211yescharacteristic
32#16837157C2212yes—
33#16837156C2212yes—
34#16837156C2212yes—
35C2×#97×C59C2211yescharacteristic
36C2×#97×C59C2212yes—
37C2×#97×C59C2211yescharacteristic
38C2×C4×C8×C59C2212yes—
39C43×C59C2211yescharacteristic
40C22×Dic118C2×C411yescharacteristic
41C22×C59⋊C8C2312yes—
42C22×C59⋊C8C2312yes—
43C4×Dic118—28no—
44C4×Dic118C2×C418yes—
45C2×C4⋊Dic59C2×C412yes—
46C2×C4⋊Dic59C2×C411yescharacteristic
47C2×Dic59⋊C4C2×C412yes—
48C2×Dic59⋊C4C2×C411yescharacteristic
49C2×Dic59⋊C4C2×C411yescharacteristic
50C2×Dic59⋊C4C2×C412yes—
51C2×C4×Dic59C2×C412yes—
52C2×C4×Dic59C2×C412yes—
53C2×C4×Dic59C2×C412yes—
54C2×C4×Dic59C2×C412yes—
55C2×C4×Dic59C2311yescharacteristic
56C2×C4×Dic59C2311yescharacteristic
57C2×C4×Dic59C2311yescharacteristic
58C2×C4×Dic59C2311yescharacteristic
59C4⋊(C59⋊C8)C2×C418yes—
60Dic59⋊C8C2×C418yes—
61Dic59⋊C8C2×C418yes—
62#14273638—22no—
63#14273638—22no—
64#14273637—24no—
65#14273637C2×C418yes—
66C8×Dic59—24no—
67C8×Dic59C2×C418yes—
68C59⋊#95C2×C412yes—
69C59⋊#95C2×C412yes—
70C59⋊#95C2×C412yes—
71C22×C8×C59C2312yes—
72C22×C8×C59C2312yes—
73C2×C42×C59C2×C412yes—
74C2×C42×C59C2×C412yes—
75C2×C42×C59C2311yescharacteristic
76C2×C42×C59C2311yescharacteristic
77C2×C42×C59C2311yescharacteristic
78#97×C59—22no—
79#97×C59C2×C418yes—
80#97×C59Q814yescomplemented
81C4×C8×C59—24no—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C22×C412yes—
85C22×C4×C59C22×C412yes—
86C22×C4×C59C2411yescharacteristic
87C22×Dic59C22×C412yes—
88C22×Dic59C22×C411yescharacteristic
89C22×Dic59C22×C411yescharacteristic
90C22×Dic59C22×C411yescharacteristic
91C22×Dic59C22×C412yes—
92C22×Dic59C22×C411yescharacteristic
93C2×Dic118—24no—
94C2×Dic118C4214yes—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59C22×C414yes—
98C2×C8×C59C22×C414yes—
99C2×C8×C59C2×Q814yes—
100C2×C8×C59#4114yes—
101C42×C59—24no—
102C42×C59—22no—
103C42×C59—42no—
104C42×C59C22×C414yes—
105C42×C59C2×Q812yes—
106C42×C59#4112yes—
107C4⋊Dic59—24no—
108C4⋊Dic59C4214yes—
109Dic59⋊C4—24no—
110Dic59⋊C4C4214yes—
111Dic59⋊C4C4214yes—
112Dic59⋊C4M1618yes—
113C4×Dic59—22no—
114C4×Dic59—22no—
115C4×Dic59—24no—
116C4×Dic59—24no—
117C4×Dic59—42no—
118C4×Dic59—42no—
119C4×Dic59C22×C414yes—
120C4×Dic59C22×C414yes—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8C22×C414yes—
126C2×C59⋊C8C22×C414yes—
127C23×C59C23×C411yescharacteristic
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59C2×C4212yes—
135C2×C4×C59C2×C4212yes—
136C2×C4×C59C4×Q814yes—
137C2×C4×C59C2×M1614yes—
138C2×C4×C59C23×C412yes—
139C2×C4×C59C22×Q811yescharacteristic
140C2×C4×C59C2×#4112yes—
141C2×C4×C59C2×#4111yescharacteristic
142C2×Dic59—22no—
143C2×Dic59—22no—
144C2×Dic59—22no—
145C2×Dic59—22no—
146C2×Dic59—22no—
147C2×Dic59—22no—
148C2×Dic59—22no—
149C2×Dic59—22no—
150C2×Dic59C2×C4212yes—
151C2×Dic59C2×C4212yes—
152C2×Dic59C2×C4212yes—
153C2×Dic59C2×C4212yes—
154C2×Dic59C2×M1614yes—
155C2×Dic59C2×M1614yes—
156Dic118—28no—
157C8×C59—24no—
158C8×C59C4×Q818yescomplemented
159C59⋊C8—24no—
160C59⋊C8—24no—
161#20520—591nomaximal, Sylow
162C22×C59C22×C4211yescharacteristic
163C22×C59C2×C4×Q811yescharacteristic
164C22×C59C4×#4111yescharacteristic
165C22×C59C22×M1612yes—
166C22×C59#56812yes—
167C4×C59—22no—
168C4×C59—22no—
169C4×C59—24no—
170C4×C59—24no—
171C4×C59C2×C4×Q812yes—
172C4×C59C4×#4112yes—
173Dic59—22no—
174Dic59—22no—
175Dic59—22no—
176Dic59—24no—
177Dic59—42no—
178Dic59C4×M1614yes—
179C2×#446—592no—
180C42×Q8—591no—
181#1771—592no—
182#1771—592no—
183C8×C4⋊C4—592no—
184#1767—591no—
185#1767—591no—
186#1767—591no—
187C4×#97—591no—
188C4×#97—591no—
189C4×#97D5911yescharacteristic
190C2×C59C42×Q811yescharacteristic
191C2×C59C2×C4×M1611yescharacteristic
192C2×C59#287511yescharacteristic
193C2×C59Q8×M1612yes—
194C2×C59#297012yes—
195C2×C4×Q8—592no—
196C2×C4×Q8—591no—
197#446—5916no—
198C2×C4⋊C8—592no—
199C2×C4⋊C8—592no—
200C2×C4⋊C8—592no—
201C2×#97—592no—
202C2×#97—592no—
203C2×#97—591no—
204C2×#97—591no—
205C2×#97C2×D5911yescharacteristic
206C2×#97C2×D5911yescharacteristic
207C2×#97C2×D5912yes—
208C2×C4×C8—592no—
209C2×C4×C8—592no—
210C2×C4×C8C2×D5912yes—
211C4×C4⋊C4—592no—
212C4×C4⋊C4—592no—
213C4×C4⋊C4—592no—
214C4×C4⋊C4—591no—
215C4×C4⋊C4—591no—
216C4×C4⋊C4—591no—
217C43—591no—
218C43—591no—
219C43C2×D5911yescharacteristic
220#336—592no—
221#336—592no—
222#336—592no—
223C59#2052011yescharacteristic, complemented, Sylow
224C22×Q8—591no—
225C22×C8—592no—
226C22×C8—592no—
227C22×C8C22×D5912yes—
228C22×C8C22×D5912yes—
229C4×Q8—598no—
230C4×Q8—1188no—
231C2×C4⋊C4—591no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—592no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4—592no—
236C2×C4⋊C4—592no—
237C2×C42—591no—
238C2×C42—591no—
239C2×C42—592no—
240C2×C42—592no—
241C2×C42—591no—
242C2×C42—591no—
243C2×C42—592no—
244C2×C42—592no—
245C2×C42C4×D5912yes—
246C2×C42C4×D5912yes—
247C2×C42C22×D5911yescharacteristic
248C2×C42C22×D5911yescharacteristic
249C2×C42C22×D5911yescharacteristic
250C4⋊C8—598no—
251C4⋊C8—598no—
252C4⋊C8—598no—
253#97—22no—
254#97—598no—
255#97—1184no—
256#97—1182no—
257#97—1182no—
258#97Dic11814yescomplemented
259#97C4×D5918yes—
260C4×C8—24no—
261C4×C8—598no—
262C4×C8—1184no—
263#95—592no—
264#95—592no—
265#95—592no—
266C2×Q8—594no—
267C2×Q8—1184no—
268C22×C4—591no—
269C22×C4—592no—
270C22×C4—591no—
271C22×C4—591no—
272C22×C4—591no—
273C22×C4—592no—
274C22×C4C2×C4×D5911yescharacteristic
275C22×C4C2×C4×D5911yescharacteristic
276C22×C4C2×C4×D5912yes—
277C22×C4C2×C4×D5912yes—
278C22×C4C23×D5911yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—594no—
282C2×C8—594no—
283C2×C8—1182no—
284C2×C8—1182no—
285C2×C8—1182no—
286C2×C8—1182no—
287C2×C8C2×Dic11814yes—
288C2×C8C2×C4×D5914yes—
289C2×C8C2×C4×D5914yes—
290C2×C8#1174568514yes—
291C4⋊C4—594no—
292C4⋊C4—594no—
293C4⋊C4—594no—
294C4⋊C4—598no—
295C4⋊C4—1184no—
296C4⋊C4—1184no—
297C42—24no—
298C42—22no—
299C42—42no—
300C42—594no—
301C42—594no—
302C42—1184no—
303C42—1184no—
304C42—1182no—
305C42—1182no—
306C42—2362no—
307C42—2362no—
308C42C2×Dic11812yes—
309C42C2×C4×D5914yes—
310C42#1174568512yes—
311C23C22×C4×D5911yescharacteristic
312Q8—1188no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—594no—
320C2×C4—592no—
321C2×C4—594no—
322C2×C4—592no—
323C2×C4—592no—
324C2×C4—592no—
325C2×C4—1182no—
326C2×C4—1182no—
327C2×C4—1182no—
328C2×C4—1182no—
329C2×C4—1182no—
330C2×C4—1182no—
331C2×C4—1182no—
332C2×C4—1182no—
333C2×C4C42×D5912yes—
334C2×C4C42×D5912yes—
335C2×C4C4×Dic11814yes—
336C2×C4M16×D5914yes—
337C2×C4C22×C4×D5912yes—
338C2×C4C22×Dic11811yescharacteristic
339C2×C4C2×#1174568512yes—
340C2×C4C2×#1174568511yescharacteristic
341C8—24no—
342C8—1184no—
343C8—1184no—
344C8C4×Dic11818yescomplemented
345C22C2×C42×D5911yescharacteristic
346C22C2×C4×Dic11811yescharacteristic
347C22C4×#1174568511yescharacteristic
348C22C2×M16×D5912yes—
349C22#1683816812yes—
350C4—24no—
351C4—22no—
352C4—22no—
353C4—24no—
354C4—594no—
355C4—1184no—
356C4—1182no—
357C4—1182no—
358C4—1182no—
359C4—2362no—
360C4C2×C4×Dic11812yes—
361C4C4×#1174568512yes—
362C2C42×Dic11811yescharacteristic
363C2C4×M16×D5911yescharacteristic
364C2#2143252511yescharacteristic
365C2M16×Dic11812yes—
366C2#2143310712yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C22×C42
FittingC4×#97×C59C2
socle, prime radicalC23×C59C23×C4
cosocleC22×C4×C59C24
hypercenterC4×#97D59
hypocenterC59#20520
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
14. #16837530C4semidirect
80. #97×C59Q8semidirect
158. C8×C59C4×Q8semidirect
223. C59#20520semidirect
258. #97Dic118semidirect
344. C8C4×Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168375302
C42×Dic1182
#214239702
#214239682
C8×Dic59⋊C42
#214239502
#214239492
#97×Dic592
C4×#97×C592
#2052059