Finite Group Databasebeta

← #28494605

Subgroups — #28494605

Order 15104.

746 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837438C212yesmaximal
2C2×#16837360C211yesmaximal, characteristic
3#21423828C212yesmaximal
4C2×#16837159C211yesmaximal, characteristic
5C2×C59×H(3,ℤ/4)C211yesmaximal, characteristic
6C23×C59⋊C8C2211yescharacteristic
7C2×#14273686C2212yes—
8C2×#14273686C2211yescharacteristic
9#16837438—24no—
10#16837438C418yescomplemented
11C2×#14273660—22no—
12C2×#14273658—22no—
13#16837360—22no—
14#16837360—22no—
15C2×#14273637C412yescomplemented
16C2×C8×Dic59C412yescomplemented
17#16837168—24no—
18#16837159—22no—
19C22×M16×C59C2211yescharacteristic
20C2×#117×C59C2212yes—
21C59×H(3,ℤ/4)—22no—
22C22×C59⋊C8C2×C414yes—
23C22×C59⋊C8C2×C411yescharacteristic
24C22×C59⋊C8C2×C411yescharacteristic
25C22×C59⋊C8D412yes—
26C22×C59⋊C8D412yes—
27C22×C59⋊C8D412yes—
28C22×C59⋊C8Q812yes—
29#14273686—22no—
30#14273686—22no—
31#14273686—22no—
32#14273686—22no—
33#14273686—22no—
34#14273686C2×C414yes—
35C2×C4⋊Dic59—22no—
36C2×C4⋊Dic59Q812yes—
37C2×C22⋊Dic59—22no—
38C2×C22⋊Dic59D412yes—
39C2×C4×Dic59—22no—
40C2×C4×Dic59C2×C412yes—
41#14273660—28no—
42#14273658—28no—
43#14273637—24no—
44C8×Dic59—22no—
45C8×Dic59—22no—
46C23×C4×C59C2311yescharacteristic
47C2×M16×C59—22no—
48C2×M16×C59—22no—
49C2×M16×C59D412yes—
50C2×M16×C59Q812yes—
51C22×C8×C59C2×C412yes—
52#117×C59—22no—
53#117×C59—22no—
54#117×C59—22no—
55#117×C59C2×C414yes—
56C2×C4⋊C4×C59—22no—
57C2×C22⋊C4×C59—22no—
58C2×C42×C59—22no—
59C24×C59C2×D411yescharacteristic
60C22×C4×C59—42no—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C22×C412yes—
63C22×C4×C59C2×D412yes—
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×Q811yescharacteristic
67C22×C4×C59C2×Q811yescharacteristic
68C22×C4×C59C2×Q811yescharacteristic
69C22×C4×C59#4112yes—
70C22×C4×C59#4112yes—
71C22×Dic59—42no—
72C22×Dic59C4⋊C414yes—
73M16×C59—24no—
74M16×C59—24no—
75C2×C8×C59—22no—
76C2×C8×C59—22no—
77C2×C8×C59C4⋊C414yes—
78C4⋊C4×C59—24no—
79C4⋊C4×C59—42no—
80C22⋊C4×C59—24no—
81C22⋊C4×C59—42no—
82C42×C59—24no—
83C42×C59—42no—
84C22⋊Dic59—24no—
85C22⋊Dic59—24no—
86C22⋊Dic59—42no—
87C4⋊Dic59—24no—
88C4⋊Dic59—24no—
89C4⋊Dic59—42no—
90C4×Dic59—22no—
91C4×Dic59—22no—
92C4×Dic59—24no—
93C4×Dic59—42no—
94C2×C59⋊C8C4212yes—
95C2×C59⋊C8C4212yes—
96C2×C59⋊C8C22⋊C411yescharacteristic
97C2×C59⋊C8C22⋊C411yescharacteristic
98C2×C59⋊C8C22⋊C412yes—
99C2×C59⋊C8C22⋊C414yes—
100C2×C59⋊C8C22⋊C414yes—
101C2×C59⋊C8C4⋊C412yes—
102C2×C59⋊C8C4⋊C412yes—
103C2×C59⋊C8C4⋊C414yes—
104C2×C59⋊C8C4⋊C414yes—
105C23×C59—42no—
106C23×C59C2×C4⋊C411yescharacteristic
107C23×C59C4×D412yes—
108C23×C59#12112yes—
109C23×C59#12711yescharacteristic
110C23×C59C4⋊Q811yescharacteristic
111C2×C4×C59—44no—
112C2×C4×C59—84no—
113C2×C4×C59C2×C4211yescharacteristic
114C2×C4×C59C2×C22⋊C411yescharacteristic
115C2×C4×C59C2×C22⋊C412yes—
116C2×C4×C59C2×C4⋊C411yescharacteristic
117C2×C4×C59C2×C4⋊C412yes—
118C2×C4×C59C2×C4⋊C411yescharacteristic
119C2×C4×C59#11711yescharacteristic
120C2×C4×C59#11711yescharacteristic
121C2×C4×C59C4×D412yes—
122C2×C4×C59C4×Q812yes—
123C2×C4×C59C4×Q812yes—
124C2×C4×C59#12112yes—
125C2×C4×C59C22⋊Q812yes—
126C2×C4×C59C22⋊Q812yes—
127C2×C4×C59#12412yes—
128C2×C4×C59#12512yes—
129C2×C4×C59C4⋊Q811yescharacteristic
130C2×C4×C59C4⋊Q811yescharacteristic
131C2×Dic59—24no—
132C2×Dic59—44no—
133C2×Dic59—44no—
134C2×Dic59—84no—
135C8×C59—24no—
136C59⋊C8#9512yes—
137C59⋊C8#9512yes—
138C59⋊C8#9514yes—
139#10866—591nomaximal, Sylow
140C22×C59—22no—
141C22×C59—24no—
142C22×C59—42no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59C4×C4⋊C411yescharacteristic
146C22×C59#38011yescharacteristic
147C22×C59#38411yescharacteristic
148C22×C59#38912yes—
149C22×C59#39012yes—
150C4×C59—84no—
151C4×C59C2×#9511yescharacteristic
152C4×C59C4×C4⋊C411yescharacteristic
153C4×C59#38011yescharacteristic
154C4×C59#38211yescharacteristic
155C4×C59#38912yes—
156C4×C59#39112yes—
157Dic59—44no—
158Dic59—84no—
159C2×#427—592no—
160C2×#405—591no—
161#1740—592no—
162C2×H(3,ℤ/4)—591no—
163C2×H(3,ℤ/4)D5911yescharacteristic
164C2×C59—22no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—22no—
168C2×C59C4⋊#9511yescharacteristic
169C2×C59#181212yes—
170C22×M16C2×D5911yescharacteristic
171C23×C8—591no—
172C2×#117—591no—
173C2×#117—592no—
174C2×#117C2×D5912yes—
175#427—598no—
176#427—1184no—
177C2×#107—1182no—
178C2×#106—1182no—
179#405—1182no—
180#405—1182no—
181C2×#97—592no—
182C2×C4×C8—592no—
183#342—1184no—
184H(3,ℤ/4)—22no—
185H(3,ℤ/4)—1182no—
186C59#1086611yescharacteristic, complemented, Sylow
187C23×C4C22×D5911yescharacteristic
188C2×M16—22no—
189C2×M16—22no—
190C2×M16Dic11812yes—
191C2×M16C22⋊D5912yes—
192C22×C8—591no—
193C22×C8—591no—
194C22×C8—594no—
195C22×C8—592no—
196C22×C8—592no—
197C22×C8—592no—
198C22×C8—592no—
199C22×C8C4×D5912yes—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—594no—
204#117—1182no—
205#117—1182no—
206#117—1182no—
207#117—1182no—
208#117—1182no—
209#117C4×D5914yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—592no—
212C2×C4⋊C4—1182no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—592no—
215C2×C22⋊C4—1182no—
216C2×C42—22no—
217C2×C42—592no—
218C2×C42—1182no—
219#107—1188no—
220#106—1188no—
221#97—1184no—
222C4×C8—1182no—
223C4×C8—1182no—
224C24D4×D5911yescharacteristic
225C22×C4—42no—
226C22×C4—594no—
227C22×C4—2362no—
228C22×C4C2×Dic11811yescharacteristic
229C22×C4C2×C4×D5911yescharacteristic
230C22×C4C2×C4×D5912yes—
231C22×C4#1174568512yes—
232C22×C4D4×D5912yes—
233C22×C4D4×D5911yescharacteristic
234C22×C4#1174568712yes—
235C22×C4Q8×D5911yescharacteristic
236C22×C4Q8×D5911yescharacteristic
237C22×C4C2×C22⋊D5911yescharacteristic
238M16—24no—
239M16—24no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—592no—
243C2×C8—592no—
244C2×C8—591no—
245C2×C8—591no—
246C2×C8—592no—
247C2×C8—594no—
248C2×C8—594no—
249C2×C8—592no—
250C2×C8—592no—
251C2×C8—594no—
252C2×C8—594no—
253C2×C8Dic59⋊C414yes—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—1184no—
257C4⋊C4—1184no—
258C4⋊C4—2362no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—1184no—
262C22⋊C4—1184no—
263C22⋊C4—2362no—
264C42—24no—
265C42—42no—
266C42—1182no—
267C42—1182no—
268C42—1184no—
269C42—2362no—
270C23—42no—
271C23C4⋊C4×D5911yescharacteristic
272C23(C22⋊D59)⋊C412yes—
273C23#1427370012yes—
274C23#1427372311yescharacteristic
275C23#1427372611yescharacteristic
276C2×C4—44no—
277C2×C4—84no—
278C2×C4—1184no—
279C2×C4—4724no—
280C2×C4—2364no—
281C2×C4—2364no—
282C2×C4C42×D5911yescharacteristic
283C2×C4C22⋊C4×D5911yescharacteristic
284C2×C4C22⋊C4×D5912yes—
285C2×C4C4⋊C4×D5911yescharacteristic
286C2×C4C4⋊C4×D5912yes—
287C2×C4C2×Dic59⋊C411yescharacteristic
288C2×C4#1427368311yescharacteristic
289C2×C4#1427368411yescharacteristic
290C2×C4C4×C22⋊D5912yes—
291C2×C4#1427369212yes—
292C2×C4C4×Dic11812yes—
293C2×C4#1427369912yes—
294C2×C4#1427370412yes—
295C2×C4#1427370512yes—
296C2×C4#1427371312yes—
297C2×C4#1427371712yes—
298C2×C4#1427372511yescharacteristic
299C2×C4#1427372611yescharacteristic
300C8—24no—
301C8—592no—
302C8—592no—
303C8—594no—
304C22—24no—
305C22—22no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22#1683725311yescharacteristic
310C22#1683726111yescharacteristic
311C22#1683727111yescharacteristic
312C22#1683729912yes—
313C22#1683730212yes—
314C4—84no—
315C4—4724no—
316C4—2364no—
317C4#95×D5911yescharacteristic
318C4C4×Dic59⋊C411yescharacteristic
319C4#1683726011yescharacteristic
320C4#1683726611yescharacteristic
321C4#1683729712yes—
322C4#1683730612yes—
323C2—22no—
324C2—21no—
325C2—21no—
326C2—22no—
327C2Dic59⋊#9511yescharacteristic
328C2#2142414112yes—
subgroupG/H
centerC2×C4#14273684
commutatorC2×C4×C59C2×C42
FrattiniC23×C4C22×D59
FittingC2×C59×H(3,ℤ/4)C2
socleC22×C59#380
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC2×H(3,ℤ/4)D59
hypocenterC59#10866
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
10. #16837438C4semidirect
15. C2×#14273637C4semidirect
16. C2×C8×Dic59C4semidirect
186. C59#10866semidirect

Maximal subgroups

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

subgroupindex
C2×#168374382
C2×#168373602
#214238282
C2×#168371592
C2×C59×H(3,ℤ/4)2
#1086659