Finite Group Databasebeta

← #28484449

Subgroups — #28484449

Order 15104.

630 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 147 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#21432435C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic
3C2×#16837138C212yesmaximal
4#21421975C212yesmaximal
5C2×C59×#382C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837308C412yescomplemented
9#16837300C412yescomplemented
10#16837286—22no—
11#16837275—21no—
12#16837265—21no—
13#16837262C2211yescharacteristic
14#16837157—22no—
15#16837138—24no—
16C22×C4⋊C4×C59—21no—
17C22×C42×C59C2211yescharacteristic
18C59×#382—24no—
19C2×#95×C59—21no—
20C2×#95×C59C2212yes—
21C22×Dic118C2×C411yescharacteristic
22C22×C59⋊C8—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C23×C4×C59—22no—
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C2×C4⋊C4×C59—24no—
42C2×C4⋊C4×C59—42no—
43C2×C4⋊C4×C59—42no—
44C2×C42×C59—22no—
45C2×C42×C59—22no—
46C2×C42×C59C2×C412yes—
47C2×C42×C59D412yes—
48#95×C59—24no—
49#95×C59—22no—
50#95×C59D414yes—
51C24×C59C2×D411yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—24no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59—44no—
62C22×C4×C59—44no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59C22⋊C412yes—
66C22×C4×C59C22⋊C412yes—
67C22×C4×C59SD1612yes—
68C22×C4×C59Q1612yes—
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D412yes—
72C22×C4×C59#4112yes—
73C22×Dic59—22no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C2×Dic118—22no—
77C2×Dic118—42no—
78C4⋊C4×C59—48no—
79C42×C59—24no—
80C42×C59—24no—
81C22⋊Dic59—41no—
82C22⋊Dic59—41no—
83C22⋊Dic59—41no—
84C22⋊Dic59—41no—
85C4⋊Dic59—22no—
86C4⋊Dic59—42no—
87Dic59⋊C4—44no—
88Dic59⋊C4—42no—
89C2×C59⋊C8—44no—
90C2×C59⋊C8—42no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59C2×C22⋊C411yescharacteristic
94C23×C59#12011yescharacteristic
95C23×C59#12311yescharacteristic
96C23×C59C2×SD1611yescharacteristic
97C23×C59C2×Q1611yescharacteristic
98C23×C59AGL(1,ℤ/8)11yescharacteristic
99C23×C59#13711yescharacteristic
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—44no—
116C2×C4×C59—44no—
117C2×C4×C59—44no—
118C2×C4×C59—84no—
119C2×C4×C59—84no—
120C2×C4×C59#10314yes—
121C2×C4×C59C4×D412yes—
122C2×C4×C59#12112yes—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130Dic118—82no—
131C59⋊C8—82no—
132#9152—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—22no—
138C22×C59—22no—
139C22×C59—22no—
140C22×C59—22no—
141C22×C59#38611yescharacteristic
142C22×C59#41012yes—
143C22×C59C2×#10311yescharacteristic
144C22×C59#41711yescharacteristic
145C22×C59#42112yes—
146C22×C59#45011yescharacteristic
147C22×C59#45111yescharacteristic
148C22×C59#48311yescharacteristic
149C22×C59#48411yescharacteristic
150C4×C59—24no—
151C4×C59—22no—
152C4×C59—22no—
153C4×C59—42no—
154C4×C59—84no—
155Dic59—81no—
156Dic59—81no—
157Dic59—82no—
158#2855—591no—
159C2×#382D5911yescharacteristic
160#1817—591no—
161C2×#328—592no—
162#1294—592no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59#151212yes—
168C2×C59#151812yes—
169C2×C59#187211yescharacteristic
170C2×C59#189711yescharacteristic
171C2×C59#189911yescharacteristic
172C2×C22⋊Q8—591no—
173C22×C4⋊C4—21no—
174C22×C42C2×D5911yescharacteristic
175C2×C22⋊C8—592no—
176#391—592no—
177#389—592no—
178#387—1182no—
179#385—1181no—
180#382—24no—
181#381—1181no—
182#380—591no—
183C2×#95—21no—
184C2×#95C2×D5912yes—
185#336—1182no—
186#328—1184no—
187C59#915211yescharacteristic, complemented, Sylow
188C22×Q8—591no—
189C23×C4—22no—
190C23×C4C22⋊D5911yescharacteristic
191C23×C4C22⋊D5911yescharacteristic
192C23×C4C22×D5911yescharacteristic
193C22×C8—1182no—
194C22⋊Q8—1182no—
195C22⋊Q8—1182no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—591no—
200C2×C4⋊C4—592no—
201C2×C4⋊C4—1182no—
202C2×C4⋊C4—1181no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4—1181no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D5912yes—
208C2×C42D23612yes—
209C22⋊C8—1182no—
210C22⋊C8—1182no—
211#95—24no—
212#95—22no—
213#95—1182no—
214#95—1182no—
215#95—1182no—
216#95—1181no—
217#95C22⋊D5914yes—
218C24C2×C22⋊D5911yescharacteristic
219C2×Q8—1182no—
220C2×Q8—2362no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1182no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4(C2×D59)⋊C412yes—
238C22×C4(C2×D59)⋊C412yes—
239C22×C4#1174567012yes—
240C22×C4#1174567212yes—
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4C2×D23611yescharacteristic
243C22×C4#1174568512yes—
244C22×C4C2×C22⋊D5912yes—
245C2×C8—2364no—
246C2×C8—2362no—
247C4⋊C4—48no—
248C4⋊C4—1182no—
249C4⋊C4—2362no—
250C4⋊C4—2364no—
251C4⋊C4—2362no—
252C22⋊C4—2361no—
253C22⋊C4—2361no—
254C22⋊C4—2361no—
255C22⋊C4—2361no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×(C2×D59)⋊C411yescharacteristic
261C23#1427369611yescharacteristic
262C23#1427371011yescharacteristic
263C23C2×#1174567011yescharacteristic
264C23C2×#1174567211yescharacteristic
265C23#1427375311yescharacteristic
266C23#1427375811yescharacteristic
267Q8—4722no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2362no—
294C2×C4—2362no—
295C2×C4#1427365014yes—
296C2×C4C4×C22⋊D5912yes—
297C2×C4#1427370112yes—
298C8—4722no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1683728111yescharacteristic
308C22#1683737512yes—
309C22C2×#1427365011yescharacteristic
310C22#1683740111yescharacteristic
311C22#1683742012yes—
312C22#1683755311yescharacteristic
313C22#1683755811yescharacteristic
314C22#1683773211yescharacteristic
315C22#1683773711yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—84no—
321C4—4722no—
322C4—4721no—
323C4—4721no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2142263212yes—
329C2#2142265912yes—
330C2#2142440311yescharacteristic
331C2#2142454411yescharacteristic
332C2#2142455411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#382C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC2×#382D59
hypocenterC59#9152
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
8. #16837308C4semidirect
9. #16837300C4semidirect
187. C59#9152semidirect

Maximal subgroups

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

subgroupindex
#214324352
#214241642
C2×#168371382
#214219752
C2×C59×#3822
#915259