Finite Group Databasebeta

← #28499275

Subgroups — #28499275

Order 15104.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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#21428404C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422101C214yesmaximal, complemented
4C4×#100×C59C211yesmaximal, characteristic
5C2×#14273708—21no—
6C22×C4⋊Dic59C411yescharacteristic
7#16837366—21no—
8#16837332C2212yes—
9#16837327C412yes—
10#16837300—21no—
11#16837263—21no—
12#16837262—21no—
13#16837262C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#100×C59—21no—
18C4×M16×C59—21no—
19C59×#343C2212yes—
20C23×Dic59—21no—
21C2×#11745664—41no—
22#14273708—44no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—41no—
26C2×C4⋊Dic59—41no—
27C2×C4⋊Dic59C2×C412yes—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—21no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—41no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59C2×C411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C2×M16×C59—22no—
40C4×D4×C59—41no—
41C4×D4×C59—41no—
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C22⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C42×C59D411yescharacteristic
46#100×C59—24no—
47#97×C59—41no—
48C4×C8×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C24×C59C22⋊C411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—42no—
68C22×Dic59—42no—
69C22×Dic59—42no—
70M16×C59—24no—
71M16×C59—42no—
72C2×C8×C59—42no—
73C4⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C42×C59—21no—
77C42×C59—21no—
78C22⋊Dic59—42no—
79C22⋊Dic59—44no—
80C4⋊Dic59—42no—
81C4⋊Dic59—41no—
82C4⋊Dic59—41no—
83C4⋊Dic59—42no—
84Dic59⋊C4—44no—
85#11745664—82no—
86C2×C59⋊C8—42no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59#10412yes—
97C23×C59C2×C22⋊C411yescharacteristic
98D4×C59—44no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—81no—
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59AGL(1,ℤ/8)12yes—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—84no—
122C8×C59—44no—
123C59⋊C8—82no—
124#11780—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—22no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—42no—
131C22×C59—41no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59C2×#9911yescharacteristic
138C22×C59#41811yescharacteristic
139C22×C59C2×#10411yescharacteristic
140C4×C59—21no—
141C4×C59—22no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59#35212yes—
146Dic59—82no—
147Dic59—82no—
148#2486—591no—
149C4×#100D5911yescharacteristic
150#1465—591no—
151#1346—594no—
152C2×C59—21no—
153C2×C59—21no—
154C2×C59—42no—
155C2×C59—81no—
156C2×C59#150711yescharacteristic
157C2×C59C2×#35211yescharacteristic
158C2×C59#213511yescharacteristic
159C2×#123—1181no—
160C2×C4×D4C2×D5911yescharacteristic
161C22×C4⋊C4—591no—
162C2×#100—21no—
163#407—1181no—
164C4×M16—21no—
165#398—592no—
166#397—592no—
167#389—1181no—
168#381—1181no—
169#380—591no—
170#380—1181no—
171#343C2×D5912yes—
172#324—592no—
173#323—1181no—
174C59#1178011yescharacteristic, complemented, Sylow
175C22×D4C22⋊D5911yescharacteristic
176C23×C4—1181no—
177C23×C4C4×D5911yescharacteristic
178C23×C4D23611yescharacteristic
179C2×M16—22no—
180C2×M16—2361no—
181#123—2364no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—592no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1182no—
188C2×C4⋊C4—2361no—
189C2×C4⋊C4—2361no—
190C2×C4⋊C4C22×D5911yescharacteristic
191C2×C22⋊C4—1181no—
192C2×C22⋊C4—1182no—
193C2×C22⋊C4C4×D5911yescharacteristic
194C2×C22⋊C4D23611yescharacteristic
195C2×C42C22⋊D5911yescharacteristic
196#100—24no—
197C22⋊C8—1182no—
198C22⋊C8—2361no—
199#97—41no—
200C4×C8—41no—
201#95—592no—
202#95—1182no—
203#95—1182no—
204C24(C2×D59)⋊C411yescharacteristic
205C24(C2×D59)⋊C411yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—1182no—
216C22×C4—1182no—
217C22×C4—2362no—
218C22×C4—2362no—
219C22×C4—2362no—
220C22×C4(C2×D59)⋊C411yescharacteristic
221C22×C4(C2×D59)⋊C411yescharacteristic
222C22×C4C2×C4×D5911yescharacteristic
223C22×C4C2×D23611yescharacteristic
224C22×C4C2×C22⋊D5911yescharacteristic
225M16—24no—
226M16—42no—
227M16—4722no—
228C2×C8—42no—
229C2×C8—2362no—
230C4⋊C4—41no—
231C4⋊C4—2362no—
232C4⋊C4—2362no—
233C4⋊C4—2361no—
234C4⋊C4—2361no—
235C4⋊C4—2364no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—2362no—
239C22⋊C4—2364no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1427365512yes—
252C23C2×(C2×D59)⋊C411yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—4724no—
268C2×C4—2362no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2362no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4#1427364111yescharacteristic
275C2×C4#1427364111yescharacteristic
276C2×C4#1427375212yes—
277C8—44no—
278C8—4722no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#1427364111yescharacteristic
292C22#1683740611yescharacteristic
293C22C2×#1427365511yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—4722no—
300C4—4722no—
301C4#1683719312yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2142261111yescharacteristic
307C2C2×#1683719311yescharacteristic
308C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC4×#100×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C4⋊C4×C59C23
hypercenterC4×#100D59
hypocenterC59#11780
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #21422101C2semidirect
174. C59#11780semidirect

Maximal subgroups

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

subgroupindex
#214284042
#214224342
#214221012
C4×#100×C592
#1178059