Finite Group Databasebeta

← #28499686

Subgroups — #28499686

Order 15104.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#21433751C211yesmaximal, characteristic
2#21425717C211yesmaximal, characteristic
3#21422083C214yesmaximal, complemented
4C59×#1812C211yesmaximal, characteristic
5C22×Dic236C411yescharacteristic, complemented
6#16837973—21no—
7C2×#14273667—21no—
8#16837660C412yescomplemented
9#16837660C2212yes—
10#16837643—21no—
11#16837643—21no—
12#16837440C411yescharacteristic, complemented
13#16837412—21no—
14#16837184—21no—
15#16837184C2212yes—
16C2×#131×C59C2211yescharacteristic
17C59×#427—21no—
18C59×#410—21no—
19C59×H(3,ℤ/4)C2212yes—
20C22×Dic118—21no—
21C2×Dic236—21no—
22C2×Dic236—41no—
23C2×Dic236—41no—
24C2×Dic236C2×C412yes—
25#14273725—41no—
26#14273714—41no—
27#14273708—41no—
28#14273708—41no—
29#14273706—22no—
30#14273706—22no—
31#14273686—21no—
32#14273667—22no—
33#14273667—41no—
34C2×C59⋊C16—41no—
35#14273661—21no—
36#14273659C2×C412yes—
37#14273652—22no—
38#14273652—22no—
39C2×#41×C59D411yescharacteristic
40#131×C59—41no—
41#131×C59—41no—
42C2×M16×C59C2×C411yescharacteristic
43C2×M16×C59D411yescharacteristic
44C2×M16×C59C2311yescharacteristic
45C22×C8×C59C2×C411yescharacteristic
46C22×C8×C59D411yescharacteristic
47C22×C8×C59D411yescharacteristic
48#117×C59—22no—
49#107×C59—41no—
50#106×C59—41no—
51#99×C59—41no—
52#99×C59—41no—
53#41×C59—41no—
54#41×C59—41no—
55C2×Q8×C59SD1611yescharacteristic
56C2×D4×C59D811yescharacteristic
57C2×D4×C59D811yescharacteristic
58C2×D4×C59SD1611yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×Dic59—41no—
63C2×Dic118—22no—
64C2×Dic118—42no—
65C2×Dic118—42no—
66C2×Dic118—41no—
67M16×C59—41no—
68M16×C59—41no—
69M16×C59—41no—
70C2×C8×C59—21no—
71C2×C8×C59—21no—
72C2×C8×C59—21no—
73C2×C8×C59—21no—
74C2×C8×C59—21no—
75C2×C8×C59—21no—
76C2×C8×C59C22⋊C411yescharacteristic
77C2×C8×C59C22⋊C411yescharacteristic
78C2×C8×C59C22×C411yescharacteristic
79C2×C8×C59C2×D411yescharacteristic
80C4⋊C4×C59—42no—
81C22⋊C4×C59—42no—
82C42×C59—42no—
83C22⋊Dic59—41no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C4⋊Dic59—22no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89C4×Dic59—41no—
90Dic236—42no—
91Dic236—41no—
92Dic236—41no—
93C59⋊C16—42no—
94C23×C59#10211yescharacteristic
95C23×C59#10211yescharacteristic
96C23×C59C2×D811yescharacteristic
97Q8×C59—41no—
98D4×C59—41no—
99D4×C59—41no—
100D4×C59—41no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—82no—
108C2×C4×C59#10211yescharacteristic
109C2×C4×C59#10211yescharacteristic
110C2×C4×C59C2×C22⋊C411yescharacteristic
111C2×C4×C59C2×SD1611yescharacteristic
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—82no—
115Dic118—42no—
116Dic118—42no—
117Dic118—82no—
118C8×C59—21no—
119C8×C59—21no—
120C8×C59—21no—
121C8×C59#9911yescharacteristic
122C8×C59#9911yescharacteristic
123#11869—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59C2×#10211yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—21no—
134C4×C59—162no—
135C4×C59C2×#9911yescharacteristic
136C4×C59#42111yescharacteristic
137Dic59—82no—
138Dic59—82no—
139#3059—591no—
140#2115—591no—
141#1812D5911yescharacteristic
142#1340—594no—
143C2×C59—21no—
144C2×C59—41no—
145C2×C59—41no—
146C2×C59—41no—
147C2×C59#151911yescharacteristic
148C22×Q16—591no—
149C2×#131C2×D5911yescharacteristic
150#536—1181no—
151C2×M32—1181no—
152#470—592no—
153#470—592no—
154#467—1181no—
155#467—1181no—
156#427—21no—
157#427—591no—
158#419—1181no—
159#410—21no—
160#349—592no—
161#349—1181no—
162H(3,ℤ/4)C2×D5912yes—
163C59#1186911yescharacteristic, complemented, Sylow
164C2×#41C22⋊D5911yescharacteristic
165C22×Q8—1181no—
166C2×Q16—592no—
167C2×Q16—1181no—
168C2×Q16—2361no—
169C2×Q16—2361no—
170#131—41no—
171#131—41no—
172C2×M16C4×D5911yescharacteristic
173C2×M16D23611yescharacteristic
174C2×M16C22×D5911yescharacteristic
175C22×C8C4×D5911yescharacteristic
176C22×C8D23611yescharacteristic
177C22×C8C22⋊D5911yescharacteristic
178C4⋊Q8—2361no—
179#124—2361no—
180#123—2361no—
181#123—2361no—
182C22⋊Q8—1182no—
183C22⋊Q8—1182no—
184#117—22no—
185#117—1181no—
186M32—1182no—
187M32—2361no—
188C2×C16—2361no—
189#107—41no—
190#107—1181no—
191#106—41no—
192#106—592no—
193#103—1182no—
194#103—1182no—
195#99—41no—
196#99—41no—
197#41—41no—
198#41—41no—
199C2×Q8—1182no—
200C2×Q8—2362no—
201C2×Q8—2362no—
202C2×Q8—2361no—
203C2×Q8#1174567111yescharacteristic
204C2×D4#1174566911yescharacteristic
205C2×D4#1174566911yescharacteristic
206C2×D4#1174567111yescharacteristic
207C22×C4—2361no—
208C22×C4(C2×D59)⋊C411yescharacteristic
209C22×C4(C2×D59)⋊C411yescharacteristic
210C22×C4C2×C22⋊D5911yescharacteristic
211Q16—2362no—
212Q16—2361no—
213Q16—2361no—
214M16—41no—
215M16—41no—
216M16—41no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8(C2×D59)⋊C411yescharacteristic
224C2×C8(C2×D59)⋊C411yescharacteristic
225C2×C8C2×C4×D5911yescharacteristic
226C2×C8C2×D23611yescharacteristic
227C4⋊C4—42no—
228C4⋊C4—1182no—
229C4⋊C4—2362no—
230C4⋊C4—2362no—
231C22⋊C4—42no—
232C22⋊C4—2361no—
233C22⋊C4—2362no—
234C22⋊C4—2362no—
235C42—42no—
236C42—2361no—
237C16—2362no—
238C23#1427364711yescharacteristic
239C23#1427364711yescharacteristic
240C23C2×#1174566911yescharacteristic
241Q8—41no—
242Q8—4722no—
243Q8—2362no—
244Q8—2362no—
245D4—41no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—82no—
255C2×C4—4722no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4#1427364711yescharacteristic
259C2×C4#1427364711yescharacteristic
260C2×C4C2×(C2×D59)⋊C411yescharacteristic
261C2×C4C2×#1174567111yescharacteristic
262C8—21no—
263C8—21no—
264C8—21no—
265C8#1427364111yescharacteristic
266C8#1427364111yescharacteristic
267C22—21no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22C2×#1427364711yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—162no—
278C4—4722no—
279C4—4722no—
280C4C2×#1427364111yescharacteristic
281C4#1683742011yescharacteristic
282C2—21no—
283C2—41no—
284C2—41no—
285C2—41no—
286C2#2142266311yescharacteristic
subgroupG/H
centerC2#21422663
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1812C2
socleC2×C59#1519
prime radicalC2×#41×C59D4
cosocleC2×M16×C59C23
hypercenter#1812D59
hypocenterC59#11869
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #1812

Complements

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

NH ≅ G/Ntype
3. #21422083C2semidirect
5. C22×Dic236C4semidirect
8. #16837660C4semidirect
12. #16837440C4semidirect
163. C59#11869semidirect

Maximal subgroups

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

subgroupindex
#214337512
#214257172
#214220832
C59×#18122
#1186959