Finite Group Databasebeta

← #28497923

Subgroups — #28497923

Order 15104.

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

normal subgroups only

This group has 398 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#21432793C211yesmaximal, characteristic
2C22⋊C16×D59C212yesmaximal, complemented
3#21422037C212yesmaximal, complemented
4#21422022C211yesmaximal, characteristic, complemented
5C59×#1315C211yesmaximal, characteristic, complemented
6C2×#14273734C411yescharacteristic
7C22×C8×D59C2211yescharacteristic
8#16837909C411yescharacteristic
9C2×C16×D59—22no—
10#16837476—22no—
11(C22⋊D59)⋊C8—22no—
12#16837428C2211yescharacteristic
13#16837365C411yescharacteristic
14#16837360C411yescharacteristic
15C22⋊(C59⋊C16)C2212yes—
16(C2×D59)⋊C16—22no—
17C59×#408C2211yescharacteristic
18C59×C22⋊C16C2212yes—
19C22×C4×D59C2×C411yescharacteristic
20#14273734—42no—
21C2×#11745664—21no—
22C2×#11745664C2×C411yescharacteristic
23C22×C59⋊C8—21no—
24C22×C59⋊C8C2×C411yescharacteristic
25C2×C8×D59—42no—
26C2×C8×D59D412yes—
27C2×C8×D59D412yes—
28#14273708—21no—
29#14273703C812yes—
30#14273698C812yes—
31#14273695—21no—
32C4×C22⋊D59—22no—
33#14273686C2×C411yescharacteristic
34C2×C59⋊C16—22no—
35C16×D59—44no—
36Dic59⋊C8—22no—
37(C2×D59)⋊C8—22no—
38#14273637—21no—
39C8×Dic59—21no—
40C2×#41×C59C2×C411yescharacteristic
41C2×M16×C59C2×C411yescharacteristic
42C22×C8×C59C2311yescharacteristic
43C2×C16×C59—22no—
44C22⋊C8×C59—22no—
45C23×D59M1611yescharacteristic
46#41×C59—41no—
47#41×C59—41no—
48C2×Q8×C59C2×C811yescharacteristic
49C2×D4×C59—21no—
50C2×D4×C59C2×C811yescharacteristic
51C22×C4×C59—21no—
52C22×C4×C59C22×C411yescharacteristic
53C2×C22⋊D59—22no—
54C22×Dic59M1611yescharacteristic
55C2×C4×D59—41no—
56C2×C4×D59—41no—
57C2×C4×D59C22⋊C412yes—
58C2×C4×D59C2×C812yes—
59M16×C59—41no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C2×C8×C59C2×D412yes—
63C22⋊Dic59—21no—
64C22⋊Dic59—22no—
65(C2×D59)⋊C4—22no—
66C4⋊Dic59C2×C812yes—
67Dic59⋊C4—22no—
68C4×Dic59—21no—
69#11745664—42no—
70#11745664—41no—
71C2×C59⋊C8—22no—
72C2×C59⋊C8—21no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8C22⋊C411yescharacteristic
75C2×C59⋊C8C22⋊C411yescharacteristic
76C8×D59—24no—
77C8×D59—42no—
78C16×C59—42no—
79C59⋊C16—42no—
80C23×C59—21no—
81C23×C59C2×M1611yescharacteristic
82C22×D59—24no—
83C22×D59—41no—
84C22×D59C22⋊C812yes—
85Q8×C59—41no—
86D4×C59—41no—
87D4×C59—42no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59C2×C22⋊C411yescharacteristic
94C2×C4×C59C22×C811yescharacteristic
95C22⋊D59—44no—
96C2×Dic59—22no—
97C2×Dic59—41no—
98C2×Dic59C22⋊C812yes—
99C4×D59—22no—
100C4×D59—22no—
101C4×D59—42no—
102C8×C59—22no—
103C8×C59—41no—
104C59⋊C8—21no—
105C59⋊C8—41no—
106C59⋊C8#9912yes—
107#11468—591nomaximal, Sylow
108C22×C59—22no—
109C22×C59—21no—
110C22×C59—42no—
111C22×C59—41no—
112C22×C59C2×C22⋊C811yescharacteristic
113C2×D59—22no—
114C2×D59—22no—
115C2×D59—44no—
116C2×D59#35014yes—
117C4×C59—21no—
118C4×C59—41no—
119C4×C59C2×#9911yescharacteristic
120C4×C59#41311yescharacteristic
121Dic59—22no—
122Dic59—42no—
123#2929—591no—
124C2×C22⋊C16—592no—
125#1323—592no—
126#1315—591no—
127#1315D5911yescharacteristic, complemented
128C2×C59—41no—
129C2×C59—81no—
130C2×C59#145911yescharacteristic
131C2×C59C2×#35012yes—
132D59—24no—
133C2×#131—591no—
134C23×C8—591no—
135#525—591no—
136C22×C16—1182no—
137#435—1182no—
138C8×D4—1182no—
139#424—591no—
140#408C2×D5911yescharacteristic
141#407—591no—
142#405—591no—
143C22⋊C16—592no—
144C22⋊C16—1182no—
145C22⋊C16C2×D5912yes—
146C59#1146811yescharacteristic, complemented, Sylow
147C2×#41C4×D5911yescharacteristic
148C23×C4—591no—
149#131—2362no—
150C2×M16—591no—
151C2×M16—1181no—
152C2×M16C4×D5911yescharacteristic
153C22×C8—591no—
154C22×C8—592no—
155C22×C8—592no—
156C22×C8—1181no—
157C22×C8—2362no—
158C22×C8C22×D5911yescharacteristic
159#123—1181no—
160C22⋊Q8—592no—
161#121—592no—
162#120—1181no—
163C4×D4—1182no—
164#117—591no—
165C2×C16—22no—
166C2×C16—1182no—
167C2×C16—2364no—
168C4⋊C8—1182no—
169C22⋊C8—22no—
170C22⋊C8—1182no—
171#97—1181no—
172C4×C8—1181no—
173C24—591no—
174#41—41no—
175#41—41no—
176C2×Q8C8×D5911yescharacteristic
177C2×D4—21no—
178C2×D4—1182no—
179C2×D4C8×D5911yescharacteristic
180C22×C4—21no—
181C22×C4—591no—
182C22×C4—592no—
183C22×C4—592no—
184C22×C4—2361no—
185C22×C4—2361no—
186C22×C4C2×C4×D5911yescharacteristic
187M16—41no—
188M16—2362no—
189M16—2361no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—591no—
193C2×C8—591no—
194C2×C8—1184no—
195C2×C8—1182no—
196C2×C8—1181no—
197C2×C8—2362no—
198C2×C8—2362no—
199C2×C8D4×D5912yes—
200C4⋊C4—592no—
201C4⋊C4—1182no—
202C22⋊C4—1182no—
203C22⋊C4—1182no—
204C22⋊C4—1181no—
205C42—1181no—
206C16—42no—
207C16—2362no—
208C23—21no—
209C23—592no—
210C23—1184no—
211C23—2361no—
212C23M16×D5911yescharacteristic
213Q8—41no—
214D4—41no—
215D4—42no—
216D4—2364no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—592no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—1182no—
226C2×C4—2362no—
227C2×C4—2361no—
228C2×C4C22⋊C4×D5911yescharacteristic
229C2×C4C2×C8×D5911yescharacteristic
230C8—22no—
231C8—41no—
232C8—592no—
233C8—1181no—
234C8—2361no—
235C22—22no—
236C22—21no—
237C22—42no—
238C22—41no—
239C22—594no—
240C22—1182no—
241C22—1182no—
242C22—2364no—
243C22C22⋊C8×D5911yescharacteristic
244C4—21no—
245C4—41no—
246C4—1182no—
247C4—2362no—
248C4#99×D5911yescharacteristic
249C4#1683738311yescharacteristic
250C2—41no—
251C2—81no—
252C2—1184no—
253C2#2142241111yescharacteristic
254C2#350×D5912yes—
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#1315C2
socleC22×C59C2×C22⋊C8
prime radical#16837909C4
cosocleC22×C8×C59C23
hypercenter#1315D59
hypocenterC59#11468
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D59C2semidirect
3. #21422037C2semidirect
4. #21422022C2semidirect
5. C59×#1315C2semidirect
127. #1315D59semidirect
146. C59#11468semidirect

Maximal subgroups

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

subgroupindex
#214327932
C22⋊C16×D592
#214220372
#214220222
C59×#13152
#1146859