Finite Group Databasebeta

← #28500951

Subgroups — #28500951

Order 15104.

398 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425142C211yesmaximal, characteristic
2#21423132C212yesmaximal
3#21422123C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#2809C211yesmaximal, characteristic, complemented
6C2×#14273758—21no—
7#16837983C2211yescharacteristic
8#16837720—22no—
9C2×#14273655—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C59×#400C2212yes—
14C59×#398—22no—
15C59×#391—21no—
16C59×#391C2211yescharacteristic
17C59×#389—21no—
18C59×#387—22no—
19C2×#11745685—21no—
20#14273758—42no—
21#14273758—42no—
22C2×#11745672—41no—
23C2×#11745671—41no—
24C2×#11745664—21no—
25#14273707—41no—
26#14273702—41no—
27#14273701—22no—
28#14273658—42no—
29#14273655—42no—
30#14273647—42no—
31C22⋊(C59⋊C8)—22no—
32C22×Q8×C59D411yescharacteristic
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59D412yes—
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59—21no—
40C2×C42×C59D411yescharacteristic
41#95×C59—22no—
42#95×C59—22no—
43#95×C59—22no—
44#95×C59—22no—
45#95×C59D412yes—
46C2×Q8×C59—21no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59SD1612yes—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D412yes—
56C2×C22⋊D59—41no—
57#11745685—41no—
58#11745685—41no—
59#11745685—41no—
60#11745685—41no—
61C2×D236—21no—
62C2×D236—22no—
63C2×C4×D59—41no—
64C2×Dic118—21no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—44no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—81no—
70C4⋊C4×C59—81no—
71C42×C59—41no—
72C42×C59—81no—
73#11745672—44no—
74#11745671—44no—
75(C2×D59)⋊C4—42no—
76C4⋊Dic59—22no—
77#11745664—22no—
78#11745664—41no—
79C2×C59⋊C8—41no—
80C2×C59⋊C8—42no—
81C23×C59#12011yescharacteristic
82C22×D59—41no—
83Q8×C59—42no—
84Q8×C59—81no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59C2×SD1611yescharacteristic
102C2×C4×C59#13711yescharacteristic
103C22⋊D59—81no—
104C22⋊D59—81no—
105C2×Dic59—41no—
106D236—41no—
107D236—41no—
108D236—82no—
109C4×D59—81no—
110C4×D59—81no—
111Dic118—41no—
112Dic118—41no—
113C59⋊C8—42no—
114C59⋊C8—82no—
115#12131—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59#44811yescharacteristic
119C22×C59#45311yescharacteristic
120C22×C59B(4,𝔽2)11yescharacteristic
121C2×D59—81no—
122C2×D59—81no—
123C2×D59—81no—
124C4×C59—21no—
125C4×C59—21no—
126C4×C59—42no—
127C4×C59—42no—
128C4×C59—42no—
129C4×C59—82no—
130C4×C59—82no—
131Dic59—81no—
132#2809D5911yescharacteristic, complemented
133#2007—591no—
134#1616—592no—
135#1353—592no—
136#1319—591no—
137C2×C59—21no—
138C2×C59—21no—
139C2×C59#159611yescharacteristic
140C2×C59#220111yescharacteristic
141C2×C59#220411yescharacteristic
142D59—161no—
143C2×#137—1181no—
144#537—591no—
145#481—1182no—
146C2×#104—1181no—
147#400C2×D5912yes—
148#398—22no—
149#391—21no—
150#391C2×D5911yescharacteristic
151#389—21no—
152#387—22no—
153#343—591no—
154#327—1182no—
155#324—592no—
156C59#1213111yescharacteristic, complemented, Sylow
157C2×#41—1181no—
158C22×Q8C22⋊D5911yescharacteristic
159#137—2362no—
160#137—2362no—
161C2×Q16—2361no—
162C2×SD16—2361no—
163C2×M16—1181no—
164C22⋊Q8—2361no—
165#121—1182no—
166#121—2361no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C22⋊D5912yes—
172C2×C4⋊C4C22×D5911yescharacteristic
173C2×C42—21no—
174C2×C42C22⋊D5911yescharacteristic
175#106—2362no—
176#104—2362no—
177#102—2362no—
178C22⋊C8—1182no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95C22⋊D5912yes—
184#41—2361no—
185#41—2361no—
186#41—2361no—
187#41—2361no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1181no—
192C2×Q8#1174567112yes—
193C2×D4—1182no—
194C2×D4—1181no—
195C2×D4—2361no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—2361no—
201C22×C4C2×C22⋊D5911yescharacteristic
202C22×C4C2×C22⋊D5912yes—
203Q16—2364no—
204SD16—2364no—
205M16—1182no—
206M16—2361no—
207C2×C8—2362no—
208C2×C8—2361no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—1182no—
216C22⋊C4—2362no—
217C42—41no—
218C42—81no—
219C23—2361no—
220C23#1427369611yescharacteristic
221Q8—42no—
222Q8—81no—
223Q8—2361no—
224Q8—2361no—
225D4—4722no—
226D4—4721no—
227D4—4721no—
228D4—2361no—
229D4—2361no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—4721no—
247C2×C4—4721no—
248C2×C4—2361no—
249C2×C4C2×#1174567111yescharacteristic
250C2×C4#1427375811yescharacteristic
251C8—4722no—
252C8—2362no—
253C22—21no—
254C22—21no—
255C22—4721no—
256C22—4721no—
257C22—4721no—
258C22#1683754111yescharacteristic
259C22#1683757011yescharacteristic
260C22#1683758911yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4—82no—
268C4—4721no—
269C2—21no—
270C2—21no—
271C2—9441no—
272C2#2142301911yescharacteristic
273C2#2142610911yescharacteristic
274C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2809C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2809D59
hypocenterC59#12131
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
3. #21422123C2semidirect
4. #21422030C2semidirect
5. C59×#2809C2semidirect
132. #2809D59semidirect
156. C59#12131semidirect

Maximal subgroups

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

subgroupindex
#214251422
#214231322
#214221232
#214220302
C59×#28092
#1213159