Finite Group Databasebeta

← #28501125

Subgroups — #28501125

Order 15104.

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

normal subgroups only

This group has 372 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#21425325C211yesmaximal, characteristic
2#21423098C212yesmaximal
3#21422102C212yesmaximal, complemented
4#21422040C211yesmaximal, characteristic, complemented
5C59×#2822C211yesmaximal, characteristic, complemented
6#16837983C2211yescharacteristic
7#16837715—22no—
8C2×#14273655—21no—
9#16837401—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C2×#125×C59—21no—
14C59×#400—22no—
15C59×#398—21no—
16C59×#398—22no—
17C59×#398C2212yes—
18C59×#382C2211yescharacteristic
19C2×#11745685—21no—
20C2×#11745664—21no—
21#14273707—41no—
22#14273702—41no—
23#14273701—22no—
24#14273660—42no—
25#14273655—42no—
26#14273650—41no—
27#14273647—41no—
28#14273647—42no—
29C22⋊(C59⋊C8)—22no—
30#125×C59—42no—
31#125×C59—42no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C4⋊C4×C59D412yes—
39C2×C4⋊C4×C59C2311yescharacteristic
40C2×C42×C59D411yescharacteristic
41#95×C59—22no—
42#95×C59—21no—
43#95×C59—22no—
44#95×C59—22no—
45#95×C59D412yes—
46C22×C4×C59—21no—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D412yes—
52C2×C22⋊D59—41no—
53#11745685—41no—
54#11745685—41no—
55#11745685—41no—
56#11745685—41no—
57C2×D236—21no—
58C2×D236—22no—
59C2×C4×D59—41no—
60C2×Dic118—21no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59—44no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—42no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—81no—
70C42×C59—21no—
71C42×C59D812yes—
72(C2×D59)⋊C4—42no—
73C4⋊Dic59—22no—
74#11745664—41no—
75#11745664—41no—
76C2×C59⋊C8—42no—
77C2×C59⋊C8—41no—
78C23×C59#12011yescharacteristic
79C22×D59—41no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59C2×D811yescharacteristic
96C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
97C22⋊D59—81no—
98C22⋊D59—81no—
99C2×Dic59—41no—
100D236—41no—
101D236—41no—
102D236—82no—
103C4×D59—81no—
104C4×D59—81no—
105Dic118—41no—
106Dic118—41no—
107C59⋊C8—81no—
108C59⋊C8—82no—
109#12157—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—21no—
112C22×C59#44711yescharacteristic
113C22×C59#45411yescharacteristic
114C22×C59B(4,𝔽2)11yescharacteristic
115C2×D59—81no—
116C2×D59—81no—
117C2×D59—81no—
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—42no—
121C4×C59—42no—
122C4×C59—82no—
123C4×C59—82no—
124C4×C59—81no—
125Dic59—81no—
126#2822D5911yescharacteristic, complemented
127#2040—591no—
128#1610—592no—
129#1346—592no—
130#1324—591no—
131C2×C59—21no—
132C2×C59—21no—
133C2×C59#159811yescharacteristic
134C2×C59#220111yescharacteristic
135C2×C59#220311yescharacteristic
136D59—161no—
137#537—591no—
138C2×#125—21no—
139#480—1182no—
140C2×#104—1181no—
141#417—1181no—
142#400—22no—
143#398—21no—
144#398—22no—
145#398C2×D5912yes—
146#382C2×D5911yescharacteristic
147#343—591no—
148#327—1182no—
149#324—592no—
150C59#1215711yescharacteristic, complemented, Sylow
151C2×#41—1181no—
152C2×M16—1181no—
153#125—42no—
154#125—42no—
155C22⋊Q8—2361no—
156#121—1182no—
157#121—2361no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D5911yescharacteristic
164C2×C4⋊C4C22⋊D5912yes—
165C2×C4⋊C4C22×D5911yescharacteristic
166C2×C42C22⋊D5911yescharacteristic
167#107—2362no—
168#104—2362no—
169#103—2361no—
170#102—2361no—
171#102—2362no—
172C22⋊C8—1182no—
173#95—22no—
174#95—21no—
175#95—22no—
176#95—22no—
177#95C22⋊D5912yes—
178#41—2361no—
179#41—2361no—
180#41—2361no—
181#41—2361no—
182C2×Q8—1181no—
183C2×D4—1182no—
184C2×D4—1181no—
185C2×D4—2361no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2361no—
191C22×C4C2×C22⋊D5911yescharacteristic
192C22×C4C2×C22⋊D5912yes—
193M16—2361no—
194M16—2361no—
195C2×C8—2362no—
196C2×C8—2361no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—1182no—
207C22⋊C4—2362no—
208C42—21no—
209C42#1174566912yes—
210C23—2361no—
211C23#1427369611yescharacteristic
212Q8—2361no—
213Q8—2361no—
214D4—4722no—
215D4—4721no—
216D4—4721no—
217D4—2361no—
218D4—2361no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—4721no—
235C2×C4—4721no—
236C2×C4—2361no—
237C2×C4C2×#1174566911yescharacteristic
238C2×C4#1427375311yescharacteristic
239C8—4721no—
240C8—4722no—
241C22—21no—
242C22—21no—
243C22—4721no—
244C22—4721no—
245C22—4721no—
246C22#1683753611yescharacteristic
247C22#1683757511yescharacteristic
248C22#1683758911yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—4721no—
257C2—21no—
258C2—21no—
259C2—9441no—
260C2#2142303111yescharacteristic
261C2#2142610911yescharacteristic
262C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2822C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2822D59
hypocenterC59#12157
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
3. #21422102C2semidirect
4. #21422040C2semidirect
5. C59×#2822C2semidirect
126. #2822D59semidirect
150. C59#12157semidirect

Maximal subgroups

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

subgroupindex
#214253252
#214230982
#214221022
#214220402
C59×#28222
#1215759