Finite Group Databasebeta

← #28491713

Subgroups — #28491713

Order 15104.

500 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 125 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#21425023C212yesmaximal
2#21424901C211yesmaximal, characteristic
3#21421979C211yesmaximal, characteristic, complemented
4#21421937C212yesmaximal, complemented
5C59×#2571C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7C2×C4⋊(C59⋊C8)C2212yes—
8C2×#14273647—22no—
9C2×#14273647—22no—
10C2×#14273647—22no—
11C2×C4×C59⋊C8C2211yescharacteristic
12C59×#400—22no—
13C59×#389—22no—
14C59×#384C2211yescharacteristic
15C59×#382—22no—
16C4×C4⋊C4×C59C2212yes—
17C22×D236—21no—
18C22×D236—22no—
19C22×C59⋊C8—21no—
20C22×C59⋊C8—22no—
21#14273724—22no—
22#14273724—21no—
23#14273724Q812yes—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273647—44no—
27#14273647—44no—
28#14273647—44no—
29C4×C59⋊C8—22no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59—22no—
32C2×C4⋊C4×C59D412yes—
33C2×C4⋊C4×C59D412yes—
34C2×C4⋊C4×C59D412yes—
35C2×C42×C59—22no—
36C2×C42×C59C2311yescharacteristic
37#95×C59—22no—
38#95×C59—24no—
39#95×C59—22no—
40C23×D59—41no—
41C22×C4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59—22no—
44C22×C4×C59C2×D411yescharacteristic
45C22×C4×C59C2×D412yes—
46C2×D236—22no—
47C2×D236—24no—
48C2×D236—42no—
49C2×D236—42no—
50C2×D236—42no—
51C2×D236—42no—
52C2×D236—42no—
53C2×D236—42no—
54C2×D236—41no—
55C2×D236—42no—
56C4⋊C4×C59—22no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59—82no—
64C42×C59—42no—
65C42×C59—42no—
66C42×C59C2×Q811yescharacteristic
67C42×C59#4112yes—
68C42×C59#4111yescharacteristic
69C2×C59⋊C8—22no—
70C2×C59⋊C8—41no—
71C2×C59⋊C8—41no—
72C2×C59⋊C8—42no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8—42no—
75C23×C59#12011yescharacteristic
76C22×D59—42no—
77C22×D59—82no—
78C22×D59—82no—
79C22×D59—81no—
80C22×D59—81no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—44no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59C22⋊Q811yescharacteristic
92C2×C4×C59C22⋊Q812yes—
93C2×C4×C59#12411yescharacteristic
94C2×C4×C59#12412yes—
95C2×C4×C59#13512yes—
96C2×C4×C59#13512yes—
97C2×C4×C59AGL(1,ℤ/8)12yes—
98C2×C4×C59AGL(1,ℤ/8)12yes—
99D236—44no—
100D236—44no—
101D236—44no—
102D236—82no—
103D236—82no—
104D236—82no—
105C59⋊C8—42no—
106C59⋊C8—82no—
107#10351—591nomaximal, Sylow
108C22×C59#39311yescharacteristic
109C22×C59#44912yes—
110C22×C59#44912yes—
111C22×C59#45311yescharacteristic
112C22×C59#45411yescharacteristic
113C2×D59—84no—
114C2×D59—82no—
115C2×D59—161no—
116C2×D59—161no—
117C2×D59—162no—
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—82no—
121C4×C59—82no—
122C4×C59—82no—
123C4×C59#47812yes—
124C4×C59#47812yes—
125C4×C59#48812yes—
126C4×C59#49012yes—
127#2571D5911yescharacteristic, complemented
128#1982—592no—
129#1957—591no—
130#1296—591no—
131#1279—592no—
132C2×C59#162011yescharacteristic
133C2×C59#162411yescharacteristic
134C2×C59#162812yes—
135C2×C59#202512yes—
136C2×C59#202711yescharacteristic
137D59—162no—
138C2×#127—591no—
139C2×C4⋊C8—592no—
140C2×#102—1182no—
141C2×#102—1182no—
142C2×#102—1182no—
143C2×C4×C8—591no—
144#400—22no—
145#389—22no—
146#384C2×D5911yescharacteristic
147#382—22no—
148C4×C4⋊C4C2×D5912yes—
149C59#1035111yescharacteristic, complemented, Sylow
150C22×D4—1181no—
151C22×D4—1182no—
152C22×C8—1181no—
153C22×C8—1182no—
154#127—592no—
155#127—1182no—
156#127—1181no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊D5912yes—
160C2×C4⋊C4C22⋊D5912yes—
161C2×C4⋊C4C22⋊D5912yes—
162C2×C42—22no—
163C2×C42C22×D5911yescharacteristic
164C4⋊C8—1182no—
165C4⋊C8—1182no—
166#102—2364no—
167#102—2364no—
168#102—2364no—
169C4×C8—1182no—
170#95—22no—
171#95—24no—
172#95—22no—
173C24—2361no—
174C2×D4—1182no—
175C2×D4—1184no—
176C2×D4—2362no—
177C2×D4—2362no—
178C2×D4—2361no—
179C2×D4—2362no—
180C2×D4—2362no—
181C2×D4—2362no—
182C2×D4—2362no—
183C2×D4—2362no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C2×C22⋊D5911yescharacteristic
188C22×C4C2×C22⋊D5912yes—
189C2×C8—1182no—
190C2×C8—2362no—
191C2×C8—2362no—
192C2×C8—2362no—
193C2×C8—2361no—
194C2×C8—2361no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C42—42no—
204C42—42no—
205C42Q8×D5911yescharacteristic
206C42#1174568912yes—
207C42#1174568911yescharacteristic
208C23—4722no—
209C23—4722no—
210C23—4721no—
211C23—4721no—
212C23—2362no—
213C23#1427369611yescharacteristic
214D4—4722no—
215D4—4722no—
216D4—4722no—
217D4—2364no—
218D4—2364no—
219D4—2364no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#1427370311yescharacteristic
231C2×C4#1427370312yes—
232C2×C4#1427371611yescharacteristic
233C2×C4#1427371612yes—
234C2×C4#1427374712yes—
235C2×C4#1427374712yes—
236C2×C4#1427375212yes—
237C2×C4#1427375212yes—
238C8—4722no—
239C8—2362no—
240C22—9441no—
241C22—9441no—
242C22—9442no—
243C22—4724no—
244C22—4722no—
245C22#1683731311yescharacteristic
246C22#1683754712yes—
247C22#1683754712yes—
248C22#1683757011yescharacteristic
249C22#1683757511yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4#1683769912yes—
256C4#1683769912yes—
257C4#1683775912yes—
258C4#1683776812yes—
259C2—9442no—
260C2#2142315511yescharacteristic
261C2#2142317911yescharacteristic
262C2#2142320612yes—
263C2#2142523912yes—
264C2#2142524911yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2571C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenter#2571D59
hypocenterC59#10351
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #21421979C2semidirect
4. #21421937C2semidirect
5. C59×#2571C2semidirect
127. #2571D59semidirect
149. C59#10351semidirect

Maximal subgroups

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

subgroupindex
#214250232
#214249012
#214219792
#214219372
C59×#25712
#1035159