Finite Group Databasebeta

← #28490936

Subgroups — #28490936

Order 15104.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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#21435034C212yesmaximal
2#21424920C211yesmaximal, characteristic
3#21422660C212yesmaximal, complemented
4#21421941C211yesmaximal, characteristic, complemented
5C59×#3162C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7#16837776—22no—
8#16837750—22no—
9#16837602—24no—
10#16837552—24no—
11#16837455C2212yes—
12C2×#14273649—22no—
13#16837268C2211yescharacteristic
14#16837139—24no—
15C59×#533C2212yes—
16C59×#501—22no—
17C59×#501—22no—
18C59×#427—22no—
19C2×#97×C59C2211yescharacteristic
20C22×D236—21no—
21C22×D236—22no—
22C2×#11745671—28no—
23#14273724—22no—
24#14273724—21no—
25#14273724D412yes—
26C2×C4⋊Dic59—21no—
27C4⋊(C59⋊C8)—22no—
28#14273649—44no—
29#14273647—28no—
30C22⋊(C59⋊C8)D414yes—
31C4×C59⋊C8—22no—
32C59⋊#95—22no—
33C22×C8×C59D412yes—
34C4⋊Q8×C59D412yes—
35C4⋊Q8×C59D412yes—
36#125×C59—22no—
37C22⋊Q8×C59—22no—
38C22⋊Q8×C59—22no—
39C4×Q8×C59—22no—
40C4×Q8×C59—22no—
41#117×C59—22no—
42C2×C42×C59C2311yescharacteristic
43#107×C59—24no—
44#106×C59—22no—
45#106×C59—22no—
46#97×C59—22no—
47C23×D59—41no—
48C2×Q8×C59—22no—
49C2×Q8×C59—22no—
50C22×C4×C59C2×D412yes—
51C22×C4×C59C2×D411yescharacteristic
52C22×Dic59—41no—
53C2×D236—24no—
54C2×D236—22no—
55C2×D236—42no—
56C2×D236—42no—
57C2×D236—41no—
58C2×D236—42no—
59C2×D236—84no—
60C2×C8×C59—22no—
61C2×C8×C59—22no—
62C2×C8×C59SD1614yescomplemented
63C4⋊C4×C59—22no—
64C4⋊C4×C59—22no—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C42×C59—42no—
71C42×C59C2×D411yescharacteristic
72C42×C59C2×D412yes—
73C42×C59#4111yescharacteristic
74#11745671—88no—
75C4⋊Dic59—42no—
76C2×C59⋊C8—24no—
77C23×C59#12711yescharacteristic
78C22×D59—42no—
79C22×D59—84no—
80C22×D59—81no—
81C22×D59—81no—
82Q8×C59—44no—
83Q8×C59—82no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59#12012yes—
88C2×C4×C59#12011yescharacteristic
89C2×C4×C59#12111yescharacteristic
90C2×C4×C59#12112yes—
91C2×C4×C59C2×SD1612yes—
92C2×C4×C59#13712yes—
93C2×Dic59—82no—
94C2×Dic59—81no—
95D236—48no—
96D236—44no—
97D236—84no—
98D236—82no—
99C8×C59—24no—
100C59⋊C8—44no—
101#10220—591nomaximal, Sylow
102C22×C59—22no—
103C22×C59#39211yescharacteristic
104C22×C59#49211yescharacteristic
105C22×C59#49711yescharacteristic
106C2×D59—84no—
107C2×D59—82no—
108C2×D59—162no—
109C2×D59—162no—
110C4×C59—22no—
111C4×C59—22no—
112C4×C59—82no—
113C4×C59—82no—
114C4×C59#44812yes—
115C4×C59#46112yes—
116Dic59—161no—
117#3208—592no—
118#3162D5911yescharacteristic, complemented
119#1961—591no—
120#1518—592no—
121#1281—591no—
122C2×C59—21no—
123C2×C59—21no—
124C2×C59#200812yes—
125C2×C59#201311yescharacteristic
126D59—162no—
127#533C2×D5912yes—
128C2×#127—591no—
129#501—22no—
130#501—22no—
131#492—1182no—
132#486—1182no—
133#460—1184no—
134#450—1184no—
135#431—592no—
136#427—22no—
137C2×#102—1182no—
138C2×#97C2×D5911yescharacteristic
139#382—591no—
140#328—1184no—
141C59#1022011yescharacteristic, complemented, Sylow
142C22×D4—1182no—
143C22×D4—1181no—
144C2×SD16—1188no—
145C22×C8C22⋊D5912yes—
146C4⋊Q8C22⋊D5912yes—
147C4⋊Q8C22⋊D5912yes—
148#127—592no—
149#127—1181no—
150#127—1182no—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117—22no—
157C2×C4⋊C4—1181no—
158C2×C42C22×D5911yescharacteristic
159#107—24no—
160#106—22no—
161#106—22no—
162C4⋊C8—1182no—
163#102—1188no—
164#102—2364no—
165C22⋊C8—594no—
166#97—22no—
167C4×C8—1182no—
168#95—1182no—
169C24—2361no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—1184no—
173C2×D4—1182no—
174C2×D4—4724no—
175C2×D4—2361no—
176C2×D4—2362no—
177C2×D4—2362no—
178C2×D4—2362no—
179C22×C4—2361no—
180C22×C4D4×D5912yes—
181C22×C4C2×C22⋊D5911yescharacteristic
182SD16—4728no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—1184no—
186C2×C8#1174567114yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2362no—
194C22⋊C4—42no—
195C42—42no—
196C42D4×D5911yescharacteristic
197C42#1174568711yescharacteristic
198C42C2×C22⋊D5912yes—
199C23—4724no—
200C23—4721no—
201C23—4721no—
202C23—2362no—
203C23#1427372311yescharacteristic
204Q8—44no—
205Q8—82no—
206D4—4724no—
207D4—4722no—
208D4—2368no—
209D4—2364no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—4722no—
214C2×C4—4721no—
215C2×C4#1427369512yes—
216C2×C4#1427369611yescharacteristic
217C2×C4#1427369811yescharacteristic
218C2×C4#1427369912yes—
219C2×C4C2×#1174567112yes—
220C2×C4#1427375812yes—
221C8—24no—
222C8—2364no—
223C22—22no—
224C22—9442no—
225C22—9442no—
226C22—4722no—
227C22—4724no—
228C22#1683731011yescharacteristic
229C22#1683777611yescharacteristic
230C22#1683779411yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—9441no—
236C4#1683754112yes—
237C4#1683760412yes—
238C2—21no—
239C2—21no—
240C2—9442no—
241C2#2142514612yes—
242C2#2142517911yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3162C2
socleC22×C59#392
prime radicalC2×#14273724C22
hypercenter#3162D59
hypocenterC59#10220
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #21422660C2semidirect
4. #21421941C2semidirect
5. C59×#3162C2semidirect
62. C2×C8×C59SD16semidirect
118. #3162D59semidirect
141. C59#10220semidirect
186. C2×C8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214350342
#214249202
#214226602
#214219412
C59×#31622
#1022059