Finite Group Databasebeta

← #28490981

Subgroups — #28490981

Order 15104.

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

normal subgroups only

This group has 466 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#21435237C212yesmaximal
2#21424916C211yesmaximal, characteristic
3#21422660C212yesmaximal
4#21421941C211yesmaximal, characteristic
5C59×#3162C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837798—22no—
8#16837752—22no—
9#16837730—24no—
10#16837687—24no—
11#16837455C2212yes—
12C2×#14273652—22no—
13#16837268C2211yescharacteristic
14#16837139—24no—
15C59×#533C2212yes—
16C59×#501—22no—
17C59×#501—22no—
18C59×#427—22no—
19C2×#97×C59C2211yescharacteristic
20C22×Dic118—21no—
21#14273727—22no—
22#14273727—21no—
23#14273727D412yes—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26#14273658—28no—
27C4⋊(C59⋊C8)—22no—
28#14273652—44no—
29#14273651—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—
47C2×Q8×C59—22no—
48C2×Q8×C59—22no—
49C22×C4×C59C2×D412yes—
50C22×C4×C59C2×D411yescharacteristic
51C22×Dic59—41no—
52C22×Dic59—41no—
53C2×Dic118—22no—
54C2×Dic118—42no—
55C2×Dic118—42no—
56C2×Dic118—41no—
57C2×C8×C59—22no—
58C2×C8×C59—22no—
59C2×C8×C59SD1614yes—
60C4⋊C4×C59—22no—
61C4⋊C4×C59—22no—
62C4⋊C4×C59—22no—
63C4⋊C4×C59—22no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—42no—
68C42×C59C2×D411yescharacteristic
69C42×C59C2×D412yes—
70C42×C59#4111yescharacteristic
71C4⋊Dic59—24no—
72C4⋊Dic59—42no—
73C4⋊Dic59—42no—
74C2×C59⋊C8—24no—
75C23×C59#12711yescharacteristic
76Q8×C59—44no—
77Q8×C59—82no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59—42no—
81C2×C4×C59#12012yes—
82C2×C4×C59#12011yescharacteristic
83C2×C4×C59#12111yescharacteristic
84C2×C4×C59#12112yes—
85C2×C4×C59C2×SD1612yes—
86C2×C4×C59#13712yes—
87C2×Dic59—42no—
88C2×Dic59—81no—
89C2×Dic59—81no—
90C2×Dic59—82no—
91C2×Dic59—81no—
92Dic118—44no—
93Dic118—82no—
94C8×C59—24no—
95C59⋊C8—44no—
96#10227—591nomaximal, Sylow
97C22×C59—22no—
98C22×C59#39211yescharacteristic
99C22×C59#49211yescharacteristic
100C22×C59#49711yescharacteristic
101C4×C59—22no—
102C4×C59—22no—
103C4×C59—82no—
104C4×C59—82no—
105C4×C59#44812yes—
106C4×C59#46112yes—
107Dic59—82no—
108Dic59—161no—
109#3229—592no—
110#3162D5911yescharacteristic
111#1960—591no—
112#1518—592no—
113#1281—591no—
114C2×C59—21no—
115C2×C59—21no—
116C2×C59#200912yes—
117C2×C59#201311yescharacteristic
118#533C2×D5912yes—
119C2×C4⋊Q8—591no—
120#501—22no—
121#501—22no—
122#498—1182no—
123#487—1182no—
124#483—1184no—
125#475—1184no—
126#431—592no—
127#427—22no—
128C2×#103—1182no—
129C2×#97C2×D5911yescharacteristic
130#382—591no—
131#328—1184no—
132C59#1022711yescharacteristic, complemented, Sylow
133C22×Q8—1181no—
134C22×C8C22⋊D5912yes—
135C4⋊Q8—592no—
136C4⋊Q8—1181no—
137C4⋊Q8—1182no—
138C4⋊Q8C22⋊D5912yes—
139C4⋊Q8C22⋊D5912yes—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—1182no—
147C2×C4⋊C4—1181no—
148C2×C42C22×D5911yescharacteristic
149#107—24no—
150#106—22no—
151#106—22no—
152#106—1188no—
153C4⋊C8—1182no—
154#103—1188no—
155#103—2364no—
156C22⋊C8—594no—
157#97—22no—
158C4×C8—1182no—
159#95—1182no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1182no—
163C2×Q8—2362no—
164C2×Q8—2362no—
165C2×Q8—2361no—
166C22×C4—2361no—
167C22×C4—2361no—
168C22×C4D4×D5912yes—
169C22×C4C2×C22⋊D5911yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1184no—
173C2×C8#1174567114yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—1184no—
181C4⋊C4—2362no—
182C4⋊C4—2362no—
183C22⋊C4—42no—
184C42—42no—
185C42D4×D5911yescharacteristic
186C42#1174568711yescharacteristic
187C42C2×C22⋊D5912yes—
188C23#1427372311yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—4722no—
192Q8—2364no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—4722no—
197C2×C4—4721no—
198C2×C4—4721no—
199C2×C4—4721no—
200C2×C4—2362no—
201C2×C4#1427369512yes—
202C2×C4#1427369611yescharacteristic
203C2×C4#1427369811yescharacteristic
204C2×C4#1427369912yes—
205C2×C4C2×#1174567112yes—
206C2×C4#1427375812yes—
207C8—24no—
208C8—2364no—
209C22—22no—
210C22#1683731011yescharacteristic
211C22#1683777611yescharacteristic
212C22#1683779411yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—9441no—
218C4—4722no—
219C4#1683754112yes—
220C4#1683760412yes—
221C2—21no—
222C2—21no—
223C2#2142515312yes—
224C2#2142517911yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3162C2
socleC22×C59#392
prime radicalC23×C59#127
hypercenter#3162D59
hypocenterC59#10227
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
132. C59#10227semidirect

Maximal subgroups

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

subgroupindex
#214352372
#214249162
#214226602
#214219412
C59×#31622
#1022759