Finite Group Databasebeta

← #28484013

Subgroups — #28484013

Order 15104.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#21434421C211yesmaximal, characteristic
2#21433198C211yesmaximal, characteristic, complemented
3#21422514C212yesmaximal
4#21421934C212yesmaximal, complemented
5C59×#1526C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837759—22no—
8#16837754C412yes—
9#16837750C412yes—
10#16837533—22no—
11Q8×C59⋊C8—22no—
12#16837463—22no—
13#16837455C2212yes—
14#16837396—22no—
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837131—22no—
17#16837131—22no—
18C59×#533C2211yescharacteristic
19C4×M16×C59C2212yes—
20C59×#330—22no—
21C2×#11745685—21no—
22C22×C59⋊C8—21no—
23#14273727C2×C411yescharacteristic
24#14273724C2×C411yescharacteristic
25#14273715—21no—
26#14273701—22no—
27C4×D236—22no—
28C4⋊(C59⋊C8)—24no—
29C4⋊(C59⋊C8)D412yes—
30C4⋊(C59⋊C8)D412yes—
31#14273650—24no—
32#14273647—24no—
33C22⋊(C59⋊C8)—22no—
34#14273638—22no—
35C4×C59⋊C8—22no—
36C4×C59⋊C8—22no—
37C4×C59⋊C8—21no—
38C4×C59⋊C8C2×C412yes—
39C2×M16×C59—22no—
40C4⋊Q8×C59C2×C412yes—
41#125×C59D412yes—
42C22⋊Q8×C59—22no—
43C4×Q8×C59—22no—
44#117×C59D412yes—
45C2×C42×C59C2311yescharacteristic
46#97×C59—22no—
47C4×C8×C59—22no—
48C2×Q8×C59—22no—
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59#4112yes—
51C2×C22⋊D59—41no—
52#11745685—44no—
53C2×D236—21no—
54C2×D236—22no—
55C2×C4×D59—41no—
56C2×Dic118—21no—
57M16×C59—24no—
58M16×C59—42no—
59C2×C8×C59—42no—
60C4⋊C4×C59—22no—
61C4⋊C4×C59—24no—
62C4⋊C4×C59C22⋊C414yes—
63C22⋊C4×C59—22no—
64C42×C59—22no—
65C42×C59C22×C411yescharacteristic
66C42×C59C2×D411yescharacteristic
67C42×C59C2×D412yes—
68(C2×D59)⋊C4—42no—
69C4⋊Dic59—22no—
70C2×C59⋊C8—22no—
71C2×C59⋊C8—22no—
72C2×C59⋊C8—22no—
73C2×C59⋊C8—42no—
74C23×C59#12311yescharacteristic
75C22×D59—41no—
76Q8×C59—42no—
77Q8×C59—42no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—24no—
82C2×C4×C59C2×C22⋊C411yescharacteristic
83C2×C4×C59C4×D412yes—
84C2×C4×C59#12011yescharacteristic
85C2×C4×C59#12112yes—
86C22⋊D59—82no—
87C2×Dic59—41no—
88D236—42no—
89D236—42no—
90D236—42no—
91C4×D59—82no—
92Dic118—42no—
93C8×C59—44no—
94C59⋊C8—42no—
95C59⋊C8—42no—
96C59⋊C8—42no—
97#9070—591nomaximal, Sylow
98C22×C59—22no—
99C22×C59—21no—
100C22×C59#38611yescharacteristic
101C2×D59—82no—
102C2×D59—81no—
103C4×C59—22no—
104C4×C59—21no—
105C4×C59—44no—
106C4×C59#44312yes—
107C4×C59#44412yes—
108C4×C59#45412yes—
109Dic59—81no—
110#3131—591no—
111#2986—591no—
112#1526D5911yescharacteristic, complemented
113#1484—592no—
114#1278—592no—
115C2×C59—41no—
116C2×C59#187612yes—
117C2×C59#188811yescharacteristic
118D59—161no—
119#533C2×D5911yescharacteristic
120#532—591no—
121#488—1182no—
122#487—592no—
123#486—592no—
124#446—1182no—
125C8×Q8—1182no—
126#433—1182no—
127#431—592no—
128#416—1182no—
129C4×M16C2×D5912yes—
130C2×C4×C8—591no—
131#330—22no—
132D4⋊C8—1182no—
133D4⋊C8—1182no—
134C59#907011yescharacteristic, complemented, Sylow
135C2×#41—1181no—
136C2×M16—22no—
137C22×C8—1181no—
138C4⋊Q8—591no—
139C4⋊Q8C4×D5912yes—
140#127—591no—
141#125C22⋊D5912yes—
142#124—1181no—
143C22⋊Q8—22no—
144#121—1182no—
145C4×Q8—22no—
146C4×D4—1182no—
147#117D23612yes—
148C2×C42C22×D5911yescharacteristic
149C4⋊C8—592no—
150C4⋊C8—592no—
151C4⋊C8—1184no—
152#103—1184no—
153#102—1184no—
154C22⋊C8—1182no—
155#97—22no—
156#97—1182no—
157C4×C8—22no—
158C4×C8—592no—
159C4×C8—1182no—
160C4×C8—1182no—
161C4×C8—1181no—
162#41—2364no—
163C2×Q8—22no—
164C2×Q8—1181no—
165C2×D4—1181no—
166C2×D4—1182no—
167C2×D4—2361no—
168C22×C4—2361no—
169C22×C4C2×D23611yescharacteristic
170C22×C4#1174568712yes—
171M16—24no—
172M16—42no—
173C2×C8—42no—
174C2×C8—1182no—
175C2×C8—1182no—
176C2×C8—1182no—
177C2×C8—2362no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—1182no—
181C4⋊C4(C2×D59)⋊C414yes—
182C22⋊C4—22no—
183C22⋊C4—2362no—
184C42—22no—
185C42C2×C4×D5911yescharacteristic
186C42D4×D5912yes—
187C42C2×C22⋊D5911yescharacteristic
188C23—2361no—
189C23#1427371111yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—2362no—
193D4—4722no—
194D4—2362no—
195D4—2362no—
196D4—2362no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—4722no—
202C2×C4—2361no—
203C2×C4C2×(C2×D59)⋊C411yescharacteristic
204C2×C4(C22⋊D59)⋊C412yes—
205C2×C4#1427369711yescharacteristic
206C2×C4#1427369912yes—
207C8—44no—
208C8—2362no—
209C8—2362no—
210C8—2362no—
211C22—22no—
212C22—21no—
213C22—4722no—
214C22—4721no—
215C22#1683727911yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—4721no—
220C4#1683751612yes—
221C4#1683752112yes—
222C4#1683757712yes—
223C2—41no—
224C2—9441no—
225C2#2142442112yes—
226C2#2142449711yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1526C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1526D59
hypocenterC59#9070
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #21433198C2semidirect
4. #21421934C2semidirect
5. C59×#1526C2semidirect
112. #1526D59semidirect
134. C59#9070semidirect

Maximal subgroups

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

subgroupindex
#214344212
#214331982
#214225142
#214219342
C59×#15262
#907059