Finite Group Databasebeta

← #28483075

Subgroups — #28483075

Order 15104.

396 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 119 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#21427282C211yesmaximal, characteristic
2#21425066C211yesmaximal, characteristic
3#21424230C212yesmaximal
4#21421978C212yesmaximal
5C2×C59×#330C211yesmaximal, characteristic
6C2×#14273638C2211yescharacteristic
7#16837301—22no—
8#16837288—22no—
9#16837288—22no—
10#16837272C2211yescharacteristic
11#16837268C2212yes—
12#16837165—22no—
13#16837165—22no—
14#16837164—22no—
15C2×#125×C59C2211yescharacteristic
16C2×#97×C59C2212yes—
17C59×#330C414yescomplemented
18C22×C59⋊C8—21no—
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—22no—
21C2×C4⋊Dic59—22no—
22C2×C4⋊Dic59D412yes—
23C2×C4⋊Dic59D412yes—
24C2×C4×Dic59—22no—
25#14273638—22no—
26C59⋊#95—22no—
27C59⋊#95—22no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C22×C8×C59—22no—
31#125×C59—22no—
32#125×C59C2×C412yes—
33#125×C59Q812yes—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59D412yes—
36C2×C42×C59C2311yescharacteristic
37#97×C59—22no—
38#97×C59C2×C414yes—
39C22×C4×C59—22no—
40C22×C4×C59C2×D411yescharacteristic
41C22×C4×C59C2×D412yes—
42C22×Dic59—22no—
43C22×Dic59—22no—
44C22×Dic59—41no—
45C2×C8×C59—24no—
46C2×C8×C59—42no—
47C2×C8×C59—42no—
48C4⋊C4×C59—24no—
49C4⋊C4×C59—24no—
50C4⋊C4×C59—22no—
51C4⋊C4×C59C4⋊C414yes—
52C42×C59C22×C411yescharacteristic
53C42×C59C2×Q811yescharacteristic
54C42×C59#4112yes—
55C4⋊Dic59—42no—
56C4⋊Dic59—42no—
57C4⋊Dic59—42no—
58C4⋊Dic59—44no—
59C4⋊Dic59—82no—
60C4×Dic59—82no—
61C2×C59⋊C8—22no—
62C2×C59⋊C8—41no—
63C2×C59⋊C8—41no—
64C23×C59#12011yescharacteristic
65C2×C4×C59—21no—
66C2×C4×C59—22no—
67C2×C4×C59—22no—
68C2×C4×C59—21no—
69C2×C4×C59—24no—
70C2×C4×C59—42no—
71C2×C4×C59—42no—
72C2×C4×C59C2×C4⋊C411yescharacteristic
73C2×C4×C59C4×D412yes—
74C2×C4×C59C22⋊Q812yes—
75C2×C4×C59#12311yescharacteristic
76C2×Dic59—42no—
77C2×Dic59—42no—
78C2×Dic59—42no—
79C2×Dic59—42no—
80C2×Dic59—42no—
81C2×Dic59—42no—
82C2×Dic59—82no—
83C2×Dic59—81no—
84C8×C59—44no—
85C59⋊C8—42no—
86#8915—591nomaximal, Sylow
87C22×C59#38511yescharacteristic
88C22×C59#45412yes—
89C22×C59#45411yescharacteristic
90C22×C59#45512yes—
91C22×C59#45611yescharacteristic
92C4×C59—21no—
93C4×C59—21no—
94C4×C59—22no—
95C4×C59—22no—
96C4×C59—44no—
97Dic59—82no—
98Dic59—82no—
99Dic59—161no—
100#2382—591no—
101#1992—591no—
102#1836—592no—
103C2×#330D5911yescharacteristic
104#1295—592no—
105C2×C59#165811yescharacteristic
106C2×C59#166211yescharacteristic
107C2×C59#166412yes—
108C2×C59#185711yescharacteristic
109C2×C59#186012yes—
110C2×#125C2×D5911yescharacteristic
111C2×#97—591no—
112C2×#97C2×D5912yes—
113#389—1182no—
114#387—1182no—
115#387—1182no—
116#384—591no—
117#382—592no—
118#340—1182no—
119#340—1182no—
120#340—1182no—
121#330Dic5914yescomplemented
122C59#891511yescharacteristic, complemented, Sylow
123C22×C8—22no—
124C22×C8—1181no—
125#125—22no—
126#125Dic11812yes—
127#125C2×Dic5912yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—592no—
130C2×C4⋊C4—592no—
131C2×C4⋊C4—1182no—
132C2×C4⋊C4—1181no—
133C2×C4⋊C4—1182no—
134C2×C4⋊C4D23612yes—
135C2×C42—1182no—
136C2×C42C22×D5911yescharacteristic
137#97—22no—
138#97—1182no—
139#97C2×Dic5914yes—
140#95—1182no—
141#95—1182no—
142#95—1182no—
143#95—1182no—
144C22×C4—22no—
145C22×C4—1182no—
146C22×C4—1182no—
147C22×C4—2361no—
148C22×C4C2×D23611yescharacteristic
149C22×C4D4×D5912yes—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—1182no—
154C2×C8—2361no—
155C2×C8—2361no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—4722no—
160C4⋊C4—2364no—
161C4⋊C4—2362no—
162C4⋊C4—2362no—
163C4⋊C4—2362no—
164C4⋊C4C4⋊Dic5914yes—
165C42—4722no—
166C42C2×Dic11811yescharacteristic
167C42#1174568712yes—
168C42C22×Dic5911yescharacteristic
169C23#1427369711yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—4722no—
178C2×C4—4721no—
179C2×C4—2362no—
180C2×C4—2362no—
181C2×C4—2362no—
182C2×C4—2362no—
183C2×C4—2362no—
184C2×C4—2362no—
185C2×C4C2×C4⋊Dic5911yescharacteristic
186C2×C4D4×Dic5912yes—
187C2×C4#1427370412yes—
188C2×C4#1427371111yescharacteristic
189C8—44no—
190C8—2362no—
191C22#1683727711yescharacteristic
192C22#1683757612yes—
193C22#1683757711yescharacteristic
194C22#1683758112yes—
195C22#1683758711yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—9441no—
202C4—4722no—
203C4—4722no—
204C2#2142339911yescharacteristic
205C2#2142342011yescharacteristic
206C2#2142343212yes—
207C2#2142431311yescharacteristic
208C2#2142433112yes—
subgroupG/H
centerC23#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#330C2
socle, prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenterC2×#330D59
hypocenterC59#8915
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
17. C59×#330C4semidirect
121. #330Dic59semidirect
122. C59#8915semidirect

Maximal subgroups

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

subgroupindex
#214272822
#214250662
#214242302
#214219782
C2×C59×#3302
#891559