Finite Group Databasebeta

← #28492832

Subgroups — #28492832

Order 15104.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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#21435477C211yesmaximal, characteristic
2#21422816C212yesmaximal, complemented
3#21422706C212yesmaximal, complemented
4#21422007C211yesmaximal, characteristic, complemented
5C59×#1838C211yesmaximal, characteristic, complemented
6#16837952C2212yes—
7#16837693—22no—
8#16837655—22no—
9#16837598—22no—
10#16837459C2212yes—
11#16837147—22no—
12#16837145—22no—
13#16837141—24no—
14C59×#423C2211yescharacteristic
15C59×#423C2211yescharacteristic
16C2×#97×C59C2211yescharacteristic
17C59×#341—22no—
18C2×#11745685—22no—
19C2×D472—22no—
20#14273727D412yes—
21#14273724D412yes—
22#14273715—22no—
23#14273701—22no—
24#14273701—22no—
25C4×D236—22no—
26C4×D236—22no—
27#14273659—22no—
28C4⋊(C59⋊C8)—22no—
29#14273649—24no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32C2×M16×C59—21no—
33C2×M16×C59D412yes—
34C22×C8×C59—21no—
35C2×C42×C59C2311yescharacteristic
36C4⋊C8×C59—22no—
37C4⋊C8×C59—22no—
38#97×C59—21no—
39#97×C59D412yes—
40C22×C4×C59C2×D411yescharacteristic
41C22×C4×C59#4112yes—
42C2×C22⋊D59—42no—
43#11745685—84no—
44C2×D236—22no—
45C2×D236—22no—
46C2×D236—22no—
47C2×C4×D59—42no—
48C2×Dic118—22no—
49M16×C59—42no—
50M16×C59—42no—
51C2×C8×C59—22no—
52C2×C8×C59—22no—
53C2×C8×C59—41no—
54C2×C8×C59—41no—
55C2×C8×C59—41no—
56C42×C59C2×D412yes—
57C42×C59C2×D411yescharacteristic
58C42×C59#4111yescharacteristic
59(C2×D59)⋊C4—42no—
60(C2×D59)⋊C4—42no—
61C4⋊Dic59—22no—
62C4⋊Dic59—22no—
63C2×C59⋊C8—42no—
64D472—82no—
65C23×C59#12411yescharacteristic
66C22×D59—42no—
67C2×C4×C59—21no—
68C2×C4×C59—21no—
69C2×C4×C59—42no—
70C2×C4×C59#12011yescharacteristic
71C2×C4×C59#12111yescharacteristic
72C2×C4×C59#12112yes—
73C2×C4×C59#12312yes—
74C22⋊D59—84no—
75C2×Dic59—42no—
76D236—42no—
77D236—42no—
78D236—82no—
79D236—82no—
80C4×D59—84no—
81Dic118—82no—
82C8×C59—42no—
83C8×C59—42no—
84C8×C59—81no—
85C59⋊C8—82no—
86#10525—591nomaximal, Sylow
87C22×C59—22no—
88C22×C59—21no—
89C22×C59#39411yescharacteristic
90C2×D59—82no—
91C2×D59—82no—
92C2×D59—82no—
93C4×C59—22no—
94C4×C59—22no—
95C4×C59—21no—
96C4×C59#45411yescharacteristic
97C4×C59#45511yescharacteristic
98Dic59—82no—
99#3253—591no—
100#1838D5911yescharacteristic, complemented
101#1557—592no—
102#1530—592no—
103#1307—591no—
104C2×C59—41no—
105C2×C59#204011yescharacteristic
106C2×C59#204812yes—
107D59—162no—
108#532—592no—
109#476—1182no—
110#469—1182no—
111#459—1182no—
112#432—592no—
113#423C2×D5911yescharacteristic
114#423C2×D5911yescharacteristic
115C2×#97C2×D5911yescharacteristic
116#341—22no—
117#332—1182no—
118#331—1182no—
119#329—1184no—
120C59#1052511yescharacteristic, complemented, Sylow
121C2×#41—1182no—
122C2×D8—1182no—
123C2×M16—21no—
124C2×M16C22⋊D5912yes—
125C22×C8—21no—
126C4⋊Q8—592no—
127#127—592no—
128#124—1182no—
129#121—1182no—
130#121—1182no—
131C4×D4—1182no—
132C4×D4—1182no—
133C2×C42C22×D5911yescharacteristic
134#106—1182no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1182no—
138#102—1184no—
139C22⋊C8—1182no—
140#97—21no—
141#97—1182no—
142#97D23612yes—
143#41—4724no—
144C2×Q8—1182no—
145C2×D4—1182no—
146C2×D4—1182no—
147C2×D4—1182no—
148C2×D4—2362no—
149C22×C4—2362no—
150C22×C4#1174568912yes—
151C22×C4C2×C22⋊D5911yescharacteristic
152D8—4722no—
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—2362no—
161C4⋊C4—1182no—
162C4⋊C4—1182no—
163C22⋊C4—2362no—
164C22⋊C4—2362no—
165C42C2×D23611yescharacteristic
166C42#1174568511yescharacteristic
167C42D4×D5912yes—
168C23—2362no—
169C23#1427371611yescharacteristic
170Q8—4722no—
171D4—4722no—
172D4—4722no—
173D4—4724no—
174D4—2362no—
175D4—2362no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—4724no—
180C2×C4—2362no—
181C2×C4#1427369511yescharacteristic
182C2×C4#1427370111yescharacteristic
183C2×C4#1427370212yes—
184C2×C4#1427371212yes—
185C8—42no—
186C8—42no—
187C8—81no—
188C8—4722no—
189C22—21no—
190C22—22no—
191C22—4722no—
192C22—4722no—
193C22—4722no—
194C22#1683731911yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—4722no—
199C4#1683757411yescharacteristic
200C4#1683757911yescharacteristic
201C2—41no—
202C2—9442no—
203C2#2142532511yescharacteristic
204C2#2142537112yes—
subgroupG/H
centerC22#16837319
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1838C2
socleC22×C59#394
prime radical#21435477C2
hypercenter#1838D59
hypocenterC59#10525
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1838

Complements

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

NH ≅ G/Ntype
2. #21422816C2semidirect
3. #21422706C2semidirect
4. #21422007C2semidirect
5. C59×#1838C2semidirect
100. #1838D59semidirect
120. C59#10525semidirect

Maximal subgroups

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

subgroupindex
#214354772
#214228162
#214227062
#214220072
C59×#18382
#1052559