Finite Group Databasebeta

← #28492827

Subgroups — #28492827

Order 15104.

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

normal subgroups only

This group has 386 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×#1849C211yesmaximal, characteristic, complemented
6#16837952C2212yes—
7#16837693—22no—
8#16837655—22no—
9#16837598—22no—
10#16837459C2212yes—
11#16837147—22no—
12#16837145—22no—
13#16837141—24no—
14C2×#108×C59—22no—
15C59×#423C2211yescharacteristic
16C59×#423C2211yescharacteristic
17C2×#97×C59C2211yescharacteristic
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
36#108×C59—44no—
37C4⋊C8×C59—22no—
38C4⋊C8×C59—22no—
39#97×C59—21no—
40#97×C59D412yes—
41C22×C4×C59C2×D411yescharacteristic
42C22×C4×C59#4112yes—
43C2×C22⋊D59—42no—
44#11745685—84no—
45C2×D236—22no—
46C2×D236—22no—
47C2×D236—22no—
48C2×C4×D59—42no—
49C2×Dic118—22no—
50M16×C59—42no—
51M16×C59—42no—
52C2×C8×C59—22no—
53C2×C8×C59—22no—
54C2×C8×C59—22no—
55C2×C8×C59—22no—
56C2×C8×C59—41no—
57C42×C59C2×D412yes—
58C42×C59C2×D411yescharacteristic
59C42×C59#4111yescharacteristic
60(C2×D59)⋊C4—42no—
61(C2×D59)⋊C4—42no—
62C4⋊Dic59—22no—
63C4⋊Dic59—22no—
64C2×C59⋊C8—42no—
65D472—82no—
66C23×C59#12411yescharacteristic
67C22×D59—42no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—42no—
71C2×C4×C59#12011yescharacteristic
72C2×C4×C59#12111yescharacteristic
73C2×C4×C59#12112yes—
74C2×C4×C59#12312yes—
75C22⋊D59—84no—
76C2×Dic59—42no—
77D236—42no—
78D236—42no—
79D236—82no—
80D236—82no—
81C4×D59—84no—
82Dic118—82no—
83C8×C59—24no—
84C8×C59—42no—
85C8×C59—81no—
86C59⋊C8—82no—
87#10524—591nomaximal, Sylow
88C22×C59—22no—
89C22×C59—21no—
90C22×C59#39411yescharacteristic
91C2×D59—82no—
92C2×D59—82no—
93C2×D59—82no—
94C4×C59—22no—
95C4×C59—22no—
96C4×C59—21no—
97C4×C59#45311yescharacteristic
98C4×C59#45611yescharacteristic
99Dic59—82no—
100#3253—591no—
101#1849D5911yescharacteristic, complemented
102#1557—592no—
103#1530—592no—
104#1307—591no—
105C2×C59—41no—
106C2×C59#203911yescharacteristic
107C2×C59#204812yes—
108D59—162no—
109#532—592no—
110#476—1182no—
111#469—1182no—
112#459—1182no—
113#432—592no—
114C2×#108—22no—
115#423C2×D5911yescharacteristic
116#423C2×D5911yescharacteristic
117C2×#97C2×D5911yescharacteristic
118#332—1182no—
119#331—1182no—
120#329—1184no—
121C59#1052411yescharacteristic, complemented, Sylow
122C2×#41—1182no—
123C2×D8—1182no—
124C2×M16—21no—
125C2×M16C22⋊D5912yes—
126C22×C8—21no—
127C4⋊Q8—592no—
128#127—592no—
129#124—1182no—
130#121—1182no—
131#121—1182no—
132C4×D4—1182no—
133C4×D4—1182no—
134C2×C42C22×D5911yescharacteristic
135#108—44no—
136#106—1182no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—1182no—
140#102—1184no—
141C22⋊C8—1182no—
142#97—21no—
143#97—1182no—
144#97D23612yes—
145#41—4724no—
146C2×Q8—1182no—
147C2×D4—1182no—
148C2×D4—1182no—
149C2×D4—1182no—
150C2×D4—2362no—
151C22×C4—2362no—
152C22×C4#1174568912yes—
153C22×C4C2×C22⋊D5911yescharacteristic
154D8—4722no—
155M16—42no—
156M16—42no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—2362no—
163C4⋊C4—1182no—
164C4⋊C4—1182no—
165C22⋊C4—2362no—
166C22⋊C4—2362no—
167C42C2×D23611yescharacteristic
168C42#1174568511yescharacteristic
169C42D4×D5912yes—
170C23—2362no—
171C23#1427371611yescharacteristic
172Q8—4722no—
173D4—4722no—
174D4—4722no—
175D4—4724no—
176D4—2362no—
177D4—2362no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—4724no—
182C2×C4—2362no—
183C2×C4#1427369511yescharacteristic
184C2×C4#1427370111yescharacteristic
185C2×C4#1427370212yes—
186C2×C4#1427371212yes—
187C8—24no—
188C8—42no—
189C8—81no—
190C8—4722no—
191C22—21no—
192C22—22no—
193C22—4722no—
194C22—4722no—
195C22—4722no—
196C22#1683731911yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—4722no—
201C4#1683756911yescharacteristic
202C4#1683758411yescharacteristic
203C2—41no—
204C2—9442no—
205C2#2142532011yescharacteristic
206C2#2142537112yes—
subgroupG/H
centerC22#16837319
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1849C2
socleC22×C59#394
prime radical#21435477C2
hypercenter#1849D59
hypocenterC59#10524
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
2. #21422816C2semidirect
3. #21422706C2semidirect
4. #21422007C2semidirect
5. C59×#1849C2semidirect
101. #1849D59semidirect
121. C59#10524semidirect

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×#18492
#1052459