Finite Group Databasebeta

← #28492829

Subgroups — #28492829

Order 15104.

370 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 77 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#21424236C211yesmaximal, characteristic, complemented
2#21422818C212yesmaximal, complemented
3#21422707C212yesmaximal, complemented
4#21422007C211yesmaximal, characteristic, complemented
5C59×#3253C211yesmaximal, characteristic
6#16837459C2212yes—
7#16837426C2211yescharacteristic
8C2×#14273638C2211yescharacteristic
9#16837166—22no—
10#16837148—22no—
11#16837146—22no—
12#16837142—24no—
13C59×#532C2212yes—
14C59×#476—22no—
15C59×#469—22no—
16C59×#459—22no—
17C59×#423C2211yescharacteristic
18C2×#11745664—21no—
19C22×C59⋊C8—21no—
20C4⋊(C59⋊C8)—22no—
21C4⋊(C59⋊C8)—22no—
22C22⋊(C59⋊C8)—22no—
23#14273638—21no—
24#14273638—22no—
25#14273638D412yes—
26C2×#41×C59—22no—
27C2×D8×C59—22no—
28C2×M16×C59D412yes—
29C4⋊Q8×C59D412yes—
30#127×C59D412yes—
31#124×C59—22no—
32#121×C59—22no—
33#121×C59—22no—
34C4×D4×C59—22no—
35C4×D4×C59—22no—
36C2×C42×C59C2311yescharacteristic
37#106×C59—22no—
38C4⋊C8×C59—22no—
39#102×C59—24no—
40#41×C59—84no—
41C2×Q8×C59—22no—
42C2×D4×C59—22no—
43C2×D4×C59—22no—
44C2×D4×C59—22no—
45C2×D4×C59—42no—
46C22×C4×C59—42no—
47C22×C4×C59C2×D411yescharacteristic
48C22×C4×C59#4112yes—
49D8×C59—82no—
50M16×C59—42no—
51C2×C8×C59—22no—
52C4⋊C4×C59—22no—
53C4⋊C4×C59—22no—
54C22⋊C4×C59—42no—
55C22⋊C4×C59—42no—
56C42×C59C2×D412yes—
57C42×C59C2×D411yescharacteristic
58C42×C59#4111yescharacteristic
59#11745664—42no—
60C2×C59⋊C8—22no—
61C2×C59⋊C8—41no—
62C2×C59⋊C8—41no—
63C2×C59⋊C8—41no—
64C2×C59⋊C8—42no—
65C23×C59—42no—
66C23×C59#12411yescharacteristic
67Q8×C59—82no—
68D4×C59—42no—
69D4×C59—42no—
70D4×C59—82no—
71D4×C59—82no—
72D4×C59—84no—
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—42no—
76C2×C4×C59—42no—
77C2×C4×C59—84no—
78C2×C4×C59#12011yescharacteristic
79C2×C4×C59#12111yescharacteristic
80C2×C4×C59#12112yes—
81C2×C4×C59#12312yes—
82C8×C59—42no—
83C59⋊C8—42no—
84C59⋊C8—82no—
85C59⋊C8—81no—
86#10525—591nomaximal, Sylow
87C22×C59—22no—
88C22×C59—21no—
89C22×C59—82no—
90C22×C59—82no—
91C22×C59—82no—
92C22×C59#39411yescharacteristic
93C4×C59—22no—
94C4×C59—22no—
95C4×C59—21no—
96C4×C59—82no—
97C4×C59#45411yescharacteristic
98C4×C59#45511yescharacteristic
99#3253D5911yescharacteristic
100#1838—591no—
101#1557—592no—
102#1530—592no—
103#1307—591no—
104C2×C59—41no—
105C2×C59—162no—
106C2×C59#204011yescharacteristic
107C2×C59#204812yes—
108#532C2×D5912yes—
109#476—22no—
110#469—22no—
111#459—22no—
112#432—592no—
113#423—591no—
114#423C2×D5911yescharacteristic
115C2×#97—591no—
116#341—1182no—
117#332—1182no—
118#331—1182no—
119#329—1184no—
120C59#1052511yescharacteristic, complemented, Sylow
121C2×#41—22no—
122C2×D8—22no—
123C2×M16—1181no—
124C2×M16C22⋊D5912yes—
125C22×C8—1181no—
126C4⋊Q8C22⋊D5912yes—
127#127C22⋊D5912yes—
128#124—22no—
129#121—22no—
130#121—22no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C42C22×D5911yescharacteristic
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—1182no—
137C4⋊C8—1182no—
138#102—24no—
139C22⋊C8—1182no—
140#97—592no—
141#97—1182no—
142#97—1181no—
143#41—84no—
144C2×Q8—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—42no—
149C22×C4—42no—
150C22×C4#1174568712yes—
151C22×C4C2×C22⋊D5911yescharacteristic
152D8—82no—
153M16—42no—
154M16—2362no—
155C2×C8—22no—
156C2×C8—1182no—
157C2×C8—2361no—
158C2×C8—2362no—
159C2×C8—2361no—
160C2×C8—2361no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42D4×D5911yescharacteristic
166C42#1174568711yescharacteristic
167C42C2×C22⋊D5912yes—
168C23—42no—
169C23#1427371411yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—82no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4#1427369611yescharacteristic
182C2×C4#1427369811yescharacteristic
183C2×C4#1427369912yes—
184C2×C4#1427370812yes—
185C8—42no—
186C8—4722no—
187C8—4721no—
188C8—2362no—
189C22—21no—
190C22—22no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1683731611yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—82no—
199C4#1683757511yescharacteristic
200C4#1683758011yescharacteristic
201C2—41no—
202C2—162no—
203C2#2142532211yescharacteristic
204C2#2142536612yes—
subgroupG/H
centerC22#16837316
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3253C2
socleC22×C59#394
hypercenter#3253D59
hypocenterC59#10525
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #21424236C2semidirect
2. #21422818C2semidirect
3. #21422707C2semidirect
4. #21422007C2semidirect
120. C59#10525semidirect

Maximal subgroups

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

subgroupindex
#214242362
#214228182
#214227072
#214220072
C59×#32532
#1052559