Finite Group Databasebeta

← #28492824

Subgroups — #28492824

Order 15104.

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

normal subgroups only

This group has 386 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#21424270C211yesmaximal, characteristic, complemented
2#21422818C212yesmaximal, complemented
3#21422707C212yesmaximal, complemented
4#21422007C211yesmaximal, characteristic, complemented
5C59×#3253C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×#14273662—22no—
8#16837426C2211yescharacteristic
9C2×#14273638C2211yescharacteristic
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—
20#14273662—44no—
21C4⋊(C59⋊C8)—22no—
22C4⋊(C59⋊C8)—22no—
23C22⋊(C59⋊C8)—22no—
24#14273638—21no—
25#14273638—22no—
26#14273638D412yes—
27C2×#41×C59—22no—
28C2×D8×C59—22no—
29C2×M16×C59D412yes—
30C4⋊Q8×C59D412yes—
31#127×C59D412yes—
32#124×C59—22no—
33#121×C59—22no—
34#121×C59—22no—
35C4×D4×C59—22no—
36C4×D4×C59—22no—
37C2×C42×C59C2311yescharacteristic
38#106×C59—22no—
39C4⋊C8×C59—22no—
40#102×C59—24no—
41#41×C59—84no—
42C2×Q8×C59—22no—
43C2×D4×C59—22no—
44C2×D4×C59—22no—
45C2×D4×C59—22no—
46C2×D4×C59—42no—
47C22×C4×C59—42no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59#4112yes—
50D8×C59—82no—
51M16×C59—42no—
52C2×C8×C59—22no—
53C4⋊C4×C59—22no—
54C4⋊C4×C59—22no—
55C22⋊C4×C59—42no—
56C22⋊C4×C59—42no—
57C42×C59C2×D412yes—
58C42×C59C2×D411yescharacteristic
59C42×C59#4111yescharacteristic
60#11745664—42no—
61C2×C59⋊C8—22no—
62C2×C59⋊C8—22no—
63C2×C59⋊C8—22no—
64C2×C59⋊C8—41no—
65C2×C59⋊C8—42no—
66C23×C59—42no—
67C23×C59#12411yescharacteristic
68Q8×C59—82no—
69D4×C59—42no—
70D4×C59—42no—
71D4×C59—82no—
72D4×C59—82no—
73D4×C59—84no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—42no—
77C2×C4×C59—42no—
78C2×C4×C59—84no—
79C2×C4×C59#12011yescharacteristic
80C2×C4×C59#12111yescharacteristic
81C2×C4×C59#12112yes—
82C2×C4×C59#12312yes—
83C8×C59—42no—
84C59⋊C8—24no—
85C59⋊C8—82no—
86C59⋊C8—81no—
87#10524—591nomaximal, Sylow
88C22×C59—22no—
89C22×C59—21no—
90C22×C59—82no—
91C22×C59—82no—
92C22×C59—82no—
93C22×C59#39411yescharacteristic
94C4×C59—22no—
95C4×C59—22no—
96C4×C59—21no—
97C4×C59—82no—
98C4×C59#45311yescharacteristic
99C4×C59#45611yescharacteristic
100#3253D5911yescharacteristic
101#1849—591no—
102#1557—592no—
103#1530—592no—
104#1307—591no—
105C2×C59—41no—
106C2×C59—162no—
107C2×C59#203911yescharacteristic
108C2×C59#204812yes—
109#532C2×D5912yes—
110#476—22no—
111#469—22no—
112#459—22no—
113#432—592no—
114C2×#108—1182no—
115#423—591no—
116#423C2×D5911yescharacteristic
117C2×#97—591no—
118#332—1182no—
119#331—1182no—
120#329—1184no—
121C59#1052411yescharacteristic, complemented, Sylow
122C2×#41—22no—
123C2×D8—22no—
124C2×M16—1181no—
125C2×M16C22⋊D5912yes—
126C22×C8—1181no—
127C4⋊Q8C22⋊D5912yes—
128#127C22⋊D5912yes—
129#124—22no—
130#121—22no—
131#121—22no—
132C4×D4—22no—
133C4×D4—22no—
134C2×C42C22×D5911yescharacteristic
135#108—2364no—
136#106—22no—
137C4⋊C8—22no—
138C4⋊C8—1182no—
139C4⋊C8—1182no—
140#102—24no—
141C22⋊C8—1182no—
142#97—592no—
143#97—1182no—
144#97—1181no—
145#41—84no—
146C2×Q8—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—42no—
151C22×C4—42no—
152C22×C4#1174568712yes—
153C22×C4C2×C22⋊D5911yescharacteristic
154D8—82no—
155M16—42no—
156M16—2362no—
157C2×C8—22no—
158C2×C8—1182no—
159C2×C8—1182no—
160C2×C8—1182no—
161C2×C8—2361no—
162C2×C8—2362no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C42D4×D5911yescharacteristic
168C42#1174568711yescharacteristic
169C42C2×C22⋊D5912yes—
170C23—42no—
171C23#1427371411yescharacteristic
172Q8—82no—
173D4—42no—
174D4—42no—
175D4—82no—
176D4—82no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#1427369611yescharacteristic
184C2×C4#1427369811yescharacteristic
185C2×C4#1427369912yes—
186C2×C4#1427370812yes—
187C8—42no—
188C8—1184no—
189C8—4722no—
190C8—4721no—
191C22—21no—
192C22—22no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22#1683731611yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—82no—
201C4#1683757011yescharacteristic
202C4#1683758511yescharacteristic
203C2—41no—
204C2—162no—
205C2#2142531711yescharacteristic
206C2#2142536612yes—
subgroupG/H
centerC22#16837316
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3253C2
socleC22×C59#394
hypercenter#3253D59
hypocenterC59#10524
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. #21424270C2semidirect
2. #21422818C2semidirect
3. #21422707C2semidirect
4. #21422007C2semidirect
121. C59#10524semidirect

Maximal subgroups

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

subgroupindex
#214242702
#214228182
#214227072
#214220072
C59×#32532
#1052459