Finite Group Databasebeta

← #28493188

Subgroups — #28493188

Order 15104.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#21424265C211yesmaximal, characteristic
2#21424229C212yesmaximal
3#21421977C212yesmaximal
4#21421977C211yesmaximal, characteristic
5C59×#2844C211yesmaximal, characteristic
6C2×#14273660—21no—
7C2×#14273658—21no—
8C2×#14273638C2212yes—
9C2×#14273638C2211yescharacteristic
10#16837272C2211yescharacteristic
11#16837164—22no—
12#16837164—22no—
13C59×#400—22no—
14C59×#400—21no—
15C59×#395—22no—
16C59×#395—21no—
17C59×#382C2212yes—
18C59×#382C2211yescharacteristic
19C22×C59⋊C8—21no—
20C22×C59⋊C8—22no—
21C2×C4⋊Dic59—22no—
22C2×C4⋊Dic59—21no—
23#14273660—42no—
24#14273658—42no—
25#14273638—21no—
26#14273638—22no—
27#14273638Q812yescomplemented
28#14273638Q814yescomplemented
29C22×Q8×C59—21no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—22no—
32C2×C4⋊C4×C59D412yes—
33C2×C4⋊C4×C59D412yes—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C4⋊C4×C59D411yescharacteristic
36C2×C42×C59C2311yescharacteristic
37#95×C59—22no—
38#95×C59—21no—
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59—22no—
43C2×Q8×C59—81no—
44C2×Q8×C59—82no—
45C22×C4×C59—21no—
46C22×C4×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59C2×D412yes—
51C22×Dic59—41no—
52C4⋊C4×C59—42no—
53C4⋊C4×C59—42no—
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C4⋊C4×C59—81no—
57C4⋊C4×C59—82no—
58C42×C59C2×Q811yescharacteristic
59C42×C59C2×Q812yes—
60C42×C59#4111yescharacteristic
61C4⋊Dic59—41no—
62C4⋊Dic59—41no—
63C4⋊Dic59—44no—
64C2×C59⋊C8—22no—
65C2×C59⋊C8—22no—
66C2×C59⋊C8—22no—
67C2×C59⋊C8—24no—
68C2×C59⋊C8—42no—
69C2×C59⋊C8—42no—
70C23×C59#12011yescharacteristic
71Q8×C59—84no—
72C2×C4×C59—21no—
73C2×C4×C59—22no—
74C2×C4×C59—22no—
75C2×C4×C59—21no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59—42no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59C22⋊Q811yescharacteristic
87C2×C4×C59C22⋊Q812yes—
88C2×C4×C59C4⋊Q811yescharacteristic
89C2×C4×C59C4⋊Q812yes—
90C2×Dic59—81no—
91C2×Dic59—82no—
92C59⋊C8—24no—
93C59⋊C8—44no—
94#10584—591nomaximal, Sylow
95C22×C59#39511yescharacteristic
96C22×C59#45311yescharacteristic
97C22×C59#45412yes—
98C22×C59#45512yes—
99C22×C59#45611yescharacteristic
100C4×C59—21no—
101C4×C59—21no—
102C4×C59—22no—
103C4×C59—22no—
104C4×C59—81no—
105C4×C59—81no—
106C4×C59—82no—
107C4×C59—82no—
108Dic59—161no—
109#2844D5911yescharacteristic
110#1847—591no—
111#1836—592no—
112#1295—592no—
113#1295—591no—
114C2×C59#165711yescharacteristic
115C2×C59#165911yescharacteristic
116C2×C59#166412yes—
117C2×C59#206011yescharacteristic
118C2×C59#206112yes—
119C2×#107—1181no—
120C2×#106—1181no—
121C2×#97—592no—
122C2×#97—591no—
123#400—22no—
124#400—21no—
125#395—22no—
126#395—21no—
127#384—591no—
128#382C2×D5912yes—
129#382C2×D5911yescharacteristic
130#340—1182no—
131#340—1182no—
132C59#1058411yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C22×C8—1181no—
135C22×C8—1182no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—1182no—
139C2×C4⋊C4—1181no—
140C2×C4⋊C4C22⋊D5912yes—
141C2×C4⋊C4C22⋊D5912yes—
142C2×C4⋊C4C22⋊D5911yescharacteristic
143C2×C4⋊C4C22⋊D5911yescharacteristic
144C2×C42C22×D5911yescharacteristic
145#107—2362no—
146#106—2362no—
147#97—592no—
148#97—594no—
149#97—1181no—
150#97—1182no—
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—2361no—
164C22×C4C2×C22⋊D5911yescharacteristic
165C22×C4C2×C22⋊D5912yes—
166C2×C8—1182no—
167C2×C8—1182no—
168C2×C8—1182no—
169C2×C8—1184no—
170C2×C8—2362no—
171C2×C8—2362no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—2361no—
179C4⋊C4—2361no—
180C4⋊C4—2364no—
181C42Q8×D5911yescharacteristic
182C42Q8×D5912yes—
183C42#1174568911yescharacteristic
184C23#1427369611yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—4721no—
201C2×C4—4722no—
202C2×C4#1427370311yescharacteristic
203C2×C4#1427370312yes—
204C2×C4#1427372511yescharacteristic
205C2×C4#1427372512yes—
206C8—1184no—
207C8—2364no—
208C22#1683732111yescharacteristic
209C22#1683757011yescharacteristic
210C22#1683757512yes—
211C22#1683758012yes—
212C22#1683758511yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—9441no—
222C2#2142339411yescharacteristic
223C2#2142340211yescharacteristic
224C2#2142343012yes—
225C2#2142543811yescharacteristic
226C2#2142544312yes—
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2844C2
socle, prime radicalC23×C59#120
hypercenter#2844D59
hypocenterC59#10584
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
27. #14273638Q8semidirect
28. #14273638Q8semidirect
132. C59#10584semidirect

Maximal subgroups

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

subgroupindex
#214242652
#214242292
#214219772
C59×#28442
#1058459