Finite Group Databasebeta

← #28491824

Subgroups — #28491824

Order 15104.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#21424928C211yesmaximal, characteristic
2#21422590C212yesmaximal
3#21421999C212yesmaximal, complemented
4#21421925C211yesmaximal, characteristic, complemented
5C59×#2402C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837455C2212yes—
8#16837426C2211yescharacteristic
9C2×#14273655—22no—
10#16837135—22no—
11#16837135—22no—
12C59×#400—22no—
13C59×#398—22no—
14C4×C4⋊C4×C59—22no—
15C4×C4⋊C4×C59C2212yes—
16C59×#376C2211yescharacteristic
17C2×#11745685—21no—
18C2×#11745664—21no—
19#14273727Q811yescharacteristic
20#14273724Q811yescharacteristic
21#14273715—21no—
22#14273701—22no—
23C4×D236—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273655—44no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—22no—
29C2×C4⋊C4×C59—22no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59—22no—
32C2×C42×C59—22no—
33C2×C42×C59D412yes—
34C2×C42×C59C2311yescharacteristic
35#95×C59—22no—
36#95×C59—22no—
37#95×C59—22no—
38#95×C59D412yes—
39#95×C59D412yes—
40C22×C4×C59—22no—
41C22×C4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59C2×D412yes—
44C22×C4×C59C2×D411yescharacteristic
45C2×C22⋊D59—41no—
46#11745685—42no—
47#11745685—42no—
48C2×D236—21no—
49C2×D236—22no—
50C2×C4×D59—41no—
51C2×Dic118—21no—
52C4⋊C4×C59—44no—
53C4⋊C4×C59—44no—
54C4⋊C4×C59—44no—
55C42×C59—22no—
56C42×C59—22no—
57C42×C59—42no—
58C42×C59—42no—
59C42×C59C2×Q811yescharacteristic
60C42×C59#4112yes—
61C42×C59#4111yescharacteristic
62(C2×D59)⋊C4—42no—
63C4⋊Dic59—22no—
64#11745664—42no—
65C2×C59⋊C8—42no—
66C2×C59⋊C8—41no—
67C23×C59#12011yescharacteristic
68C22×D59—41no—
69C2×C4×C59—24no—
70C2×C4×C59—42no—
71C2×C4×C59—42no—
72C2×C4×C59—42no—
73C2×C4×C59—42no—
74C2×C4×C59—42no—
75C2×C4×C59—42no—
76C2×C4×C59—42no—
77C2×C4×C59—42no—
78C2×C4×C59—42no—
79C2×C4×C59C22⋊Q812yes—
80C2×C4×C59C22⋊Q811yescharacteristic
81C2×C4×C59#12412yes—
82C2×C4×C59#12411yescharacteristic
83C2×C4×C59#13512yes—
84C2×C4×C59AGL(1,ℤ/8)12yes—
85C22⋊D59—82no—
86C2×Dic59—41no—
87D236—42no—
88D236—82no—
89C4×D59—81no—
90C4×D59—81no—
91Dic118—42no—
92C59⋊C8—82no—
93C59⋊C8—81no—
94#10372—591nomaximal, Sylow
95C22×C59—22no—
96C22×C59#39311yescharacteristic
97C22×C59#44912yes—
98C2×D59—81no—
99C2×D59—82no—
100C4×C59—22no—
101C4×C59—22no—
102C4×C59—44no—
103C4×C59—82no—
104C4×C59—82no—
105C4×C59#47812yes—
106C4×C59#48811yescharacteristic
107C4×C59#49011yescharacteristic
108Dic59—81no—
109#2402D5911yescharacteristic, complemented
110#1963—591no—
111#1502—592no—
112#1303—592no—
113#1274—591no—
114C2×C59—22no—
115C2×C59#202511yescharacteristic
116C2×C59#203112yes—
117D59—161no—
118#532—591no—
119#431—592no—
120#423—591no—
121C2×#104—1182no—
122#400—22no—
123#398—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×D5912yes—
126#376C2×D5911yescharacteristic
127#327—1182no—
128#327—1182no—
129C59#1037211yescharacteristic, complemented, Sylow
130C2×#41—1181no—
131C2×M16—1181no—
132C4⋊Q8—591no—
133#127—591no—
134#124—1181no—
135#121—1182no—
136C4×D4—1182no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C42—22no—
141C2×C42C22⋊D5912yes—
142C2×C42C22×D5911yescharacteristic
143C4⋊C8—1182no—
144C4⋊C8—1182no—
145#104—2364no—
146C22⋊C8—1182no—
147C4×C8—1182no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95C22⋊D5912yes—
152#95C22⋊D5912yes—
153#41—2362no—
154#41—2362no—
155C2×Q8—1181no—
156C2×D4—1182no—
157C2×D4—1181no—
158C2×D4—2361no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2361no—
163C22×C4C2×C22⋊D5912yes—
164C22×C4C2×C22⋊D5911yescharacteristic
165M16—2362no—
166C2×C8—2362no—
167C2×C8—2361no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—1182no—
172C22⋊C4—2362no—
173C42—22no—
174C42—22no—
175C42—42no—
176C42—42no—
177C42Q8×D5911yescharacteristic
178C42#1174568911yescharacteristic
179C42#1174568912yes—
180C23—2361no—
181C23#1427369611yescharacteristic
182Q8—2362no—
183D4—4722no—
184D4—4722no—
185D4—2362no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—4721no—
197C2×C4—4721no—
198C2×C4—2361no—
199C2×C4#1427370312yes—
200C2×C4#1427370311yescharacteristic
201C2×C4#1427371612yes—
202C2×C4#1427371611yescharacteristic
203C2×C4#1427374712yes—
204C2×C4#1427375212yes—
205C8—4722no—
206C8—4721no—
207C22—22no—
208C22—4721no—
209C22—4722no—
210C22#1683731311yescharacteristic
211C22#1683754712yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—82no—
216C4—82no—
217C4—4721no—
218C4#1683769912yes—
219C4#1683775911yescharacteristic
220C4#1683776811yescharacteristic
221C2—22no—
222C2—9441no—
223C2#2142523911yescharacteristic
224C2#2142526512yes—
subgroupG/H
centerC2×C4#14273716
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2402C2
socleC22×C59#393
prime radical#16837952C22
hypercenter#2402D59
hypocenterC59#10372
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2402

Complements

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

NH ≅ G/Ntype
3. #21421999C2semidirect
4. #21421925C2semidirect
5. C59×#2402C2semidirect
109. #2402D59semidirect
129. C59#10372semidirect

Maximal subgroups

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

subgroupindex
#214249282
#214225902
#214219992
#214219252
C59×#24022
#1037259