Finite Group Databasebeta

← #28482272

Subgroups — #28482272

Order 15104.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#21427061C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic
3#21421997C212yesmaximal
4#21421956C212yesmaximal
5C59×#1801C211yesmaximal, characteristic
6#16837455C2212yes—
7#16837309—21no—
8#16837288—22no—
9#16837270C412yes—
10#16837268C2211yescharacteristic
11C4⋊C4×Dic59—21no—
12C4×C4⋊Dic59C2211yescharacteristic
13#16837248C412yes—
14#16837128—22no—
15C2×C4×Q8×C59C2211yescharacteristic
16C59×#423C2212yes—
17C2×#101×C59—21no—
18C59×#343—21no—
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×C4×Dic59—22no—
22C2×C4×Dic59—21no—
23C4⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—22no—
25C4×C59⋊C8—22no—
26C59⋊#95—22no—
27C59⋊#95—22no—
28C59⋊#95—22no—
29C59⋊#95—21no—
30C59⋊#95C2×C411yescharacteristic
31C59⋊#95C2×C412yes—
32C59⋊#95C2×C411yescharacteristic
33C22×Q8×C59D411yescharacteristic
34C2×M16×C59—22no—
35C4×Q8×C59—22no—
36C4×Q8×C59—22no—
37C2×C4⋊C4×C59D412yes—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C42×C59C2×C412yes—
40C2×C42×C59C2311yescharacteristic
41C4⋊C8×C59—22no—
42C4⋊C8×C59—22no—
43#101×C59—42no—
44C2×Q8×C59—22no—
45C2×Q8×C59—22no—
46C2×Q8×C59—21no—
47C2×Q8×C59—21no—
48C22×C4×C59C22⋊C412yes—
49C22×C4×C59C22⋊C412yes—
50C22×C4×C59C22×C411yescharacteristic
51C22×C4×C59C2×D411yescharacteristic
52C22×C4×C59C2×D411yescharacteristic
53C22×Dic59—21no—
54C22×Dic59—22no—
55C22×Dic59—21no—
56M16×C59—42no—
57M16×C59—42no—
58C2×C8×C59—42no—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—24no—
61C42×C59—22no—
62C42×C59—22no—
63C42×C59#4112yes—
64C42×C59#4112yes—
65C4⋊Dic59—42no—
66C4⋊Dic59—42no—
67C4×Dic59—44no—
68C4×Dic59—41no—
69C4×Dic59—41no—
70C2×C59⋊C8—42no—
71C23×C59C2×C22⋊C411yescharacteristic
72Q8×C59—42no—
73Q8×C59—42no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—41no—
79C2×C4×C59—41no—
80C2×C4×C59—42no—
81C2×C4×C59#11711yescharacteristic
82C2×C4×C59#11711yescharacteristic
83C2×C4×C59#12312yes—
84C2×C4×C59#12312yes—
85C2×C4×C59#13512yes—
86C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C59#13711yescharacteristic
88C2×Dic59—22no—
89C2×Dic59—41no—
90C2×Dic59—41no—
91C2×Dic59—42no—
92C2×Dic59—41no—
93C2×Dic59—41no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—41no—
97C8×C59—82no—
98C59⋊C8—82no—
99#8768—591nomaximal, Sylow
100C22×C59—21no—
101C22×C59—21no—
102C22×C59#38111yescharacteristic
103C22×C59#41611yescharacteristic
104C22×C59#41711yescharacteristic
105C4×C59—22no—
106C4×C59—21no—
107C4×C59—21no—
108C4×C59—42no—
109C4×C59—42no—
110C4×C59#48212yes—
111C4×C59#48512yes—
112Dic59—42no—
113Dic59—81no—
114Dic59—82no—
115#2356—591no—
116#1801D5911yescharacteristic
117#1461—591no—
118#1303—592no—
119#1286—592no—
120C2×C59—21no—
121C2×C59—21no—
122C2×C59#182511yescharacteristic
123C2×C59#182711yescharacteristic
124C2×C59#183011yescharacteristic
125C2×C4×Q8C2×D5911yescharacteristic
126#431—592no—
127#423C2×D5912yes—
128C2×#101—21no—
129#391—1181no—
130#387—1182no—
131#383—592no—
132#382—591no—
133C4×C4⋊C4—591no—
134C4×C4⋊C4—1181no—
135#376—592no—
136#343—21no—
137#324—1182no—
138C59#876811yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5911yescharacteristic
140C2×M16—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—1181no—
144C2×C4⋊C4—1181no—
145C2×C4⋊C4D23612yes—
146C2×C4⋊C4C22⋊D5911yescharacteristic
147C2×C42—1182no—
148C2×C42—1181no—
149C2×C42C4×D5912yes—
150C2×C42C22×D5911yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—1182no—
154#101—42no—
155C22⋊C8—1182no—
156C4×C8—1182no—
157#95—591no—
158#95—592no—
159#95—591no—
160#95—1182no—
161#95—1182no—
162#95—1182no—
163#95—1181no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—1181no—
169C22×C4—1182no—
170C22×C4—1181no—
171C22×C4(C2×D59)⋊C412yes—
172C22×C4(C2×D59)⋊C412yes—
173C22×C4C2×C4×D5911yescharacteristic
174C22×C4C2×D23611yescharacteristic
175C22×C4C2×C22⋊D5911yescharacteristic
176M16—42no—
177M16—42no—
178C2×C8—42no—
179C2×C8—2362no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—2362no—
183C4⋊C4—2362no—
184C42—22no—
185C42—22no—
186C42—2364no—
187C42—2361no—
188C42—2361no—
189C42#1174568712yes—
190C42#1174568712yes—
191C23C2×(C2×D59)⋊C411yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—1182no—
202C2×C4—2362no—
203C2×C4—2361no—
204C2×C4—2362no—
205C2×C4—2361no—
206C2×C4—2362no—
207C2×C4—2361no—
208C2×C4—2361no—
209C2×C4—2361no—
210C2×C4#1427368311yescharacteristic
211C2×C4#1427368311yescharacteristic
212C2×C4#1427370812yes—
213C2×C4#1427371112yes—
214C2×C4#1427374712yes—
215C2×C4#1427375311yescharacteristic
216C2×C4#1427375811yescharacteristic
217C8—82no—
218C8—4722no—
219C22—21no—
220C22—21no—
221C22#1683726311yescharacteristic
222C22#1683739611yescharacteristic
223C22#1683740111yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—4722no—
230C4—4721no—
231C4—2362no—
232C4#1683772312yes—
233C4#1683774012yes—
234C2—21no—
235C2—21no—
236C2#2142418811yescharacteristic
237C2#2142420111yescharacteristic
238C2#2142420811yescharacteristic
subgroupG/H
centerC22#16837263
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#381
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1801D59
hypocenterC59#8768
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
138. C59#8768semidirect

Maximal subgroups

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

subgroupindex
#214270612
#214224202
#214219972
#214219562
C59×#18012
#876859