Finite Group Databasebeta

← #28501100

Subgroups — #28501100

Order 15104.

424 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 59 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#21429469C211yesmaximal, characteristic, complemented
2#21423099C212yesmaximal
3#21422101C212yesmaximal, complemented
4#21422042C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic
6C2×C4×Dic118—21no—
7#16837713—22no—
8#16837332—21no—
9#16837332C2212yes—
10#16837307—22no—
11#16837301—22no—
12#16837136—22no—
13#16837128C2212yes—
14C59×#537C2211yescharacteristic
15C2×#104×C59—21no—
16C59×#417—21no—
17C59×#384C2211yescharacteristic
18C59×#343C2211yescharacteristic
19C22×Dic118—21no—
20C4×Dic118—42no—
21C4×Dic118—42no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59D412yes—
25C2×Dic59⋊C4—22no—
26C2×C4×Dic59—22no—
27#14273660—42no—
28#14273648—42no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C59⋊#95D412yes—
34C2×#41×C59—21no—
35C2×M16×C59—21no—
36C22⋊Q8×C59—41no—
37#121×C59—22no—
38#121×C59—41no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C42×C59D411yescharacteristic
43#104×C59—42no—
44#103×C59—41no—
45#102×C59—41no—
46#41×C59—41no—
47#41×C59—41no—
48#41×C59—41no—
49#41×C59—41no—
50C2×Q8×C59—21no—
51C2×D4×C59—21no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—41no—
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D412yes—
59C22×Dic59—22no—
60C22×Dic59—22no—
61C2×Dic118—24no—
62C2×Dic118—42no—
63C2×Dic118—42no—
64M16×C59—41no—
65M16×C59—41no—
66C2×C8×C59—41no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—44no—
70C4⋊C4×C59—41no—
71C22⋊C4×C59—42no—
72C42×C59—21no—
73C42×C59D812yes—
74C4⋊Dic59—42no—
75C4⋊Dic59—42no—
76C4⋊Dic59—81no—
77Dic59⋊C4—42no—
78Dic59⋊C4—42no—
79C4×Dic59—44no—
80C2×C59⋊C8—42no—
81C23×C59—41no—
82C23×C59#12011yescharacteristic
83Q8×C59—41no—
84Q8×C59—41no—
85D4×C59—41no—
86D4×C59—41no—
87D4×C59—82no—
88D4×C59—81no—
89D4×C59—81no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—81no—
101C2×C4×C59—81no—
102C2×C4×C59C2×D811yescharacteristic
103C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
104C2×Dic59—24no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—42no—
109C2×Dic59—42no—
110Dic118—48no—
111C8×C59—81no—
112C59⋊C8—82no—
113#12153—591nomaximal, Sylow
114C22×C59—21no—
115C22×C59—21no—
116C22×C59—81no—
117C22×C59—81no—
118C22×C59—81no—
119C22×C59#44711yescharacteristic
120C22×C59#45411yescharacteristic
121C22×C59B(4,𝔽2)11yescharacteristic
122C4×C59—24no—
123C4×C59—21no—
124C4×C59—21no—
125C4×C59—42no—
126C4×C59—81no—
127C4×C59—81no—
128Dic59—44no—
129Dic59—82no—
130#2582—591no—
131#2039D5911yescharacteristic
132#1610—592no—
133#1346—592no—
134#1325—591no—
135C2×C59—21no—
136C2×C59—21no—
137C2×C59—161no—
138C2×C59#159811yescharacteristic
139C2×C59#219811yescharacteristic
140C2×C59#220511yescharacteristic
141#537C2×D5911yescharacteristic
142C2×C4×Q8—1181no—
143#480—1182no—
144C2×#104—21no—
145#417—21no—
146#398—592no—
147#398—1181no—
148#391—1182no—
149#389—1182no—
150#384C2×D5911yescharacteristic
151#343C2×D5911yescharacteristic
152#327—1182no—
153#324—592no—
154C59#1215311yescharacteristic, complemented, Sylow
155C2×#41—21no—
156C22×Q8—1181no—
157C2×M16—21no—
158C22⋊Q8—41no—
159#121—22no—
160#121—41no—
161C4×Q8—2362no—
162C4×Q8—2362no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4—1181no—
166C2×C4⋊C4—1181no—
167C2×C4⋊C4—1182no—
168C2×C4⋊C4C22⋊D5911yescharacteristic
169C2×C4⋊C4C22×D5911yescharacteristic
170C2×C42—1182no—
171C2×C42C22⋊D5911yescharacteristic
172#107—2362no—
173#104—42no—
174#103—41no—
175#102—41no—
176#102—2362no—
177C22⋊C8—1182no—
178#95—592no—
179#95—1182no—
180#95—1181no—
181#95—1182no—
182#41—41no—
183#41—41no—
184#41—41no—
185#41—41no—
186C2×Q8—21no—
187C2×Q8—1184no—
188C2×Q8—2362no—
189C2×Q8—2362no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—41no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—1182no—
197C22×C4—1182no—
198C22×C4D4×D5912yes—
199C22×C4C2×C22⋊D5911yescharacteristic
200M16—41no—
201M16—41no—
202C2×C8—41no—
203C2×C8—2362no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—44no—
208C4⋊C4—4721no—
209C4⋊C4—2362no—
210C4⋊C4—2362no—
211C4⋊C4—2362no—
212C4⋊C4—2362no—
213C22⋊C4—42no—
214C42—21no—
215C42—2364no—
216C42#1174566912yes—
217C23—41no—
218C23#1427369511yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—2368no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—1184no—
240C2×C4—2362no—
241C2×C4—2362no—
242C2×C4—2362no—
243C2×C4—2362no—
244C2×C4—2362no—
245C2×C4C2×#1174566911yescharacteristic
246C2×C4#1427375211yescharacteristic
247C8—81no—
248C8—4722no—
249C22—21no—
250C22—21no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#1683753511yescharacteristic
255C22#1683757411yescharacteristic
256C22#1683758811yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4—4722no—
264C4—2364no—
265C2—21no—
266C2—21no—
267C2—161no—
268C2#2142303011yescharacteristic
269C2#2142609111yescharacteristic
270C2#2142612511yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2039C2
socleC22×C59B(4,𝔽2)
prime radicalC59×#537C22
hypercenter#2039D59
hypocenterC59#12153
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #21429469C2semidirect
3. #21422101C2semidirect
4. #21422042C2semidirect
154. C59#12153semidirect

Maximal subgroups

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

subgroupindex
#214294692
#214230992
#214221012
#214220422
C59×#20392
#1215359