Finite Group Databasebeta

← #28489679

Subgroups — #28489679

Order 15104.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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#21434620C211yesmaximal, characteristic, complemented
2#21434104C212yesmaximal
3#21422498C212yesmaximal, complemented
4#21421927C211yesmaximal, characteristic, complemented
5C59×#1955C211yesmaximal, characteristic
6#16837959C2212yes—
7#16837811—22no—
8#16837811—22no—
9#16837732—22no—
10#16837692—22no—
11#16837607—22no—
12#16837553—22no—
13C4×#11745670—24no—
14#16837455C2212yes—
15#16837440—22no—
16#16837132—24no—
17C2×C4×D4×C59C2211yescharacteristic
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×C22⋊C8×C59—22no—
20C2×#97×C59C2211yescharacteristic
21C2×#11745670—24no—
22#14273727D412yes—
23#14273727D412yes—
24#14273719—22no—
25#14273706—22no—
26#14273706—22no—
27C4×Dic118—22no—
28C4×Dic118—22no—
29#14273686—22no—
30#14273661—22no—
31#14273661—22no—
32#14273659—24no—
33#14273658—24no—
34C4⋊(C59⋊C8)—22no—
35#14273650—24no—
36#14273648—24no—
37C22⋊(C59⋊C8)—22no—
38C4×C59⋊C8D414yes—
39C22×D4×C59D411yescharacteristic
40C23×C4×C59—21no—
41C22×C8×C59—21no—
42C22×C8×C59D412yes—
43C4×D4×C59—22no—
44C4×D4×C59—22no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59—21no—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49C22⋊C8×C59—44no—
50#97×C59—22no—
51C24×C59—21no—
52C2×D4×C59—22no—
53C2×D4×C59—42no—
54C22×C4×C59—21no—
55C22×C4×C59—41no—
56C22×C4×C59—41no—
57C22×C4×C59—42no—
58C22×C4×C59—42no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×C4×C59#4111yescharacteristic
62C2×Dic118—22no—
63C2×Dic118—22no—
64C2×C8×C59—22no—
65C2×C8×C59—22no—
66C2×C8×C59—41no—
67C2×C8×C59—42no—
68C2×C8×C59SD1614yescomplemented
69C4⋊C4×C59—22no—
70C22⋊C4×C59—42no—
71C42×C59C2×D412yes—
72C42×C59C2×D412yes—
73C22⋊Dic59—42no—
74#11745670—48no—
75C4⋊Dic59—22no—
76C4⋊Dic59—22no—
77C4⋊Dic59—22no—
78C4⋊Dic59—22no—
79Dic59⋊C4—42no—
80Dic59⋊C4—42no—
81C4×Dic59—42no—
82C2×C59⋊C8—24no—
83C23×C59—22no—
84C23×C59—41no—
85C23×C59—42no—
86C23×C59—41no—
87C23×C59—42no—
88C23×C59#12111yescharacteristic
89D4×C59—24no—
90D4×C59—42no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—82no—
96C2×C4×C59—81no—
97C2×C4×C59—81no—
98C2×C4×C59#12011yescharacteristic
99C2×C4×C59#12012yes—
100C2×C4×C59#12112yes—
101C2×C4×C59#12711yescharacteristic
102C2×C4×C59C2×SD1612yes—
103C2×C4×C59#13712yes—
104C2×Dic59—42no—
105C2×Dic59—42no—
106Dic118—44no—
107Dic118—44no—
108C8×C59—24no—
109C8×C59—81no—
110C59⋊C8—44no—
111#10036—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—44no—
115C22×C59—42no—
116C22×C59—82no—
117C22×C59—81no—
118C22×C59—81no—
119C22×C59#39211yescharacteristic
120C22×C59#46112yes—
121C4×C59—22no—
122C4×C59—22no—
123C4×C59—81no—
124C4×C59#44812yes—
125C4×C59#49211yescharacteristic
126C4×C59#49711yescharacteristic
127Dic59—82no—
128Dic59—82no—
129#3162—591no—
130#3101—592no—
131#1955D5911yescharacteristic
132#1481—592no—
133#1275—591no—
134C2×C59—22no—
135C2×C59—82no—
136C2×C59#201011yescharacteristic
137C2×C59#201311yescharacteristic
138C2×C59#201711yescharacteristic
139#533—592no—
140C2×C4×D4C2×D5911yescharacteristic
141#501—1182no—
142#501—1182no—
143#483—1182no—
144#476—1182no—
145#461—1182no—
146#450—1182no—
147C4×SD16—1184no—
148#431—592no—
149#427—1182no—
150C2×C4⋊C8C2×D5911yescharacteristic
151C2×C22⋊C8—22no—
152C2×#97C2×D5911yescharacteristic
153Q8⋊C8—1184no—
154C59#1003611yescharacteristic, complemented, Sylow
155C22×D4D23611yescharacteristic
156C23×C4—21no—
157C2×SD16—1184no—
158C22×C8—21no—
159C22×C8C22⋊D5912yes—
160C4⋊Q8—592no—
161C4⋊Q8—592no—
162#125—1182no—
163C22⋊Q8—1182no—
164C22⋊Q8—1182no—
165C4×Q8—1182no—
166C4×Q8—1182no—
167C4×D4—22no—
168C4×D4—22no—
169#117—1182no—
170C2×C4⋊C4D23611yescharacteristic
171C2×C22⋊C4—21no—
172C2×C42C22×D5911yescharacteristic
173#107—1182no—
174#107—1182no—
175#106—1184no—
176#106—1184no—
177C4⋊C8—22no—
178C4⋊C8—1182no—
179#103—1184no—
180#102—1184no—
181C22⋊C8—44no—
182C22⋊C8—1182no—
183#97—22no—
184C4×C8—594no—
185C24—21no—
186C2×Q8—1182no—
187C2×Q8—1182no—
188C2×D4—22no—
189C2×D4—42no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×D23611yescharacteristic
196C22×C4#1174568511yescharacteristic
197C22×C4C2×C22⋊D5911yescharacteristic
198SD16—2368no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—1184no—
204C2×C8#1174567014yescomplemented
205C4⋊C4—22no—
206C4⋊C4—1182no—
207C4⋊C4—1182no—
208C4⋊C4—1182no—
209C4⋊C4—1182no—
210C4⋊C4—2362no—
211C4⋊C4—2362no—
212C22⋊C4—42no—
213C22⋊C4—2362no—
214C42—2362no—
215C42D4×D5912yes—
216C42D4×D5912yes—
217C23—22no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23#1427370111yescharacteristic
223Q8—2364no—
224Q8—2364no—
225D4—24no—
226D4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4#1427369511yescharacteristic
237C2×C4#1427369712yes—
238C2×C4#1427370012yes—
239C2×C4#1427372311yescharacteristic
240C2×C4C2×#1174567012yes—
241C2×C4#1427375712yes—
242C8—24no—
243C8—81no—
244C8—2364no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1683731111yescharacteristic
253C22#1683760712yes—
254C4—22no—
255C4—22no—
256C4—81no—
257C4—4722no—
258C4—4722no—
259C4#1683754012yes—
260C4#1683777411yescharacteristic
261C4#1683779711yescharacteristic
262C2—22no—
263C2—82no—
264C2#2142516311yescharacteristic
265C2#2142517811yescharacteristic
266C2#2142520011yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1955C2
socleC22×C59#392
prime radicalC22×D4×C59D4
hypercenter#1955D59
hypocenterC59#10036
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #21434620C2semidirect
3. #21422498C2semidirect
4. #21421927C2semidirect
68. C2×C8×C59SD16semidirect
154. C59#10036semidirect
204. C2×C8#11745670semidirect

Maximal subgroups

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

subgroupindex
#214346202
#214341042
#214224982
#214219272
C59×#19552
#1003659