Finite Group Databasebeta

← #28489684

Subgroups — #28489684

Order 15104.

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

normal subgroups only

This group has 476 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
2#21434135C212yesmaximal
3#21422498C212yesmaximal
4#21421927C211yesmaximal, characteristic
5C59×#1985C211yesmaximal, characteristic
6#16837959C2212yes—
7#16837811—22no—
8#16837811—22no—
9#16837737—22no—
10#16837697—22no—
11#16837612—22no—
12#16837558—22no—
13C4×#11745672—24no—
14#16837455C2212yes—
15#16837440—22no—
16#16837132—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×#97×C59C2211yescharacteristic
20C59×#336—22no—
21C2×#11745672—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#14273660—24no—
33#14273659—24no—
34C4⋊(C59⋊C8)—22no—
35#14273651—24no—
36#14273650—24no—
37C22⋊(C59⋊C8)—22no—
38C4×C59⋊C8D414yes—
39C22×Q8×C59D411yescharacteristic
40C22×C8×C59—21no—
41C22×C8×C59D412yes—
42C4×Q8×C59—22no—
43C4×Q8×C59—22no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C42×C59—21no—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49#97×C59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—42no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59#4111yescharacteristic
57C2×Dic118—22no—
58C2×Dic118—22no—
59C2×C8×C59—22no—
60C2×C8×C59—22no—
61C2×C8×C59—41no—
62C2×C8×C59—42no—
63C2×C8×C59Q1614yescomplemented
64C4⋊C4×C59—22no—
65C4⋊C4×C59—42no—
66C42×C59—41no—
67C42×C59—41no—
68C42×C59C2×D412yes—
69C42×C59C2×D412yes—
70C22⋊Dic59—42no—
71#11745672—48no—
72C4⋊Dic59—22no—
73C4⋊Dic59—22no—
74C4⋊Dic59—22no—
75C4⋊Dic59—22no—
76Dic59⋊C4—42no—
77Dic59⋊C4—42no—
78C4×Dic59—42no—
79C2×C59⋊C8—24no—
80C23×C59#12111yescharacteristic
81Q8×C59—24no—
82Q8×C59—42no—
83C2×C4×C59—22no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59—41no—
90C2×C4×C59#12011yescharacteristic
91C2×C4×C59#12012yes—
92C2×C4×C59#12112yes—
93C2×C4×C59#12711yescharacteristic
94C2×C4×C59C2×Q1612yes—
95C2×C4×C59#13712yes—
96C2×Dic59—42no—
97C2×Dic59—42no—
98Dic118—44no—
99Dic118—44no—
100C8×C59—24no—
101C8×C59—81no—
102C59⋊C8—44no—
103#10037—591nomaximal, Sylow
104C22×C59—21no—
105C22×C59—21no—
106C22×C59#39211yescharacteristic
107C22×C59#46212yes—
108C4×C59—22no—
109C4×C59—22no—
110C4×C59—42no—
111C4×C59—81no—
112C4×C59#45112yes—
113C4×C59#49411yescharacteristic
114C4×C59#49711yescharacteristic
115Dic59—82no—
116Dic59—82no—
117#3162—591no—
118#3104—592no—
119#1985D5911yescharacteristic
120#1481—592no—
121#1275—591no—
122C2×C59—22no—
123C2×C59#201011yescharacteristic
124C2×C59#201611yescharacteristic
125C2×C59#201711yescharacteristic
126#533—592no—
127C2×C4×Q8C2×D5911yescharacteristic
128#501—1182no—
129#501—1182no—
130#484—1182no—
131#477—1182no—
132#462—1182no—
133#451—1182no—
134C4×Q16—1184no—
135#431—592no—
136#427—1182no—
137C2×C4⋊C8C2×D5911yescharacteristic
138C2×#97C2×D5911yescharacteristic
139#336—22no—
140Q8⋊C8—1184no—
141C59#1003711yescharacteristic, complemented, Sylow
142C22×Q8D23611yescharacteristic
143C2×Q16—1184no—
144C22×C8—21no—
145C22×C8C22⋊D5912yes—
146C4⋊Q8—592no—
147C4⋊Q8—592no—
148#125—1182no—
149C22⋊Q8—1182no—
150C22⋊Q8—1182no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1182no—
154C4×Q8—1182no—
155#117—1182no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D23611yescharacteristic
158C2×C42—21no—
159C2×C42C22×D5911yescharacteristic
160#107—1184no—
161#107—1182no—
162#107—1182no—
163#106—1184no—
164C4⋊C8—22no—
165C4⋊C8—1182no—
166#103—1184no—
167#103—1184no—
168C22⋊C8—1182no—
169#97—22no—
170C4×C8—594no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×Q8—1182no—
174C2×Q8—1182no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C2×D23611yescharacteristic
178C22×C4#1174568511yescharacteristic
179C22×C4C2×C22⋊D5911yescharacteristic
180Q16—2368no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—1184no—
186C2×C8#1174567214yescomplemented
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—1182no—
190C4⋊C4—1182no—
191C4⋊C4—1182no—
192C4⋊C4—1182no—
193C4⋊C4—2362no—
194C4⋊C4—2362no—
195C22⋊C4—2362no—
196C42—41no—
197C42—41no—
198C42—2362no—
199C42D4×D5912yes—
200C42D4×D5912yes—
201C23#1427370111yescharacteristic
202Q8—24no—
203Q8—42no—
204Q8—2364no—
205Q8—2364no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—2362no—
214C2×C4—2362no—
215C2×C4#1427369511yescharacteristic
216C2×C4#1427369712yes—
217C2×C4#1427370012yes—
218C2×C4#1427372311yescharacteristic
219C2×C4C2×#1174567212yes—
220C2×C4#1427375712yes—
221C8—24no—
222C8—81no—
223C8—2364no—
224C22—21no—
225C22—21no—
226C22#1683731111yescharacteristic
227C22#1683761212yes—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—81no—
232C4—4722no—
233C4—4722no—
234C4#1683755712yes—
235C4#1683778111yescharacteristic
236C4#1683779711yescharacteristic
237C2—22no—
238C2#2142516311yescharacteristic
239C2#2142519511yescharacteristic
240C2#2142520011yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1985C2
socleC22×C59#392
prime radicalC23×C59#121
hypercenter#1985D59
hypocenterC59#10037
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
63. C2×C8×C59Q16semidirect
141. C59#10037semidirect
186. C2×C8#11745672semidirect

Maximal subgroups

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

subgroupindex
#214346202
#214341352
#214224982
#214219272
C59×#19852
#1003759