Finite Group Databasebeta

← #28489685

Subgroups — #28489685

Order 15104.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434136C212yesmaximal
3#21425036C211yesmaximal, characteristic
4#21422500C212yesmaximal
5C59×#1275C211yesmaximal, characteristic
6#16837959C2212yes—
7C2×C4×Dic118C2211yescharacteristic
8#16837811—22no—
9#16837811—22no—
10#16837738—22no—
11#16837698—22no—
12#16837613—22no—
13#16837560—22no—
14C4×Dic236—24no—
15#16837440—22no—
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17#16837156—22no—
18#16837134—24no—
19C59×#431C2212yes—
20C2×#97×C59C2211yescharacteristic
21C22×Dic118D411yescharacteristic
22C2×Dic236—24no—
23C22×C59⋊C8—21no—
24#14273727D412yes—
25#14273727D412yes—
26#14273719—22no—
27#14273706—22no—
28#14273706—22no—
29C4×Dic118—22no—
30C4×Dic118—22no—
31C4×Dic118—22no—
32C4×Dic118—22no—
33#14273686—22no—
34C2×C4⋊Dic59D411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×C4×Dic59—21no—
37#14273661—22no—
38#14273661—22no—
39#14273661—24no—
40#14273659—24no—
41C4⋊(C59⋊C8)—22no—
42#14273652—24no—
43#14273652—24no—
44C22×C8×C59D412yes—
45C2×C42×C59C2311yescharacteristic
46C4⋊C8×C59—22no—
47C22⋊C8×C59—22no—
48#97×C59—22no—
49C4×C8×C59D414yes—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D411yescharacteristic
52C22×C4×C59#4111yescharacteristic
53C22×Dic59—21no—
54C22×Dic59—21no—
55C2×Dic118—22no—
56C2×Dic118—22no—
57C2×Dic118—22no—
58C2×Dic118—42no—
59C2×C8×C59—22no—
60C2×C8×C59—22no—
61C2×C8×C59—24no—
62C2×C8×C59Q1614yescomplemented
63C42×C59C2×D412yes—
64C42×C59C2×D412yes—
65C22⋊Dic59—42no—
66C4⋊Dic59—22no—
67C4⋊Dic59—22no—
68C4⋊Dic59—22no—
69C4⋊Dic59—22no—
70C4⋊Dic59—22no—
71Dic59⋊C4—42no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74C4×Dic59—41no—
75C4×Dic59—41no—
76C4×Dic59—42no—
77C2×C59⋊C8—41no—
78C2×C59⋊C8—42no—
79Dic236—48no—
80C23×C59#12111yescharacteristic
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59#12012yes—
84C2×C4×C59#12011yescharacteristic
85C2×C4×C59#12112yes—
86C2×C4×C59#12711yescharacteristic
87C2×C4×C59C2×Q1612yes—
88C2×C4×C59#13712yes—
89C2×Dic59—22no—
90C2×Dic59—41no—
91C2×Dic59—41no—
92C2×Dic59—42no—
93C2×Dic59—41no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96Dic118—24no—
97Dic118—42no—
98Dic118—44no—
99Dic118—44no—
100C8×C59—24no—
101C8×C59—44no—
102C59⋊C8—81no—
103#10037—591nomaximal, Sylow
104C22×C59—21no—
105C22×C59—21no—
106C22×C59#39211yescharacteristic
107C22×C59#46212yes—
108C4×C59—22no—
109C4×C59—22no—
110C4×C59#45112yes—
111C4×C59#49411yescharacteristic
112C4×C59#49711yescharacteristic
113Dic59—42no—
114Dic59—82no—
115Dic59—82no—
116Dic59—81no—
117#3162—591no—
118#3104—592no—
119#1985—591no—
120#1481—592no—
121#1275D5911yescharacteristic
122C2×C59—22no—
123C2×C59#201011yescharacteristic
124C2×C59#201611yescharacteristic
125C2×C59#201711yescharacteristic
126#533—592no—
127C2×C4×Q8—591no—
128#501—1182no—
129#501—1182no—
130#484—1182no—
131#477—1182no—
132#462—1182no—
133#451—1182no—
134C4×Q16—1184no—
135#431C2×D5912yes—
136#427—1182no—
137C2×C4⋊C8—591no—
138C2×#97C2×D5911yescharacteristic
139#336—1182no—
140Q8⋊C8—1184no—
141C59#1003711yescharacteristic, complemented, Sylow
142C22×Q8—591no—
143C2×Q16—1184no—
144C22×C8—1181no—
145C22×C8D23612yes—
146C4⋊Q8—592no—
147C4⋊Q8—592no—
148#125—1182no—
149C22⋊Q8—1182no—
150C22⋊Q8—1182no—
151C4×Q8—1182no—
152C4×Q8—1182no—
153C4×Q8—1182no—
154C4×Q8—1182no—
155#117—1182no—
156C2×C4⋊C4—591no—
157C2×C4⋊C4—1181no—
158C2×C42—1181no—
159C2×C42C22×D5911yescharacteristic
160#107—1182no—
161#107—1182no—
162#107—1184no—
163#106—1184no—
164C4⋊C8—22no—
165C4⋊C8—1182no—
166#103—1184no—
167#103—1184no—
168C22⋊C8—22no—
169#97—22no—
170C4×C8D23614yes—
171C2×Q8—1182no—
172C2×Q8—1182no—
173C2×Q8—1182no—
174C2×Q8—2362no—
175C22×C4—1181no—
176C22×C4—1181no—
177C22×C4C2×D23611yescharacteristic
178C22×C4D4×D5911yescharacteristic
179C22×C4#1174568911yescharacteristic
180Q16—2368no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—2361no—
185C2×C8—2362no—
186C2×C8Dic23614yescomplemented
187C4⋊C4—1182no—
188C4⋊C4—1182no—
189C4⋊C4—1182no—
190C4⋊C4—1182no—
191C4⋊C4—1182no—
192C4⋊C4—2362no—
193C4⋊C4—2362no—
194C4⋊C4—2362no—
195C22⋊C4—2362no—
196C42—2361no—
197C42—2361no—
198C42—2362no—
199C42C2×D23612yes—
200C42D4×D5912yes—
201C23#1427370211yescharacteristic
202Q8—1184no—
203Q8—2364no—
204Q8—2364no—
205Q8—2362no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1182no—
209C2×C4—2362no—
210C2×C4—2361no—
211C2×C4—2362no—
212C2×C4—2362no—
213C2×C4—2361no—
214C2×C4—2361no—
215C2×C4#1427369712yes—
216C2×C4#1427369711yescharacteristic
217C2×C4#1427370212yes—
218C2×C4#1427372411yescharacteristic
219C2×C4C2×Dic23612yes—
220C2×C4#1427376012yes—
221C8—24no—
222C8—44no—
223C8—4721no—
224C22—21no—
225C22—21no—
226C22#1683731211yescharacteristic
227C22#1683761312yes—
228C4—22no—
229C4—22no—
230C4—4722no—
231C4—4721no—
232C4—4722no—
233C4—2362no—
234C4#1683756012yes—
235C4#1683778211yescharacteristic
236C4#1683779611yescharacteristic
237C2—22no—
238C2#2142516411yescharacteristic
239C2#2142519811yescharacteristic
240C2#2142520111yescharacteristic
subgroupG/H
centerC2×C4#14273724
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1275C2
socleC22×C59#392
prime radicalC23×C59#121
hypercenter#1275D59
hypocenterC59#10037
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
62. C2×C8×C59Q16semidirect
141. C59#10037semidirect
186. C2×C8Dic236semidirect

Maximal subgroups

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

subgroupindex
#214346202
#214341362
#214250362
#214225002
C59×#12752
#1003759