Finite Group Databasebeta

← #28488352

Subgroups — #28488352

Order 15104.

418 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#16837798C211yesmaximal, characteristic
2#21425085C211yesmaximal, characteristic
3#21422854C211yesmaximal, characteristic
4#21422782C211yesmaximal, characteristic
5#21422768C211yesmaximal, characteristic
6#21422383C211yesmaximal, characteristic
7C59×#1966C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9#16837798—21no—
10#16837798—21no—
11#16837798—22no—
12#16837455C2211yescharacteristic
13C2×#14273658C414yes—
14#16837426C2211yescharacteristic
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837170—22no—
17#16837154—22no—
18#16837148—22no—
19#16837147—22no—
20#16837124—21no—
21#16837124—21no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C59×#423C2211yescharacteristic
24C2×#101×C59—22no—
25C4×M16×C59C2211yescharacteristic
26C22×Dic118—21no—
27C2×#11745664—21no—
28C22×C59⋊C8C2×C412yes—
29#14273727—21no—
30#14273727—21no—
31#14273727—21no—
32#14273727—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59C2×C412yes—
35#14273658—24no—
36#14273658—24no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D412yes—
40C22⋊(C59⋊C8)—21no—
41C4×C59⋊C8—21no—
42C4×C59⋊C8D411yescharacteristic
43C4×C59⋊C8D411yescharacteristic
44C4×C59⋊C8D412yes—
45C22×Q8×C59—21no—
46C2×M16×C59—21no—
47C2×M16×C59D412yes—
48C4⋊Q8×C59—21no—
49C4⋊Q8×C59—21no—
50C4⋊Q8×C59—21no—
51C4⋊Q8×C59—21no—
52C2×C4⋊C4×C59—21no—
53C2×C4⋊C4×C59C2×C412yes—
54C2×C42×C59C2311yescharacteristic
55C4⋊C8×C59—21no—
56C4⋊C8×C59—21no—
57#101×C59—44no—
58#97×C59—21no—
59C4×C8×C59—21no—
60C2×Q8×C59—41no—
61C2×Q8×C59—41no—
62C2×Q8×C59—42no—
63C2×Q8×C59—42no—
64C22×C4×C59—22no—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59#4111yescharacteristic
68C22×Dic59—22no—
69C2×Dic118—41no—
70C2×Dic118—41no—
71C2×Dic118—82no—
72M16×C59—22no—
73M16×C59—22no—
74M16×C59—42no—
75C2×C8×C59—22no—
76C2×C8×C59—41no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C42×C59C2×D411yescharacteristic
82C42×C59C2×D411yescharacteristic
83C42×C59C2×D411yescharacteristic
84C42×C59#4111yescharacteristic
85C4⋊Dic59—22no—
86C4⋊Dic59—22no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89#11745664—42no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—22no—
92C2×C59⋊C8—41no—
93C2×C59⋊C8C22⋊C412yes—
94C2×C59⋊C8C22⋊C412yes—
95C23×C59C4×D411yescharacteristic
96Q8×C59—82no—
97Q8×C59—81no—
98Q8×C59—81no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—42no—
102C2×C4×C59—44no—
103C2×C4×C59C2×C22⋊C411yescharacteristic
104C2×C4×C59C4×D411yescharacteristic
105C2×C4×C59#12111yescharacteristic
106C2×C4×C59#12111yescharacteristic
107C2×C4×C59#12411yescharacteristic
108C2×C4×C59#12711yescharacteristic
109C2×C4×C59AGL(1,ℤ/8)12yes—
110C2×C4×C59#13712yes—
111C2×Dic59—42no—
112C2×Dic59—82no—
113Dic118—82no—
114Dic118—82no—
115C8×C59—42no—
116C8×C59—81no—
117C59⋊C8—22no—
118C59⋊C8—42no—
119C59⋊C8—81no—
120C59⋊C8#10114yescomplemented
121#9806—591nomaximal, Sylow
122C22×C59—21no—
123C22×C59—21no—
124C22×C59#39011yescharacteristic
125C22×C59#44012yes—
126C4×C59—21no—
127C4×C59—21no—
128C4×C59—82no—
129C4×C59C2×#10112yes—
130C4×C59#46812yes—
131C4×C59#47112yes—
132C4×C59#49611yescharacteristic
133C4×C59#49711yescharacteristic
134Dic59—82no—
135C2×#498—591no—
136#1997—591no—
137#1966D5911yescharacteristic
138#1567—591no—
139#1549—591no—
140#1546—591no—
141#1451—591no—
142C2×C59—22no—
143C2×C59#196611yescharacteristic
144C2×C59#197311yescharacteristic
145C2×C59#197711yescharacteristic
146C2×C4⋊Q8—591no—
147C2×C4⋊Q8C2×D5911yescharacteristic
148#498—1181no—
149#498—1181no—
150#498—1182no—
151#431—591no—
152C2×#106—594no—
153#423—591no—
154#423C2×D5911yescharacteristic
155C2×#101—22no—
156C4×M16C2×D5911yescharacteristic
157C2×C4×C8—591no—
158#343—1182no—
159#335—1182no—
160#332—1182no—
161#332—1182no—
162#322—1181no—
163#322—1181no—
164C59#980611yescharacteristic, complemented, Sylow
165C22×Q8—21no—
166C22×Q8—1181no—
167C2×M16—21no—
168C2×M16—1181no—
169C2×M16C22⋊D5912yes—
170C22×C8—592no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—1181no—
176C4⋊Q8—1181no—
177C4⋊Q8—1181no—
178C4⋊Q8—1181no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—592no—
181C2×C4⋊C4—1181no—
182C2×C4⋊C4C4×D5912yes—
183C2×C42C22×D5911yescharacteristic
184#106—1184no—
185#106—1184no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—592no—
189C4⋊C8—1181no—
190C4⋊C8—1181no—
191#101—44no—
192C22⋊C8—1181no—
193#97—21no—
194C4×C8—21no—
195C4×C8—591no—
196C4×C8—591no—
197C4×C8—592no—
198C4×C8—1181no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—4722no—
204C2×Q8—2361no—
205C2×Q8—2361no—
206C22×C4—22no—
207C22×C4—1182no—
208C22×C4C2×C4×D5911yescharacteristic
209C22×C4#1174568511yescharacteristic
210C22×C4C2×C22⋊D5911yescharacteristic
211M16—22no—
212M16—22no—
213M16—42no—
214M16—2362no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—592no—
218C2×C8—592no—
219C2×C8—1184no—
220C2×C8—1182no—
221C2×C8—2361no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—1182no—
227C4⋊C4—1182no—
228C4⋊C4—2361no—
229C4⋊C4—2361no—
230C42D4×D5911yescharacteristic
231C42D4×D5911yescharacteristic
232C42D4×D5911yescharacteristic
233C42#1174568711yescharacteristic
234C23C4×C22⋊D5911yescharacteristic
235Q8—82no—
236Q8—81no—
237Q8—81no—
238Q8—4722no—
239Q8—4722no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—4722no—
245C2×C4—2362no—
246C2×C4C22⋊C4×D5911yescharacteristic
247C2×C4(C22⋊D59)⋊C411yescharacteristic
248C2×C4#1427369911yescharacteristic
249C2×C4#1427370011yescharacteristic
250C2×C4#1427371311yescharacteristic
251C2×C4#1427372311yescharacteristic
252C2×C4#1427375312yes—
253C2×C4#1427375712yes—
254C8—42no—
255C8—81no—
256C8—594no—
257C8—1182no—
258C8—4721no—
259C8—2362no—
260C22—21no—
261C22—21no—
262C22#1683730211yescharacteristic
263C22#1683750212yes—
264C4—21no—
265C4—21no—
266C4—82no—
267C4—4722no—
268C4#101×D5912yes—
269C4#1683764612yes—
270C4#1683766812yes—
271C4#1683778911yescharacteristic
272C4#1683779711yescharacteristic
273C2—22no—
274C2#2142493811yescharacteristic
275C2#2142497611yescharacteristic
276C2#2142500011yescharacteristic
subgroupG/H
centerC22#16837302
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1966C2
socleC22×C59#390
prime radicalC23×C59C4×D4
cosocleC2×C42×C59C23
hypercenter#1966D59
hypocenterC59#9806
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
120. C59⋊C8#101semidirect
164. C59#9806semidirect

Maximal subgroups

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

subgroupindex
C2×#168377982
#214250852
#214228542
#214227822
#214227682
#214223832
C59×#19662
#980659