Finite Group Databasebeta

← #28502566

Subgroups — #28502566

Order 15104.

288 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 67 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#21437101C211yesmaximal, characteristic
2#21426325C211yesmaximal, characteristic
3#21423487C211yesmaximal, characteristic
4#21422146C211yesmaximal, characteristic
5#21422074C211yesmaximal, characteristic
6#21422060C211yesmaximal, characteristic
7C59×#2225C211yesmaximal, characteristic
8#16838130—21no—
9#16837782C2211yescharacteristic
10#16837780—21no—
11#16837612—21no—
12#16837611—22no—
13#16837609—21no—
14C8×Dic118C2211yescharacteristic
15#16837496—21no—
16#16837205—22no—
17C4×C59⋊C16C2211yescharacteristic
18#16837155C2211yescharacteristic
19#16837148—21no—
20C59×#500C2211yescharacteristic
21C4×Q16×C59C2211yescharacteristic
22C59×#367—21no—
23C59×C4⋊C16C2211yescharacteristic
24C59×#358—21no—
25C2×Dic236—21no—
26C2×Dic236D412yes—
27C2×#11745672—22no—
28C2×#11745672—41no—
29#14273727—21no—
30#14273725—22no—
31#14273719—41no—
32#14273717—42no—
33C4×Dic118—41no—
34C4×Dic118D411yescharacteristic
35#14273692—22no—
36C2×C59⋊C16—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)D411yescharacteristic
39Dic59⋊C8—21no—
40#14273651—41no—
41#14273650—22no—
42C8×Dic59—21no—
43C2×Q16×C59D411yescharacteristic
44C4⋊Q8×C59—21no—
45C4×Q8×C59—21no—
46C2×C16×C59—21no—
47#107×C59—21no—
48#107×C59—21no—
49#107×C59D411yescharacteristic
50#103×C59—21no—
51C4×C8×C59C2311yescharacteristic
52C2×Q8×C59—21no—
53C2×Q8×C59—41no—
54C2×Dic118—22no—
55C2×Dic118D811yescharacteristic
56Q16×C59—21no—
57Q16×C59—21no—
58C2×C8×C59C2×D411yescharacteristic
59C2×C8×C59C2×D411yescharacteristic
60C4⋊C4×C59—21no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C42×C59—41no—
65C42×C59C2×D411yescharacteristic
66#11745672—82no—
67#11745672—82no—
68C4⋊Dic59—41no—
69C4⋊Dic59—41no—
70C4⋊Dic59D811yescharacteristic
71Dic59⋊C4—21no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74Dic59⋊C4—42no—
75C4×Dic59—21no—
76C4×Dic59—42no—
77C2×C59⋊C8—21no—
78C2×C59⋊C8—41no—
79Dic236—42no—
80Dic236—41no—
81Dic236—41no—
82C16×C59—41no—
83C59⋊C16—41no—
84Q8×C59—41no—
85Q8×C59—41no—
86Q8×C59—81no—
87Q8×C59—81no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—81no—
91C2×C4×C59#12011yescharacteristic
92C2×C4×C59C2×D811yescharacteristic
93C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—42no—
97Dic118—21no—
98Dic118—42no—
99Dic118—42no—
100Dic118SD3212yes—
101C8×C59—21no—
102C8×C59#13511yescharacteristic
103C8×C59#13711yescharacteristic
104C59⋊C8—41no—
105C59⋊C8—81no—
106#12382—591nomaximal, Sylow
107C22×C59#44711yescharacteristic
108C4×C59—21no—
109C4×C59—81no—
110C4×C59—81no—
111C4×C59—161no—
112C4×C59#45211yescharacteristic
113C4×C59#45511yescharacteristic
114C4×C59C2×SD3211yescharacteristic
115C4×C59#51011yescharacteristic
116Dic59—22no—
117Dic59—41no—
118Dic59—82no—
119#3395—591no—
120#2241—591no—
121#2225D5911yescharacteristic
122#1674—591no—
123#1362—591no—
124Q8⋊C16—591no—
125#1332—591no—
126C2×C59#164111yescharacteristic
127C2×C59#218511yescharacteristic
128C2×C59#219511yescharacteristic
129#557—1181no—
130#500C2×D5911yescharacteristic
131#494—591no—
132#494—1181no—
133#462—1182no—
134#462—1181no—
135#462—1181no—
136C8×Q8—591no—
137C4×Q16—1181no—
138C4×Q16C2×D5911yescharacteristic
139#367—21no—
140C4⋊C16C2×D5911yescharacteristic
141#358—21no—
142#358—1182no—
143C4×C16—591no—
144#335—591no—
145#332—1181no—
146C59#1238211yescharacteristic, complemented, Sylow
147C2×Q16—592no—
148C2×Q16—1182no—
149C2×Q16—1181no—
150C2×Q16—2361no—
151C2×Q16C22⋊D5911yescharacteristic
152C4⋊Q8—21no—
153C4⋊Q8—1182no—
154C4⋊Q8—1181no—
155#125—2361no—
156#125—2362no—
157C4×Q8—21no—
158C4×Q8—591no—
159C4×Q8—1182no—
160C4×Q8—2361no—
161C2×C16—21no—
162C2×C16—1181no—
163#107—21no—
164#107—21no—
165#107C22⋊D5911yescharacteristic
166C4⋊C8—591no—
167C4⋊C8—1181no—
168C4⋊C8—1181no—
169#103—21no—
170#103—1182no—
171#103—2361no—
172C4×C8—1181no—
173C4×C8C22×D5911yescharacteristic
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—591no—
177C2×Q8—1182no—
178Q16—21no—
179Q16—21no—
180Q16—4722no—
181Q16—4722no—
182Q16—2362no—
183Q16—2361no—
184Q16—2361no—
185C2×C8—1181no—
186C2×C8—2361no—
187C2×C8D4×D5911yescharacteristic
188C2×C8C2×C22⋊D5911yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—591no—
194C4⋊C4—1181no—
195C4⋊C4—2362no—
196C4⋊C4—2362no—
197C4⋊C4—2362no—
198C4⋊C4—2361no—
199C4⋊C4—2361no—
200C42—41no—
201C42—1181no—
202C42—2362no—
203C42D4×D5911yescharacteristic
204C16—41no—
205C16—2361no—
206Q8—41no—
207Q8—41no—
208Q8—81no—
209Q8—81no—
210Q8—592no—
211Q8—1181no—
212Q8—2362no—
213Q8—2362no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—1181no—
218C2×C4—1181no—
219C2×C4—2362no—
220C2×C4#1427369511yescharacteristic
221C2×C4D8×D5911yescharacteristic
222C2×C4#1427375111yescharacteristic
223C8—21no—
224C8—4721no—
225C8—2361no—
226C8#1427374711yescharacteristic
227C8#1427375811yescharacteristic
228C22#1683753411yescharacteristic
229C4—21no—
230C4—81no—
231C4—81no—
232C4—161no—
233C4—1182no—
234C4—4722no—
235C4—2361no—
236C4#1683756211yescharacteristic
237C4#1683757811yescharacteristic
238C4SD32×D5911yescharacteristic
239C4#1683784111yescharacteristic
240C2#2142328811yescharacteristic
241C2#2142601111yescharacteristic
242C2#2142607111yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2225C2
socle, prime radicalC22×C59#447
hypercenter#2225D59
hypocenterC59#12382
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
146. C59#12382semidirect

Maximal subgroups

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

subgroupindex
#214371012
#214263252
#214234872
#214221462
#214220742
#214220602
C59×#22252
#1238259