Finite Group Databasebeta

← #28502244

Subgroups — #28502244

Order 15104.

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

normal subgroups only

This group has 312 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#21437085C211yesmaximal, characteristic
2#21426348C211yesmaximal, characteristic, complemented
3#21426138C211yesmaximal, characteristic
4#21422144C211yesmaximal, characteristic, complemented
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422056C211yesmaximal, characteristic
7C59×#1670C211yesmaximal, characteristic, complemented
8#16838125—21no—
9C2×#14273673—22no—
10C2×#14273670—21no—
11#16837786—21no—
12#16837782C2211yescharacteristic
13#16837625—22no—
14#16837608—21no—
15C4×Dic236—21no—
16C4×D472C2211yescharacteristic
17C4⋊(C59⋊C16)C2211yescharacteristic
18#16837206—21no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#494C2211yescharacteristic
21C59×#475—21no—
22C8×Q8×C59C2211yescharacteristic
23C59×#335C2211yescharacteristic
24C59×#332—21no—
25C2×Dic236—21no—
26C2×Dic236D412yes—
27C2×#11745660—22no—
28C2×D472D411yescharacteristic
29#14273727—21no—
30#14273722—42no—
31#14273715—22no—
32#14273703—41no—
33C4×Dic118—22no—
34C4×D236—21no—
35#14273685—41no—
36#14273673—44no—
37#14273670—42no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40#14273661D411yescharacteristic
41#14273652—22no—
42#14273649—21no—
43C2×Q16×C59—21no—
44C2×Q16×C59—21no—
45C4⋊Q8×C59—21no—
46C4×Q8×C59D411yescharacteristic
47#106×C59—41no—
48C4⋊C8×C59—21no—
49C4⋊C8×C59—21no—
50C4⋊C8×C59D411yescharacteristic
51#103×C59—41no—
52C4×C8×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—41no—
55C2×Q8×C59D811yescharacteristic
56C2×D236—21no—
57C2×C4×D59—41no—
58C2×Dic118—22no—
59Q16×C59—42no—
60Q16×C59—41no—
61Q16×C59—41no—
62C2×C8×C59—21no—
63C2×C8×C59—41no—
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—21no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59D811yescharacteristic
70C42×C59—21no—
71C42×C59C2×D411yescharacteristic
72(C2×D59)⋊C4—42no—
73(C2×D59)⋊C4—42no—
74(C2×D59)⋊C4—41no—
75C4⋊Dic59—21no—
76C4⋊Dic59—41no—
77C4⋊Dic59—41no—
78Dic59⋊C4—42no—
79C4×Dic59—42no—
80Dic236—42no—
81Dic236—41no—
82Dic236—41no—
83D472—21no—
84D472—21no—
85#11745660—82no—
86C59⋊C16—22no—
87C59⋊C16—41no—
88C22×D59—41no—
89Q8×C59—21no—
90Q8×C59—21no—
91Q8×C59—81no—
92Q8×C59—81no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—81no—
96C2×C4×C59#12011yescharacteristic
97C2×C4×C59C2×D811yescharacteristic
98C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
99C2×Dic59—41no—
100C2×Dic59—42no—
101D236—41no—
102D236—41no—
103C4×D59—81no—
104C4×D59—81no—
105Dic118—42no—
106Dic118—42no—
107C8×C59—21no—
108C8×C59—41no—
109C8×C59—81no—
110C8×C59#13511yescharacteristic
111C8×C59AGL(1,ℤ/8)11yescharacteristic
112#12335—591nomaximal, Sylow
113C22×C59#44711yescharacteristic
114C2×D59—81no—
115C2×D59—81no—
116C2×D59—81no—
117C4×C59—21no—
118C4×C59—41no—
119C4×C59—41no—
120C4×C59—161no—
121C4×C59#44911yescharacteristic
122C4×C59#45311yescharacteristic
123C4×C59#50811yescharacteristic
124C4×C59#51011yescharacteristic
125Dic59—82no—
126Dic59—81no—
127#3393—591no—
128#2247—591no—
129#2208—591no—
130#1670D5911yescharacteristic, complemented
131#1362—591no—
132Q8⋊C16—591no—
133#1331—591no—
134C2×C59#162111yescharacteristic
135C2×C59#218911yescharacteristic
136C2×C59#219511yescharacteristic
137D59—161no—
138#556—1181no—
139C2×Q32—1182no—
140C2×SD32—1181no—
141#495—1181no—
142#494—591no—
143#494C2×D5911yescharacteristic
144#475—21no—
145#464—1182no—
146#461—1181no—
147C8×Q8C2×D5911yescharacteristic
148C4×Q16—1181no—
149C4×D8—591no—
150C4⋊C16—591no—
151#358—1181no—
152C4×C16—591no—
153#335C2×D5911yescharacteristic
154#332—21no—
155C59#1233511yescharacteristic, complemented, Sylow
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—592no—
159C2×Q16—1181no—
160C2×SD16—1182no—
161C2×D8—591no—
162C4⋊Q8—21no—
163C4⋊Q8—1181no—
164#126—2362no—
165#124—1182no—
166C22⋊Q8—2361no—
167C4×Q8—1182no—
168C4×Q8C22⋊D5911yescharacteristic
169C4×D4—1181no—
170#117—2361no—
171Q32—2364no—
172SD32—2362no—
173C2×C16—1181no—
174C2×C16—1181no—
175#107—591no—
176#106—41no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8C22⋊D5911yescharacteristic
180#103—41no—
181#103—1182no—
182#102—1181no—
183C4×C8—21no—
184C4×C8C22×D5911yescharacteristic
185C2×Q8—41no—
186C2×Q8—1182no—
187C2×Q8#1174566911yescharacteristic
188C2×D4—1181no—
189C22×C4—2361no—
190Q16—42no—
191Q16—41no—
192Q16—41no—
193Q16—2362no—
194Q16—2361no—
195Q16—2361no—
196SD16—4722no—
197D8—1181no—
198D8—1181no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8D4×D5911yescharacteristic
202C2×C8D4×D5911yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—1181no—
207C4⋊C4—2362no—
208C4⋊C4—2361no—
209C4⋊C4—2361no—
210C4⋊C4#1174566911yescharacteristic
211C22⋊C4—2362no—
212C22⋊C4—2362no—
213C22⋊C4—2361no—
214C42—21no—
215C42—2362no—
216C42C2×C22⋊D5911yescharacteristic
217C16—1182no—
218C16—2361no—
219C23—2361no—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223Q8—81no—
224Q8—2362no—
225Q8—2362no—
226D4—2361no—
227D4—2361no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—81no—
231C2×C4—4721no—
232C2×C4—4721no—
233C2×C4—2361no—
234C2×C4—2362no—
235C2×C4#1427369511yescharacteristic
236C2×C4C2×#1174566911yescharacteristic
237C2×C4#1427375211yescharacteristic
238C8—21no—
239C8—41no—
240C8—81no—
241C8#1427374811yescharacteristic
242C8#1427375411yescharacteristic
243C22—4721no—
244C22—4721no—
245C22—4721no—
246C22#1683753511yescharacteristic
247C4—21no—
248C4—41no—
249C4—41no—
250C4—161no—
251C4—4722no—
252C4—4721no—
253C4#1683754611yescharacteristic
254C4#1683756811yescharacteristic
255C4#1683783211yescharacteristic
256C4#1683784311yescharacteristic
257C2—9441no—
258C2#2142315911yescharacteristic
259C2#2142603411yescharacteristic
260C2#2142607211yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1670C2
socleC22×C59#447
prime radicalC2×D472D4
hypercenter#1670D59
hypocenterC59#12335
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. #21426348C2semidirect
4. #21422144C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
7. C59×#1670C2semidirect
130. #1670D59semidirect
155. C59#12335semidirect

Maximal subgroups

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

subgroupindex
#214370852
#214263482
#214261382
#214221442
Q8⋊(C59⋊C16)2
#214220562
C59×#16702
#1233559