Finite Group Databasebeta

← #28502509

Subgroups — #28502509

Order 15104.

344 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 73 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
1D4×Dic236C211yesmaximal, characteristic
2#21426325C211yesmaximal, characteristic, complemented
3#21426245C211yesmaximal, characteristic, complemented
4#21422144C211yesmaximal, characteristic, complemented
5#21422059C211yesmaximal, characteristic, complemented
6D4⋊(C59⋊C16)C211yesmaximal, characteristic
7C59×#1675C211yesmaximal, characteristic
8C22×Dic236—21no—
9D4×Dic118—21no—
10#16837782C2211yescharacteristic
11#16837643—21no—
12#16837613—21no—
13#16837560—22no—
14C4×Dic236C2211yescharacteristic
15#16837227—21no—
16C4⋊(C59⋊C16)C2211yescharacteristic
17#16837205—21no—
18#16837205—22no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×#476—21no—
22C8×D4×C59C2211yescharacteristic
23C59×#335C2211yescharacteristic
24C59×#333—21no—
25C22×Dic118—22no—
26C2×Dic236—22no—
27C2×Dic236—21no—
28C2×Dic236—42no—
29C2×Dic236—42no—
30C2×Dic236D411yescharacteristic
31C2×Dic236D412yes—
32#14273727—21no—
33#14273726—41no—
34#14273706—22no—
35#14273704—42no—
36C4×Dic118—21no—
37D4×Dic59—41no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40#14273661D411yescharacteristic
41#14273652—22no—
42#14273652—21no—
43C22×C8×C59—21no—
44C4⋊Q8×C59—21no—
45C4×D4×C59D411yescharacteristic
46#107×C59—21no—
47#107×C59—21no—
48#106×C59—41no—
49C4⋊C8×C59—21no—
50C4⋊C8×C59D411yescharacteristic
51#102×C59—41no—
52C22⋊C8×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—41no—
55C2×D4×C59D811yescharacteristic
56C22×C4×C59—21no—
57C22×Dic59—42no—
58C2×Dic118—21no—
59C2×Dic118—22no—
60C2×Dic118—44no—
61C2×Dic118—42no—
62C2×Dic118—82no—
63C2×C8×C59—21no—
64C2×C8×C59—41no—
65C2×C8×C59—41no—
66C2×C8×C59—41no—
67C2×C8×C59C2×D411yescharacteristic
68C2×C8×C59C2×D411yescharacteristic
69C4⋊C4×C59—41no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59D811yescharacteristic
72C22⋊C4×C59—21no—
73C42×C59C2×D411yescharacteristic
74C22⋊Dic59—42no—
75C4⋊Dic59—21no—
76C4⋊Dic59—41no—
77C4⋊Dic59—41no—
78Dic59⋊C4—42no—
79Dic59⋊C4—41no—
80C4×Dic59—41no—
81Dic236—21no—
82Dic236—42no—
83Dic236—44no—
84Dic236—41no—
85Dic236—41no—
86Dic236SD1612yes—
87C59⋊C16—41no—
88C59⋊C16—41no—
89C23×C59—21no—
90Q8×C59—161no—
91D4×C59—21no—
92D4×C59Q3212yes—
93C2×C4×C59—21no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—81no—
97C2×C4×C59#12011yescharacteristic
98C2×C4×C59C2×D811yescharacteristic
99C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
100C2×Dic59—41no—
101C2×Dic59—41no—
102C2×Dic59—42no—
103C2×Dic59—84no—
104Dic118—22no—
105Dic118—41no—
106Dic118—42no—
107Dic118—42no—
108Dic118—84no—
109C8×C59—21no—
110C8×C59—41no—
111C8×C59—81no—
112C8×C59C2×SD1611yescharacteristic
113C8×C59AGL(1,ℤ/8)11yescharacteristic
114#12374—591nomaximal, Sylow
115C22×C59—22no—
116C22×C59—41no—
117C22×C59—41no—
118C22×C59#44711yescharacteristic
119C4×C59—21no—
120C4×C59—41no—
121C4×C59—161no—
122C4×C59#45011yescharacteristic
123C4×C59#45511yescharacteristic
124C4×C59C2×Q3211yescharacteristic
125C4×C59#51011yescharacteristic
126Dic59—42no—
127Dic59—82no—
128Dic59—81no—
129D4×Q16—591no—
130#2241—591no—
131#2227—591no—
132#1675D5911yescharacteristic
133#1362—591no—
134#1332—591no—
135D4⋊C16—591no—
136C2×C59—42no—
137C2×C59#163911yescharacteristic
138C2×C59#218811yescharacteristic
139C2×C59#219311yescharacteristic
140C22×Q16—1181no—
141D4×Q8—1181no—
142#500C2×D5911yescharacteristic
143#494—591no—
144#476—21no—
145#467—1181no—
146#462—1181no—
147#451—1182no—
148C4×Q16—591no—
149C8×D4C2×D5911yescharacteristic
150#366—1181no—
151C4⋊C16—591no—
152#358—1181no—
153#358—1182no—
154C4×C16—591no—
155#335C2×D5911yescharacteristic
156#333—21no—
157C59#1237411yescharacteristic, complemented, Sylow
158C22×Q8—1182no—
159C2×Q16—591no—
160C2×Q16—592no—
161C2×Q16—1182no—
162C2×Q16—1181no—
163C2×Q16—2362no—
164C2×Q16—2362no—
165C22×C8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—1181no—
168C4⋊Q8—2361no—
169C22⋊Q8—1182no—
170C22⋊Q8—2362no—
171C4×Q8—1181no—
172C4×D4—2361no—
173C4×D4C22⋊D5911yescharacteristic
174C2×C16—1181no—
175C2×C16—1181no—
176#107—21no—
177#107—21no—
178#107—591no—
179#106—41no—
180C4⋊C8—21no—
181C4⋊C8C22⋊D5911yescharacteristic
182#103—1182no—
183#103—1181no—
184#102—41no—
185C22⋊C8—21no—
186C4×C8C22×D5911yescharacteristic
187C2×Q8—41no—
188C2×Q8—1181no—
189C2×Q8—1182no—
190C2×Q8—4722no—
191C2×Q8—2362no—
192C2×Q8—2364no—
193C2×D4#1174566911yescharacteristic
194C22×C4—21no—
195C22×C4—2362no—
196Q16—592no—
197Q16—1181no—
198Q16—2362no—
199Q16—2364no—
200Q16—2361no—
201Q16—2361no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8D4×D5911yescharacteristic
207C2×C8D4×D5911yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—1181no—
211C4⋊C4—2362no—
212C4⋊C4—2361no—
213C4⋊C4—2361no—
214C4⋊C4—2361no—
215C4⋊C4#1174566911yescharacteristic
216C22⋊C4—21no—
217C22⋊C4—2362no—
218C42—2361no—
219C42C2×C22⋊D5911yescharacteristic
220C16—2361no—
221C16—2361no—
222C23—21no—
223Q8—161no—
224Q8—1182no—
225Q8—4724no—
226Q8—2362no—
227Q8—2362no—
228Q8—2361no—
229D4—21no—
230D4#1427367312yes—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—4724no—
236C2×C4—2361no—
237C2×C4—2362no—
238C2×C4—2361no—
239C2×C4#1427369511yescharacteristic
240C2×C4C2×#1174566911yescharacteristic
241C2×C4#1427375211yescharacteristic
242C8—21no—
243C8—41no—
244C8—81no—
245C8SD16×D5911yescharacteristic
246C8#1427375411yescharacteristic
247C22—22no—
248C22—41no—
249C22—41no—
250C22#1683753511yescharacteristic
251C4—21no—
252C4—41no—
253C4—161no—
254C4—4721no—
255C4—4722no—
256C4—2362no—
257C4#1683755111yescharacteristic
258C4#1683757811yescharacteristic
259C4C2×#1427367311yescharacteristic
260C4#1683784211yescharacteristic
261C2—42no—
262C2#2142327311yescharacteristic
263C2#2142602911yescharacteristic
264C2#2142605811yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1675C2
socleC22×C59#447
prime radicalC2×D4×C59D8
hypercenter#1675D59
hypocenterC59#12374
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. #21426325C2semidirect
3. #21426245C2semidirect
4. #21422144C2semidirect
5. #21422059C2semidirect
157. C59#12374semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2362
#214263252
#214262452
#214221442
#214220592
D4⋊(C59⋊C16)2
C59×#16752
#1237459