Finite Group Databasebeta

← #28502287

Subgroups — #28502287

Order 15104.

306 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 65 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#21436528C211yesmaximal, characteristic, complemented
2#21426319C211yesmaximal, characteristic, complemented
3#21423448C211yesmaximal, characteristic
4#21422150C211yesmaximal, characteristic, complemented
5#21422060C211yesmaximal, characteristic, complemented
6#21422050C211yesmaximal, characteristic, complemented
7C59×#2207C211yesmaximal, characteristic, complemented
8#16838125—21no—
9#16837807C2211yescharacteristic
10#16837742—22no—
11#16837687—21no—
12#16837640—21no—
13#16837594—21no—
14C8×D236C2211yescharacteristic
15#16837439—21no—
16#16837203—22no—
17C4×C59⋊C16C2211yescharacteristic
18#16837155C2211yescharacteristic
19#16837146—21no—
20C2×SD32×C59—21no—
21C59×#493C2211yescharacteristic
22C4×Q16×C59C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#357—21no—
25C2×#11745669—41no—
26C2×C8×D59—21no—
27#14273727—21no—
28#14273722—42no—
29#14273715—41no—
30#14273703—22no—
31C4×Dic118—41no—
32C4×D236D411yescharacteristic
33#14273685—22no—
34C2×C59⋊C16—21no—
35#14273661—21no—
36#14273661D412yes—
37#14273658—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40#14273651—22no—
41#14273648—41no—
42(C2×D59)⋊C8—21no—
43C2×Q16×C59D411yescharacteristic
44C2×D8×C59—21no—
45C2×D8×C59—21no—
46#127×C59—21no—
47C4×Q8×C59—21no—
48SD32×C59—42no—
49C2×C16×C59—21no—
50#107×C59D411yescharacteristic
51#103×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—21no—
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D236D811yescharacteristic
58C2×C4×D59—21no—
59C2×Dic118—41no—
60Q16×C59—21no—
61Q16×C59—21no—
62D8×C59—42no—
63D8×C59—41no—
64D8×C59—41no—
65C2×C8×C59C2×D411yescharacteristic
66C2×C8×C59C2×D411yescharacteristic
67C4⋊C4×C59—21no—
68C4⋊C4×C59—41no—
69C42×C59—41no—
70C42×C59C2×D411yescharacteristic
71#11745669—82no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—42no—
74(C2×D59)⋊C4—42no—
75C4⋊Dic59—22no—
76C4⋊Dic59—41no—
77C4⋊Dic59D811yescharacteristic
78Dic59⋊C4—42no—
79C4×Dic59—42no—
80C2×C59⋊C8—21no—
81C2×C59⋊C8—41no—
82C8×D59—41no—
83C8×D59—41no—
84C16×C59—41no—
85C59⋊C16—41no—
86C23×C59—81no—
87C22×D59—21no—
88Q8×C59—41no—
89Q8×C59—41no—
90D4×C59—81no—
91D4×C59—81no—
92D4×C59—81no—
93D4×C59—81no—
94D4×C59—161no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59#12011yescharacteristic
98C2×C4×C59C2×D811yescharacteristic
99C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
100C2×Dic59—21no—
101C2×Dic59—42no—
102D236—21no—
103D236—21no—
104C4×D59—41no—
105C4×D59—41no—
106Dic118—81no—
107Dic118—81no—
108C8×C59—21no—
109C8×C59#13511yescharacteristic
110C8×C59AGL(1,ℤ/8)11yescharacteristic
111C59⋊C8—41no—
112C59⋊C8—81no—
113#12341—591nomaximal, Sylow
114C22×C59—161no—
115C22×C59—161no—
116C22×C59—161no—
117C22×C59#44711yescharacteristic
118C2×D59—41no—
119C2×D59—41no—
120C2×D59—41no—
121C4×C59—21no—
122C4×C59—81no—
123C4×C59—81no—
124C4×C59#44911yescharacteristic
125C4×C59#45411yescharacteristic
126C4×C59#50811yescharacteristic
127C4×C59#50911yescharacteristic
128Dic59—41no—
129Dic59—82no—
130#3342—591no—
131#2240—591no—
132#2207D5911yescharacteristic, complemented
133#1667—591no—
134#1364—591no—
135#1332—591no—
136D4⋊C16—591no—
137C2×C59—321no—
138C2×C59#162511yescharacteristic
139C2×C59#218611yescharacteristic
140C2×C59#219311yescharacteristic
141D59—81no—
142#556—1181no—
143C2×SD32—21no—
144#500—591no—
145#493C2×D5911yescharacteristic
146#485—1182no—
147#475—1181no—
148#467—1181no—
149#459—1181no—
150C4×Q16C2×D5911yescharacteristic
151C8×D4—591no—
152#427—1181no—
153C4⋊C16C2×D5911yescharacteristic
154#357—21no—
155#357—1182no—
156C4×C16—591no—
157#335—591no—
158#331—1181no—
159C59#1234111yescharacteristic, complemented, Sylow
160C2×Q16C22⋊D5911yescharacteristic
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—2361no—
164C22×C8—1181no—
165C4⋊Q8—1181no—
166#127—21no—
167#126—2362no—
168#124—2361no—
169C22⋊Q8—1182no—
170C4×Q8—21no—
171C4×Q8—2361no—
172C4×D4—591no—
173#117—1182no—
174SD32—42no—
175C2×C16—21no—
176C2×C16—1181no—
177#107—592no—
178#107—1181no—
179#107C22⋊D5911yescharacteristic
180#106—1182no—
181C4⋊C8—591no—
182C4⋊C8—1181no—
183#103—21no—
184#103—1182no—
185#102—2361no—
186C22⋊C8—1181no—
187C4×C8C22×D5911yescharacteristic
188C2×Q8—21no—
189C2×Q8—2361no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—591no—
194C22×C4—1181no—
195Q16—21no—
196Q16—21no—
197D8—42no—
198D8—41no—
199D8—41no—
200D8—4722no—
201C2×C8—1181no—
202C2×C8—2361no—
203C2×C8—2361no—
204C2×C8—2361no—
205C2×C8D4×D5911yescharacteristic
206C2×C8C2×C22⋊D5911yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—591no—
210C4⋊C4—1182no—
211C4⋊C4—2362no—
212C4⋊C4—2361no—
213C22⋊C4—1181no—
214C22⋊C4—2362no—
215C22⋊C4—2362no—
216C42—41no—
217C42—2362no—
218C42D4×D5911yescharacteristic
219C16—41no—
220C16—2361no—
221C23—81no—
222C23—1181no—
223Q8—41no—
224Q8—41no—
225Q8—4721no—
226Q8—4721no—
227D4—81no—
228D4—81no—
229D4—81no—
230D4—81no—
231D4—161no—
232D4—1181no—
233D4—1181no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—1181no—
237C2×C4—2362no—
238C2×C4—2361no—
239C2×C4—2361no—
240C2×C4#1427369511yescharacteristic
241C2×C4D8×D5911yescharacteristic
242C2×C4#1427375111yescharacteristic
243C8—21no—
244C8—4721no—
245C8—2361no—
246C8#1427374711yescharacteristic
247C8#1427375311yescharacteristic
248C22—161no—
249C22—161no—
250C22—161no—
251C22—2361no—
252C22—2361no—
253C22—2361no—
254C22#1683753411yescharacteristic
255C4—21no—
256C4—81no—
257C4—81no—
258C4—4722no—
259C4—2361no—
260C4#1683754511yescharacteristic
261C4#1683757311yescharacteristic
262C4#1683783311yescharacteristic
263C4#1683783911yescharacteristic
264C2—321no—
265C2—4721no—
266C2#2142318411yescharacteristic
267C2#2142601811yescharacteristic
268C2#2142605711yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2207C2
socleC22×C59#447
prime radical#21423448C2
hypercenter#2207D59
hypocenterC59#12341
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21436528C2semidirect
2. #21426319C2semidirect
4. #21422150C2semidirect
5. #21422060C2semidirect
6. #21422050C2semidirect
7. C59×#2207C2semidirect
132. #2207D59semidirect
159. C59#12341semidirect

Maximal subgroups

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

subgroupindex
#214365282
#214263192
#214234482
#214221502
#214220602
#214220502
C59×#22072
#1234159