Finite Group Databasebeta

← #28502906

Subgroups — #28502906

Order 15104.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#21426352C211yesmaximal, characteristic, complemented
2C16×D236C211yesmaximal, characteristic, complemented
3#21423462C211yesmaximal, characteristic, complemented
4#21423452C211yesmaximal, characteristic
5#21422144C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1372C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9C2×#14273672—21no—
10C2×C16×D59—21no—
11#16837782C2211yescharacteristic
12#16837779C2211yescharacteristic
13#16837683—21no—
14#16837598—21no—
15C8×D236C2211yescharacteristic
16C4⋊(C59⋊C16)C2211yescharacteristic
17(C2×D59)⋊C16—21no—
18#16837147—21no—
19#16837145—21no—
20C4×C16×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×D472—21no—
26C2×D472—21no—
27C2×D472—41no—
28C2×C8×D59—21no—
29#14273727—21no—
30#14273724—21no—
31C4×D236D411yescharacteristic
32#14273672—42no—
33#14273672—42no—
34C2×C59⋊C16—21no—
35C16×D59—42no—
36#14273661—41no—
37C4⋊(C59⋊C8)D411yescharacteristic
38#14273649—41no—
39#14273649—41no—
40(C2×D59)⋊C8—21no—
41C2×C16×C59D411yescharacteristic
42C2×C16×C59D411yescharacteristic
43C4⋊C8×C59—21no—
44C4⋊C8×C59—21no—
45C4×C8×C59C2311yescharacteristic
46C2×D236—41no—
47C2×D236—41no—
48C2×D236—41no—
49C2×D236D811yescharacteristic
50C2×C4×D59—21no—
51C2×Dic118—41no—
52C2×C8×C59—41no—
53C2×C8×C59—41no—
54C2×C8×C59C2×D411yescharacteristic
55C2×C8×C59#4111yescharacteristic
56C42×C59C2×D411yescharacteristic
57(C2×D59)⋊C4—21no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C4⋊Dic59D811yescharacteristic
61C2×C59⋊C8—21no—
62Dic236—41no—
63Dic236—41no—
64Dic236—81no—
65D472—41no—
66D472—41no—
67D472—81no—
68D472—82no—
69C8×D59—41no—
70C8×D59—41no—
71C16×C59—21no—
72C16×C59D812yescomplemented
73C59⋊C16—41no—
74C22×D59—21no—
75C22×D59—81no—
76C2×C4×C59#12111yescharacteristic
77C2×C4×C59C2×D811yescharacteristic
78C2×C4×C59#13511yescharacteristic
79C2×Dic59—21no—
80C2×Dic59—81no—
81D236—21no—
82D236—81no—
83D236—81no—
84D236—81no—
85D236—81no—
86D236—161no—
87D236SD3212yescomplemented
88C4×D59—41no—
89C4×D59—41no—
90Dic118—81no—
91Dic118—81no—
92C8×C59—21no—
93C8×C59—81no—
94C8×C59—81no—
95C8×C59C2×D811yescharacteristic
96C8×C59AGL(1,ℤ/8)11yescharacteristic
97C59⋊C8—41no—
98#12432—591nomaximal, Sylow
99C22×C59#46611yescharacteristic
100C2×D59—22no—
101C2×D59—41no—
102C2×D59—41no—
103C2×D59—161no—
104C2×D59—161no—
105C2×D59—161no—
106C4×C59—21no—
107C4×C59#45911yescharacteristic
108C4×C59#47211yescharacteristic
109C4×C59C2×SD3211yescharacteristic
110C4×C59#50811yescharacteristic
111Dic59—41no—
112Dic59—161no—
113#2248—591no—
114D4×C16—591no—
115#1669—591no—
116#1667—591no—
117#1372D5911yescharacteristic, complemented
118#1362—591no—
119#1361—591no—
120C2×C59#166711yescharacteristic
121C2×C59#221711yescharacteristic
122C2×C59#222311yescharacteristic
123D59—42no—
124D59—321no—
125C2×SD32—1181no—
126C2×SD32—1181no—
127C22×C16—1181no—
128#494—591no—
129#493—591no—
130#474—1181no—
131#459—1181no—
132C8×D4—591no—
133C4⋊C16—591no—
134C22⋊C16—1181no—
135C4×C16C2×D5911yescharacteristic
136#335C2×D5911yescharacteristic
137#335C2×D5911yescharacteristic
138#332—1181no—
139#331—1181no—
140C59#1243211yescharacteristic, complemented, Sylow
141C2×Q16—1181no—
142C2×Q16—1181no—
143C2×D8—1181no—
144C2×D8—1181no—
145C2×D8—2361no—
146C22×C8—1181no—
147C4⋊Q8—1181no—
148#127—1181no—
149C4×D4—591no—
150SD32—2362no—
151SD32—2362no—
152C2×C16—1181no—
153C2×C16—2362no—
154C2×C16D23611yescharacteristic
155C2×C16D23611yescharacteristic
156#107—2361no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—591no—
160#102—2361no—
161#102—2361no—
162C22⋊C8—1181no—
163C4×C8C22×D5911yescharacteristic
164C2×Q8—2361no—
165C2×D4—591no—
166C2×D4—2361no—
167C2×D4—2361no—
168C2×D4—2361no—
169C22×C4—1181no—
170Q16—4721no—
171Q16—2361no—
172Q16—2361no—
173D8—4721no—
174D8—4722no—
175D8—2361no—
176D8—2361no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—1181no—
180C2×C8—2361no—
181C2×C8—2361no—
182C2×C8C2×D23611yescharacteristic
183C2×C8#1174568911yescharacteristic
184C4⋊C4—591no—
185C4⋊C4—2361no—
186C4⋊C4—2361no—
187C22⋊C4—1181no—
188C42D4×D5911yescharacteristic
189C16—21no—
190C16—2361no—
191C16D47212yescomplemented
192C23—1181no—
193C23—4721no—
194Q8—4721no—
195Q8—4721no—
196D4—592no—
197D4—1181no—
198D4—9441no—
199D4—4721no—
200D4—4721no—
201D4—4721no—
202D4—4721no—
203C2×C4—1181no—
204C2×C4—4721no—
205C2×C4—2361no—
206C2×C4—2361no—
207C2×C4#1427370211yescharacteristic
208C2×C4D8×D5911yescharacteristic
209C2×C4#1427375011yescharacteristic
210C8—21no—
211C8—81no—
212C8—81no—
213C8—2361no—
214C8C2×D47211yescharacteristic
215C8#1427375511yescharacteristic
216C22—1182no—
217C22—9441no—
218C22—9441no—
219C22—9441no—
220C22—2361no—
221C22—2361no—
222C22#1683763911yescharacteristic
223C4—21no—
224C4—9441no—
225C4—2361no—
226C4#1683759811yescharacteristic
227C4#1683767311yescharacteristic
228C4SD32×D5911yescharacteristic
229C4#1683783311yescharacteristic
230C2—18881no—
231C2—2362no—
232C2#2142345211yescharacteristic
233C2#2142619311yescharacteristic
234C2#2142622711yescharacteristic
subgroupG/H
centerC22#16837639
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1372C2
socleC22×C59#466
prime radical#21423452C2
hypercenter#1372D59
hypocenterC59#12432
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #21426352C2semidirect
2. C16×D236C2semidirect
3. #21423462C2semidirect
5. #21422144C2semidirect
6. #21422141C2semidirect
7. C59×#1372C2semidirect
72. C16×C59D8semidirect
87. D236SD32semidirect
117. #1372D59semidirect
140. C59#12432semidirect
191. C16D472semidirect

Maximal subgroups

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

subgroupindex
#214263522
C16×D2362
#214234622
#214234522
#214221442
#214221412
C59×#13722
#1243259