Finite Group Databasebeta

← #28502908

Subgroups — #28502908

Order 15104.

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

normal subgroups only

This group has 252 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#21426353C211yesmaximal, characteristic, complemented
2#21425915C211yesmaximal, characteristic, complemented
3#21423458C211yesmaximal, characteristic, complemented
4#21423452C211yesmaximal, characteristic
5#21422169C211yesmaximal, characteristic, complemented
6#21422143C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9C2×#14273670—21no—
10C2×C16×D59—21no—
11#16837779C2211yescharacteristic
12#16837679—21no—
13#16837598—21no—
14C8×D236C2211yescharacteristic
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837148—21no—
19#16837145—21no—
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×D472—21no—
24C2×D472—21no—
25C2×D472—41no—
26C2×C8×D59—21no—
27#14273724—21no—
28C4×D236D411yescharacteristic
29#14273670—42no—
30#14273670—42no—
31C2×C59⋊C16D411yescharacteristic
32C2×C59⋊C16D411yescharacteristic
33C16×D59—42no—
34#14273660—41no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)D411yescharacteristic
37#14273649—41no—
38#14273647—41no—
39(C2×D59)⋊C8—21no—
40C2×Q16×C59—21no—
41C2×Q16×C59—21no—
42C4⋊Q8×C59—21no—
43C2×C16×C59—21no—
44C4⋊C8×C59—21no—
45C4×C8×C59C2311yescharacteristic
46C2×Q8×C59—41no—
47C2×D236—41no—
48C2×D236—41no—
49C2×D236—41no—
50C2×D236D811yescharacteristic
51C2×C4×D59—21no—
52Q16×C59—41no—
53Q16×C59—41no—
54Q16×C59—81no—
55C2×C8×C59—41no—
56C2×C8×C59C2×D411yescharacteristic
57C2×C8×C59#4111yescharacteristic
58C4⋊C4×C59—41no—
59C4⋊C4×C59—41no—
60C42×C59C2×D411yescharacteristic
61(C2×D59)⋊C4—21no—
62C4⋊Dic59D811yescharacteristic
63C2×C59⋊C8—21no—
64C2×C59⋊C8—41no—
65D472—41no—
66D472—41no—
67D472—81no—
68D472—82no—
69C8×D59—41no—
70C8×D59—41no—
71C16×C59—41no—
72C59⋊C16—21no—
73C59⋊C16D812yescomplemented
74C22×D59—21no—
75C22×D59—81no—
76Q8×C59—81no—
77Q8×C59—81no—
78C2×C4×C59—81no—
79C2×C4×C59#12111yescharacteristic
80C2×C4×C59C2×D811yescharacteristic
81C2×C4×C59#13511yescharacteristic
82C2×Dic59—21no—
83D236—21no—
84D236—81no—
85D236—81no—
86D236—81no—
87D236—81no—
88D236—161no—
89D236SD3212yescomplemented
90C4×D59—41no—
91C4×D59—41no—
92C8×C59—21no—
93C8×C59—81no—
94C8×C59C2×D811yescharacteristic
95C8×C59AGL(1,ℤ/8)11yescharacteristic
96C59⋊C8—41no—
97C59⋊C8—81no—
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—161no—
108C4×C59#45911yescharacteristic
109C4×C59#47211yescharacteristic
110C4×C59C2×SD3211yescharacteristic
111C4×C59#50811yescharacteristic
112Dic59—41no—
113#2248—591no—
114D4×C16—591no—
115#1669—591no—
116#1667—591no—
117#1372—591no—
118#1362D5911yescharacteristic, complemented
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#494C2×D5911yescharacteristic
129#493—591no—
130#474—1181no—
131#459—1181no—
132C8×D4—591no—
133C4⋊C16C2×D5911yescharacteristic
134C22⋊C16—1181no—
135C4×C16—591no—
136#335—591no—
137#335C2×D5911yescharacteristic
138#332—1181no—
139#331—1181no—
140C59#1243211yescharacteristic, complemented, Sylow
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—1181no—
144C2×D8—1181no—
145C2×D8—2361no—
146C22×C8—1181no—
147C4⋊Q8—21no—
148#127—1181no—
149C4×D4—591no—
150SD32—2362no—
151SD32—2362no—
152C2×C16—21no—
153C2×C16—591no—
154C2×C16—591no—
155C2×C16—2362no—
156#107—2361no—
157C4⋊C8—21no—
158C4⋊C8—591no—
159C4⋊C8—1181no—
160#102—2361no—
161#102—2361no—
162C22⋊C8—1181no—
163C4×C8C22×D5911yescharacteristic
164C2×Q8—41no—
165C2×D4—591no—
166C2×D4—2361no—
167C2×D4—2361no—
168C2×D4—2361no—
169C22×C4—1181no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—4721no—
174D8—4722no—
175D8—2361no—
176D8—2361no—
177C2×C8—41no—
178C2×C8—1181no—
179C2×C8—2361no—
180C2×C8—2361no—
181C2×C8—2361no—
182C2×C8#1174568511yescharacteristic
183C2×C8D4×D5911yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—591no—
187C22⋊C4—1181no—
188C42D4×D5911yescharacteristic
189C16—41no—
190C16—592no—
191C16—1181no—
192C23—1181no—
193C23—4721no—
194Q8—81no—
195Q8—81no—
196D4—592no—
197D4—1181no—
198D4—9441no—
199D4—4721no—
200D4—4721no—
201D4—4721no—
202D4—4721no—
203C2×C4—81no—
204C2×C4—1181no—
205C2×C4—2361no—
206C2×C4—2361no—
207C2×C4#1427370011yescharacteristic
208C2×C4D8×D5911yescharacteristic
209C2×C4#1427374811yescharacteristic
210C8—21no—
211C8—81no—
212C8—4721no—
213C8—2361no—
214C8D8×D5911yescharacteristic
215C8#1427375411yescharacteristic
216C22—1182no—
217C22—9441no—
218C22—9441no—
219C22—9441no—
220C22—2361no—
221C22—2361no—
222C22#1683763711yescharacteristic
223C4—21no—
224C4—161no—
225C4—2361no—
226C4#1683759611yescharacteristic
227C4#1683767111yescharacteristic
228C4SD32×D5911yescharacteristic
229C4#1683783511yescharacteristic
230C2—18881no—
231C2—2362no—
232C2#2142345011yescharacteristic
233C2#2142619511yescharacteristic
234C2#2142622911yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59#466
prime radical#21423452C2
hypercenter#1362D59
hypocenterC59#12432
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #21426353C2semidirect
2. #21425915C2semidirect
3. #21423458C2semidirect
5. #21422169C2semidirect
6. #21422143C2semidirect
7. C59×#1362C2semidirect
73. C59⋊C16D8semidirect
89. D236SD32semidirect
118. #1362D59semidirect
140. C59#12432semidirect

Maximal subgroups

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

subgroupindex
#214263532
#214259152
#214234582
#214234522
#214221692
#214221432
C59×#13622
#1243259