Finite Group Databasebeta

← #28502891

Subgroups — #28502891

Order 15104.

240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426143C211yesmaximal, characteristic, complemented
2#21426137C211yesmaximal, characteristic
3#21423703C211yesmaximal, characteristic
4#21422902C211yesmaximal, characteristic, complemented
5#21422165C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C59×#1362C211yesmaximal, characteristic, complemented
8C2×#14273673—21no—
9C2×#14273672—21no—
10#16837779C2211yescharacteristic
11#16837750—21no—
12#16837750—21no—
13#16837526—21no—
14C4×Dic236C2211yescharacteristic
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837206—21no—
17#16837204—21no—
18#16837132—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236D411yescharacteristic
24C2×D472—21no—
25C2×D472—21no—
26#14273724—21no—
27C4×Dic118—21no—
28#14273673—42no—
29#14273672—42no—
30C2×C59⋊C16—21no—
31#14273661D411yescharacteristic
32Dic59⋊C8—41no—
33#14273652—21no—
34#14273647—42no—
35#14273647—41no—
36#14273647—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
39C8×Dic59—41no—
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×Dic118—21no—
51Q16×C59—41no—
52Q16×C59—41no—
53Q16×C59—81no—
54C2×C8×C59—41no—
55C2×C8×C59C2×D411yescharacteristic
56C2×C8×C59#4111yescharacteristic
57C4⋊C4×C59—41no—
58C4⋊C4×C59—41no—
59C42×C59C2×D411yescharacteristic
60C4⋊Dic59—21no—
61Dic59⋊C4—41no—
62C4×Dic59—41no—
63C2×C59⋊C8—21no—
64C2×C59⋊C8D811yescharacteristic
65C2×C59⋊C8D811yescharacteristic
66Dic236—21no—
67Dic236D812yes—
68D472—41no—
69D472—41no—
70D472—81no—
71C16×C59—41no—
72C59⋊C16—41no—
73C22×D59—81no—
74Q8×C59—81no—
75Q8×C59—81no—
76C2×C4×C59—81no—
77C2×C4×C59#12111yescharacteristic
78C2×C4×C59C2×D811yescharacteristic
79C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
80C2×Dic59—41no—
81C2×Dic59—41no—
82D236—82no—
83D236—81no—
84D236—81no—
85D236—81no—
86D236—81no—
87Dic118—22no—
88Dic118—41no—
89C8×C59—21no—
90C8×C59—81no—
91C8×C59C2×D811yescharacteristic
92C8×C59#13511yescharacteristic
93C59⋊C8—21no—
94C59⋊C8—21no—
95C59⋊C8—41no—
96#12429—591nomaximal, Sylow
97C22×C59#45911yescharacteristic
98C2×D59—161no—
99C2×D59—161no—
100C2×D59—161no—
101C4×C59—21no—
102C4×C59—161no—
103C4×C59#46611yescharacteristic
104C4×C59#47211yescharacteristic
105C4×C59#50811yescharacteristic
106C4×C59#51011yescharacteristic
107Dic59—42no—
108Dic59—81no—
109#2209—591no—
110#2207—591no—
111#1709—591no—
112C8×Q16—591no—
113#1370—591no—
114#1362D5911yescharacteristic, complemented
115#1361—591no—
116C2×C59#166711yescharacteristic
117C2×C59#221111yescharacteristic
118C2×C59#221211yescharacteristic
119D59—321no—
120C2×Q32—1181no—
121C2×SD32—1181no—
122#494C2×D5911yescharacteristic
123#493—591no—
124#486—1181no—
125#486—1181no—
126C8×Q8—1181no—
127C4×Q16—591no—
128C4⋊C16—591no—
129C4⋊C16C2×D5911yescharacteristic
130#358—1181no—
131#357—1181no—
132#335C2×D5911yescharacteristic
133Q8⋊C8—1181no—
134C82—591no—
135C59#1242911yescharacteristic, complemented, Sylow
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—591no—
139C2×D8—1181no—
140C2×D8—1181no—
141C4⋊Q8—21no—
142#127—1181no—
143C4×Q8—1181no—
144Q32—2362no—
145SD32—2362no—
146C2×C16—21no—
147C2×C16—1181no—
148#107—591no—
149C4⋊C8—21no—
150C4⋊C8—2361no—
151#103—1181no—
152#102—2361no—
153#102—2361no—
154#102—2362no—
155C4×C8—591no—
156C4×C8—591no—
157C4×C8—2361no—
158C4×C8C22×D5911yescharacteristic
159C2×Q8—41no—
160C2×Q8—1181no—
161C2×D4—2361no—
162C2×D4—2361no—
163C2×D4—2361no—
164Q16—41no—
165Q16—41no—
166Q16—81no—
167Q16—592no—
168Q16—1181no—
169D8—4721no—
170D8—2361no—
171D8—2361no—
172C2×C8—41no—
173C2×C8—591no—
174C2×C8—591no—
175C2×C8—1181no—
176C2×C8#1174568511yescharacteristic
177C2×C8D4×D5911yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—1181no—
181C4⋊C4—2361no—
182C42—2361no—
183C42D4×D5911yescharacteristic
184C16—41no—
185C16—2361no—
186C23—4721no—
187Q8—81no—
188Q8—81no—
189Q8—1182no—
190Q8—2361no—
191D4—4722no—
192D4—4721no—
193D4—4721no—
194D4—4721no—
195D4—4721no—
196C2×C4—81no—
197C2×C4—2361no—
198C2×C4—2361no—
199C2×C4#1427370011yescharacteristic
200C2×C4D8×D5911yescharacteristic
201C2×C4#1427375411yescharacteristic
202C8—21no—
203C8—81no—
204C8—1181no—
205C8—1181no—
206C8—2361no—
207C8D8×D5911yescharacteristic
208C8#1427374811yescharacteristic
209C22—9441no—
210C22—9441no—
211C22—9441no—
212C22#1683759611yescharacteristic
213C4—21no—
214C4—161no—
215C4—4721no—
216C4—2362no—
217C4#1683763711yescharacteristic
218C4#1683767111yescharacteristic
219C4#1683783311yescharacteristic
220C4#1683784411yescharacteristic
221C2—18881no—
222C2#2142345011yescharacteristic
223C2#2142616111yescharacteristic
224C2#2142616811yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59#459
prime radical#16837779C22
hypercenter#1362D59
hypocenterC59#12429
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. #21426143C2semidirect
4. #21422902C2semidirect
5. #21422165C2semidirect
7. C59×#1362C2semidirect
114. #1362D59semidirect
135. C59#12429semidirect

Maximal subgroups

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

subgroupindex
#214261432
#214261372
#214237032
#214229022
#214221652
#214221412
C59×#13622
#1242959