Finite Group Databasebeta

← #28500208

Subgroups — #28500208

Order 15104.

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

normal subgroups only

This group has 268 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#21434752C211yesmaximal, characteristic
2#21425911C211yesmaximal, characteristic
3#21425860C211yesmaximal, characteristic
4#21422158C211yesmaximal, characteristic, complemented
5#21422156C211yesmaximal, characteristic, complemented
6#21422135C211yesmaximal, characteristic, complemented
7C59×#1358C211yesmaximal, characteristic, complemented
8#16838013C412yes—
9M32×D59—21no—
10#16837804C2211yescharacteristic
11#16837770C412yes—
12#16837657—21no—
13#16837640—21no—
14#16837509—21no—
15Q16×Dic59—21no—
16#16837356C2211yescharacteristic
17#16837223C2211yescharacteristic
18#16837223C2211yescharacteristic
19#16837222—21no—
20#16837183—21no—
21#16837179—21no—
22C59×#494C2211yescharacteristic
23C59×#364C2211yescharacteristic
24C59×#364C2211yescharacteristic
25C2×#11745659D411yescharacteristic
26C2×C8×D59D411yescharacteristic
27#14273722—42no—
28#14273719C2×C412yes—
29#14273717—42no—
30#14273703—22no—
31Q8×Dic59—22no—
32#14273684C2×C411yescharacteristic
33#14273667—21no—
34#14273667—21no—
35#14273666—41no—
36C16×D59—41no—
37#14273661—21no—
38#14273659—21no—
39#14273651—22no—
40#14273651—22no—
41#14273638C2×C411yescharacteristic
42#14273637D411yescharacteristic
43C8×Dic59D411yescharacteristic
44C2×Q16×C59—21no—
45C2×Q16×C59—21no—
46C4⋊Q8×C59C2×C412yes—
47M32×C59—21no—
48M32×C59—21no—
49C4×C8×C59C2311yescharacteristic
50C2×Q8×C59—22no—
51C2×C4×D59C22⋊C411yescharacteristic
52Q16×C59—41no—
53Q16×C59—41no—
54Q16×C59—41no—
55Q16×C59—41no—
56C2×C8×C59C2×D411yescharacteristic
57C2×C8×C59C2×D411yescharacteristic
58C4⋊C4×C59—42no—
59C42×C59C22×C411yescharacteristic
60(C2×D59)⋊C4—21no—
61(C2×D59)⋊C4—42no—
62C4⋊Dic59—22no—
63C4⋊Dic59—42no—
64Dic59⋊C4—21no—
65Dic59⋊C4—42no—
66Dic59⋊C4—42no—
67C4×Dic59—42no—
68C4×Dic59C22⋊C411yescharacteristic
69C2×C59⋊C8C22⋊C411yescharacteristic
70C2×C59⋊C8C22⋊C411yescharacteristic
71#11745659—21no—
72#11745659—21no—
73C8×D59—21no—
74C8×D59—21no—
75C16×C59—41no—
76C16×C59—41no—
77C59⋊C16—41no—
78C59⋊C16—41no—
79C22×D59#10011yescharacteristic
80Q8×C59—42no—
81Q8×C59—42no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59C2×C22⋊C411yescharacteristic
85C2×Dic59—21no—
86C2×Dic59—42no—
87C2×Dic59#10011yescharacteristic
88C4×D59—21no—
89C4×D59—21no—
90C8×C59#13511yescharacteristic
91C8×C59#13511yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C8×C59#13711yescharacteristic
94C59⋊C8—21no—
95C59⋊C8—21no—
96#11990—591nomaximal, Sylow
97C22×C59C2×#10011yescharacteristic
98C2×D59—21no—
99C2×D59—41no—
100C4×C59—41no—
101C4×C59—82no—
102C4×C59#41611yescharacteristic
103C4×C59#41711yescharacteristic
104Dic59—21no—
105Dic59—41no—
106Dic59—82no—
107#3177—591no—
108#2166—591no—
109#2153—591no—
110#1366—591no—
111#1366—591no—
112#1358—591no—
113#1358D5911yescharacteristic, complemented
114C2×C59—21no—
115C2×C59#155111yescharacteristic
116D59—41no—
117#541—592no—
118C2×M32—1181no—
119#499—591no—
120#494C2×D5911yescharacteristic
121#491—592no—
122#470—1181no—
123#467—1181no—
124#441—1181no—
125C4×Q16—1181no—
126#405—591no—
127#364—591no—
128#364—591no—
129#364—1181no—
130#364C2×D5911yescharacteristic
131#364C2×D5911yescharacteristic
132#349—1181no—
133#347—1181no—
134C59#1199011yescharacteristic, complemented, Sylow
135C2×Q16—21no—
136C2×Q16—21no—
137C2×M16—591no—
138C22×C8—591no—
139C4⋊Q8C4×D5912yes—
140#126—2362no—
141#125—592no—
142#125—2362no—
143C22⋊Q8—1182no—
144C4×Q8—1182no—
145#117—591no—
146M32—21no—
147M32—21no—
148M32—1181no—
149M32—1181no—
150M32—2361no—
151C2×C16—2361no—
152#107—1181no—
153#106—1181no—
154#103—1182no—
155#103—1182no—
156#97—591no—
157#97—591no—
158C4×C8—591no—
159C4×C8C22×D5911yescharacteristic
160C2×Q8—22no—
161C22×C4—591no—
162Q16—41no—
163Q16—41no—
164Q16—41no—
165Q16—41no—
166M16—1181no—
167M16—1181no—
168C2×C8—591no—
169C2×C8—591no—
170C2×C8—1181no—
171C2×C8—1181no—
172C2×C8D4×D5911yescharacteristic
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—42no—
175C4⋊C4—1181no—
176C4⋊C4—1182no—
177C4⋊C4—2362no—
178C4⋊C4—2362no—
179C4⋊C4—2362no—
180C22⋊C4—1181no—
181C22⋊C4—2362no—
182C42—591no—
183C42—2362no—
184C42C2×C4×D5911yescharacteristic
185C16—41no—
186C16—41no—
187C16—2361no—
188C16—2361no—
189C23—591no—
190Q8—42no—
191Q8—42no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—591no—
195C2×C4—1181no—
196C2×C4—1181no—
197C2×C4—1181no—
198C2×C4—2362no—
199C2×C4C22⋊C4×D5911yescharacteristic
200C8—1181no—
201C8—1181no—
202C8#1427374611yescharacteristic
203C8#1427374811yescharacteristic
204C8#1427375111yescharacteristic
205C8#1427375611yescharacteristic
206C22—1181no—
207C22—2361no—
208C22#100×D5911yescharacteristic
209C4—41no—
210C4—82no—
211C4—1181no—
212C4—4722no—
213C4—2361no—
214C4#1683739511yescharacteristic
215C4#1683740011yescharacteristic
216C2—21no—
217C2—2361no—
218C2#2142278811yescharacteristic
subgroupG/H
centerC2#21422788
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1358C2
socleC2×C59#1551
prime radicalC22×D59#100
cosocleC4×C8×C59C23
hypercenter#1358D59
hypocenterC59#11990
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
4. #21422158C2semidirect
5. #21422156C2semidirect
6. #21422135C2semidirect
7. C59×#1358C2semidirect
113. #1358D59semidirect
134. C59#11990semidirect

Maximal subgroups

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

subgroupindex
#214347522
#214259112
#214258602
#214221582
#214221562
#214221352
C59×#13582
#1199059