Finite Group Databasebeta

← #28502463

Subgroups — #28502463

Order 15104.

336 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426267C212yesmaximal, complemented
2#21423727C211yesmaximal, characteristic, complemented
3#21422135C211yesmaximal, characteristic, complemented
4#21422072C212yesmaximal, complemented
5C59×#3293C211yesmaximal, characteristic
6#16837800C2211yescharacteristic
7#16837798—21no—
8#16837754—21no—
9#16837226—22no—
10#16837223C2212yes—
11#16837212—22no—
12#16837123C2211yescharacteristic
13C59×#578—22no—
14C59×#494C2211yescharacteristic
15C59×#473—22no—
16C59×#456—21no—
17C59×#456—21no—
18C59×#443C2212yes—
19#14273727—21no—
20#14273667—22no—
21#14273659—22no—
22#14273658—42no—
23#14273650—42no—
24C4×C59⋊C8—21no—
252−1+4×C59—21no—
262−1+4×C59—21no—
27#137×C59—22no—
28#137×C59—22no—
29#137×C59—42no—
30#137×C59—42no—
31#135×C59—42no—
32#135×C59D412yes—
33C2×Q16×C59—22no—
34C2×Q16×C59—42no—
35#131×C59—22no—
36C4⋊Q8×C59D411yescharacteristic
37C4⋊Q8×C59D411yescharacteristic
38#108×C59D412yes—
39#104×C59—22no—
40#101×C59—21no—
41#101×C59—21no—
42C4×C8×C59C2311yescharacteristic
43#41×C59—22no—
44#41×C59—42no—
45#41×C59—42no—
46#41×C59—81no—
47#41×C59—81no—
48C2×Q8×C59—21no—
49C2×Q8×C59—21no—
50C2×Q8×C59—42no—
51C2×Q8×C59—42no—
52C2×Dic118—41no—
53Q16×C59—42no—
54Q16×C59—42no—
55Q16×C59—42no—
56Q16×C59—42no—
57Q16×C59SD1612yes—
58SD16×C59—22no—
59SD16×C59—42no—
60SD16×C59—42no—
61D8×C59SD1612yes—
62M16×C59—22no—
63M16×C59—42no—
64C2×C8×C59—42no—
65C2×C8×C59C2×D412yes—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C42×C59C2×D411yescharacteristic
69C4⋊Dic59—41no—
70C4⋊Dic59—41no—
71C2×C59⋊C8—22no—
72C59⋊C16—42no—
73Q8×C59—22no—
74Q8×C59—42no—
75Q8×C59—42no—
76Q8×C59—81no—
77Q8×C59—81no—
78D4×C59—22no—
79D4×C59—42no—
80D4×C59—81no—
81D4×C59—81no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—82no—
87C2×C4×C59—82no—
88C2×C4×C59#12011yescharacteristic
89C2×Dic59—81no—
90Dic118—82no—
91C8×C59—42no—
92C8×C59C2×SD1612yes—
93C8×C59AGL(1,ℤ/8)12yes—
94C59⋊C8—42no—
95#12367—591nomaximal, Sylow
96C22×C59—42no—
97C22×C59#45311yescharacteristic
98C4×C59—22no—
99C4×C59—42no—
100C4×C59—81no—
101C4×C59—81no—
102C4×C59#45012yes—
103Dic59—161no—
104#3293D5911yescharacteristic
105#2230—592no—
106#1716—591no—
107#1358—591no—
108#1336—592no—
109C2×C59—21no—
110C2×C59—82no—
111C2×C59#162911yescharacteristic
112#578—22no—
113#498—591no—
114#498—1181no—
115#494C2×D5911yescharacteristic
116#487—1181no—
117#473—22no—
118#456—21no—
119#456—21no—
120#443C2×D5912yes—
121#365—1182no—
122#364—592no—
123#360—1182no—
124#322—591no—
125C59#1236711yescharacteristic, complemented, Sylow
1262−1+4—21no—
1272−1+4—21no—
128#137—22no—
129#137—22no—
130#137—42no—
131#137—42no—
132#135—42no—
133#135C22⋊D5912yes—
134C2×Q16—22no—
135C2×Q16—42no—
136#131—22no—
137C4⋊Q8—1181no—
138C4⋊Q8C22⋊D5911yescharacteristic
139C4⋊Q8C22⋊D5911yescharacteristic
140M32—1182no—
141#108C22⋊D5912yes—
142#106—1182no—
143#106—2362no—
144#104—22no—
145#103—2362no—
146#101—21no—
147#101—21no—
148C4×C8—1181no—
149C4×C8C22×D5911yescharacteristic
150#41—22no—
151#41—42no—
152#41—42no—
153#41—81no—
154#41—81no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—2361no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163Q16—42no—
164Q16#1174567012yes—
165SD16—22no—
166SD16—42no—
167SD16—42no—
168D8#1174567012yes—
169M16—22no—
170M16—42no—
171C2×C8—42no—
172C2×C8—1182no—
173C2×C8C2×C22⋊D5912yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2361no—
177C4⋊C4—2361no—
178C42C2×C22⋊D5911yescharacteristic
179C16—2362no—
180Q8—22no—
181Q8—42no—
182Q8—42no—
183Q8—81no—
184Q8—81no—
185Q8—4722no—
186D4—22no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—4721no—
197C2×C4#1427369611yescharacteristic
198C8—42no—
199C8—2362no—
200C8C2×#1174567012yes—
201C8#1427375312yes—
202C22—42no—
203C22#1683757011yescharacteristic
204C4—22no—
205C4—42no—
206C4—81no—
207C4—81no—
208C4—9441no—
209C4#1683755312yes—
210C2—21no—
211C2—82no—
212C2#2142321211yescharacteristic
subgroupG/H
centerC2#21423212
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
Fitting, prime radicalC59×#3293C2
socleC2×C59#1629
hypercenter#3293D59
hypocenterC59#12367
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #21426267C2semidirect
2. #21423727C2semidirect
3. #21422135C2semidirect
4. #21422072C2semidirect
125. C59#12367semidirect

Maximal subgroups

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

subgroupindex
#214262672
#214237272
#214221352
#214220722
C59×#32932
#1236759