Finite Group Databasebeta

← #28500643

Subgroups — #28500643

Order 15104.

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

normal subgroups only

This group has 220 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#21425958C211yesmaximal, characteristic
2#21423005C211yesmaximal, characteristic, complemented
3#21422919C211yesmaximal, characteristic
4#21422161C211yesmaximal, characteristic, complemented
5#21422064C211yesmaximal, characteristic
6#21422016C211yesmaximal, characteristic, complemented
7C59×#1334C211yesmaximal, characteristic, complemented
8C2×#14273670C411yescharacteristic
9#16837533—21no—
10C4×D472C2211yescharacteristic
11#16837467—21no—
12#16837227C411yescharacteristic
13#16837206C411yescharacteristic
14#16837202C411yescharacteristic
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17Q8⋊(C59⋊C8)—21no—
18#16837129—21no—
19#16837123C2211yescharacteristic
20C4×Q16×C59C2211yescharacteristic
21C59×#346C2211yescharacteristic
22C59×#334C2211yescharacteristic
23C2×D472C2×C411yescharacteristic
24C2×C8×D59—41no—
25C4×D236—21no—
26#14273670—22no—
27C2×C59⋊C16C2×C411yescharacteristic
28C2×C59⋊C16C2×C411yescharacteristic
29#14273661C2×C411yescharacteristic
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)—41no—
32#14273649—21no—
33(C2×D59)⋊C8—41no—
34#14273638—41no—
35C4×C59⋊C8D411yescharacteristic
36C4×C59⋊C8D411yescharacteristic
37C2×Q16×C59C2×C411yescharacteristic
38C4×Q8×C59—21no—
39C2×C16×C59—21no—
40#107×C59C2×C411yescharacteristic
41C4⋊C8×C59—21no—
42#103×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×Q8×C59—21no—
45C2×D236—21no—
46C2×C4×D59—41no—
47Q16×C59—21no—
48Q16×C59—21no—
49C2×C8×C59—41no—
50C2×C8×C59C22×C411yescharacteristic
51C2×C8×C59#4111yescharacteristic
52C4⋊C4×C59—21no—
53C4⋊C4×C59—41no—
54C42×C59—41no—
55C42×C59C2×D411yescharacteristic
56(C2×D59)⋊C4—41no—
57C4⋊Dic59—21no—
58C2×C59⋊C8—21no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8—41no—
61D472—21no—
62D472M1612yes—
63C8×D59—82no—
64C16×C59—22no—
65C59⋊C16—21no—
66C59⋊C16—21no—
67C22×D59—41no—
68Q8×C59—41no—
69Q8×C59—41no—
70C2×C4×C59—41no—
71C2×C4×C59—41no—
72C2×C4×C59C4×D411yescharacteristic
73C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
74C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
75C2×Dic59—41no—
76D236—22no—
77D236—41no—
78C4×D59—81no—
79C4×D59—81no—
80C8×C59—21no—
81C8×C59—81no—
82C8×C59C2×M1611yescharacteristic
83C8×C59#13111yescharacteristic
84C59⋊C8—41no—
85C59⋊C8—41no—
86C59⋊C8—81no—
87#12077—591nomaximal, Sylow
88C22×C59#44211yescharacteristic
89C2×D59—42no—
90C2×D59—81no—
91C2×D59—81no—
92C4×C59—21no—
93C4×C59—21no—
94C4×C59—81no—
95C4×C59—81no—
96C4×C59#43511yescharacteristic
97C4×C59#44311yescharacteristic
98Dic59—81no—
99#2175—591no—
100#1593—591no—
101#1580—591no—
102#1368—591no—
103#1334D5911yescharacteristic, complemented
104#1333—591no—
105#1313—591no—
106C2×C59#158611yescharacteristic
107C2×C59#218011yescharacteristic
108C2×C59AGL(1,ℤ/16)11yescharacteristic
109D59—82no—
110C2×SD32—591no—
111#446—1181no—
112C4×Q16C2×D5911yescharacteristic
113C4×D8—591no—
114C8×D4—1181no—
115#366—591no—
116#358—591no—
117#357—591no—
118#346—591no—
119#346C2×D5911yescharacteristic
120#334—591no—
121#334C2×D5911yescharacteristic
122Q8⋊C8—1181no—
123D4⋊C8—1181no—
124#322—591no—
125C59#1207711yescharacteristic, complemented, Sylow
126C2×Q16C4×D5911yescharacteristic
127C2×D8—591no—
128C22×C8—2361no—
129C4×Q8—21no—
130C4×D4—1181no—
131SD32—1182no—
132C2×C16—21no—
133C2×C16—591no—
134C2×C16—591no—
135#107—591no—
136#107C4×D5911yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—1181no—
139C4⋊C8—2361no—
140#103—21no—
141#102—1181no—
142C22⋊C8—2361no—
143#97—2361no—
144C4×C8—591no—
145C4×C8—591no—
146C4×C8C22×D5911yescharacteristic
147C2×Q8—21no—
148C2×D4—1181no—
149C22×C4—2361no—
150Q16—21no—
151Q16—21no—
152D8—592no—
153D8—1181no—
154C2×C8—41no—
155C2×C8—1181no—
156C2×C8—1181no—
157C2×C8—4722no—
158C2×C8—2361no—
159C2×C8C2×C4×D5911yescharacteristic
160C2×C8#1174568711yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—1181no—
164C22⋊C4—2361no—
165C42—41no—
166C42D4×D5911yescharacteristic
167C16—22no—
168C16—1181no—
169C16—1181no—
170C23—2361no—
171Q8—41no—
172Q8—41no—
173D4—1182no—
174D4—2361no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—4721no—
178C2×C4—4721no—
179C2×C4—2361no—
180C2×C4(C22⋊D59)⋊C411yescharacteristic
181C2×C4#1427375111yescharacteristic
182C2×C4#1427375411yescharacteristic
183C8—21no—
184C8—81no—
185C8—4721no—
186C8—2361no—
187C8—2361no—
188C8M16×D5911yescharacteristic
189C8#1427373311yescharacteristic
190C22—4721no—
191C22—4721no—
192C22—2362no—
193C22#1683751111yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—81no—
198C4—4721no—
199C4#1683747611yescharacteristic
200C4#1683751611yescharacteristic
201C2—4722no—
202C2#2142296311yescharacteristic
203C2#2142599011yescharacteristic
204C2#2142599111yescharacteristic
subgroupG/H
centerC22#16837511
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1334C2
socleC22×C59#442
prime radicalC2×D472C2×C4
cosocleC4×C8×C59C23
hypercenter#1334D59
hypocenterC59#12077
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1334

Complements

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

NH ≅ G/Ntype
2. #21423005C2semidirect
4. #21422161C2semidirect
6. #21422016C2semidirect
7. C59×#1334C2semidirect
103. #1334D59semidirect
125. C59#12077semidirect

Maximal subgroups

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

subgroupindex
#214259582
#214230052
#214229192
#214221612
#214220642
#214220162
C59×#13342
#1207759