Finite Group Databasebeta

← #28500637

Subgroups — #28500637

Order 15104.

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

normal subgroups only

This group has 220 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#21425960C211yesmaximal, characteristic
2#21423004C211yesmaximal, characteristic, complemented
3#21422920C211yesmaximal, characteristic
4#21422161C211yesmaximal, characteristic, complemented
5#21422067C211yesmaximal, characteristic, complemented
6#21422016C211yesmaximal, characteristic, complemented
7C59×#1333C211yesmaximal, characteristic
8C2×#14273671C411yescharacteristic
9#16837529—21no—
10C4×Dic236C2211yescharacteristic
11D4×C59⋊C8—21no—
12#16837227C411yescharacteristic
13#16837205C411yescharacteristic
14#16837203C411yescharacteristic
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17#16837132—21no—
18D4⋊(C59⋊C8)—21no—
19#16837123C2211yescharacteristic
20C4×D8×C59C2211yescharacteristic
21C59×#346C2211yescharacteristic
22C59×#334C2211yescharacteristic
23C2×Dic236C2×C411yescharacteristic
24C22×C59⋊C8—41no—
25C4×Dic118—21no—
26#14273671—22no—
27C2×C59⋊C16C2×C411yescharacteristic
28C2×C59⋊C16C2×C411yescharacteristic
29#14273661C2×C411yescharacteristic
30C4⋊(C59⋊C8)—21no—
31Dic59⋊C8—41no—
32#14273652—21no—
33C22⋊(C59⋊C8)—41no—
34#14273637—41no—
35C4×C59⋊C8D411yescharacteristic
36C4×C59⋊C8D411yescharacteristic
37C2×D8×C59C2×C411yescharacteristic
38C4×D4×C59—21no—
39C2×C16×C59—21no—
40#107×C59C2×C411yescharacteristic
41C4⋊C8×C59—21no—
42#102×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D4×C59—21no—
45C22×C4×C59—41no—
46C2×Dic118—21no—
47D8×C59—21no—
48D8×C59M1612yes—
49C2×C8×C59—41no—
50C2×C8×C59C22×C411yescharacteristic
51C2×C8×C59#4111yescharacteristic
52C4⋊C4×C59—21no—
53C22⋊C4×C59—41no—
54C42×C59C2×D411yescharacteristic
55C4⋊Dic59—21no—
56Dic59⋊C4—41no—
57C4×Dic59—41no—
58C2×C59⋊C8—21no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8—41no—
61C2×C59⋊C8—82no—
62Dic236—21no—
63Dic236—21no—
64C16×C59—22no—
65C59⋊C16—21no—
66C59⋊C16—21no—
67C23×C59—41no—
68D4×C59—22no—
69D4×C59—41no—
70C2×C4×C59—41no—
71C2×C4×C59—81no—
72C2×C4×C59—81no—
73C2×C4×C59C4×D411yescharacteristic
74C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
75C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
76C2×Dic59—41no—
77C2×Dic59—41no—
78Dic118—41no—
79Dic118—41no—
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—42no—
89C22×C59—81no—
90C22×C59—81no—
91C22×C59#44211yescharacteristic
92C4×C59—21no—
93C4×C59—21no—
94C4×C59—81no—
95C4×C59#43511yescharacteristic
96C4×C59#44311yescharacteristic
97Dic59—81no—
98Dic59—81no—
99#2175—591no—
100#1593—591no—
101#1580—591no—
102#1368—591no—
103#1334—591no—
104#1333D5911yescharacteristic
105#1313—591no—
106C2×C59—82no—
107C2×C59#158611yescharacteristic
108C2×C59#218011yescharacteristic
109C2×C59AGL(1,ℤ/16)11yescharacteristic
110C2×SD32—591no—
111#446—1181no—
112C4×Q16—591no—
113C4×D8C2×D5911yescharacteristic
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×Q16—591no—
127C2×D8C4×D5911yescharacteristic
128C22×C8—2361no—
129C4×Q8—1181no—
130C4×D4—21no—
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—1181no—
141#102—21no—
142C22⋊C8—2361no—
143#97—2361no—
144C4×C8—591no—
145C4×C8—591no—
146C4×C8C22×D5911yescharacteristic
147C2×Q8—1181no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—1181no—
151Q16—1181no—
152D8—21no—
153D8#1174565912yes—
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—1181no—
163C4⋊C4—2361no—
164C22⋊C4—41no—
165C42—2361no—
166C42D4×D5911yescharacteristic
167C16—22no—
168C16—1181no—
169C16—1181no—
170C23—41no—
171Q8—2361no—
172Q8—2361no—
173D4—22no—
174D4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—2361no—
179C2×C4—2361no—
180C2×C4(C22⋊D59)⋊C411yescharacteristic
181C2×C4#1427375111yescharacteristic
182C2×C4#1427375411yescharacteristic
183C8—21no—
184C8—81no—
185C8—4721no—
186C8—2361no—
187C8—2361no—
188C8C2×#1174565911yescharacteristic
189C8#1427373511yescharacteristic
190C22—42no—
191C22—81no—
192C22—81no—
193C22#1683751111yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—4721no—
198C4—4721no—
199C4#1683747011yescharacteristic
200C4#1683751611yescharacteristic
201C2—82no—
202C2#2142295711yescharacteristic
203C2#2142598411yescharacteristic
204C2#2142599711yescharacteristic
subgroupG/H
centerC22#16837511
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1333C2
socleC22×C59#442
prime radicalC2×D8×C59C2×C4
cosocleC4×C8×C59C23
hypercenter#1333D59
hypocenterC59#12077
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #21423004C2semidirect
4. #21422161C2semidirect
5. #21422067C2semidirect
6. #21422016C2semidirect
125. C59#12077semidirect

Maximal subgroups

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

subgroupindex
#214259602
#214230042
#214229202
#214221612
#214220672
#214220162
C59×#13332
#1207759