Finite Group Databasebeta

← #28506747

Subgroups — #28506747

Order 15104.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16837232C211yesmaximal, characteristic
2#21425692C211yesmaximal, characteristic
3#21424879C212yesmaximal
4#21422204C212yesmaximal
5C2×C59×#366C211yesmaximal, characteristic
6C2×#14273667C2211yescharacteristic
7C2×#14273663C2212yes—
8#16837442—22no—
9#16837357C2211yescharacteristic
10#16837232C414yescomplemented
11#16837177—21no—
12C16×Dic59—21no—
13#16837171—22no—
14C22×C16×C59C2211yescharacteristic
15C2×#107×C59C2212yes—
16C59×#366—21no—
17C59×#366—21no—
18C2×#11745664—22no—
19C2×#11745664D411yescharacteristic
20#14273686D411yescharacteristic
21#14273667—21no—
22#14273667Q812yes—
23C2×C59⋊C16—21no—
24#14273663—22no—
25#14273663C2×C414yes—
26#14273660—42no—
27#14273658—42no—
28#14273637—21no—
29C8×Dic59—21no—
30C22×C8×C59C2311yescharacteristic
31C2×C4⋊C4×C59—22no—
32C2×C16×C59C2×C412yes—
33C2×C16×C59D411yescharacteristic
34C2×C16×C59D411yescharacteristic
35C2×C16×C59Q812yes—
36#107×C59—22no—
37#107×C59—22no—
38C22×C4×C59—42no—
39C22×C4×C59C2×D411yescharacteristic
40C2×C8×C59C22×C411yescharacteristic
41C2×C8×C59C2×D411yescharacteristic
42C2×C8×C59C2×Q811yescharacteristic
43C2×C8×C59C2×Q811yescharacteristic
44C2×C8×C59#4111yescharacteristic
45C2×C8×C59#4111yescharacteristic
46C4⋊C4×C59—42no—
47C4⋊C4×C59—42no—
48C22⋊Dic59—21no—
49C4⋊Dic59D811yescharacteristic
50C4⋊Dic59D811yescharacteristic
51C4×Dic59—21no—
52#11745664—21no—
53#11745664—24no—
54#11745664—42no—
55#11745664Q1612yes—
56C2×C59⋊C8—21no—
57C2×C59⋊C8—42no—
58C16×C59C4⋊C412yes—
59C16×C59C4⋊C412yes—
60C59⋊C16—22no—
61C23×C59#13511yescharacteristic
62C2×C4×C59—82no—
63C2×C4×C59—162no—
64C2×C4×C59C4×D411yescharacteristic
65C2×C4×C59#12111yescharacteristic
66C2×C4×C59C4⋊Q811yescharacteristic
67C2×C4×C59C2×D811yescharacteristic
68C2×C4×C59C2×Q1611yescharacteristic
69C2×C4×C59#13511yescharacteristic
70C2×Dic59—21no—
71C2×Dic59—21no—
72C8×C59C2×C4⋊C411yescharacteristic
73C8×C59C4×Q811yescharacteristic
74C8×C59C22⋊Q811yescharacteristic
75C8×C59#12511yescharacteristic
76C59⋊C8—22no—
77C59⋊C8—44no—
78#13171—591nomaximal, Sylow
79C22×C59—21no—
80C22×C59—21no—
81C22×C59C4×Q1611yescharacteristic
82C22×C59#46611yescharacteristic
83C22×C59#49911yescharacteristic
84C4×C59—162no—
85C4×C59#38911yescharacteristic
86C4×C59C4×D811yescharacteristic
87C4×C59#46711yescharacteristic
88C4×C59#50011yescharacteristic
89Dic59—41no—
90Dic59—41no—
91C2×#368—591no—
92C2×#366D5911yescharacteristic
93#2108—591no—
94#1952—592no—
95#1387—592no—
96C2×C59—21no—
97C2×C59—21no—
98C2×C59#194211yescharacteristic
99C2×C59#222111yescharacteristic
100C2×C59#222211yescharacteristic
101C2×M32—591no—
102C22×C16C2×D5911yescharacteristic
103C2×#108—592no—
104#428—1182no—
105C2×#107C2×D5912yes—
106#405—591no—
107#368—594no—
108#366—21no—
109#366—21no—
110#346—1181no—
111C4×C16—1181no—
112#343—1182no—
113C59#1317111yescharacteristic, complemented, Sylow
114C2×M16—591no—
115C2×M16—1182no—
116C22×C8C22×D5911yescharacteristic
117#117—591no—
118C2×C4⋊C4—22no—
119M32—592no—
120M32—1181no—
121C2×C16—1181no—
122C2×C16Dic11812yes—
123C2×C16C4×D5912yes—
124C2×C16C22⋊D5911yescharacteristic
125C2×C16C22⋊D5911yescharacteristic
126#108—594no—
127#108—1182no—
128#107—22no—
129#107—22no—
130#107—2362no—
131#106—2362no—
132#97—1181no—
133C4×C8—1181no—
134C22×C4—42no—
135C22×C4D4×D5911yescharacteristic
136M16—592no—
137M16—1181no—
138M16—1184no—
139M16—2362no—
140C2×C8—1181no—
141C2×C8—2362no—
142C2×C8C2×Dic11811yescharacteristic
143C2×C8C2×C4×D5911yescharacteristic
144C2×C8#1174568711yescharacteristic
145C2×C8Q8×D5911yescharacteristic
146C2×C8#1174568911yescharacteristic
147C2×C8C2×C22⋊D5911yescharacteristic
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—591no—
151C4⋊C4—591no—
152C22⋊C4—1181no—
153C42—1181no—
154C16—1182no—
155C16Dic59⋊C412yes—
156C16Dic59⋊C412yes—
157C23#1427374611yescharacteristic
158C2×C4—82no—
159C2×C4—162no—
160C2×C4—1181no—
161C2×C4—1181no—
162C2×C4#1427368911yescharacteristic
163C2×C4#1427369811yescharacteristic
164C2×C4#1427372611yescharacteristic
165C2×C4D8×D5911yescharacteristic
166C2×C4Q16×D5911yescharacteristic
167C2×C4#1427375011yescharacteristic
168C8—1182no—
169C8—2364no—
170C8C2×Dic59⋊C411yescharacteristic
171C8#1427369211yescharacteristic
172C8#1427370311yescharacteristic
173C8#1427371811yescharacteristic
174C22—21no—
175C22—21no—
176C22#1683749611yescharacteristic
177C22#1683763511yescharacteristic
178C22#1683780311yescharacteristic
179C4—162no—
180C4—2361no—
181C4—2361no—
182C4#1683729811yescharacteristic
183C4#1683748411yescharacteristic
184C4#1683764011yescharacteristic
185C4#1683780611yescharacteristic
186C2—21no—
187C2—21no—
188C2#2142482711yescharacteristic
189C2#2142621311yescharacteristic
190C2#2142621811yescharacteristic
subgroupG/H
centerC22#16837635
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#366C2
socleC22×C59#466
prime radicalC23×C59#135
cosocleC22×C8×C59C23
hypercenterC2×#366D59
hypocenterC59#13171
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#366

Complements

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

NH ≅ G/Ntype
10. #16837232C4semidirect
113. C59#13171semidirect

Maximal subgroups

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

subgroupindex
C2×#168372322
#214256922
#214248792
#214222042
C2×C59×#3662
#1317159