Finite Group Databasebeta

← #27304956

Subgroups — #27304956

Order 13568.

188 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 51 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×#16810458C211yesmaximal, characteristic, complemented
2#20277010C211yesmaximal, characteristic, complemented
3#20276836C211yesmaximal, characteristic, complemented
4#20276832C211yesmaximal, characteristic, complemented
5#20272918C211yesmaximal, characteristic, complemented
6#20272906C211yesmaximal, characteristic, complemented
7C53×#1419C211yesmaximal, characteristic, complemented
8C2×D848C2211yescharacteristic
9#16811039C2211yescharacteristic
10#16810894—21no—
11#16810890—21no—
12#16810458—21no—
13#16810458—21no—
14#16810456—21no—
15C2×#13133123C2211yescharacteristic
16C32×D53—21no—
17#16810453C2211yescharacteristic
18#16810436—21no—
19#16810435—21no—
20C2×D16×C53C2211yescharacteristic
21C53×M64C2211yescharacteristic
22C53×#368C2211yescharacteristic
23#13133245—41no—
24#13133243—41no—
25C2×D424—21no—
26#13133223D411yescharacteristic
27D848—21no—
28D848—21no—
29#13133157D411yescharacteristic
30#13133156—21no—
31C16×D53—21no—
32#13133153—21no—
33#13133134—41no—
34#13133133—41no—
35#13133123D411yescharacteristic
36#13133123D411yescharacteristic
37C2×D8×C53—21no—
38D16×C53—21no—
39D16×C53—21no—
40C2×C16×C53C2311yescharacteristic
41#108×C53—21no—
42C32×C53—21no—
43C2×D4×C53—41no—
44#11719945D811yescharacteristic
45C2×D212—41no—
46D8×C53—41no—
47D8×C53—41no—
48M16×C53—41no—
49C2×C8×C53C2×D411yescharacteristic
50#11719923—81no—
51#11719922—81no—
52#11719917—41no—
53#11719917D811yescharacteristic
54D424—41no—
55D424—41no—
56D424—81no—
57#11719913—81no—
58#11719912—21no—
59C8×D53—21no—
60C16×C53C2×D411yescharacteristic
61C16×C53#4111yescharacteristic
62#11719908—21no—
63C23×C53—81no—
64C22×D53—81no—
65D4×C53—81no—
66D4×C53—81no—
67C2×C4×C53C2×D811yescharacteristic
68C22⋊D53—21no—
69D212—81no—
70D212—81no—
71D212D1611yescharacteristic
72C4×D53—21no—
73Dic106D1611yescharacteristic
74C8×C53—81no—
75C8×C53#12111yescharacteristic
76C8×C53#13511yescharacteristic
77#88322—21no—
78#88322—81no—
79#13700—531nomaximal, Sylow
80C22×C53—161no—
81C22×C53—321no—
82C22×C53#50811yescharacteristic
83C2×D53—21no—
84C2×D53—161no—
85C2×D53—321no—
86C4×C53#46611yescharacteristic
87C4×C53C2×D1611yescharacteristic
88Dic53—21no—
89C2×D32—531no—
90#2258—531no—
91#2221—531no—
92#2221—531no—
93#1425—531no—
94#1419—531no—
95#1419D5311yescharacteristic, complemented
96C2×C53—21no—
97C2×C53—321no—
98C2×C53#221511yescharacteristic
99D53—41no—
100D53—321no—
101C2×D16—531no—
102C2×D16C2×D5311yescharacteristic
103#504—531no—
104#472—1061no—
105#472—1061no—
106D32—1061no—
107D32—1061no—
108M64—1061no—
109M64C2×D5311yescharacteristic
110C2×C32—531no—
111C2×C32—1061no—
112#368—531no—
113#368C2×D5311yescharacteristic
114#361—1061no—
115#361—1061no—
116C53#1370011yescharacteristic, complemented, Sylow
117AGL(1,ℤ/8)—2121no—
118AGL(1,ℤ/8)—2121no—
119C2×D8—21no—
120C2×D8—1061no—
121#131—531no—
122D16—21no—
123D16—21no—
124D16—1061no—
125D16—1061no—
126M32—531no—
127M32—1061no—
128C2×C16—1061no—
129C2×C16C22×D5311yescharacteristic
130#108—21no—
131#108—1061no—
132#100—2121no—
133#100—2121no—
134C32—21no—
135C32—531no—
136C32—531no—
137#41—531no—
138C2×D4—41no—
139C2×D4—2121no—
140SD16—4241no—
141SD16—4241no—
142D8—41no—
143D8—41no—
144D8—4241no—
145D8—4241no—
146D8—2121no—
147D8—2121no—
148M16—41no—
149M16—531no—
150M16—1061no—
151M16—2121no—
152C2×C8—1061no—
153C2×C8D4×D5311yescharacteristic
154C16—1061no—
155C16#1171994511yescharacteristic
156C16D4×D5311yescharacteristic
157C23—81no—
158C23—4241no—
159Q8—531no—
160D4—81no—
161D4—81no—
162D4—531no—
163D4—1061no—
164D4—4241no—
165D4—4241no—
166C2×C4—1061no—
167C2×C4D8×D5311yescharacteristic
168C8—81no—
169C8—1061no—
170C8—4241no—
171C8#1313319011yescharacteristic
172C8#1313323811yescharacteristic
173C22—161no—
174C22—321no—
175C22—8481no—
176C22—1061no—
177C22—16961no—
178C22#1681105411yescharacteristic
179C4—1061no—
180C4#1681085811yescharacteristic
181C4D16×D5311yescharacteristic
182C2—21no—
183C2—321no—
184C2—16961no—
185C2—2121no—
186C2#2027680011yescharacteristic
subgroupG/H
centerC2#20276800
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1419C2
socleC2×C53#2215
hypercenter#1419D53
hypocenterC53#13700
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
1. C2×#16810458C2semidirect
2. #20277010C2semidirect
3. #20276836C2semidirect
4. #20276832C2semidirect
5. #20272918C2semidirect
6. #20272906C2semidirect
7. C53×#1419C2semidirect
95. #1419D53semidirect
116. C53#13700semidirect

Maximal subgroups

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

subgroupindex
C2×#168104582
#202770102
#202768362
#202768322
#202729182
#202729062
C53×#14192
#1370053