Finite Group Databasebeta

← #27271472

Subgroups — #27271472

Order 13568.

204 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 43 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#20275534C211yesmaximal, characteristic
2#20272659C211yesmaximal, characteristic
3C53×#1324C211yesmaximal, characteristic
4C2×#13133217C411yescharacteristic
5C2×#13133140—21no—
6C2×#13133140—21no—
7C2×#13133128C411yescharacteristic
8#16810349—21no—
9C53×#382C2211yescharacteristic
10C53×#324—21no—
11C22×Dic106—21no—
12C22×#88322—21no—
13#13133217—21no—
14#13133217—41no—
15#13133217C812yescomplemented
16C2×C4⋊Dic53—41no—
17#13133140—42no—
18#13133140—42no—
19C22⋊#88322—41no—
20#13133128—41no—
21C2×C4⋊C4×C53D411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C42×C53C2×C411yescharacteristic
24C22⋊C8×C53—41no—
25#95×C53—41no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C22×Dic53—41no—
31C2×Dic106—22no—
32C2×Dic106—41no—
33C2×Dic106—41no—
34C2×Dic106—41no—
35C2×Dic106—41no—
36C2×Dic106—41no—
37C2×C8×C53—81no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—41no—
40C42×C53—21no—
41C42×C53C2×C811yescharacteristic
42C42×C53M1611yescharacteristic
43C4⋊Dic53—42no—
44C4⋊Dic53—42no—
45C2×#88322—22no—
46C2×#88322—41no—
47C2×#88322—41no—
48C2×#88322—81no—
49C23×C53#9911yescharacteristic
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—81no—
59C2×C4×C53—81no—
60C2×C4×C53C22⋊C811yescharacteristic
61C2×C4×C53#10011yescharacteristic
62C2×Dic53—42no—
63C2×Dic53—81no—
64C2×Dic53—81no—
65Dic106—42no—
66Dic106—42no—
67Dic106—81no—
68Dic106—81no—
69C8×C53—161no—
70#88322—42no—
71#88322—161no—
72#7353—531nomaximal, Sylow
73C22×C53—21no—
74C22×C53—21no—
75C22×C53#32311yescharacteristic
76C22×C53#35311yescharacteristic
77C22×C53#35511yescharacteristic
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—41no—
81C4×C53—41no—
82C4×C53—81no—
83C4×C53—81no—
84Dic53—82no—
85#1960—531no—
86#1324—531no—
87#1324D5311yescharacteristic
88C2×C53—21no—
89C2×C53—21no—
90C2×C53#131811yescharacteristic
91C2×C53#140711yescharacteristic
92C2×C53#140711yescharacteristic
93C2×C4⋊Q8—531no—
94C2×#103—1061no—
95C2×#103—1061no—
96C2×#97—531no—
97#382C2×D5311yescharacteristic
98#324—21no—
99#324—1061no—
100C53#735311yescharacteristic, complemented, Sylow
101C22×Q8—1061no—
102C22×C8—1061no—
103C4⋊Q8—532no—
104C4⋊Q8—1061no—
105C4⋊Q8—2121no—
106C2×C4⋊C4—2121no—
107C2×C4⋊C4D21211yescharacteristic
108C2×C4⋊C4C22⋊D5311yescharacteristic
109C2×C42C4×D5311yescharacteristic
110#103—2122no—
111#103—2122no—
112C22⋊C8—41no—
113C22⋊C8—2121no—
114#97—2121no—
115#95—41no—
116C2×Q8—1062no—
117C2×Q8—2121no—
118C2×Q8—2121no—
119C2×Q8—2121no—
120C2×Q8—2121no—
121C2×Q8—2121no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—2121no—
126C22×C4#1171992111yescharacteristic
127C2×C8—81no—
128C2×C8—1062no—
129C2×C8—4241no—
130C2×C8—2121no—
131C2×C8—2121no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—2122no—
135C4⋊C4—2122no—
136C42—21no—
137C42C8×D5311yescharacteristic
138C42#1171991211yescharacteristic
139C23#1313313111yescharacteristic
140Q8—4241no—
141Q8—4241no—
142Q8—2122no—
143Q8—2122no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—4241no—
155C2×C4—4241no—
156C2×C4—2122no—
157C2×C4#1313312911yescharacteristic
158C2×C4#1313313311yescharacteristic
159C8—161no—
160C8—8481no—
161C8—2122no—
162C22—21no—
163C22—21no—
164C22#1681034611yescharacteristic
165C22#1681041511yescharacteristic
166C22#1681041911yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—81no—
173C4—4242no—
174C2—21no—
175C2—21no—
176C2#2027264611yescharacteristic
177C2#2027287411yescharacteristic
178C2#2027287611yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC42×C53C2×C8
Frattini#382C2×D53
FittingC53×#1324C2
socleC22×C53#353
prime radicalC23×C53#99
cosocleC53×#382C22
hypercenter#1324D53
hypocenterC53#7353
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1324

Complements

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

NH ≅ G/Ntype
15. #13133217C8semidirect
100. C53#7353semidirect

Maximal subgroups

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

subgroupindex
#202755342
#202726592
C53×#13242
#735353