Finite Group Databasebeta

← #27271520

Subgroups — #27271520

Order 13568.

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

normal subgroups only

This group has 202 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#20274929C211yesmaximal, characteristic
2#20272541C211yesmaximal, characteristic
3C53×#1319C211yesmaximal, characteristic
4#16811101C411yescharacteristic
5#16810663—21no—
6C2×#13133136C411yescharacteristic
7#16810383—21no—
8#16810378—21no—
9C53×#391C2211yescharacteristic
10C53×#324—21no—
11C2×#11719917—21no—
12C22×#88322—41no—
13Q8×Dic53—41no—
14Q8×Dic53—41no—
15#13133176—21no—
16#13133176—41no—
17#13133150—41no—
18#13133148—41no—
19#13133136—21no—
20#13133136C812yescomplemented
21C22×Q8×C53C2×C411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C42×C53D411yescharacteristic
24C22⋊C8×C53—41no—
25#95×C53—41no—
26C2×Q8×C53—21no—
27C2×Q8×C53—41no—
28C2×Q8×C53—41no—
29C2×Q8×C53C2×C811yescharacteristic
30C2×Q8×C53M1611yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53C22⋊C411yescharacteristic
35C2×C8×C53—81no—
36C4⋊C4×C53—41no—
37C42×C53—41no—
38C22⋊Dic53—41no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—41no—
42C4⋊Dic53—41no—
43C4×Dic53—41no—
44C4×Dic53—42no—
45#11719917—22no—
46#11719917—41no—
47C2×#88322—41no—
48C2×#88322—42no—
49C2×#88322—81no—
50C2×#88322—81no—
51C23×C53#9911yescharacteristic
52Q8×C53—42no—
53Q8×C53—41no—
54Q8×C53—41no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—81no—
67C2×C4×C53C22⋊C811yescharacteristic
68C2×C4×C53#10111yescharacteristic
69C2×Dic53—41no—
70C2×Dic53—41no—
71C8×C53—161no—
72#88322—42no—
73#88322—82no—
74#7376—531nomaximal, Sylow
75C22×C53—21no—
76C22×C53—21no—
77C22×C53#32411yescharacteristic
78C22×C53AΣL(1,𝔽16)11yescharacteristic
79C22×C53#35211yescharacteristic
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53—81no—
87Dic53—81no—
88Dic53—81no—
89#1857—531no—
90#1319D5311yescharacteristic
91#1272—531no—
92C2×C53—21no—
93C2×C53—21no—
94C2×C53#132611yescharacteristic
95C2×C53#140411yescharacteristic
96C2×C53#140511yescharacteristic
97#519—531no—
98#428—1061no—
99C2×#101—531no—
100#391C2×D5311yescharacteristic
101#339—1061no—
102#336—1061no—
103#324—21no—
104C53#737611yescharacteristic, complemented, Sylow
105C22×Q8C4×D5311yescharacteristic
106C2×M16—1061no—
107C22×C8—2121no—
108C4×Q8—2121no—
109C4×Q8—2121no—
110#117—1061no—
111#117—2121no—
112C2×C4⋊C4C22⋊D5311yescharacteristic
113C2×C42D21211yescharacteristic
114#107—2121no—
115#106—2121no—
116#101—532no—
117#101—1061no—
118C22⋊C8—41no—
119#95—41no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8C8×D5311yescharacteristic
124C2×Q8#1171991211yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4#1171992111yescharacteristic
129M16—1062no—
130M16—2121no—
131C2×C8—81no—
132C2×C8—4241no—
133C2×C8—4241no—
134C2×C8—2122no—
135C2×C8—2121no—
136C4⋊C4—41no—
137C4⋊C4—1061no—
138C4⋊C4—1061no—
139C4⋊C4—2121no—
140C4⋊C4—2121no—
141C22⋊C4—2121no—
142C42—41no—
143C42—2121no—
144C42—2122no—
145C23#1313313111yescharacteristic
146Q8—41no—
147Q8—41no—
148Q8—42no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4#1313312911yescharacteristic
164C2×C4#1313313511yescharacteristic
165C8—161no—
166C8—4242no—
167C8—2122no—
168C22—21no—
169C22—21no—
170C22#1681034811yescharacteristic
171C22#1681040911yescharacteristic
172C22#1681041411yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—81no—
180C4—4241no—
181C4—4241no—
182C2—21no—
183C2—21no—
184C2#2027266211yescharacteristic
185C2#2027286511yescharacteristic
186C2#2027286811yescharacteristic
subgroupG/H
centerC22#16810409
commutatorC2×Q8×C53C2×C8
Frattini#391C2×D53
FittingC53×#1319C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC53×#391C22
hypercenter#1319D53
hypocenterC53#7376
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1319

Complements

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

NH ≅ G/Ntype
20. #13133136C8semidirect
104. C53#7376semidirect

Maximal subgroups

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

subgroupindex
#202749292
#202725412
C53×#13192
#737653