Finite Group Databasebeta

← #27271875

Subgroups — #27271875

Order 13568.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276400C211yesmaximal, characteristic
2#20275544C211yesmaximal, characteristic
3C53×#2132C211yesmaximal, characteristic
4#16810647C411yescharacteristic
5C2×#13133144—21no—
6C2×#13133144—21no—
7#16810607—21no—
8C42×Dic53C411yescharacteristic
9#16810422—21no—
10#16810420C412yescomplemented
11C53×#532C2211yescharacteristic
12C53×#413—21no—
13C53×#356—21no—
14C53×#355C412yescomplemented
15C2×#11719917—21no—
16C2×#11719917—41no—
17C2×C4×Dic53—21no—
18C2×C4×Dic53—41no—
19C4⋊#88322—41no—
20#13133144—42no—
21#13133144—42no—
22#13133136—22no—
23#13133134—41no—
24C2×#41×C53D411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C2×M16×C53—41no—
27C4⋊Q8×C53D411yescharacteristic
28#127×C53Q811yescharacteristic
29#124×C53C2×C411yescharacteristic
30#124×C53C2×C411yescharacteristic
31#121×C53—41no—
32C4×D4×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
34#101×C53—22no—
35#100×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C22⋊C411yescharacteristic
41C2×Q8×C53C22⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×Dic53—41no—
52M16×C53—42no—
53M16×C53—81no—
54C2×C8×C53—81no—
55C4⋊C4×C53—41no—
56C22⋊C4×C53—41no—
57C22⋊C4×C53—41no—
58C42×C53—21no—
59C42×C53C4211yescharacteristic
60C42×C53C4⋊C411yescharacteristic
61C4×Dic53—22no—
62C4×Dic53—41no—
63C4×Dic53—44no—
64#11719917—42no—
65#11719917—41no—
66#11719917—41no—
67#11719917—81no—
68C2×#88322—41no—
69C2×#88322—81no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73Q8×C53—41no—
74Q8×C53—41no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—82no—
78D4×C53—81no—
79D4×C53—81no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×Dic53—42no—
94C2×Dic53—44no—
95C8×C53—82no—
96#88322—82no—
97#88322—81no—
98#7501—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—81no—
104C22×C53—81no—
105C22×C53#34211yescharacteristic
106C4×C53—22no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—41no—
111C4×C53#35311yescharacteristic
112C4×C53#35411yescharacteristic
113Dic53—44no—
114#2132—531no—
115#2132D5311yescharacteristic
116#1962—531no—
117C2×C53—21no—
118C2×C53—41no—
119C2×C53—81no—
120C2×C53—81no—
121C2×C53#139311yescharacteristic
122#532C2×D5311yescharacteristic
123#423—531no—
124C2×#104—1061no—
125C2×#104—1061no—
126#413—21no—
127#413—1061no—
128C43—531no—
129#356—21no—
130#356—1061no—
131#355—532no—
132#355Dic5312yescomplemented
133C53#750111yescharacteristic, complemented, Sylow
134C2×#41D21211yescharacteristic
135C2×#41C22⋊D5311yescharacteristic
136C2×M16—41no—
137C2×M16—1061no—
138C2×M16—2121no—
139C4⋊Q8C22⋊D5311yescharacteristic
140#127Dic10611yescharacteristic
141#124C4×D5311yescharacteristic
142#124C2×Dic5311yescharacteristic
143#121—41no—
144C4×D4—41no—
145C2×C42—1061no—
146C2×C42—2121no—
147C2×C42C4×D5311yescharacteristic
148C4⋊C8—2121no—
149#104—2122no—
150#104—2122no—
151#101—22no—
152#101—1062no—
153#100—41no—
154#100—2121no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8#1171992111yescharacteristic
160C2×Q8C22⋊Dic5311yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4#1171991911yescharacteristic
165C2×D4C4⋊Dic5311yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—2121no—
170C22×C4#1171992111yescharacteristic
171M16—42no—
172M16—81no—
173M16—4241no—
174M16—2122no—
175M16—2121no—
176M16—2121no—
177C2×C8—81no—
178C2×C8—4241no—
179C2×C8—2121no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C42—21no—
184C42—1062no—
185C42—2121no—
186C42—2124no—
187C42C4×Dic5311yescharacteristic
188C42#1171991911yescharacteristic
189C23—21no—
190C23—21no—
191C23#1313313111yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—82no—
197D4—81no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—2122no—
211C2×C4—2124no—
212C2×C4#1313312411yescharacteristic
213C2×C4#1313313111yescharacteristic
214C8—82no—
215C8—4242no—
216C8—4241no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—81no—
222C22—81no—
223C22#1681038911yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—2124no—
230C4#1681041511yescharacteristic
231C4#1681041711yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2027283911yescharacteristic
subgroupG/H
centerC4#16810417
commutatorC42×C53C42
Frattini#532C2×D53
FittingC53×#2132C2
socleC2×C53#1393
prime radical, cosocleC53×#532C22
hypercenter#2132D53
hypocenterC53#7501
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #16810420C4semidirect
14. C53×#355C4semidirect
132. #355Dic53semidirect
133. C53#7501semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202755442
C53×#21322
#750153