Finite Group Databasebeta

← #27271889

Subgroups — #27271889

Order 13568.

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

normal subgroups only

This group has 266 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#20275548C211yesmaximal, characteristic
3C53×#2132C211yesmaximal, characteristic
4#16810647C411yescharacteristic
5C2×#13133144—21no—
6C2×#13133144—21no—
7#16810607—21no—
8#16810469C411yescharacteristic
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—
18C4⋊#88322—41no—
19#13133144—42no—
20#13133144—42no—
21#13133136—22no—
22#13133134—41no—
23#13133124—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—
63#11719917—42no—
64#11719917—41no—
65#11719917—41no—
66#11719917—81no—
67C2×#88322—41no—
68C2×#88322—81no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53#9911yescharacteristic
72Q8×C53—41no—
73Q8×C53—41no—
74D4×C53—41no—
75D4×C53—41no—
76D4×C53—81no—
77D4×C53—81no—
78D4×C53—82no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—81no—
90C2×C4×C53#9511yescharacteristic
91C2×C4×C53#9911yescharacteristic
92C2×Dic53—42no—
93C2×Dic53—82no—
94C8×C53—82no—
95#88322—82no—
96#88322—81no—
97#7507—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53#34211yescharacteristic
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53#35311yescharacteristic
111C4×C53#35411yescharacteristic
112Dic53—82no—
113#2132—531no—
114#2132D5311yescharacteristic
115#1963—531no—
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—81no—
119C2×C53—81no—
120C2×C53#139311yescharacteristic
121#532C2×D5311yescharacteristic
122#423—531no—
123C2×#104—1061no—
124C2×#104—1061no—
125#413—21no—
126#413—1061no—
127#376—531no—
128#356—21no—
129#356—1061no—
130#355—532no—
131#355Dic5312yescomplemented
132C53#750711yescharacteristic, complemented, Sylow
133C2×#41D21211yescharacteristic
134C2×#41C22⋊D5311yescharacteristic
135C2×M16—41no—
136C2×M16—1061no—
137C2×M16—2121no—
138C4⋊Q8C22⋊D5311yescharacteristic
139#127Dic10611yescharacteristic
140#124C4×D5311yescharacteristic
141#124C2×Dic5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144C2×C42—1061no—
145C2×C42C4×D5311yescharacteristic
146C4⋊C8—2121no—
147#104—2122no—
148#104—2122no—
149#101—22no—
150#101—1062no—
151#100—41no—
152#100—2121no—
153#95—2121no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8#1171992111yescharacteristic
159C2×Q8C22⋊Dic5311yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4#1171991911yescharacteristic
164C2×D4C4⋊Dic5311yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—2121no—
169C22×C4#1171992111yescharacteristic
170M16—42no—
171M16—81no—
172M16—4241no—
173M16—2122no—
174M16—2121no—
175M16—2121no—
176C2×C8—81no—
177C2×C8—4241no—
178C2×C8—2121no—
179C4⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—1062no—
184C42—2121no—
185C42C4×Dic5311yescharacteristic
186C42#1171991911yescharacteristic
187C23—21no—
188C23—21no—
189C23#1313313111yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—4242no—
209C2×C4—2122no—
210C2×C4#1313312411yescharacteristic
211C2×C4#1313313111yescharacteristic
212C8—82no—
213C8—4242no—
214C8—4241no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22#1681038911yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—4242no—
228C4#1681041511yescharacteristic
229C4#1681041711yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2027283911yescharacteristic
subgroupG/H
centerC4#16810415
commutatorC42×C53C42
Frattini#532C2×D53
FittingC53×#2132C2
socleC2×C53#1393
prime radical, cosocleC53×#532C22
hypercenter#2132D53
hypocenterC53#7507
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
131. #355Dic53semidirect
132. C53#7507semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202755482
C53×#21322
#750753