Finite Group Databasebeta

← #28298840

Subgroups — #28298840

Order 13568.

256 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 54 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#20290889C211yesmaximal, characteristic
2#20289978C211yesmaximal, characteristic
3#20274709C211yesmaximal, characteristic
4#16811643—21no—
5C2×#13133284C411yescharacteristic
6C2×#13133272C411yescharacteristic
7#16811489—21no—
8#16811489—21no—
9#16811468—21no—
10#16811468C412yes—
11C2×Q8×Dic53C2211yescharacteristic
12#16810646—21no—
13C2×#13133136C411yescharacteristic
14#16810391—21no—
15C2×#101×C53C411yescharacteristic
16C22×#88324—21no—
17C22×#88324—21no—
18#13133284—41no—
19#13133284—41no—
20C22⋊#88324—22no—
21#13133272—41no—
22#13133272—41no—
23C2×#11719917—21no—
24Q8×Dic53—41no—
25Q8×Dic53—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53D411yescharacteristic
28C2×C4⋊Dic53Q811yescharacteristic
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32#13133146—41no—
33#13133146—41no—
34#13133136—41no—
35C22×Q8×C53C2×C411yescharacteristic
36C2×M16×C53—21no—
37#101×C53—41no—
38C2×Q8×C53—21no—
39C2×Q8×C53—21no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C22×C4×C53C4211yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C22×C4×C53C4⋊C411yescharacteristic
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C4⋊C411yescharacteristic
48C22×Dic53C4⋊C411yescharacteristic
49C2×#88324—22no—
50C2×#88324—41no—
51C2×#88324—41no—
52C2×#88324—42no—
53C2×#88324—42no—
54C2×#88324—41no—
55M16×C53—41no—
56M16×C53—41no—
57C2×C8×C53—41no—
58C4⋊Dic53—41no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61C4×Dic53—21no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C4×Dic53—21no—
67#11719917—41no—
68#11719917—41no—
69C2×#88322—41no—
70C23×C53#9511yescharacteristic
71Q8×C53—42no—
72Q8×C53—42no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53#10011yescharacteristic
82C2×C4×C53#10111yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—41no—
89C2×Dic53—41no—
90C2×Dic53—41no—
91C2×Dic53#9911yescharacteristic
92C2×Dic53#9911yescharacteristic
93#88324—42no—
94#88324—82no—
95#88324—81no—
96C8×C53—81no—
97#88322—81no—
98#7323—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53#34211yescharacteristic
102C22×C53#34311yescharacteristic
103C22×C53#34411yescharacteristic
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108Dic53—21no—
109Dic53—41no—
110Dic53—41no—
111Dic53—41no—
112Dic53#35511yescharacteristic
113Dic53#35611yescharacteristic
114#1985—531no—
115#1801—531no—
116#1467—531no—
117C2×C53—21no—
118C2×C53—21no—
119C2×C53#130511yescharacteristic
120C2×C53#139411yescharacteristic
121C2×C53#139511yescharacteristic
122C2×C4×Q8—531no—
123#432—1061no—
124#423—1061no—
125C2×C4⋊C8—531no—
126C2×#101—531no—
127C2×#101#615711yescharacteristic
128C2×#97—531no—
129#343—1061no—
130#336—1061no—
131#336—1061no—
132#324—532no—
133#324—1061no—
134C53#732311yescharacteristic, complemented, Sylow
135C22×Q8C2×#615711yescharacteristic
136C2×M16—21no—
137C2×M16—1061no—
138C22×C8—1061no—
139C22×C8—1061no—
140C4×Q8—2121no—
141C4×Q8—2121no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C42—531no—
146C2×C42—531no—
147C2×C42—531no—
148C4⋊C8—2121no—
149C4⋊C8—2121no—
150C4⋊C8—2121no—
151C4⋊C8—2121no—
152#101—41no—
153#101—2121no—
154C22⋊C8—1062no—
155#97—2121no—
156#97—2121no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—41no—
160C2×Q8—41no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4C4×#615711yescharacteristic
166C22×C4C4⋊#615711yescharacteristic
167C22×C4C22⋊#615711yescharacteristic
168M16—41no—
169M16—41no—
170M16—2121no—
171M16—2121no—
172C2×C8—41no—
173C2×C8—1062no—
174C2×C8—2121no—
175C2×C8—2121no—
176C2×C8—2122no—
177C2×C8—2122no—
178C2×C8—2121no—
179C2×C8—2121no—
180C4⋊C4—2121no—
181C4⋊C4—2121no—
182C4⋊C4—2121no—
183C42—1061no—
184C42—1061no—
185C42—1061no—
186C42—1061no—
187C42—1061no—
188C42—1061no—
189C23(C2×#6157)⋊C411yescharacteristic
190Q8—42no—
191Q8—42no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—531no—
201C2×C4—531no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—2121no—
208C2×C4—2121no—
209C2×C4—2121no—
210C2×C4#1313327711yescharacteristic
211C2×C4#1313327811yescharacteristic
212C8—81no—
213C8—4241no—
214C8—4241no—
215C8—4242no—
216C8—2122no—
217C22—21no—
218C22—21no—
219C22#1681150011yescharacteristic
220C22#1681150311yescharacteristic
221C22#1681150411yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—531no—
227C4—531no—
228C4—1061no—
229C4—2121no—
230C4—2121no—
231C4—2121no—
232C2—21no—
233C2—21no—
234C2#2028961611yescharacteristic
235C2#2028981111yescharacteristic
236C2#2028981711yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC22×Q8C2×#6157
FittingC2×#101×C53C4
socleC22×C53#342
prime radicalC23×C53#95
cosocleC2×Q8×Dic53C22
hypercenterC2×#101#6157
hypocenterC53#7323
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
134. C53#7323semidirect

Maximal subgroups

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

subgroupindex
#202908892
#202899782
#202747092
#732353