Finite Group Databasebeta

← #28298617

Subgroups — #28298617

Order 13568.

244 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 52 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#20290886C211yesmaximal, characteristic
2#20289979C211yesmaximal, characteristic
3#20273057C211yesmaximal, characteristic
4#16811643—21no—
5C2×#13133284C411yescharacteristic
6C2×C4×#88324C411yescharacteristic
7#16811489—21no—
8#16811489—21no—
9#16811469—21no—
10#16811469C412yes—
11C2×#13133182C2211yescharacteristic
12#16810677—21no—
13#16810349C411yescharacteristic
14#16810348—21no—
15C53×#324C411yescharacteristic
16C22×#88324—21no—
17C22×#88324—21no—
18#13133284—41no—
19#13133284—41no—
20C22⋊#88324—22no—
21#13133272—41no—
22C4×#88324—41no—
23C22×Dic106Q811yescharacteristic
24#13133182—41no—
25#13133182—41no—
26C2×#11719919C2×C411yescharacteristic
27C2×#11719919D411yescharacteristic
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D411yescharacteristic
30C2×C4×Dic53D411yescharacteristic
31#13133146—41no—
32C22⋊#88322—21no—
33#13133127—41no—
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C22⋊C8×C53—21no—
36C22×C4×C53C4211yescharacteristic
37C22×C4×C53C22⋊C411yescharacteristic
38C22×C4×C53C4⋊C411yescharacteristic
39C22×Dic53C22⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C4⋊C411yescharacteristic
42C22×Dic53C4⋊C411yescharacteristic
43C2×Dic106—22no—
44C2×Dic106—41no—
45C2×Dic106—41no—
46C2×#88324—22no—
47C2×#88324—41no—
48C2×#88324—41no—
49C2×#88324—42no—
50C2×#88324—42no—
51C2×#88324—41no—
52C2×C8×C53—41no—
53C4⋊C4×C53—41no—
54#11719919—41no—
55#11719919—41no—
56C4×Dic53—21no—
57C4×Dic53—21no—
58C4×Dic53—21no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C4×Dic53—21no—
62C2×#88322—41no—
63C23×C53#9511yescharacteristic
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—41no—
80C2×Dic53#9911yescharacteristic
81C2×Dic53#9911yescharacteristic
82C2×Dic53#10812yes—
83Dic106—44no—
84#88324—42no—
85#88324—82no—
86#88324—81no—
87C8×C53—81no—
88#88322—81no—
89#7248—531nomaximal, Sylow
90C22×C53—21no—
91C22×C53—21no—
92C22×C53#33811yescharacteristic
93C22×C53#34111yescharacteristic
94C22×C53#34211yescharacteristic
95C4×C53—41no—
96C4×C53—41no—
97C4×C53—41no—
98Dic53—22no—
99Dic53—21no—
100Dic53—41no—
101Dic53—41no—
102Dic53#35511yescharacteristic
103Dic53#35611yescharacteristic
104#1984—531no—
105#1467—531no—
106#1467—531no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#129311yescharacteristic
110C2×C53#139111yescharacteristic
111C2×C53#139411yescharacteristic
112C2×C4×Q8—531no—
113#432—1061no—
114#432—1061no—
115C2×C4⋊C8—531no—
116C2×C4×C8—531no—
117#336—1061no—
118#336—1061no—
119#324—532no—
120#324—531no—
121#324—1061no—
122#324—1061no—
123#324#615711yescharacteristic
124C53#724811yescharacteristic, complemented, Sylow
125C22×Q8—531no—
126C22×C8—1061no—
127C22×C8—1061no—
128C4×Q8—2121no—
129C4×Q8—2121no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4C2×#615711yescharacteristic
133C2×C42—531no—
134C2×C42—531no—
135C2×C42—531no—
136C4⋊C8—2121no—
137C4⋊C8—2121no—
138C4⋊C8—2121no—
139C22⋊C8—21no—
140C22⋊C8—1062no—
141C22⋊C8—1061no—
142#97—2121no—
143#97—2121no—
144C4×C8—2121no—
145C2×Q8—1062no—
146C2×Q8—2121no—
147C2×Q8—2121no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4C4×#615711yescharacteristic
153C22×C4C4⋊#615711yescharacteristic
154C22×C4C22⋊#615711yescharacteristic
155C2×C8—41no—
156C2×C8—1062no—
157C2×C8—2121no—
158C2×C8—2121no—
159C2×C8—2122no—
160C2×C8—2122no—
161C2×C8—2121no—
162C2×C8—2121no—
163C4⋊C4—41no—
164C4⋊C4—2121no—
165C4⋊C4—2121no—
166C42—1061no—
167C42—1061no—
168C42—1061no—
169C42—1061no—
170C42—1061no—
171C42—1061no—
172C23(C2×#6157)⋊C411yescharacteristic
173Q8—2124no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—531no—
182C2×C4—532no—
183C2×C4—531no—
184C2×C4—1061no—
185C2×C4—1061no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—2121no—
193C8—81no—
194C8—4241no—
195C8—4241no—
196C8—4242no—
197C8—2122no—
198C22—21no—
199C22—21no—
200C22#1681149211yescharacteristic
201C22#1681149811yescharacteristic
202C22#1681150011yescharacteristic
203C4—41no—
204C4—41no—
205C4—41no—
206C4—531no—
207C4—531no—
208C4—1061no—
209C4—1062no—
210C4—2121no—
211C4—2121no—
212C2—21no—
213C2—21no—
214C2#2028958611yescharacteristic
215C2#2028980211yescharacteristic
216C2#2028981111yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×C53#95
cosocleC2×#13133182C22
hypercenter#324#6157
hypocenterC53#7248
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
124. C53#7248semidirect

Maximal subgroups

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

subgroupindex
#202908862
#202899792
#202730572
#724853