Finite Group Databasebeta

← #28308208

Subgroups — #28308208

Order 13568.

418 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289966C212yesmaximal
2#20289657C214yesmaximal, complemented
3#20277338C211yesmaximal, characteristic
4#16811643—22no—
5#16811468C2214yes—
6#16811466—22no—
7C2×#13133176C2211yescharacteristic
8C22×C4⋊Dic53C411yescharacteristic
9#16810522—22no—
10#16810498—22no—
11#16810493C2212yes—
12#16810489C412yes—
13C2×#117×C53C411yescharacteristic
14#13133284—42no—
15C22⋊#88324—24no—
16#13133272—42no—
17C23×Dic53—21no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—24no—
21C2×C4⋊Dic53—42no—
22C2×C4⋊Dic53C2×C412yes—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2311yescharacteristic
25C2×C22⋊Dic53D412yes—
26C2×C4×Dic53D412yes—
27#13133124—24no—
28C23×C4×C53C2×C411yescharacteristic
29#117×C53—24no—
30C2×C4⋊C4×C53—21no—
31C2×C22⋊C4×C53—21no—
32C2×C42×C53C2×C412yes—
33C24×C53C22⋊C411yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53C22⋊C411yescharacteristic
39C22×C4×C53C22×C411yescharacteristic
40C22×Dic53—22no—
41C22×Dic53—42no—
42C22×Dic53—82no—
43C22×Dic53C22⋊C412yes—
44C22×Dic53C2×D412yes—
45C2×#88324—44no—
46C4⋊C4×C53—24no—
47C22⋊C4×C53—24no—
48C42×C53—22no—
49C42×C53M1614yes—
50C22⋊Dic53—42no—
51C4⋊Dic53—41no—
52C4⋊Dic53—41no—
53C4⋊Dic53—42no—
54C4⋊Dic53—42no—
55C4⋊Dic53—84no—
56C4×Dic53—22no—
57C4×Dic53—22no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53#10112yes—
63C23×C53C2×C22⋊C411yescharacteristic
64C2×C4×C53—28no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—24no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53#9912yes—
73C2×C4×C53C2×M1612yes—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—44no—
79C2×Dic53—42no—
80C2×Dic53—84no—
81#88324—84no—
82#11516—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—22no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53#40711yescharacteristic
91C22×C53C2×#9911yescharacteristic
92C22×C53C2×#10111yescharacteristic
93C4×C53—22no—
94C4×C53—22no—
95C4×C53—44no—
96Dic53—42no—
97Dic53—42no—
98Dic53—82no—
99#2317—531no—
100#1464—532no—
101#1325—534no—
102C2×C53—21no—
103C2×C53—21no—
104C2×C53—42no—
105C2×C53#146311yescharacteristic
106C2×C53#212811yescharacteristic
107C2×C53#213511yescharacteristic
108C2×#117—531no—
109C2×#117#615711yescharacteristic
110C22×C4⋊C4—531no—
111#432—1062no—
112#389—1062no—
113#385—1062no—
114#384—532no—
115#382—532no—
116#324—534no—
117#323—1062no—
118C53#1151611yescharacteristic, complemented, Sylow
119C23×C4—1061no—
120C23×C4C2×#615711yescharacteristic
121#117—24no—
122#117—2122no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—531no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—1064no—
128C2×C4⋊C4—1061no—
129C2×C4⋊C4—2122no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—532no—
132C2×C42—532no—
133C2×C42C2×#615712yes—
134C4⋊C8—2122no—
135C22⋊C8—1064no—
136#97—2122no—
137#95—1064no—
138C24C22⋊#615711yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—532no—
144C22×C4—532no—
145C22×C4—1062no—
146C22×C4—4242no—
147C22×C4—2122no—
148C22×C4C22⋊#615711yescharacteristic
149C22×C4C22×#615711yescharacteristic
150C2×C8—2124no—
151C4⋊C4—24no—
152C4⋊C4—4244no—
153C4⋊C4—2121no—
154C4⋊C4—2121no—
155C4⋊C4—2122no—
156C4⋊C4—2122no—
157C22⋊C4—24no—
158C22⋊C4—2122no—
159C42—22no—
160C42—1062no—
161C42—1062no—
162C42#1171994014yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#1313327812yes—
168C23C2×C22⋊#615711yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—4244no—
180C2×C4—2124no—
181C2×C4—2122no—
182C2×C4—2122no—
183C2×C4—2122no—
184C2×C4#1313327612yes—
185C2×C4C2×#1171994012yes—
186C8—4244no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#1681159311yescharacteristic
195C22C2×#1313327611yescharacteristic
196C22C2×#1313327811yescharacteristic
197C4—22no—
198C4—22no—
199C4—44no—
200C4—4242no—
201C4—2122no—
202C4—2122no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#2028996311yescharacteristic
207C2#2029099611yescharacteristic
208C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11516
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289657C2semidirect
118. C53#11516semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896572
#202773382
#1151653