Finite Group Databasebeta

← C2×#20289851 #28312445

Subgroups — C2×#20289851 #28312445

Order 13568.

450 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×#16811534C212yesmaximal, complemented
2#20289851C214yesmaximal, direct product factor
3C2×#16810996C211yesmaximal, characteristic
4C2×#13133278—22no—
5#16811534—22no—
6#16811534C2214yescomplemented
7C22×#11719913—22no—
8C2×#13133217C2211yescharacteristic
9C2×#13133214C411yescharacteristic
10#16810996—22no—
11#16810996C412yes—
12#16810996C2212yes—
13C2×C4×C8×C53C411yescharacteristic
14C2×#11719940—42no—
15#13133278—24no—
16#13133278—42no—
17C22×Dic106D412yes—
18C22×D212—21no—
19C22×D212—41no—
20C2×#11719913—24no—
21C2×#11719913—24no—
22C2×#11719913—44no—
23C2×#11719913—44no—
24#13133217—41no—
25#13133217C2×C411yescharacteristic
26#13133217D412yes—
27#13133217C2311yescharacteristic
28#13133214—21no—
29#13133214—41no—
30#13133214C2×C412yes—
31C2×C4⋊Dic53—41no—
32C22×C8×C53—21no—
33C2×C42×C53C2×C411yescharacteristic
34C4×C8×C53—21no—
35C4×C8×C53C2×C412yes—
36C23×D53—41no—
37C22×C4×C53—21no—
38C22×C4×C53C22⋊C411yescharacteristic
39C22×Dic53—22no—
40C2×D212—22no—
41C2×D212—42no—
42C2×D212—42no—
43C2×D212—42no—
44C2×D212—42no—
45C2×D212—41no—
46C2×D212—84no—
47C2×Dic106—22no—
48C2×Dic106—42no—
49C2×Dic106—42no—
50C2×Dic106C22⋊C412yes—
51C2×Dic106C2×D412yes—
52#11719940—44no—
53#11719940—82no—
54C2×#88324—82no—
55C2×C8×C53—22no—
56C2×C8×C53—22no—
57C2×C8×C53—22no—
58C42×C53—21no—
59C42×C53C22⋊C411yescharacteristic
60C42×C53C22×C411yescharacteristic
61C4⋊Dic53—41no—
62C4⋊Dic53—42no—
63C4⋊Dic53—41no—
64#11719913—48no—
65#11719913—48no—
66C23×C53#9911yescharacteristic
67C22×D53—42no—
68C22×D53—84no—
69C22×D53—81no—
70C22×D53—81no—
71C2×C4×C53—24no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53#9911yescharacteristic
77C2×C4×C53C2×C22⋊C411yescharacteristic
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82D212—44no—
83D212—82no—
84D212—84no—
85D212—84no—
86Dic106—42no—
87Dic106—42no—
88Dic106—82no—
89#88324—84no—
90C8×C53—24no—
91C2×#1410—531nomaximal, Sylow
92C22×C53—22no—
93C22×C53#35311yescharacteristic
94C22×C53#35311yescharacteristic
95C22×C53C2×#9911yescharacteristic
96C2×D53—84no—
97C2×D53—82no—
98C2×D53—162no—
99C2×D53—162no—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—21no—
103C4×C53—21no—
104Dic53—42no—
105Dic53—42no—
106C2×#492—531no—
107C2×#356—532no—
108#1410—534no—
109C2×C53—21no—
110C2×C53—21no—
111C2×C53#141012yescomplemented
112C2×C53C2×#35311yescharacteristic
113D53—162no—
114C22×SD16—1062no—
115C2×C4⋊Q8—531no—
116C2×#127—531no—
117#492—532no—
118#492—532no—
119#492—1062no—
120C2×#101—1062no—
121C2×C4×C8#615711yescharacteristic
122#356—534no—
123#356—1062no—
124C53C2×#141011yescharacteristic, complemented, Sylow
125C22×Q8—532no—
126C22×D4—1061no—
127C22×D4—2121no—
128C2×SD16—1064no—
129C2×SD16—1064no—
130C2×SD16—2124no—
131C2×SD16—2124no—
132C2×M16—2122no—
133C22×C8—21no—
134C4⋊Q8—532no—
135C4⋊Q8—531no—
136C4⋊Q8—531no—
137C4⋊Q8—2121no—
138#127—532no—
139#127—1061no—
140#127—2121no—
141C2×C4⋊C4—2121no—
142C2×C42C2×#615711yescharacteristic
143#101—1064no—
144#101—2122no—
145C4×C8—21no—
146C4×C8C2×#615712yes—
147C24—2121no—
148C2×Q8—532no—
149C2×Q8—532no—
150C2×Q8—1062no—
151C2×Q8—2122no—
152C2×Q8—2122no—
153C2×D4—1062no—
154C2×D4—4244no—
155C2×D4—2122no—
156C2×D4—2122no—
157C2×D4—2122no—
158C2×D4—2122no—
159C2×D4—2121no—
160C22×C4—21no—
161C22×C4—1062no—
162C22×C4C22⋊#615711yescharacteristic
163SD16—2128no—
164SD16—2128no—
165M16—4242no—
166M16—2124no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—4242no—
171C4⋊C4—2122no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C42—21no—
175C42C22⋊#615711yescharacteristic
176C42C22×#615711yescharacteristic
177C23—4244no—
178C23—4241no—
179C23—4241no—
180C23—2122no—
181C23#1313327611yescharacteristic
182Q8—4242no—
183Q8—2122no—
184Q8—2122no—
185D4—4244no—
186D4—4244no—
187D4—4242no—
188D4—2124no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1313327611yescharacteristic
199C2×C4C2×C22⋊#615711yescharacteristic
200C8—24no—
201C8—4244no—
202C22—22no—
203C22—8482no—
204C22—8482no—
205C22—4244no—
206C22—4242no—
207C22#1681152811yescharacteristic
208C22#1681152811yescharacteristic
209C22C2×#1313327611yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—21no—
214C4—2122no—
215C4—2122no—
216C2—21no—
217C2—21no—
218C2—8482no—
219C2#2028985112yesdirect product factor
220C2C2×#1681152811yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#353
prime radicalC2×#13133214C4
cosocle#13133217C23
hypercenterC2×C4×C8#6157
hypocenterC53C2×#1410
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#16811534C2semidirect
2. #20289851C2direct (N × H)
6. #16811534C22semidirect
111. C2×C53#1410semidirect
124. C53C2×#1410semidirect
219. C2#20289851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115342
#202898512
C2×#168109962
C2×#141053