Finite Group Databasebeta

← C2×#20272882 #27304417

Subgroups — C2×#20272882 #27304417

Order 13568.

450 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16810422C212yesmaximal, complemented
2#20272882C214yesmaximal, direct product factor
3C2×C53×#492C211yesmaximal, characteristic
4C2×#13133136—22no—
5#16810422—22no—
6#16810422C2214yescomplemented
7C22×SD16×C53—22no—
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×#127×C53C411yescharacteristic
10C53×#492—22no—
11C53×#492C412yes—
12C53×#492C2212yes—
13C2×C4×C8×C53C411yescharacteristic
14C2×#11719917—42no—
15#13133136—24no—
16#13133136—42no—
17C22×Q8×C53D412yes—
18C22×D4×C53—21no—
19C22×D4×C53—41no—
20C2×SD16×C53—24no—
21C2×SD16×C53—24no—
22C2×SD16×C53—44no—
23C2×SD16×C53—44no—
24C22×C8×C53—21no—
25C4⋊Q8×C53—41no—
26C4⋊Q8×C53C2×C411yescharacteristic
27C4⋊Q8×C53D412yes—
28C4⋊Q8×C53C2311yescharacteristic
29#127×C53—21no—
30#127×C53—41no—
31#127×C53C2×C412yes—
32C2×C4⋊C4×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
34C4×C8×C53—21no—
35C4×C8×C53C2×C412yes—
36C24×C53—41no—
37C2×Q8×C53—22no—
38C2×Q8×C53—42no—
39C2×Q8×C53—42no—
40C2×Q8×C53C22⋊C412yes—
41C2×Q8×C53C2×D412yes—
42C2×D4×C53—22no—
43C2×D4×C53—42no—
44C2×D4×C53—42no—
45C2×D4×C53—42no—
46C2×D4×C53—42no—
47C2×D4×C53—41no—
48C2×D4×C53—84no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53C22⋊C411yescharacteristic
52SD16×C53—48no—
53SD16×C53—48no—
54C2×C8×C53—22no—
55C2×C8×C53—22no—
56C2×C8×C53—22no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—41no—
60C42×C53—21no—
61C42×C53C22⋊C411yescharacteristic
62C42×C53C22×C411yescharacteristic
63#11719917—44no—
64#11719917—82no—
65C2×#88322—82no—
66C23×C53—42no—
67C23×C53—84no—
68C23×C53—81no—
69C23×C53—81no—
70C23×C53#9911yescharacteristic
71Q8×C53—42no—
72Q8×C53—42no—
73Q8×C53—82no—
74D4×C53—44no—
75D4×C53—82no—
76D4×C53—84no—
77D4×C53—84no—
78C2×C4×C53—24no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53#9911yescharacteristic
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C8×C53—24no—
90#88322—84no—
91C2×#1410—531nomaximal, Sylow
92C22×C53—22no—
93C22×C53—84no—
94C22×C53—82no—
95C22×C53—162no—
96C22×C53—162no—
97C22×C53#35311yescharacteristic
98C22×C53#35311yescharacteristic
99C22×C53C2×#9911yescharacteristic
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—42no—
106C2×#492D5311yescharacteristic
107C2×#356—532no—
108#1410—534no—
109C2×C53—21no—
110C2×C53—21no—
111C2×C53—162no—
112C2×C53#141012yescomplemented
113C2×C53C2×#35311yescharacteristic
114C22×SD16—22no—
115C2×C4⋊Q8C2×D5311yescharacteristic
116C2×#127Dic5311yescharacteristic
117#492—22no—
118#492Dic5312yes—
119#492C2×D5312yes—
120C2×#101—1062no—
121C2×C4×C8Dic5311yescharacteristic
122#356—534no—
123#356—1062no—
124C53C2×#141011yescharacteristic, complemented, Sylow
125C22×Q8C22⋊D5312yes—
126C22×D4—21no—
127C22×D4—41no—
128C2×SD16—24no—
129C2×SD16—24no—
130C2×SD16—44no—
131C2×SD16—44no—
132C2×M16—2122no—
133C22×C8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×Dic5311yescharacteristic
136C4⋊Q8C22⋊D5312yes—
137C4⋊Q8C22×D5311yescharacteristic
138#127—21no—
139#127—41no—
140#127C2×Dic5312yes—
141C2×C4⋊C4—41no—
142C2×C42C2×Dic5311yescharacteristic
143#101—1064no—
144#101—2122no—
145C4×C8—21no—
146C4×C8C2×Dic5312yes—
147C24—41no—
148C2×Q8—22no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8C22⋊Dic5312yes—
152C2×Q8C2×C22⋊D5312yes—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—84no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4C22⋊Dic5311yescharacteristic
163SD16—48no—
164SD16—48no—
165M16—4242no—
166M16—2124no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—4242no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C42—21no—
175C42C22⋊Dic5311yescharacteristic
176C42C22×Dic5311yescharacteristic
177C23—42no—
178C23—84no—
179C23—81no—
180C23—81no—
181C23#1313313211yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—82no—
185D4—44no—
186D4—82no—
187D4—84no—
188D4—84no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4#1313313211yescharacteristic
199C2×C4C2×C22⋊Dic5311yescharacteristic
200C8—24no—
201C8—4244no—
202C22—22no—
203C22—84no—
204C22—82no—
205C22—162no—
206C22—162no—
207C22#1681041611yescharacteristic
208C22#1681041611yescharacteristic
209C22C2×#1313313211yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C2—21no—
217C2—21no—
218C2—162no—
219C2#2027288212yesdirect product factor
220C2C2×#1681041611yescharacteristic
subgroupG/H
centerC22#16810416
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC2×C53×#492C2
socleC22×C53#353
prime radicalC2×#127×C53C4
cosocleC4⋊Q8×C53C23
hypercenterC2×#492D53
hypocenterC53C2×#1410
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C2×#16810422C2semidirect
2. #20272882C2direct (N × H)
6. #16810422C22semidirect
112. C2×C53#1410semidirect
124. C53C2×#1410semidirect
219. C2#20272882direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168104222
#202728822
C2×C53×#4922
C2×#141053