Finite Group Databasebeta

← C22⋊C4×C22⋊C27 #13136173

Subgroups — C22⋊C4×C22⋊C27 #13136173

Order 1728.

439 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C23×C22⋊C27C211yesmaximal, characteristic
2C2×C4×C22⋊C27C212yesmaximal, complemented
3C22×C9×C22⋊C4C311yesmaximal, characteristic
4C22×C22⋊C27—22no—
5C22×C22⋊C27C412yescomplemented
6C22×C22⋊C27C2211yescharacteristic
7C4×C22⋊C27—22no—
8C22⋊C4×C27—41nomaximal
9C25×C9C2×C311yescharacteristic
10C23×C4×C9C2×C312yes—
11C2×C9×C22⋊C4—31no—
12C2×C9×C22⋊C4—31no—
13C2×C9×C22⋊C4—32no—
14C23×C27—41no—
15C2×C22⋊C27—22no—
16C2×C22⋊C27C2×C411yescharacteristic
17C2×C22⋊C27D412yes—
18C2×C4×C27—42no—
19C22×C3×C22⋊C4C911yescharacteristic
20C24×C9—22no—
21C24×C9—31no—
22C24×C9—31no—
23C24×C9—32no—
24C24×C9—62no—
25C24×C9C3×C412yes—
26C24×C9C22×C311yescharacteristic
27C22×C4×C9—22no—
28C22×C4×C9—32no—
29C22×C4×C9—32no—
30C22×C4×C9—62no—
31C9×C22⋊C4—32no—
32C9×C22⋊C4—31no—
33C9×C22⋊C4—32no—
34C9×C22⋊C4A411yescharacteristic
35C22×C27—41no—
36C22×C27—42no—
37C22×C27—82no—
38C22⋊C27C22⋊C411yescharacteristic, direct product factor
39C4×C27—82no—
40C25×C3C2×C911yescharacteristic
41C23×C3×C4C2×C912yes—
42C2×C3×C22⋊C4—31no—
43C2×C3×C22⋊C4—31no—
44C2×C3×C22⋊C4—32no—
45C23×C9—22no—
46C23×C9—32no—
47C23×C9—31no—
48C23×C9—31no—
49C23×C9—32no—
50C23×C9—32no—
51C23×C9—32no—
52C23×C9—32no—
53C23×C9—31no—
54C23×C9—62no—
55C23×C9—62no—
56C23×C9—62no—
57C23×C9—64no—
58C23×C9—62no—
59C23×C9—62no—
60C23×C9—64no—
61C23×C9C2×C3×C411yescharacteristic
62C23×C9C3×D412yes—
63C23×C9C2×A411yescharacteristic
64C2×C4×C9—32no—
65C2×C4×C9—62no—
66C2×C4×C9—62no—
67C2×C4×C9—62no—
68C2×C4×C9—62no—
69C2×C4×C9C2×A412yes—
70C22×C22⋊C4C2711yescharacteristic, complemented, Sylow
71C2×C27—41no—
72C2×C27—42no—
73C2×C27—82no—
74C24×C3—22no—
75C24×C3—32no—
76C24×C3—31no—
77C24×C3—31no—
78C24×C3—62no—
79C24×C3C4×C912yes—
80C24×C3C22×C911yescharacteristic
81C22×C3×C4—22no—
82C22×C3×C4—32no—
83C22×C3×C4—32no—
84C22×C3×C4—62no—
85C3×C22⋊C4—32no—
86C3×C22⋊C4—31no—
87C3×C22⋊C4—32no—
88C3×C22⋊C4C22⋊C911yescharacteristic
89C22×C9—22no—
90C22×C9—31no—
91C22×C9—32no—
92C22×C9—32no—
93C22×C9—31no—
94C22×C9—31no—
95C22×C9—32no—
96C22×C9—32no—
97C22×C9—32no—
98C22×C9—62no—
99C22×C9—64no—
100C22×C9—62no—
101C22×C9—62no—
102C22×C9—62no—
103C22×C9—64no—
104C22×C9—62no—
105C22×C9C3×C22⋊C411yescharacteristic
106C22×C9C4×A412yes—
107C22×C9C22×A411yescharacteristic
108C4×C9—22no—
109C4×C9—62no—
110C25C2×C2711yescharacteristic
111C23×C4C2×C2712yescomplemented
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—32no—
115C27—41noSylow
116C23×C3—22no—
117C23×C3—32no—
118C23×C3—31no—
119C23×C3—31no—
120C23×C3—32no—
121C23×C3—32no—
122C23×C3—32no—
123C23×C3—32no—
124C23×C3—31no—
125C23×C3—62no—
126C23×C3—62no—
127C23×C3—62no—
128C23×C3—64no—
129C23×C3—62no—
130C23×C3—62no—
131C23×C3—64no—
132C23×C3C2×C4×C911yescharacteristic
133C23×C3D4×C912yes—
134C23×C3C2×C22⋊C911yescharacteristic
135C2×C3×C4—32no—
136C2×C3×C4—62no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—62no—
140C2×C3×C4C2×C22⋊C912yes—
141C2×C9—22no—
142C2×C9—31no—
143C2×C9—32no—
144C2×C9—31no—
145C2×C9—62no—
146C2×C9C2×C4×A411yescharacteristic
147C2×C9D4×A412yes—
148C24—22no—
149C24—31no—
150C24—32no—
151C24—31no—
152C24—62no—
153C24C4×C2712yescomplemented
154C24C22×C2711yescharacteristic
155C22×C4—22no—
156C22×C4—32no—
157C22×C4—32no—
158C22×C4—62no—
159C22⋊C4—32no—
160C22⋊C4—31no—
161C22⋊C4—32no—
162C22⋊C4C22⋊C2711yescharacteristic, direct product factor
163C22×C3—22no—
164C22×C3—31no—
165C22×C3—32no—
166C22×C3—32no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—32no—
170C22×C3—32no—
171C22×C3—32no—
172C22×C3—62no—
173C22×C3—64no—
174C22×C3—62no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3—64no—
178C22×C3—62no—
179C22×C3C9×C22⋊C411yescharacteristic
180C22×C3C4×C22⋊C912yes—
181C22×C3C22×C22⋊C911yescharacteristic
182C3×C4—22no—
183C3×C4—62no—
184C9A4×C22⋊C411yescharacteristic
185C23—22no—
186C23—32no—
187C23—31no—
188C23—31no—
189C23—32no—
190C23—32no—
191C23—31no—
192C23—32no—
193C23—32no—
194C23—62no—
195C23—62no—
196C23—62no—
197C23—64no—
198C23—62no—
199C23—62no—
200C23—64no—
201C23C2×C4×C2711yescharacteristic
202C23D4×C2712yes—
203C23C2×C22⋊C2711yescharacteristic
204C2×C4—32no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4C2×C22⋊C2712yescomplemented
210C2×C3—22no—
211C2×C3—31no—
212C2×C3—31no—
213C2×C3—32no—
214C2×C3—62no—
215C2×C3C2×C4×C22⋊C911yescharacteristic
216C2×C3D4×C22⋊C912yes—
217C22—22no—
218C22—31no—
219C22—32no—
220C22—32no—
221C22—31no—
222C22—31no—
223C22—32no—
224C22—32no—
225C22—32no—
226C22—62no—
227C22—62no—
228C22—64no—
229C22—62no—
230C22—62no—
231C22—64no—
232C22—62no—
233C22C22⋊C4×C2711yescharacteristic, complemented
234C22C4×C22⋊C2712yescomplemented
235C22C22×C22⋊C2711yescharacteristic
236C4—22no—
237C4—62no—
238C3C22⋊C4×C22⋊C911yescharacteristic
239C2—22no—
240C2—31no—
241C2—31no—
242C2—32no—
243C2—62no—
244C2C2×C4×C22⋊C2711yescharacteristic
245C2D4×C22⋊C2712yes—
subgroupG/H
center, FrattiniC22×C9C22×A4
commutatorC23C2×C4×C27
FittingC22×C9×C22⋊C4C3
socleC24×C3C22×C9
prime radicalC25×C3C2×C9
cosocleC24×C9C22×C3
hypercenterC9×C22⋊C4A4
hypocenterC22C22⋊C4×C27
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C22×C9 ◁ C9×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊C27C2semidirect
5. C22×C22⋊C27C4semidirect
38. C22⋊C27C22⋊C4direct (N × H)
70. C22×C22⋊C4C27semidirect
111. C23×C4C2×C27semidirect
153. C24C4×C27semidirect
162. C22⋊C4C22⋊C27direct (N × H)
209. C2×C4C2×C22⋊C27semidirect
233. C22C22⋊C4×C27semidirect
234. C22C4×C22⋊C27semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C272
C2×C4×C22⋊C272
C22×C9×C22⋊C43
C22⋊C4×C274