Finite Group Databasebeta

← #2120×C211 #41009740

Subgroups — #2120×C211 #41009740

Order 27008.

264 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 84 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#525×C211C211yesmaximal, characteristic
2#413×C211C211yesmaximal, characteristic, complemented
3#410×C211C211yesmaximal, characteristic, complemented
4#352×C211C212yesmaximal, complemented
5AΣL(1,𝔽16)×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C23×C4×C211C411yescharacteristic, complemented
8C2×M16×C211—21no—
9#123×C211C411yescharacteristic, complemented
10#123×C211C2211yescharacteristic
11C22⋊Q8×C211—21no—
12#121×C211—21no—
13#120×C211C411yescharacteristic, complemented
14#120×C211C2211yescharacteristic
15C4×D4×C211—21no—
16C4×D4×C211—21no—
17#117×C211—21no—
18#117×C211C411yescharacteristic, complemented
19#101×C211—21no—
20#100×C211C2212yes—
21#99×C211—21no—
22#99×C211C2212yes—
23C24×C211C2×C411yescharacteristic
24#41×C211—41no—
25#41×C211—41no—
26C2×Q8×C211D411yescharacteristic
27C2×D4×C211—21no—
28C2×D4×C211—21no—
29C2×D4×C211C2×C411yescharacteristic
30C2×D4×C211D411yescharacteristic
31C2×D4×C211C2311yescharacteristic
32C22×C4×C211—21no—
33C22×C4×C211—21no—
34C22×C4×C211—41no—
35C22×C4×C211—41no—
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211C2×C411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40M16×C211—22no—
41M16×C211—41no—
42C2×C8×C211—41no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—41no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C22⋊C4×C211—22no—
52C22⋊C4×C211C2×C411yescharacteristic
53C22⋊C4×C211C2×C411yescharacteristic
54C42×C211—21no—
55C42×C211—41no—
56C23×C211—21no—
57C23×C211—41no—
58C23×C211—42no—
59C23×C211C22⋊C411yescharacteristic
60C23×C211C22×C411yescharacteristic
61C23×C211C2×D411yescharacteristic
62Q8×C211—41no—
63D4×C211—41no—
64D4×C211—41no—
65D4×C211—41no—
66D4×C211—42no—
67D4×C211—42no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—21no—
75C2×C4×C211—21no—
76C2×C4×C211—21no—
77C2×C4×C211—42no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—42no—
82C2×C4×C211C22⋊C411yescharacteristic
83C2×C4×C211C22⋊C411yescharacteristic
84C2×C4×C211C22⋊C411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C8×C211—42no—
87C22×C211—22no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—21no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211—41no—
94C22×C211—41no—
95C22×C211—42no—
96C22×C211—42no—
97C22×C211—41no—
98C22×C211C2×C22⋊C411yescharacteristic
99C4×C211—21no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—41no—
103C4×C211—41no—
104C4×C211—41no—
105C4×C211—82no—
106C4×C211#9911yescharacteristic
107C4×C211#9911yescharacteristic
108C2×C211—21no—
109C2×C211—41no—
110C2×C211—41no—
111C2×C211—42no—
112C2×C211—41no—
113C2×C211C2×#9911yescharacteristic
114C211#212011yescharacteristic, direct product factor, Sylow
115#2120C21111yesmaximal, characteristic, direct product factor, Sylow
116#525C2×C21111yescharacteristic
117#413C2×C21111yescharacteristic, complemented
118#410C2×C21111yescharacteristic, complemented
119#352C2×C21112yescomplemented
120AΣL(1,𝔽16)C2×C21112yescomplemented
121C2×#41C22×C21111yescharacteristic
122C23×C4C4×C21111yescharacteristic, complemented
123C2×M16—21no—
124#123C4×C21111yescharacteristic, complemented
125#123C22×C21111yescharacteristic
126C22⋊Q8—21no—
127#121—21no—
128#120C4×C21111yescharacteristic, complemented
129#120C22×C21111yescharacteristic
130C4×D4—21no—
131C4×D4—21no—
132#117—21no—
133#117C4×C21111yescharacteristic, complemented
134#101—21no—
135#100C22×C21112yes—
136#99—21no—
137#99C22×C21112yes—
138C24C2×C4×C21111yescharacteristic
139#41—41no—
140#41—41no—
141C2×Q8D4×C21111yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×C4×C21111yescharacteristic
145C2×D4D4×C21111yescharacteristic
146C2×D4C23×C21111yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C2×C4×C21111yescharacteristic
152C22×C4C2×C4×C21111yescharacteristic
153C22×C4D4×C21111yescharacteristic
154C22×C4D4×C21111yescharacteristic
155M16—22no—
156M16—41no—
157C2×C8—41no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—41no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—22no—
167C22⋊C4C2×C4×C21111yescharacteristic
168C22⋊C4C2×C4×C21111yescharacteristic
169C42—21no—
170C42—41no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23C22⋊C4×C21111yescharacteristic
175C23C22×C4×C21111yescharacteristic
176C23C2×D4×C21111yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4C22⋊C4×C21111yescharacteristic
198C2×C4C22⋊C4×C21111yescharacteristic
199C2×C4C22⋊C4×C21111yescharacteristic
200C2×C4C2×D4×C21111yescharacteristic
201C8—42no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22C2×C22⋊C4×C21111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4#99×C21111yescharacteristic
222C4#99×C21111yescharacteristic
223C2—21no—
224C2—41no—
225C2—42no—
226C2—41no—
227C2—41no—
228C2C2×#99×C21111yescharacteristic
subgroupG/H
centerC4×C211#99
commutatorC23C22×C4×C211
Frattini, cosocleC2×D4C23×C211
socleC2×C211C2×#99
prime radical#525×C211C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ C2×C4×C211 ◁ C2×#41×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #413×C211C2semidirect
3. #410×C211C2semidirect
4. #352×C211C2semidirect
5. AΣL(1,𝔽16)×C211C2semidirect
7. C23×C4×C211C4semidirect
9. #123×C211C4semidirect
13. #120×C211C4semidirect
18. #117×C211C4semidirect
114. C211#2120direct (N × H)
115. #2120C211direct (N × H)
117. #413C2×C211semidirect
118. #410C2×C211semidirect
119. #352C2×C211semidirect
120. AΣL(1,𝔽16)C2×C211semidirect
122. C23×C4C4×C211semidirect
124. #123C4×C211semidirect
128. #120C4×C211semidirect
133. #117C4×C211semidirect

Maximal subgroups

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

subgroupindex
#525×C2112
#413×C2112
#410×C2112
#352×C2112
AΣL(1,𝔽16)×C2112
#2120211