Finite Group Databasebeta

← #2121×C167 #36320481

Subgroups — #2121×C167 #36320481

Order 21376.

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

normal subgroups only

This group has 280 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#534×C167C211yesmaximal, characteristic
2C2×#100×C167C211yesmaximal, characteristic, complemented
3C2×#99×C167C211yesmaximal, characteristic, complemented
4#352×C167C212yesmaximal, complemented
5AΣL(1,𝔽16)×C167C212yesmaximal, complemented
6C22×D4×C167C411yescharacteristic, complemented
7C22×D4×C167C2211yescharacteristic
8C2×M16×C167—21no—
9#123×C167—21no—
10#123×C167C411yescharacteristic, complemented
11#123×C167C2211yescharacteristic
12#121×C167—21no—
13#121×C167—21no—
14#120×C167—21no—
15#120×C167C411yescharacteristic, complemented
16#120×C167C2211yescharacteristic
17C2×C22⋊C4×C167—21no—
18C2×C22⋊C4×C167C411yescharacteristic, complemented
19#100×C167—21no—
20#100×C167C2212yes—
21#99×C167—21no—
22#99×C167C2212yes—
23C24×C167C2×C411yescharacteristic
24C24×C167C2×C411yescharacteristic, complemented
25C24×C167D411yescharacteristic
26C2×D4×C167—21no—
27C2×D4×C167—21no—
28C2×D4×C167—21no—
29C2×D4×C167—21no—
30C2×D4×C167—41no—
31C2×D4×C167—41no—
32C2×D4×C167—41no—
33C2×D4×C167—41no—
34C2×D4×C167C2×C411yescharacteristic
35C2×D4×C167D411yescharacteristic
36C2×D4×C167D411yescharacteristic
37C2×D4×C167C2311yescharacteristic
38C22×C4×C167—21no—
39C22×C4×C167—41no—
40C22×C4×C167C2×C411yescharacteristic, complemented
41C22×C4×C167D411yescharacteristic
42M16×C167—22no—
43M16×C167—41no—
44C2×C8×C167—41no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167—21no—
47C22⋊C4×C167—21no—
48C22⋊C4×C167—21no—
49C22⋊C4×C167—21no—
50C22⋊C4×C167—21no—
51C22⋊C4×C167—21no—
52C22⋊C4×C167—22no—
53C22⋊C4×C167—41no—
54C22⋊C4×C167C2×C411yescharacteristic, complemented
55C22⋊C4×C167C2×C411yescharacteristic, complemented
56C23×C167—21no—
57C23×C167—21no—
58C23×C167—21no—
59C23×C167—21no—
60C23×C167—21no—
61C23×C167—41no—
62C23×C167—81no—
63C23×C167—81no—
64C23×C167—81no—
65C23×C167C22⋊C411yescharacteristic
66C23×C167C22⋊C411yescharacteristic
67C23×C167C22⋊C411yescharacteristic
68C23×C167C22×C411yescharacteristic
69C23×C167C2×D411yescharacteristic
70D4×C167—41no—
71D4×C167—41no—
72D4×C167—41no—
73D4×C167—41no—
74D4×C167—42no—
75D4×C167—41no—
76D4×C167—41no—
77D4×C167—81no—
78D4×C167—81no—
79C2×C4×C167—21no—
80C2×C4×C167—21no—
81C2×C4×C167—21no—
82C2×C4×C167—21no—
83C2×C4×C167—21no—
84C2×C4×C167—41no—
85C2×C4×C167—42no—
86C2×C4×C167—81no—
87C2×C4×C167—82no—
88C2×C4×C167C22⋊C411yescharacteristic
89C2×C4×C167C2×D411yescharacteristic
90C8×C167—42no—
91C22×C167—21no—
92C22×C167—21no—
93C22×C167—21no—
94C22×C167—21no—
95C22×C167—41no—
96C22×C167—41no—
97C22×C167—41no—
98C22×C167—41no—
99C22×C167—41no—
100C22×C167—42no—
101C22×C167—41no—
102C22×C167—41no—
103C22×C167—41no—
104C22×C167—81no—
105C22×C167—81no—
106C22×C167—81no—
107C22×C167—82no—
108C22×C167#9911yescharacteristic
109C22×C167#9911yescharacteristic
110C22×C167C2×C22⋊C411yescharacteristic
111C4×C167—21no—
112C4×C167—21no—
113C4×C167—41no—
114C4×C167—41no—
115C4×C167—41no—
116C4×C167—82no—
117C2×C167—21no—
118C2×C167—21no—
119C2×C167—21no—
120C2×C167—41no—
121C2×C167—41no—
122C2×C167—41no—
123C2×C167—41no—
124C2×C167—81no—
125C2×C167C2×#9911yescharacteristic
126C167#212111yescharacteristic, direct product factor, Sylow
127#2121C16711yesmaximal, characteristic, direct product factor, Sylow
128#534C2×C16711yescharacteristic
129C2×#100C2×C16711yescharacteristic, complemented
130C2×#99C2×C16711yescharacteristic, complemented
131#352C2×C16712yescomplemented
132AΣL(1,𝔽16)C2×C16712yescomplemented
133C22×D4C4×C16711yescharacteristic, complemented
134C22×D4C22×C16711yescharacteristic
135C2×M16—21no—
136#123—21no—
137#123C4×C16711yescharacteristic, complemented
138#123C22×C16711yescharacteristic
139#121—21no—
140#121—21no—
141#120—21no—
142#120C4×C16711yescharacteristic, complemented
143#120C22×C16711yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4C4×C16711yescharacteristic, complemented
146#100—21no—
147#100C22×C16712yes—
148#99—21no—
149#99C22×C16712yes—
150C24C2×C4×C16711yescharacteristic
151C24C2×C4×C16711yescharacteristic, complemented
152C24D4×C16711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×C4×C16711yescharacteristic
162C2×D4D4×C16711yescharacteristic
163C2×D4D4×C16711yescharacteristic
164C2×D4C23×C16711yescharacteristic
165C22×C4—21no—
166C22×C4—41no—
167C22×C4C2×C4×C16711yescharacteristic, complemented
168C22×C4D4×C16711yescharacteristic
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C22⋊C4C2×C4×C16711yescharacteristic, complemented
182C22⋊C4C2×C4×C16711yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23C22⋊C4×C16711yescharacteristic
193C23C22⋊C4×C16711yescharacteristic
194C23C22⋊C4×C16711yescharacteristic
195C23C22×C4×C16711yescharacteristic
196C23C2×D4×C16711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4C22⋊C4×C16711yescharacteristic
216C2×C4C2×D4×C16711yescharacteristic
217C8—42no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—82no—
235C22#99×C16711yescharacteristic
236C22#99×C16711yescharacteristic
237C22C2×C22⋊C4×C16711yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—82no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2C2×#99×C16711yescharacteristic
subgroupG/H
center, socleC2×C167C2×#99
commutatorC23C22×C4×C167
Frattini, cosocleC2×D4C23×C167
prime radical#534×C167C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C167 ◁ C23×C167 ◁ C22×D4×C167 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#100×C167C2semidirect
3. C2×#99×C167C2semidirect
4. #352×C167C2semidirect
5. AΣL(1,𝔽16)×C167C2semidirect
6. C22×D4×C167C4semidirect
10. #123×C167C4semidirect
15. #120×C167C4semidirect
18. C2×C22⋊C4×C167C4semidirect
24. C24×C167C2×C4semidirect
40. C22×C4×C167C2×C4semidirect
54. C22⋊C4×C167C2×C4semidirect
55. C22⋊C4×C167C2×C4semidirect
126. C167#2121direct (N × H)
127. #2121C167direct (N × H)
129. C2×#100C2×C167semidirect
130. C2×#99C2×C167semidirect
131. #352C2×C167semidirect
132. AΣL(1,𝔽16)C2×C167semidirect
133. C22×D4C4×C167semidirect
137. #123C4×C167semidirect
142. #120C4×C167semidirect
145. C2×C22⋊C4C4×C167semidirect
151. C24C2×C4×C167semidirect
167. C22×C4C2×C4×C167semidirect
181. C22⋊C4C2×C4×C167semidirect
182. C22⋊C4C2×C4×C167semidirect

Maximal subgroups

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

subgroupindex
#534×C1672
C2×#100×C1672
C2×#99×C1672
#352×C1672
AΣL(1,𝔽16)×C1672
#2121167