Finite Group Databasebeta

← #13074554

Subgroups — #13074554

Order 1408.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66997C211yesmaximal, characteristic, complemented
2C2×#66938C211yesmaximal, characteristic
3C11×#393C211yesmaximal, characteristic, complemented
4#10622844C211yesmaximal, characteristic, complemented
5#10622570C212yesmaximal
6#10622565C212yesmaximal, complemented
7#10622560C211yesmaximal, characteristic, complemented
8#10622288C212yesmaximal, complemented
9#10622286C211yesmaximal, characteristic, complemented
10#10622261C211yesmaximal, characteristic, complemented
11#10622259C212yesmaximal, complemented
12C22×Q8×C11C2211yescharacteristic
13C22×C22⋊D11C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C22×Dic22C2211yescharacteristic
16C2×C11×C22⋊C4C2211yescharacteristic
17C2×C11×C22⋊C4C2212yes—
18C2×C22⋊Dic11C2211yescharacteristic
19C2×C22⋊Dic11C2212yes—
20#66997—24no—
21C2×(C2×D11)⋊C4C2211yescharacteristic
22C2×(C2×D11)⋊C4C2211yescharacteristic
23C2×(C2×D11)⋊C4C2212yes—
24C2×(C2×D11)⋊C4C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×Dic11⋊C4C2212yes—
27C2×C4×Dic11C2212yes—
28C2×C4×Dic11C2211yescharacteristic
29#66938—24no—
30C11×#95C2212yes—
31C11×#95C2211yescharacteristic
32C11⋊#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2211yescharacteristic
37C24×C11C2311yescharacteristic
38C23×D11C2311yescharacteristic
39C2×Q8×C11—22no—
40C2×Q8×C11—42no—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×C4×C11C2312yes—
44C22×C4×C11C2311yescharacteristic
45C2×C22⋊D11—22no—
46C2×C22⋊D11—24no—
47C22×Dic11C2312yes—
48C22×Dic11C2311yescharacteristic
49C22×Dic11C2312yes—
50C22×Dic11C2311yescharacteristic
51C22×Dic11C2311yescharacteristic
52C2×Dic22—22no—
53C2×Dic22—24no—
54C2×Dic22—22no—
55C2×Dic22—42no—
56C11×C22⋊C4—22no—
57C11×C22⋊C4—42no—
58C22⋊Dic11—22no—
59C22⋊Dic11—42no—
60(C2×D11)⋊C4—22no—
61(C2×D11)⋊C4—22no—
62(C2×D11)⋊C4—42no—
63(C2×D11)⋊C4—42no—
64Dic11⋊C4—24no—
65Dic11⋊C4—24no—
66C4×Dic11—24no—
67C4×Dic11D414yes—
68#2640—111nomaximal, Sylow
69C23×C11—21no—
70C23×C11—22no—
71C23×C11—22no—
72C23×C11—21no—
73C23×C11—21no—
74C23×C11C2411yescharacteristic
75C22×D11—21no—
76C22×D11—22no—
77C22×D11—22no—
78C22×D11—21no—
79C22×D11—21no—
80Q8×C11—44no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11—21no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11—22no—
88C2×C4×C11—22no—
89C2×C4×C11—22no—
90C2×C4×C11C2×D412yes—
91C22⋊D11—44no—
92C2×Dic11—21no—
93C2×Dic11—21no—
94C2×Dic11—22no—
95C2×Dic11—21no—
96C2×Dic11—22no—
97C2×Dic11—22no—
98C2×Dic11—22no—
99C2×Dic11—22no—
100C2×Dic11—22no—
101C2×Dic11—22no—
102C2×Dic11C2×D412yes—
103C2×Dic11C2×D412yes—
104C2×Dic11#4114yes—
105Dic22—44no—
106Dic22—44no—
107C2×C4⋊Q8—111no—
108C2×#124—111no—
109#397—112no—
110#397—112no—
111#393—111no—
112#393—111no—
113#393—111no—
114#393D1111yescharacteristic, complemented
115#391—112no—
116#390—112no—
117#376—111no—
118C22×C11—41no—
119C22×C11—41no—
120C22×C11—41no—
121C22×C11—42no—
122C22×C11—42no—
123C22×C11C22×D411yescharacteristic
124C22×C11C2×#4112yes—
125C22×C11H(5,𝔽2)12yes—
126C22×C112−1+411yescharacteristic
127C22×C112−1+411yescharacteristic
128C2×D11—41no—
129C2×D11—41no—
130C2×D11—41no—
131C2×D11—42no—
132C2×D11—42no—
133C4×C11—22no—
134C4×C11—41no—
135C4×C11—42no—
136C4×C11—42no—
137Dic11—22no—
138Dic11—41no—
139Dic11—42no—
140Dic11—42no—
141Dic11#12414yes—
142C22×Q8—111no—
143C22×Q8—111no—
144C22×Q8C2×D1111yescharacteristic
145C22×D4—111no—
146C4⋊Q8—224no—
147#124—224no—
148C2×C4⋊C4—112no—
149C2×C4⋊C4—112no—
150C2×C22⋊C4—111no—
151C2×C22⋊C4—111no—
152C2×C22⋊C4—111no—
153C2×C22⋊C4—112no—
154C2×C22⋊C4—112no—
155C2×C22⋊C4—112no—
156C2×C22⋊C4C2×D1111yescharacteristic
157C2×C22⋊C4C2×D1112yes—
158C2×C42—111no—
159C2×C42—112no—
160#95—111no—
161#95—112no—
162#95—112no—
163#95—112no—
164#95—112no—
165#95C2×D1111yescharacteristic
166#95C2×D1112yes—
167C2×C11—81no—
168C2×C11C2×#12411yescharacteristic
169C2×C11#53511yescharacteristic
170C2×C11#53611yescharacteristic
171C2×C11#54212yes—
172C2×C11#54212yes—
173D11—81no—
174C24—111no—
175C24C22×D1111yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—222no—
179C2×Q8—224no—
180C2×Q8—222no—
181C2×Q8—442no—
182C2×D4—222no—
183C2×D4—224no—
184C22×C4—111no—
185C22×C4—111no—
186C22×C4—112no—
187C22×C4—111no—
188C22×C4—112no—
189C22×C4C22×D1112yes—
190C22×C4C22×D1111yescharacteristic
191C22×C4C22×D1112yes—
192C22×C4C22×D1111yescharacteristic
193C4⋊C4—224no—
194C4⋊C4—224no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—222no—
198C22⋊C4—222no—
199C22⋊C4—222no—
200C22⋊C4—442no—
201C22⋊C4—442no—
202C22⋊C4—442no—
203C42—114no—
204C42—224no—
205C11#264011yescharacteristic, complemented, Sylow
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—221no—
212C23—222no—
213C23—221no—
214C23—222no—
215C23—221no—
216C23C23×D1111yescharacteristic
217Q8—44no—
218Q8—444no—
219Q8—444no—
220D4—444no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—112no—
231C2×C4—114no—
232C2×C4—112no—
233C2×C4—221no—
234C2×C4—221no—
235C2×C4—221no—
236C2×C4—222no—
237C2×C4—222no—
238C2×C4—222no—
239C2×C4—222no—
240C2×C4—222no—
241C2×C4—222no—
242C2×C4—222no—
243C2×C4D4×D1112yes—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—441no—
250C22—441no—
251C22—442no—
252C22—442no—
253C22—441no—
254C22C2×D4×D1111yescharacteristic
255C22#6703712yes—
256C22#6704011yescharacteristic
257C22#41×D1112yes—
258C22#6704311yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—114no—
264C4—222no—
265C4—442no—
266C4—442no—
267C4—441no—
268C2—81no—
269C2—881no—
270C2#1062312311yescharacteristic
271C2#1062315911yescharacteristic
272C2#1062317112yes—
273C2#1062320012yes—
274C2D11×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#393C2
prime radicalC22×C22⋊D11C22
hypercenter#393D11
hypocenterC11#2640
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#66997C2semidirect
3. C11×#393C2semidirect
4. #10622844C2semidirect
6. #10622565C2semidirect
7. #10622560C2semidirect
8. #10622288C2semidirect
9. #10622286C2semidirect
10. #10622261C2semidirect
11. #10622259C2semidirect
114. #393D11semidirect
205. C11#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#669972
C2×#669382
C11×#3932
#106228442
#106225702
#106225652
#106225602
#106222882
#106222862
#106222612
#106222592
#264011