Finite Group Databasebeta

← #13074482

Subgroups — #13074482

Order 1408.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#67000C211yesmaximal, characteristic
2C2×#66940C211yesmaximal, characteristic, complemented
3C11×#395C211yesmaximal, characteristic, complemented
4#10622842C211yesmaximal, characteristic, complemented
5#10622605C212yesmaximal, complemented
6#10622604C211yesmaximal, characteristic
7#10622603C212yesmaximal, complemented
8#10622286C212yesmaximal, complemented
9#10622285C211yesmaximal, characteristic
10#10622282C212yesmaximal, complemented
11#10622264C211yesmaximal, characteristic, complemented
12C22×Q8×C11C2211yescharacteristic
13C22×D44C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C4⋊C4C2212yes—
17#67000—24no—
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×(C2×D11)⋊C4C2212yes—
22C2×(C2×D11)⋊C4C2212yes—
23C2×(C2×D11)⋊C4C2212yes—
24C2×Dic11⋊C4C2211yescharacteristic
25C2×Dic11⋊C4C2211yescharacteristic
26C2×Dic11⋊C4C2212yes—
27C2×C4×Dic11C2211yescharacteristic
28C2×C4×Dic11C2211yescharacteristic
29C2×C4×Dic11C2211yescharacteristic
30#66940—24no—
31C11×#95C2211yescharacteristic
32C11×#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2211yescharacteristic
36C11⋊#95C2212yes—
37C23×D11C2312yes—
38C2×Q8×C11—22no—
39C2×Q8×C11—42no—
40C22×C4×C11C2312yes—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×C4×C11C2311yescharacteristic
44C22×C4×C11C2311yescharacteristic
45C22×Dic11C2312yes—
46C22×Dic11C2311yescharacteristic
47C22×Dic11C2311yescharacteristic
48C22×Dic11C2311yescharacteristic
49C22×Dic11C2311yescharacteristic
50C2×D44—22no—
51C2×D44—42no—
52C2×Dic22—22no—
53C2×Dic22—22no—
54C2×Dic22—22no—
55C11×C4⋊C4—22no—
56C11×C4⋊C4—42no—
57(C2×D11)⋊C4—24no—
58(C2×D11)⋊C4—24no—
59(C2×D11)⋊C4—24no—
60(C2×D11)⋊C4—24no—
61(C2×D11)⋊C4—42no—
62(C2×D11)⋊C4—42no—
63Dic11⋊C4—22no—
64Dic11⋊C4—22no—
65Dic11⋊C4—42no—
66C4×Dic11—22no—
67C4×Dic11—22no—
68C4×Dic11D414yes—
69#2632—111nomaximal, Sylow
70C23×C11C2411yescharacteristic
71C22×D11—22no—
72C22×D11—22no—
73C22×D11—22no—
74C22×D11—22no—
75C22×D11—22no—
76C22×D11—22no—
77C22×D11—22no—
78Q8×C11—44no—
79C2×C4×C11—21no—
80C2×C4×C11—21no—
81C2×C4×C11—22no—
82C2×C4×C11—21no—
83C2×C4×C11—22no—
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—
91C2×C4×C11#4112yes—
92C2×Dic11—21no—
93C2×Dic11—21no—
94C2×Dic11—21no—
95C2×Dic11—21no—
96C2×Dic11—22no—
97C2×Dic11—21no—
98C2×Dic11—22no—
99C2×Dic11—22no—
100C2×Dic11—22no—
101C2×Dic11—22no—
102C2×Dic11C2×D412yes—
103C2×Dic11C2×D412yes—
104C2×Dic11#4112yes—
105D44—44no—
106Dic22—28no—
107C2×#124—111no—
108C2×C4×Q8—111no—
109#399—112no—
110#395—111no—
111#395D1111yescharacteristic, complemented
112#394—111no—
113#394—111no—
114#393—112no—
115#388—112no—
116#388—112no—
117#383—111no—
118C22×C11C22×D411yescharacteristic
119C22×C11C2×#4111yescharacteristic
120C22×C11C2×#4111yescharacteristic
121C22×C11H(5,𝔽2)12yes—
122C22×C112−1+412yes—
123C2×D11—42no—
124C2×D11—42no—
125C2×D11—42no—
126C2×D11—42no—
127C2×D11—42no—
128C2×D11—42no—
129C2×D11—42no—
130C4×C11—22no—
131C4×C11—22no—
132C4×C11—41no—
133C4×C11—42no—
134C4×C11—42no—
135Dic11—22no—
136Dic11—22no—
137Dic11—22no—
138Dic11—41no—
139Dic11—42no—
140C22×Q8—111no—
141C22×Q8C2×D1111yescharacteristic
142C22×D4—111no—
143#124—224no—
144C4×Q8—224no—
145C2×C4⋊C4—111no—
146C2×C4⋊C4—111no—
147C2×C4⋊C4—112no—
148C2×C4⋊C4C2×D1111yescharacteristic
149C2×C4⋊C4C2×D1112yes—
150C2×C22⋊C4—112no—
151C2×C22⋊C4—112no—
152C2×C22⋊C4—112no—
153C2×C22⋊C4—112no—
154C2×C22⋊C4—112no—
155C2×C22⋊C4—112no—
156C2×C42—111no—
157C2×C42—111no—
158C2×C42—111no—
159#95—112no—
160#95—112no—
161#95—111no—
162#95—112no—
163#95C2×D1111yescharacteristic
164#95C2×D1112yes—
165C2×C11#52511yescharacteristic
166C2×C11#53511yescharacteristic
167C2×C11#53611yescharacteristic
168C2×C11#54212yes—
169C2×C11#54212yes—
170D11—82no—
171C24—112no—
172C2×Q8—22no—
173C2×Q8—42no—
174C2×Q8—222no—
175C2×Q8—222no—
176C2×Q8—222no—
177C2×D4—222no—
178C2×D4—442no—
179C22×C4—111no—
180C22×C4—111no—
181C22×C4—111no—
182C22×C4—112no—
183C22×C4—111no—
184C22×C4C22×D1112yes—
185C22×C4C22×D1112yes—
186C22×C4C22×D1111yescharacteristic
187C22×C4C22×D1111yescharacteristic
188C22×C4C22×D1111yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—222no—
192C4⋊C4—222no—
193C4⋊C4—442no—
194C22⋊C4—224no—
195C22⋊C4—224no—
196C22⋊C4—224no—
197C22⋊C4—224no—
198C22⋊C4—442no—
199C22⋊C4—442no—
200C42—114no—
201C42—222no—
202C42—222no—
203C11#263211yescharacteristic, complemented, Sylow
204C23—222no—
205C23—222no—
206C23—222no—
207C23—222no—
208C23—222no—
209C23—222no—
210C23—222no—
211C23C23×D1111yescharacteristic
212Q8—44no—
213Q8—228no—
214D4—444no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—112no—
227C2×C4—112no—
228C2×C4—112no—
229C2×C4—221no—
230C2×C4—221no—
231C2×C4—221no—
232C2×C4—222no—
233C2×C4—221no—
234C2×C4—221no—
235C2×C4—222no—
236C2×C4—222no—
237C2×C4—222no—
238C2×C4—222no—
239C2×C4D4×D1112yes—
240C2×C4#434312yes—
241C22—442no—
242C22—442no—
243C22—442no—
244C22—442no—
245C22—442no—
246C22—442no—
247C22—442no—
248C22C2×D4×D1111yescharacteristic
249C22#6703712yes—
250C22C2×#434311yescharacteristic
251C22#6704012yes—
252C22#41×D1111yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—222no—
259C4—222no—
260C4—222no—
261C4—442no—
262C4—441no—
263C2—882no—
264C2#1062312311yescharacteristic
265C2#1062315811yescharacteristic
266C2#1062317112yes—
267C2#1062325511yescharacteristic
268C2#1062326412yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#395C2
prime radicalC22×D44C22
hypercenter#395D11
hypocenterC11#2632
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#66940C2semidirect
3. C11×#395C2semidirect
4. #10622842C2semidirect
5. #10622605C2semidirect
7. #10622603C2semidirect
8. #10622286C2semidirect
10. #10622282C2semidirect
11. #10622264C2semidirect
111. #395D11semidirect
203. C11#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#670002
C2×#669402
C11×#3952
#106228422
#106226052
#106226042
#106226032
#106222862
#106222852
#106222822
#106222642
#263211