Finite Group Databasebeta

← #13073589

Subgroups — #13073589

Order 1408.

704 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 157 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, complemented
2C2×#67000C211yesmaximal, characteristic, complemented
3C2×#66946C211yesmaximal, characteristic, complemented
4C2×#66927C211yesmaximal, characteristic
5C11×#391C211yesmaximal, characteristic, complemented
6#10622601C211yesmaximal, characteristic, complemented
7C4×Dic11⋊C4C211yesmaximal, characteristic, complemented
8#10622285C214yesmaximal, complemented
9#10622278C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C22×D44C2211yescharacteristic
12C22×D44C2214yes—
13C22×D44C2211yescharacteristic
14C22×C4×D11C2212yes—
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C42×C11C2211yescharacteristic
17#67000—24no—
18#67000—24no—
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×(C2×D11)⋊C4C2214yes—
22C2×(C2×D11)⋊C4C2212yes—
23C2×(C2×D11)⋊C4C2214yes—
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11C2211yescharacteristic
26C2×C4×Dic11C2211yescharacteristic
27#66946—24no—
28#66927—24no—
29C11×#95C2214yes—
30C11⋊#95C2212yes—
31C23×D11C2312yes—
32C23×D11C2312yes—
33C2×Q8×C11—22no—
34C2×Q8×C11—22no—
35C2×Q8×C11—22no—
36C22×C4×C11C2312yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C22×Dic11C2312yes—
41C22×Dic11C2311yescharacteristic
42C22×Dic11C2311yescharacteristic
43C2×D44—28no—
44C2×D44—22no—
45C2×D44—22no—
46C2×D44—22no—
47C2×D44—22no—
48C2×D44—22no—
49C2×D44—28no—
50C2×D44—44no—
51C2×D44D414yes—
52C2×C4×D11—24no—
53C2×C4×D11—44no—
54C11×C4⋊C4—22no—
55C42×C11—22no—
56(C2×D11)⋊C4—24no—
57(C2×D11)⋊C4—24no—
58(C2×D11)⋊C4—24no—
59(C2×D11)⋊C4—28no—
60(C2×D11)⋊C4—44no—
61Dic11⋊C4—24no—
62C4×Dic11—22no—
63C4×Dic11D414yescomplemented
64#2544—111nomaximal, Sylow
65C23×C11C2411yescharacteristic
66C22×D11—22no—
67C22×D11—22no—
68C22×D11—24no—
69C22×D11—24no—
70C22×D11—22no—
71C22×D11—22no—
72C22×D11—22no—
73C22×D11—24no—
74C22×D11—24no—
75C22×D11C2×D414yes—
76Q8×C11—44no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11—24no—
84C2×C4×C11C2×D412yes—
85C2×C4×C11C2×D412yes—
86C2×C4×C11#4112yes—
87C2×C4×C11#4114yes—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11C2×D412yes—
94C2×Dic11C2×D412yes—
95D44—28no—
96D44—416no—
97D44—44no—
98C4×D11—48no—
99C2×#127—111no—
100C2×#124—111no—
101C2×#124—111no—
102C2×C4×D4—111no—
103#394—114no—
104#391D1111yescharacteristic, complemented
105#390—111no—
106#386—114no—
107C4×C4⋊C4—111no—
108C22×C11C22×D411yescharacteristic
109C22×C11C22×D411yescharacteristic
110C22×C11C2×#4111yescharacteristic
111C22×C11C2×#4112yes—
112C22×C11H(5,𝔽2)12yes—
113C2×D11—24no—
114C2×D11—28no—
115C2×D11—42no—
116C2×D11—42no—
117C2×D11—44no—
118C2×D11—42no—
119C2×D11—42no—
120C2×D11—42no—
121C2×D11—44no—
122C2×D11—44no—
123C4×C11—24no—
124C4×C11—42no—
125C4×C11—42no—
126C4×C11#12114yes—
127Dic11—22no—
128Dic11—22no—
129Dic11—42no—
130C22×Q8C2×D1111yescharacteristic
131C22×D4—111no—
132C22×D4—111no—
133C22×D4—114no—
134C23×C4—112no—
135#127—224no—
136#124—224no—
137#124—224no—
138C4×D4—224no—
139C2×C4⋊C4—112no—
140C2×C4⋊C4C2×D1111yescharacteristic
141C2×C22⋊C4—112no—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—114no—
144C2×C22⋊C4—112no—
145C2×C22⋊C4—114no—
146C2×C42—111no—
147C2×C42—111no—
148C2×C42C2×D1111yescharacteristic
149#95—112no—
150#95C2×D1114yes—
151C2×C11C2×#12111yescharacteristic
152C2×C11#53211yescharacteristic
153C2×C11#53511yescharacteristic
154C2×C11#54612yes—
155C2×C11#55912yes—
156D11—44no—
157D11—82no—
158C24—112no—
159C24—112no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×D4—114no—
164C2×D4—228no—
165C2×D4—222no—
166C2×D4—222no—
167C2×D4—222no—
168C2×D4—222no—
169C2×D4—222no—
170C2×D4—228no—
171C2×D4—444no—
172C22×C4—111no—
173C22×C4—111no—
174C22×C4—112no—
175C22×C4—224no—
176C22×C4—444no—
177C22×C4C22×D1112yes—
178C22×C4C22×D1112yes—
179C22×C4C22×D1112yes—
180C22×C4C22×D1111yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—224no—
183C22⋊C4—224no—
184C22⋊C4—224no—
185C22⋊C4—224no—
186C22⋊C4—228no—
187C22⋊C4—444no—
188C42—22no—
189C42—114no—
190C42—222no—
191C11#254411yescharacteristic, complemented, Sylow
192C23—114no—
193C23—222no—
194C23—222no—
195C23—224no—
196C23—224no—
197C23—222no—
198C23—222no—
199C23—222no—
200C23—224no—
201C23—224no—
202C23C23×D1111yescharacteristic
203Q8—44no—
204D4—228no—
205D4—4416no—
206D4—444no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—112no—
215C2×C4—112no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—448no—
222C2×C4#433912yes—
223C2×C4D4×D1112yes—
224C2×C4D4×D1112yes—
225C2×C4#434314yes—
226C22—224no—
227C22—228no—
228C22—442no—
229C22—444no—
230C22—442no—
231C22—442no—
232C22—442no—
233C22—444no—
234C22—444no—
235C22—442no—
236C22C2×#433911yescharacteristic
237C22C2×D4×D1111yescharacteristic
238C22C2×D4×D1111yescharacteristic
239C22C2×#434312yes—
240C22#6704212yes—
241C4—24no—
242C4—42no—
243C4—42no—
244C4—222no—
245C4—222no—
246C4—442no—
247C4#6693714yes—
248C2—444no—
249C2—882no—
250C2C2×#6693711yescharacteristic
251C2#1062311612yes—
252C2#1062316112yes—
253C2#1062320511yescharacteristic
254C2#1062325611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC2×#66927C2
hypercenter#391D11
hypocenterC11#2544
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#67000C2semidirect
2. C2×#67000C2semidirect
3. C2×#66946C2semidirect
5. C11×#391C2semidirect
6. #10622601C2semidirect
7. C4×Dic11⋊C4C2semidirect
8. #10622285C2semidirect
9. #10622278C2semidirect
63. C4×Dic11D4semidirect
104. #391D11semidirect
191. C11#2544semidirect

Maximal subgroups

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

subgroupindex
C2×#670002
C2×#669462
C2×#669272
C11×#3912
#106226012
C4×Dic11⋊C42
#106222852
#106222782
#254411