Finite Group Databasebeta

← #13073788

Subgroups — #13073788

Order 1408.

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

normal subgroups only

This group has 564 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×#66946C211yesmaximal, characteristic
2C2×#66941C211yesmaximal, characteristic, complemented
3C11×#382C211yesmaximal, characteristic, complemented
4#10622601C211yesmaximal, characteristic
5#10622590C211yesmaximal, characteristic, complemented
6#10622552C211yesmaximal, characteristic
7C4×Dic11⋊C4C211yesmaximal, characteristic, complemented
8#10622286C214yesmaximal, complemented
9#10622280C214yesmaximal, complemented
10C22×D44C2211yescharacteristic
11C22×C4×D11C2212yes—
12C22×Dic22C2212yes—
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C11×C4⋊C4C2211yescharacteristic
15C2×C42×C11C2211yescharacteristic
16C2×(C2×D11)⋊C4C2212yes—
17C2×(C2×D11)⋊C4C2214yes—
18C2×(C2×D11)⋊C4C2214yes—
19C2×C4⋊Dic11C2211yescharacteristic
20C2×Dic11⋊C4C2212yes—
21C2×Dic11⋊C4C2212yes—
22C2×C4×Dic11C2211yescharacteristic
23C2×C4×Dic11C2211yescharacteristic
24#66946—24no—
25#66941—24no—
26C11×#95C2214yes—
27C11⋊#95C2214yes—
28C11⋊#95C2212yes—
29C23×D11C2312yes—
30C22×C4×C11C2312yes—
31C22×C4×C11C2312yes—
32C22×C4×C11C2312yes—
33C22×C4×C11C2311yescharacteristic
34C22×Dic11C2312yes—
35C22×Dic11C2312yes—
36C22×Dic11C2311yescharacteristic
37C22×Dic11C2311yescharacteristic
38C2×D44—22no—
39C2×D44—22no—
40C2×D44Q814yescomplemented
41C2×C4×D11—24no—
42C2×C4×D11—28no—
43C2×Dic22—24no—
44C2×Dic22—44no—
45C11×C4⋊C4—22no—
46C11×C4⋊C4—22no—
47C42×C11—22no—
48(C2×D11)⋊C4—24no—
49(C2×D11)⋊C4—44no—
50(C2×D11)⋊C4—44no—
51C4⋊Dic11—22no—
52Dic11⋊C4—24no—
53Dic11⋊C4—24no—
54C4×Dic11—22no—
55C4×Dic11D414yes—
56#2563—111nomaximal, Sylow
57C23×C11C2411yescharacteristic
58C22×D11—22no—
59C22×D11—24no—
60C22×D11—22no—
61C22×D11—24no—
62C22×D11C2×Q814yes—
63C2×C4×C11—22no—
64C2×C4×C11—22no—
65C2×C4×C11—22no—
66C2×C4×C11—22no—
67C2×C4×C11—24no—
68C2×C4×C11—22no—
69C2×C4×C11—24no—
70C2×C4×C11C2×D412yes—
71C2×C4×C11C2×Q812yes—
72C2×C4×C11#4112yes—
73C2×Dic11—22no—
74C2×Dic11—22no—
75C2×Dic11—22no—
76C2×Dic11—22no—
77C2×Dic11—22no—
78C2×Dic11—24no—
79C2×Dic11C2×D412yes—
80C2×Dic11C2×D412yes—
81C2×Dic11#4114yes—
82D44—28no—
83C4×D11—48no—
84Dic22—48no—
85C2×C4⋊Q8—111no—
86C2×C4×D4—111no—
87#393—114no—
88#390—111no—
89#390—111no—
90#382—111no—
91#382D1111yescharacteristic, complemented
92#380—114no—
93C4×C4⋊C4—111no—
94C22×C11C22×D411yescharacteristic
95C22×C11C22×Q811yescharacteristic
96C22×C11C2×#4111yescharacteristic
97C22×C11C2×#4112yes—
98C22×C11H(5,𝔽2)12yes—
99C2×D11—24no—
100C2×D11—28no—
101C2×D11—42no—
102C2×D11—42no—
103C2×D11—44no—
104C4×C11—42no—
105C4×C11—42no—
106C4×C11—42no—
107C4×C11C22⋊Q814yes—
108Dic11—22no—
109Dic11—42no—
110Dic11—42no—
111Dic11#12414yes—
112C22×Q8—112no—
113C22×D4—111no—
114C23×C4—112no—
115C4⋊Q8—224no—
116C4×D4—224no—
117C2×C4⋊C4—112no—
118C2×C4⋊C4—111no—
119C2×C4⋊C4—112no—
120C2×C4⋊C4C2×D1111yescharacteristic
121C2×C4⋊C4C2×D1111yescharacteristic
122C2×C22⋊C4—112no—
123C2×C22⋊C4—114no—
124C2×C22⋊C4—114no—
125C2×C42—111no—
126C2×C42—111no—
127C2×C42C2×D1111yescharacteristic
128#95—112no—
129#95—114no—
130#95C2×D1114yes—
131C2×C11C2×C22⋊Q811yescharacteristic
132C2×C11C2×#12411yescharacteristic
133C2×C11#53511yescharacteristic
134C2×C11#55412yes—
135C2×C11#55512yes—
136D11—44no—
137C24—112no—
138C2×Q8—224no—
139C2×Q8—444no—
140C2×D4—114no—
141C2×D4—222no—
142C2×D4—222no—
143C22×C4—111no—
144C22×C4—111no—
145C22×C4—112no—
146C22×C4—112no—
147C22×C4—224no—
148C22×C4—228no—
149C22×C4C22×D1112yes—
150C22×C4C22×D1112yes—
151C22×C4C22×D1112yes—
152C22×C4C22×D1111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—224no—
156C4⋊C4—224no—
157C4⋊C4—222no—
158C22⋊C4—224no—
159C22⋊C4—444no—
160C22⋊C4—444no—
161C42—22no—
162C42—114no—
163C42—222no—
164C11#256311yescharacteristic, complemented, Sylow
165C23—114no—
166C23—222no—
167C23—222no—
168C23—224no—
169C23—224no—
170C23C23×D1111yescharacteristic
171Q8—448no—
172D4—228no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—112no—
181C2×C4—114no—
182C2×C4—112no—
183C2×C4—222no—
184C2×C4—222no—
185C2×C4—222no—
186C2×C4—222no—
187C2×C4—222no—
188C2×C4—224no—
189C2×C4—448no—
190C2×C4#433912yes—
191C2×C4D4×D1112yes—
192C2×C4Q8×D1112yes—
193C22—224no—
194C22—228no—
195C22—444no—
196C22—442no—
197C22—442no—
198C22C2×#433911yescharacteristic
199C22C2×D4×D1111yescharacteristic
200C22C2×Q8×D1111yescharacteristic
201C22#41×D1112yes—
202C22#6704212yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—114no—
207C4—222no—
208C4—442no—
209C4—442no—
210C4#6694914yes—
211C2—444no—
212C2C2×#6694911yescharacteristic
213C2#1062306312yes—
214C2D11×#12411yescharacteristic
215C2#1062320511yescharacteristic
216C2#1062322112yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#382C2
prime radicalC22×D44C22
hypercenter#382D11
hypocenterC11#2563
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#66941C2semidirect
3. C11×#382C2semidirect
5. #10622590C2semidirect
7. C4×Dic11⋊C4C2semidirect
8. #10622286C2semidirect
9. #10622280C2semidirect
40. C2×D44Q8semidirect
91. #382D11semidirect
164. C11#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#669462
C2×#669412
C11×#3822
#106226012
#106225902
#106225522
C4×Dic11⋊C42
#106222862
#106222802
#256311