Finite Group Databasebeta

← #13073787

Subgroups — #13073787

Order 1408.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×D4×Dic11C211yesmaximal, characteristic
2C2×#66941C211yesmaximal, characteristic, complemented
3C11×#390C211yesmaximal, characteristic
4#10622834C211yesmaximal, characteristic
5#10622590C211yesmaximal, characteristic, complemented
6#10622560C214yesmaximal, complemented
7#10622555C214yesmaximal, complemented
8#10622549C211yesmaximal, characteristic, complemented
9C4×Dic11⋊C4C211yesmaximal, characteristic, complemented
10C22×D4×C11C2211yescharacteristic
11C23×Dic11C2212yes—
12C22×Dic22C2212yes—
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C11×C22⋊C4C2214yes—
15C2×C42×C11C2211yescharacteristic
16C2×C22⋊Dic11C2212yes—
17C2×C22⋊Dic11C2214yes—
18D4×Dic11—24no—
19C2×C4⋊Dic11C2211yescharacteristic
20C2×C4⋊Dic11C2211yescharacteristic
21C2×Dic11⋊C4C2212yes—
22C2×Dic11⋊C4C2212yes—
23C2×C4×Dic11C2211yescharacteristic
24C2×C4×Dic11C2211yescharacteristic
25#66941—24no—
26C11⋊#95C2212yes—
27C11⋊#95C2214yes—
28C11⋊#95C2214yes—
29C24×C11C2312yes—
30C2×D4×C11—22no—
31C2×D4×C11—22no—
32C2×D4×C11Q814yescomplemented
33C22×C4×C11C2312yes—
34C22×C4×C11C2312yes—
35C22×C4×C11C2311yescharacteristic
36C22×Dic11—24no—
37C22×Dic11—28no—
38C22×Dic11C2312yes—
39C22×Dic11C2312yes—
40C22×Dic11C2312yes—
41C22×Dic11C2311yescharacteristic
42C22×Dic11C2311yescharacteristic
43C2×Dic22—24no—
44C2×Dic22—44no—
45C11×C4⋊C4—22no—
46C11×C22⋊C4—44no—
47C42×C11—22no—
48C22⋊Dic11—24no—
49C22⋊Dic11—44no—
50C4⋊Dic11—22no—
51C4⋊Dic11—22no—
52Dic11⋊C4—24no—
53Dic11⋊C4—24no—
54C4×Dic11—22no—
55C4×Dic11D414yes—
56#2563—111nomaximal, Sylow
57C23×C11—22no—
58C23×C11—24no—
59C23×C11—22no—
60C23×C11—24no—
61C23×C11C2×Q814yes—
62C23×C11C2411yescharacteristic
63D4×C11—28no—
64C2×C4×C11—22no—
65C2×C4×C11—22no—
66C2×C4×C11—22no—
67C2×C4×C11—22no—
68C2×C4×C11—24no—
69C2×C4×C11C2×D412yes—
70C2×C4×C11C2×Q812yes—
71C2×C4×C11#4112yes—
72C2×Dic11—22no—
73C2×Dic11—22no—
74C2×Dic11—22no—
75C2×Dic11—22no—
76C2×Dic11—22no—
77C2×Dic11—24no—
78C2×Dic11—22no—
79C2×Dic11—24no—
80C2×Dic11—48no—
81C2×Dic11C2×D412yes—
82C2×Dic11C2×D412yes—
83C2×Dic11#4114yes—
84Dic22—48no—
85C2×C4⋊Q8—111no—
86C2×C4×D4—111no—
87#393—114no—
88#390—111no—
89#390D1111yescharacteristic
90#382—111no—
91#382—111no—
92#380—114no—
93C4×C4⋊C4—111no—
94C22×C11—24no—
95C22×C11—28no—
96C22×C11—42no—
97C22×C11—42no—
98C22×C11—44no—
99C22×C11C22×D411yescharacteristic
100C22×C11C22×Q811yescharacteristic
101C22×C11C2×#4111yescharacteristic
102C22×C11C2×#4112yes—
103C22×C11H(5,𝔽2)12yes—
104C4×C11—42no—
105C4×C11—42no—
106C4×C11C22⋊Q814yes—
107Dic11—22no—
108Dic11—42no—
109Dic11—42no—
110Dic11—42no—
111Dic11#12414yes—
112C22×Q8—112no—
113C22×D4C2×D1111yescharacteristic
114C23×C4—112no—
115C4⋊Q8—224no—
116C4×D4—224no—
117C2×C4⋊C4—111no—
118C2×C4⋊C4—112no—
119C2×C4⋊C4—111no—
120C2×C4⋊C4—112no—
121C2×C4⋊C4C2×D1111yescharacteristic
122C2×C22⋊C4—112no—
123C2×C22⋊C4—114no—
124C2×C22⋊C4C2×D1114yes—
125C2×C42—111no—
126C2×C42—111no—
127C2×C42C2×D1111yescharacteristic
128#95—112no—
129#95—114no—
130#95—114no—
131C2×C11—44no—
132C2×C11C2×C22⋊Q811yescharacteristic
133C2×C11C2×#12411yescharacteristic
134C2×C11#53511yescharacteristic
135C2×C11#55412yes—
136C2×C11#55512yes—
137C24C22×D1112yes—
138C2×Q8—224no—
139C2×Q8—444no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4Dic2214yescomplemented
143C22×C4—111no—
144C22×C4—111no—
145C22×C4—112no—
146C22×C4—112no—
147C22×C4—112no—
148C22×C4—224no—
149C22×C4—228no—
150C22×C4C22×D1112yes—
151C22×C4C22×D1112yes—
152C22×C4C22×D1111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—224no—
155C4⋊C4—224no—
156C4⋊C4—222no—
157C4⋊C4—222no—
158C22⋊C4—44no—
159C22⋊C4—224no—
160C22⋊C4—444no—
161C42—22no—
162C42—114no—
163C42—222no—
164C11#256311yescharacteristic, complemented, Sylow
165C23—22no—
166C23—22no—
167C23—24no—
168C23—24no—
169C23C2×Dic2214yes—
170C23C23×D1111yescharacteristic
171Q8—448no—
172D4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—112no—
179C2×C4—114no—
180C2×C4—112no—
181C2×C4—222no—
182C2×C4—222no—
183C2×C4—222no—
184C2×C4—222no—
185C2×C4—222no—
186C2×C4—224no—
187C2×C4—222no—
188C2×C4—224no—
189C2×C4—448no—
190C2×C4C2×Dic2212yes—
191C2×C4D4×D1112yes—
192C2×C4#434112yes—
193C22—24no—
194C22—28no—
195C22—42no—
196C22—44no—
197C22—42no—
198C22C22×Dic2211yescharacteristic
199C22C2×D4×D1111yescharacteristic
200C22C2×#434111yescharacteristic
201C22#6703712yes—
202C22#41×D1112yes—
203C4—42no—
204C4—42no—
205C4—114no—
206C4—222no—
207C4—442no—
208C4—442no—
209C4—442no—
210C4#6693114yes—
211C2—44no—
212C2C2×#6693111yescharacteristic
213C2#1062307012yes—
214C2#1062314712yes—
215C2D11×#12411yescharacteristic
216C2#1062324611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#390C2
prime radicalC22×D4×C11C22
hypercenter#390D11
hypocenterC11#2563
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#66941C2semidirect
5. #10622590C2semidirect
6. #10622560C2semidirect
7. #10622555C2semidirect
8. #10622549C2semidirect
9. C4×Dic11⋊C4C2semidirect
32. C2×D4×C11Q8semidirect
142. C2×D4Dic22semidirect
164. C11#2563semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic112
C2×#669412
C11×#3902
#106228342
#106225902
#106225602
#106225552
#106225492
C4×Dic11⋊C42
#256311