Finite Group Databasebeta

← #13054335

Subgroups — #13054335

Order 1344.

316 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 88 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×C3×#1159C211yesmaximal, characteristic
2#10621090C212yesmaximal, complemented
3#10620673C211yesmaximal, characteristic, complemented
4C7×#834C211yesmaximal, characteristic, complemented
5#10620530C212yesmaximal, complemented
6#89512—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×C3×C22⋊D7C411yescharacteristic, complemented
9C2×C3×C22⋊D7C2211yescharacteristic
10C22×C3×Dic7C411yescharacteristic, complemented
11C22×C3×Dic7C2211yescharacteristic
12C3×#1159—22no—
13C3×#1159—22no—
14C2×C3×C4×D7C411yescharacteristic, complemented
15C2×C3×Dic14C411yescharacteristic, complemented
16#66636C2212yes—
17C7×#217C2212yes—
18Dic7⋊Dic3—22no—
19Dic3×Dic7—22no—
20C2×#1159D311yescharacteristic
21#6440—32no—
22C7×#99—31no—
23#6408—31no—
24#6380—32no—
25#5240—71nomaximal, Hall
26C23×C3×C7C2×C411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×C3×D7C2×C411yescharacteristic
29C3×C7×D4—22no—
30C2×C3×C4×C7C2×C411yescharacteristic
31C2×Dic21—22no—
32C2×C7×Dic3—22no—
33C3×C22⋊D7—22no—
34C3×C22⋊D7—22no—
35C2×C3×Dic7—21no—
36C2×C3×Dic7—21no—
37C2×C3×Dic7C2×C411yescharacteristic
38C2×C3×Dic7C2×C411yescharacteristic
39C2×C3×Dic7C2×C411yescharacteristic
40C2×C3×Dic7D412yes—
41C2×C3×Dic7D412yes—
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44C3×Dic14—22no—
45C2×C7×D4C2×D311yescharacteristic
46C2×C22⋊D7Dic311yescharacteristic, complemented
47C2×C22⋊D7C2×D311yescharacteristic
48C22×Dic7Dic311yescharacteristic, complemented
49C22×Dic7C2×D311yescharacteristic
50#1159—22no—
51#1159—22no—
52C2×C4×D7Dic311yescharacteristic, complemented
53C2×Dic14Dic311yescharacteristic, complemented
54C7×C22⋊C4—32no—
55C22⋊Dic7—32no—
56Dic7⋊C4—62no—
57C4×Dic7—62no—
58C2×C3×#41—71no—
59#926—72no—
60#834—71no—
61#834—72no—
62#834D711yescharacteristic, complemented
63C22×C3×C7—21no—
64C22×C3×C7—41no—
65C22×C3×C7—41no—
66C22×C3×C7C22×C411yescharacteristic
67C22×C3×C7C2×D412yes—
68C2×C3×D7—21no—
69C2×C3×D7—41no—
70C3×C4×C7—21no—
71Dic21—42no—
72C3×Dic7—21no—
73C3×Dic7C22⋊C412yes—
74C3×Dic7C22⋊C411yescharacteristic
75C3×Dic7C22⋊C411yescharacteristic
76C7×Dic3—42no—
77#410—211noSylow
78C23×C7C2×Dic311yescharacteristic
79C23×C7C22×D311yescharacteristic
80C22×D7C2×Dic311yescharacteristic
81C7×D4—22no—
82C2×C4×C7—62no—
83C2×C4×C7C2×Dic311yescharacteristic
84C22⋊D7—22no—
85C22⋊D7—22no—
86C2×Dic7—21no—
87C2×Dic7—21no—
88C2×Dic7—62no—
89C2×Dic7C2×Dic311yescharacteristic
90C2×Dic7C2×Dic311yescharacteristic
91C2×Dic7C2×Dic311yescharacteristic
92C2×Dic7#6712yes—
93C2×Dic7#6712yes—
94C4×D7—21no—
95C4×D7—21no—
96Dic14—22no—
97C3×#41—142no—
98C3×#41—142no—
99C2×C3×Q8—71no—
100C2×C3×D4—71no—
101C2×C3×D4—71no—
102C2×C3×D4C2×D711yescharacteristic
103C22×C3×C4—71no—
104C22×C3×C4—71no—
105C22×C3×C4—71no—
106#217—72no—
107#217C2×D712yes—
108C4⋊Dic3—142no—
109C4×Dic3—142no—
110C2×C3×C7—22no—
111C2×C3×C7—21no—
112C2×C3×C7—41no—
113C2×C3×C7C2×C22⋊C411yescharacteristic
114C3×D7—41no—
115C2×#41—71no—
116#117—212no—
117#99—31no—
118#99—212no—
119#99—211no—
120C22×C7—21no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7C22×Dic311yescharacteristic
124C22×C7C2×#6712yes—
125C2×D7—21no—
126C2×D7—41no—
127C4×C7—21no—
128C4×C7—122no—
129Dic7—21no—
130Dic7—122no—
131Dic7#21712yes—
132Dic7#21711yescharacteristic
133Dic7#21711yescharacteristic
134C23×C3—71no—
135C23×C3C4×D711yescharacteristic
136C23×C3C22×D711yescharacteristic
137C3×Q8—142no—
138C3×D4—22no—
139C3×D4—142no—
140C3×D4—142no—
141C2×C3×C4—71no—
142C2×C3×C4—71no—
143C2×C3×C4—72no—
144C2×C3×C4—72no—
145C2×C3×C4—71no—
146C2×C3×C4—141no—
147C2×C3×C4—141no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4C4×D711yescharacteristic
151C2×Dic3—22no—
152C2×Dic3—142no—
153C3×C7#41011yescharacteristic, complemented, Hall
154#41—142no—
155#41—142no—
156C2×Q8—71no—
157C2×D4—71no—
158C2×D4—71no—
159C2×D4D3×D711yescharacteristic
160C22×C4—71no—
161C22×C4—71no—
162C22×C4—71no—
163C4⋊C4—422no—
164C22⋊C4—32no—
165C22⋊C4—212no—
166C42—422no—
167C2×C7—22no—
168C2×C7—21no—
169C2×C7—41no—
170C2×C7C2×#21711yescharacteristic
171D7—41no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—141no—
176C22×C3—281no—
177C22×C3C2×C4×D711yescharacteristic
178C22×C3D4×D712yes—
179C3×C4—21no—
180C3×C4—71no—
181C3×C4—71no—
182C3×C4—72no—
183C3×C4—141no—
184Dic3—42no—
185Dic3—282no—
186C23—71no—
187C23Dic3×D711yescharacteristic
188C23C2×D3×D711yescharacteristic
189Q8—142no—
190D4—22no—
191D4—142no—
192D4—142no—
193C2×C4—62no—
194C2×C4—71no—
195C2×C4—71no—
196C2×C4—72no—
197C2×C4—72no—
198C2×C4—71no—
199C2×C4—141no—
200C2×C4—141no—
201C2×C4—141no—
202C2×C4—141no—
203C2×C4—422no—
204C2×C4Dic3×D711yescharacteristic
205C7#524011yescharacteristic, complemented, Sylow
206C2×C3—22no—
207C2×C3—21no—
208C2×C3—41no—
209C2×C3—281no—
210C2×C3D7×C22⋊C411yescharacteristic
211C22—21no—
212C22—41no—
213C22—41no—
214C22—141no—
215C22—281no—
216C22C2×Dic3×D711yescharacteristic
217C22D7×#6712yes—
218C4—21no—
219C4—71no—
220C4—71no—
221C4—72no—
222C4—122no—
223C4—141no—
224C4—842no—
225C3#8951211yescharacteristic, complemented, Sylow
226C2—22no—
227C2—21no—
228C2—41no—
229C2—281no—
230C2D7×#21711yescharacteristic
subgroupG/H
centerC2D7×#217
commutatorC22×C3×C7C22×C4
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×C3×#1159C2
cosocleC23×C3×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#410
LCSG ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10621090C2semidirect
3. #10620673C2semidirect
4. C7×#834C2semidirect
5. #10620530C2semidirect
8. C2×C3×C22⋊D7C4semidirect
10. C22×C3×Dic7C4semidirect
14. C2×C3×C4×D7C4semidirect
15. C2×C3×Dic14C4semidirect
46. C2×C22⋊D7Dic3semidirect
48. C22×Dic7Dic3semidirect
52. C2×C4×D7Dic3semidirect
53. C2×Dic14Dic3semidirect
62. #834D7semidirect
153. C3×C7#410semidirect
205. C7#5240semidirect
225. C3#89512semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
#106210902
#106206732
C7×#8342
#106205302
#895123
#52407