Finite Group Databasebeta

← #13054574

Subgroups — #13054574

Order 1344.

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

normal subgroups only

This group has 324 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×Q8×D21C211yesmaximal, characteristic
2#10620999C211yesmaximal, characteristic
3C2×D21⋊C8C211yesmaximal, characteristic
4C7×#835C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8D7×#103—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10Q8×D21—22no—
11Q8×D21C414yescomplemented
12C2×C4×D21—21no—
13C2×C4×D21C2211yescharacteristic
14C2×Dic42—21no—
15C2×Dic42C2211yescharacteristic
16C2×D21⋊C4—21no—
17C7×C4⋊Dic3C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×C4⋊Dic7C2211yescharacteristic
20C2×C3×C7⋊C8C2211yescharacteristic
21#66577—21no—
22D21⋊C8—22no—
23C2×Q8×D7—31no—
24C2×C8×D7—31no—
25D7×C4⋊C4—31no—
26C7×#103—31no—
27#6409D311yescharacteristic, complemented
28#6390—31no—
29#6384—31no—
30D3×#103—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8C2×C412yescomplemented
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C4×D21—24no—
39C4×D21C2×C412yes—
40C4×D21D412yes—
41Dic42—21no—
42Dic42—24no—
43Dic42C2×C412yescomplemented
44C2×C7×Dic3—21no—
45C2×C3×Dic7—21no—
46D21⋊C4—42no—
47C3×C7⋊C8—21no—
48C7×C3⋊C8—21no—
49C2×C7×Q8C2×D311yescharacteristic
50Q8×D7—34no—
51Q8×D7—62no—
52C2×C4×D7—31no—
53C2×C4×D7—61no—
54C2×C4×D7—61no—
55C2×Dic14—31no—
56C2×Dic14—61no—
57C2×C7×C8—31no—
58C7×C4⋊C4—31no—
59C4⋊Dic7C2×D311yescharacteristic
60Dic7⋊C4—61no—
61C2×C7⋊C8C2×D311yescharacteristic
62C8×D7—62no—
63C2×D3×Q8—71no—
64C2×D3×C8—71no—
65D3×C4⋊C4—71no—
66C3×#103—71no—
67#835D711yescharacteristic, complemented
68#816—71no—
69#810—71no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21C22⋊C412yes—
72C2×D21SD1612yes—
73C2×D21Q1612yes—
74C3×C4×C7—22no—
75C3×C4×C7C22×C411yescharacteristic
76C3×C4×C7C2×D411yescharacteristic
77Dic21—22no—
78Dic21C22⋊C412yes—
79C3×Dic7—41no—
80C7×Dic3—41no—
81C2×#103—211noSylow
82C22×D7—31no—
83C7×Q8—21no—
84C7×Q8C4×D312yescomplemented
85C2×C4×C7—21no—
86C2×C4×C7—61no—
87C2×C4×C7C22×D311yescharacteristic
88C2×Dic7—21no—
89C2×Dic7—31no—
90C2×Dic7—61no—
91C4×D7—32no—
92C4×D7—32no—
93C4×D7—64no—
94C4×D7—122no—
95Dic14—32no—
96Dic14—64no—
97Dic14—61no—
98C7×C8—61no—
99C7⋊C8—21no—
100C2×C3×Q8C2×D711yescharacteristic
101D3×Q8—74no—
102D3×Q8—142no—
103C2×C4×D3—71no—
104C2×C4×D3—141no—
105C2×C4×D3—141no—
106C2×Dic6—71no—
107C2×Dic6—141no—
108C2×C3×C8—71no—
109C3×C4⋊C4—71no—
110C4⋊Dic3C2×D711yescharacteristic
111Dic3⋊C4—141no—
112C2×C3⋊C8C2×D711yescharacteristic
113D3×C8—142no—
114C2×C3×C7C2×C22⋊C411yescharacteristic
115C2×C3×C7C2×SD1611yescharacteristic
116C2×C3×C7C2×Q1611yescharacteristic
117D21#10314yescomplemented
118C22×Q8—211no—
119C22×C8—211no—
120C2×C4⋊C4—211no—
121#103—31no—
122#103—71no—
123#103—211no—
124#103—211no—
125C22×C7D3×D411yescharacteristic
126C2×D7—32no—
127C2×D7—32no—
128C2×D7—32no—
129C4×C7—22no—
130C4×C7—121no—
131C4×C7C2×C4×D311yescharacteristic
132C4×C7D3×D411yescharacteristic
133Dic7—32no—
134Dic7—41no—
135Dic7—62no—
136C22×D3—71no—
137C3×Q8—21no—
138C3×Q8C4×D712yescomplemented
139C2×C3×C4—21no—
140C2×C3×C4—141no—
141C2×C3×C4C22×D711yescharacteristic
142C2×Dic3—21no—
143C2×Dic3—71no—
144C2×Dic3—141no—
145C4×D3—72no—
146C4×D3—72no—
147C4×D3—144no—
148C4×D3—282no—
149Dic6—72no—
150Dic6—144no—
151Dic6—141no—
152C3×C8—141no—
153C3⋊C8—21no—
154C3×C7C2×#10311yescharacteristic, complemented, Hall
155C2×Q8—214no—
156C2×Q8—211no—
157C2×Q8—421no—
158C2×Q8—422no—
159C2×Q8D3×D711yescharacteristic
160C22×C4—211no—
161C22×C4—421no—
162C22×C4—421no—
163C2×C8—31no—
164C2×C8—71no—
165C2×C8—422no—
166C4⋊C4—31no—
167C4⋊C4—71no—
168C4⋊C4—421no—
169C2×C7D3×C22⋊C411yescharacteristic
170C2×C7D3×SD1611yescharacteristic
171C2×C7D3×Q1611yescharacteristic
172D7—34no—
173C22×C3D4×D711yescharacteristic
174C2×D3—72no—
175C2×D3—72no—
176C2×D3—72no—
177C3×C4—22no—
178C3×C4—281no—
179C3×C4C2×C4×D711yescharacteristic
180C3×C4D4×D711yescharacteristic
181Dic3—41no—
182Dic3—72no—
183Dic3—142no—
184C23—211no—
185Q8—21no—
186Q8—212no—
187Q8—424no—
188Q8—421no—
189Q8D21⋊C412yescomplemented
190C2×C4—21no—
191C2×C4—61no—
192C2×C4—141no—
193C2×C4—211no—
194C2×C4—212no—
195C2×C4—212no—
196C2×C4—424no—
197C2×C4—421no—
198C2×C4—842no—
199C2×C4C2×D3×D711yescharacteristic
200C8—61no—
201C8—141no—
202C7D3×#10311yescharacteristic, complemented, Sylow
203C2×C3D7×C22⋊C411yescharacteristic
204C2×C3D7×SD1611yescharacteristic
205C2×C3D7×Q1611yescharacteristic
206D3—74no—
207C22—212no—
208C22—212no—
209C22—212no—
210C22#6669411yescharacteristic
211C4—22no—
212C4—121no—
213C4—212no—
214C4—281no—
215C4—422no—
216C4C2×D21⋊C411yescharacteristic
217C4#6669411yescharacteristic
218C3D7×#10311yescharacteristic, complemented, Sylow
219C2—214no—
220C2#1062106711yescharacteristic
221C2#1062107311yescharacteristic
222C2#1062112311yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7C2×#103
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. C7×#835C2semidirect
5. C3×#6409C2semidirect
6. #10620510C2semidirect
7. #10620509C2semidirect
11. Q8×D21C4semidirect
27. #6409D3semidirect
33. C3×C7×Q8C2×C4semidirect
43. Dic42C2×C4semidirect
67. #835D7semidirect
84. C7×Q8C4×D3semidirect
117. D21#103semidirect
138. C3×Q8C4×D7semidirect
154. C3×C7C2×#103semidirect
189. Q8D21⋊C4semidirect
202. C7D3×#103semidirect
218. C3D7×#103semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D212
#106209992
C2×D21⋊C82
C7×#8352
C3×#64092
#106205102
#106205092
D7×#1033
D3×#1037