Finite Group Databasebeta

← #13054750

Subgroups — #13054750

Order 1344.

304 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 68 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×#66732C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620952C211yesmaximal, characteristic, complemented
4C2×#66566C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620504C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89582—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66732—22no—
11#66732—22no—
12C2×D84—21no—
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×C4×D21C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C7×C4⋊Dic3C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×#1144—22no—
20C2×C3×C7⋊C8C2211yescharacteristic
21Dic3⋊Dic7—21no—
22#66575—21no—
23#66566—22no—
24C2×#1161—31no—
25C2×#1144D311yescharacteristic, complemented
26C2×#1132—31no—
27#6460—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6383—31no—
31#5187—71nomaximal, Hall
32C22×D21—21no—
33C22×D21D411yescharacteristic
34C3×C7×Q8—21no—
35C3×C7×Q8D412yes—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21D411yescharacteristic
39D84—24no—
40D84—21no—
41D84D412yes—
42C4×D21—24no—
43C4×D21—21no—
44C4×D21—21no—
45C2×C7×Dic3—21no—
46C2×C3×Dic7—21no—
47C3×Dic14—21no—
48C3×Dic14—21no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8C2×D311yescharacteristic
52#1161—62no—
53#1161—62no—
54C2×D28—31no—
55C2×D28—61no—
56C2×C4×D7—31no—
57C2×C4×D7—61no—
58C2×Dic14C2×D311yescharacteristic
59C2×C7×C8—31no—
60C7×C4⋊C4—31no—
61#1144—22no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8C2×D311yescharacteristic
65#1132—62no—
66C2×#239—71no—
67C2×C3×Q16—71no—
68#946—71no—
69C2×#215—71no—
70#835D711yescharacteristic, complemented
71#821—71no—
72(C2×D3)⋊C8—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—22no—
76C2×D21—21no—
77C2×D21—21no—
78C2×D21—41no—
79C2×D21—41no—
80C3×C4×C7—22no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—21no—
84C3×Dic7—41no—
85C7×Dic3—41no—
86#452—211noSylow
87C22×D7—31no—
88C22×D7—61no—
89C7×Q8—21no—
90C7×Q8#6712yes—
91C2×C4×C7—21no—
92C2×C4×C7—61no—
93C2×C4×C7C22×D311yescharacteristic
94C2×Dic7—21no—
95C2×Dic7—31no—
96D28—32no—
97D28—64no—
98D28—61no—
99C4×D7—64no—
100C4×D7—61no—
101C4×D7—61no—
102Dic14—21no—
103Dic14—21no—
104C7×C8—61no—
105C7⋊C8—21no—
106C2×C3×Q8—71no—
107C2×C3×Q8C2×D711yescharacteristic
108#239—142no—
109#239—142no—
110C2×D12—71no—
111C2×D12—141no—
112C2×C4×D3—71no—
113C2×C4×D3—141no—
114C3×Q16—142no—
115C2×C3×C8—71no—
116#215—142no—
117(C2×D3)⋊C4—141no—
118C4⋊Dic3C2×D711yescharacteristic
119Dic3⋊C4—141no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124D21—41no—
125D21—42no—
126C2×#41—211no—
127C2×Q16—71no—
128C2×SD16—211no—
129C22⋊Q8—211no—
130#103—31no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7D3×D411yescharacteristic
134C2×D7—61no—
135C2×D7—61no—
136C2×D7—62no—
137C2×D7—61no—
138C2×D7—121no—
139C2×D7—121no—
140C4×C7—22no—
141C4×C7—121no—
142C4×C7D3×D411yescharacteristic
143C4×C7C2×#6711yescharacteristic
144Dic7—41no—
145Dic7—61no—
146C22×D3—71no—
147C22×D3—141no—
148C3×Q8—21no—
149C3×Q8—141no—
150C3×Q8—141no—
151C3×Q8D2812yes—
152C2×C3×C4—21no—
153C2×C3×C4—141no—
154C2×C3×C4C22×D711yescharacteristic
155C2×Dic3—21no—
156C2×Dic3—71no—
157D12—72no—
158D12—144no—
159D12—141no—
160C4×D3—144no—
161C4×D3—141no—
162C4×D3—141no—
163C3×C8—141no—
164C3⋊C8—21no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—71no—
169C2×Q8D3×D711yescharacteristic
170C2×D4—211no—
171C2×D4—421no—
172C22×C4—211no—
173C22×C4—421no—
174Q16—142no—
175SD16—422no—
176C2×C8—31no—
177C2×C8—71no—
178C4⋊C4—31no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7#91811yescharacteristic
182C2×C7#91911yescharacteristic
183C2×C7#93711yescharacteristic
184D7—121no—
185D7—122no—
186C22×C3D4×D711yescharacteristic
187C2×D3—142no—
188C2×D3—141no—
189C2×D3—141no—
190C2×D3—141no—
191C2×D3—281no—
192C2×D3—281no—
193C3×C4—22no—
194C3×C4—281no—
195C3×C4C2×D2811yescharacteristic
196C3×C4D4×D711yescharacteristic
197Dic3—41no—
198Dic3—141no—
199C23—211no—
200C23—421no—
201Q8—21no—
202Q8—141no—
203Q8—141no—
204Q8#424612yes—
205D4—212no—
206D4—424no—
207D4—421no—
208C2×C4—21no—
209C2×C4—61no—
210C2×C4—141no—
211C2×C4—211no—
212C2×C4—424no—
213C2×C4—421no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—141no—
218C7#518711yescharacteristic, complemented, Sylow
219C2×C3#644511yescharacteristic
220C2×C3#647911yescharacteristic
221C2×C3#648111yescharacteristic
222D3—281no—
223D3—282no—
224C22—421no—
225C22—421no—
226C22—422no—
227C22—421no—
228C22—841no—
229C22—841no—
230C22#6669411yescharacteristic
231C4—22no—
232C4—121no—
233C4—281no—
234C4—421no—
235C4C2×#424611yescharacteristic
236C4#6669411yescharacteristic
237C3#8958211yescharacteristic, complemented, Sylow
238C2—841no—
239C2—842no—
240C2#1062107411yescharacteristic
241C2#1062108511yescharacteristic
242C2#1062113411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×#66732C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×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
2. C2×C3×#1144C2semidirect
3. #10620952C2semidirect
4. C2×#66566C2semidirect
5. C7×#835C2semidirect
6. #10620504C2semidirect
7. #10620492C2semidirect
25. C2×#1144D3semidirect
70. #835D7semidirect
165. C3×C7#452semidirect
218. C7#5187semidirect
237. C3#89582semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×C3×#11442
#106209522
C2×#665662
C7×#8352
#106205042
#106204922
#895823
#51877