Finite Group Databasebeta

← #13054786

Subgroups — #13054786

Order 1344.

304 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66672C211yesmaximal, characteristic
2#10621429C211yesmaximal, characteristic, complemented
3C2×C7×#215C211yesmaximal, characteristic, complemented
4C2×C3×#1144C211yesmaximal, characteristic, complemented
5#10620506C211yesmaximal, characteristic, complemented
6#10620502C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89945—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C2×#4247—21no—
14C2×D3×Dic7—21no—
15#66672—22no—
16#66672—22no—
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19#66629—21no—
20C7×#215—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×#1144—22no—
23C2×C3×C7⋊C8C2211yescharacteristic
24C2×#1159—31no—
25C2×C7×SD16—31no—
26#6509—31no—
27C2×#1144D311yescharacteristic, complemented
28#6406—31no—
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#5187—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C22×D3×C7—21no—
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40C4×D21—21no—
41C4×D21—21no—
42C7×D12—21no—
43C7×D12D412yes—
44C2×C3×Dic7—21no—
45C3×Dic14—21no—
46C3×Dic14D412yes—
47#4247—24no—
48D3×Dic7—24no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8C2×D311yescharacteristic
52C2×C7×D4—31no—
53C2×C22⋊D7—61no—
54C22×Dic7—61no—
55#1159—62no—
56#1159—62no—
57C2×C4×D7—31no—
58C2×Dic14C2×D311yescharacteristic
59C7×SD16—62no—
60C2×C7×C8—31no—
61#1144—22no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—31no—
64Dic7⋊C4—61no—
65C2×C7⋊C8C2×D311yescharacteristic
66C2×#239—71no—
67C2×C3×Q16—71no—
68#946—71no—
69C2×#215D711yescharacteristic, complemented
70#835—71no—
71#821—71no—
72(C2×D3)⋊C8—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21—21no—
77C2×D3×C7—22no—
78C2×D3×C7—41no—
79C2×D3×C7—41no—
80C3×C4×C7—41no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—21no—
84Dic21—41no—
85C3×Dic7—22no—
86#452—211noSylow
87C23×C7—61no—
88C22×D7—31no—
89C7×Q8—21no—
90C7×Q8—21no—
91C7×D4—32no—
92C7×D4—61no—
93C2×C4×C7—21no—
94C2×C4×C7C22×D311yescharacteristic
95C22⋊D7—64no—
96C2×Dic7—21no—
97C2×Dic7—31no—
98C2×Dic7—64no—
99C2×Dic7—61no—
100C4×D7—61no—
101C4×D7—61no—
102Dic14—21no—
103Dic14#6712yes—
104C7×C8—61no—
105C7⋊C8—21no—
106C2×C3×Q8—71no—
107C2×C3×Q8C2×D711yescharacteristic
108#239—142no—
109#239—142no—
110C2×D12—141no—
111C2×D12C2×D711yescharacteristic
112C2×C4×D3—71no—
113C2×C4×D3—141no—
114C3×Q16—142no—
115C2×C3×C8—71no—
116#215—22no—
117(C2×D3)⋊C4—141no—
118C4⋊Dic3—71no—
119Dic3⋊C4—141no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124D21—41no—
125D3×C7—42no—
126C2×#41—211no—
127C2×Q16—71no—
128C2×SD16—31no—
129C22⋊Q8—211no—
130#103—211no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7—62no—
134C22×C7—121no—
135C22×C7—121no—
136C22×C7D3×D411yescharacteristic
137C2×D7—61no—
138C2×D7—61no—
139C2×D7—61no—
140C4×C7—41no—
141C4×C7D3×D411yescharacteristic
142C4×C7C2×#6711yescharacteristic
143Dic7—22no—
144Dic7—61no—
145Dic7—121no—
146C22×D3—21no—
147C22×D3—71no—
148C3×Q8—21no—
149C3×Q8—21no—
150C3×Q8—72no—
151C3×Q8—141no—
152C2×C3×C4—21no—
153C2×C3×C4—141no—
154C2×C3×C4C22×D711yescharacteristic
155C2×Dic3—71no—
156C2×Dic3—141no—
157D12—21no—
158D12—144no—
159D12C22⋊D712yes—
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—31no—
171C2×D4—421no—
172C22×C4—211no—
173C22×C4—421no—
174Q16—142no—
175SD16—62no—
176C2×C8—31no—
177C2×C8—71no—
178C4⋊C4—211no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7—122no—
182C2×C7#91811yescharacteristic
183C2×C7#91911yescharacteristic
184C2×C7#93711yescharacteristic
185D7—121no—
186C22×C3D4×D711yescharacteristic
187C2×D3—22no—
188C2×D3—41no—
189C2×D3—41no—
190C2×D3—141no—
191C2×D3—141no—
192C2×D3—141no—
193C3×C4—41no—
194C3×C4—142no—
195C3×C4D4×D711yescharacteristic
196C3×C4C2×C22⋊D711yescharacteristic
197Dic3—141no—
198Dic3—281no—
199C23—61no—
200C23—211no—
201Q8—21no—
202Q8—21no—
203Q8—72no—
204Q8—141no—
205D4—32no—
206D4—61no—
207D4—424no—
208C2×C4—21no—
209C2×C4—141no—
210C2×C4—211no—
211C2×C4—424no—
212C2×C4—421no—
213C2×C4—421no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—141no—
218C7#518711yescharacteristic, complemented, Sylow
219C2×C3#647811yescharacteristic
220C2×C3#647911yescharacteristic
221C2×C3#650011yescharacteristic
222D3—42no—
223D3—281no—
224C22—62no—
225C22—121no—
226C22—121no—
227C22—421no—
228C22—421no—
229C22—421no—
230C22#6669411yescharacteristic
231C4—41no—
232C4—142no—
233C4—421no—
234C4—841no—
235C4D7×#6711yescharacteristic
236C4D3×C22⋊D711yescharacteristic
237C3#8994511yescharacteristic, complemented, Sylow
238C2—122no—
239C2—841no—
240C2#1062108511yescharacteristic
241C2#1062108611yescharacteristic
242C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×#66672C2
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. #10621429C2semidirect
3. C2×C7×#215C2semidirect
4. C2×C3×#1144C2semidirect
5. #10620506C2semidirect
6. #10620502C2semidirect
7. #10620492C2semidirect
27. C2×#1144D3semidirect
69. C2×#215D7semidirect
165. C3×C7#452semidirect
218. C7#5187semidirect
237. C3#89945semidirect

Maximal subgroups

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

subgroupindex
C2×#666722
#106214292
C2×C7×#2152
C2×C3×#11442
#106205062
#106205022
#106204922
#899453
#51877