Finite Group Databasebeta

← #13054524

Subgroups — #13054524

Order 1344.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#10621419C211yesmaximal, characteristic
2C2×C3×#1142C211yesmaximal, characteristic
3#10620893C211yesmaximal, characteristic, complemented
4C2×#66570C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic
6#10620518C211yesmaximal, characteristic, complemented
7#10620510C211yesmaximal, characteristic, complemented
8#89539—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66636—21no—
13#66636—21no—
14C4×Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1142—22no—
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic7⋊Dic3—21no—
20Dic3×Dic7—21no—
21#66570—22no—
22#6499—31no—
23C2×#1142D311yescharacteristic
24C2×Dic28—31no—
25#6450—31no—
26C7×#102—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5166—71nomaximal, Hall
30C23×C3×C7—21no—
31C3×C7×D4—21no—
32C3×C7×D4—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37Dic42—21no—
38Dic42—21no—
39C2×C7×Dic3—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14D412yes—
43C3×C7⋊C8—21no—
44C7×C3⋊C8—21no—
45C2×C7×D4C2×D311yescharacteristic
46C2×Dic14—31no—
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4—31no—
50C22⋊Dic7—61no—
51C22⋊Dic7—61no—
52#1142—22no—
53Dic7⋊C4—61no—
54C4×Dic7—31no—
55C4×Dic7—61no—
56C2×C7⋊C8C2×D311yescharacteristic
57Dic28—62no—
58C2×C3×SD16—71no—
59Q8×Dic3—71no—
60C2×#216—71no—
61#936—71no—
62#832D711yescharacteristic
63#816—71no—
64Dic3⋊C8—71no—
65C22×C3×C7—41no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7C2×D411yescharacteristic
69C3×C4×C7C2×D411yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—21no—
72Dic21—21no—
73Dic21—41no—
74C3×Dic7—22no—
75C7×Dic3—41no—
76#464—211noSylow
77C23×C7—21no—
78C7×D4—21no—
79C7×D4—21no—
80C2×C4×C7—61no—
81C2×C4×C7C22×D311yescharacteristic
82C2×Dic7—21no—
83C2×Dic7—31no—
84C2×Dic7—31no—
85C2×Dic7—61no—
86Dic14—21no—
87Dic14—61no—
88Dic14—61no—
89Dic14#6712yes—
90C7×C8—61no—
91C7⋊C8—21no—
92C2×C3×Q8—71no—
93C2×C3×D4C2×D711yescharacteristic
94C2×Dic6—71no—
95C3×SD16—142no—
96C2×C3×C8—71no—
97#217—141no—
98#217—141no—
99#216—142no—
100C4⋊Dic3—141no—
101C4⋊Dic3C2×D711yescharacteristic
102C4×Dic3—71no—
103C4×Dic3—141no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7—81no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7#13511yescharacteristic
108C2×C3×C7#13711yescharacteristic
109C2×Q16—211no—
110C2×SD16—71no—
111#124—211no—
112C4×Q8—211no—
113C4⋊C8—211no—
114#103—211no—
115#102—31no—
116C22×C7—41no—
117C22×C7—41no—
118C22×C7—41no—
119C22×C7D3×D411yescharacteristic
120C4×C7—121no—
121C4×C7#23711yescharacteristic
122C4×C7C2×#6711yescharacteristic
123Dic7—22no—
124Dic7—61no—
125Dic7—61no—
126Dic7—121no—
127C23×C3—21no—
128C3×Q8—72no—
129C3×Q8—141no—
130C3×D4—21no—
131C3×D4—21no—
132C2×C3×C4—141no—
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—71no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—141no—
141C3⋊C8—21no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8—211no—
145C2×D4D3×D711yescharacteristic
146Q16—422no—
147SD16—142no—
148C2×C8—31no—
149C2×C8—71no—
150C4⋊C4—31no—
151C4⋊C4—421no—
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—211no—
155C42—421no—
156C2×C7—81no—
157C2×C7#91511yescharacteristic
158C2×C7#91611yescharacteristic
159C2×C7#93911yescharacteristic
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3D4×D711yescharacteristic
164C3×C4—142no—
165C3×C4#115711yescharacteristic
166C3×C4D4×D711yescharacteristic
167Dic3—41no—
168Dic3—141no—
169Dic3—141no—
170Dic3—281no—
171C23—21no—
172Q8—72no—
173Q8—141no—
174Q8—421no—
175Q8—421no—
176D4—21no—
177D4—21no—
178C2×C4—61no—
179C2×C4—141no—
180C2×C4—211no—
181C2×C4—211no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—141no—
186C7#516611yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3#644711yescharacteristic
189C2×C3#647511yescharacteristic
190C2×C3#647811yescharacteristic
191C22—41no—
192C22—41no—
193C22—41no—
194C22#6669411yescharacteristic
195C4—121no—
196C4—142no—
197C4—421no—
198C4—421no—
199C4—841no—
200C4#6668311yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8953911yescharacteristic, complemented, Sylow
203C2—81no—
204C2#1062105011yescharacteristic
205C2#1062106111yescharacteristic
206C2#1062111211yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
Fitting, prime radicalC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#464
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #10620893C2semidirect
4. C2×#66570C2semidirect
6. #10620518C2semidirect
7. #10620510C2semidirect
142. C3×C7#464semidirect
186. C7#5166semidirect
202. C3#89539semidirect

Maximal subgroups

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

subgroupindex
#106214192
C2×C3×#11422
#106208932
C2×#665702
C7×#8322
#106205182
#106205102
#895393
#51667