Finite Group Databasebeta

← #13054737

Subgroups — #13054737

Order 1344.

240 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 64 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#10621427C211yesmaximal, characteristic, complemented
2C2×C3×#1143C211yesmaximal, characteristic
3#10620996C211yesmaximal, characteristic
4C2×#66565C211yesmaximal, characteristic
5C7×#835C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620510C211yesmaximal, characteristic, complemented
8#89590—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×Dic3×D7—21no—
13#66629—21no—
14#66629—21no—
15C4×Dic21C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1143—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21#66565—22no—
22#6507—31no—
23C2×#1143D311yescharacteristic
24C2×#1132—31no—
25#6456—31no—
26C7×#103—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5162—71nomaximal, Hall
30C22×C3×D7—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38Dic42—21no—
39Dic42—21no—
40C2×C7×Dic3—21no—
41C3×D28—21no—
42C3×D28D412yes—
43Dic3×D7—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28C2×D311yescharacteristic
48C2×C4×D7—61no—
49C2×Dic14—31no—
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52#1143—22no—
53(C2×D7)⋊C4—61no—
54Dic7⋊C4—61no—
55Dic7⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8C2×D311yescharacteristic
58#1132—62no—
59C2×C3×SD16—71no—
60#944—71no—
61D4×Dic3—71no—
62C2×#214—71no—
63#835D711yescharacteristic, complemented
64#816—71no—
65Dic3⋊C8—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×C3×D7—22no—
68C2×C3×D7—41no—
69C2×C3×D7—41no—
70C3×C4×C7—41no—
71C3×C4×C7C2×D411yescharacteristic
72C3×C4×C7#4111yescharacteristic
73Dic21—21no—
74Dic21—41no—
75Dic21SD1612yescomplemented
76C7×Dic3—41no—
77#461—211noSylow
78C22×D7—21no—
79C7×Q8—21no—
80C7×Q8—21no—
81C2×C4×C7—21no—
82C2×C4×C7—61no—
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—31no—
85C2×Dic7—31no—
86C2×Dic7—61no—
87D28—21no—
88D28#6712yes—
89C4×D7—122no—
90Dic14—61no—
91Dic14—61no—
92C7×C8—61no—
93C7⋊C8—21no—
94C2×C3×Q8C2×D711yescharacteristic
95C2×C3×D4—71no—
96C22×Dic3—141no—
97C2×Dic6—71no—
98C3×SD16—142no—
99C2×C3×C8—71no—
100#217—141no—
101#214—142no—
102C4⋊Dic3C2×D711yescharacteristic
103Dic3⋊C4—141no—
104Dic3⋊C4—141no—
105C4×Dic3—71no—
106C2×C3⋊C8C2×D711yescharacteristic
107C2×C3×C7#12111yescharacteristic
108C2×C3×C7C2×SD1611yescharacteristic
109C2×C3×C7#13711yescharacteristic
110C3×D7—42no—
111C2×SD16—71no—
112C2×SD16—211no—
113C4⋊Q8—211no—
114C4×D4—211no—
115C4⋊C8—211no—
116#103—31no—
117#103—211no—
118C22×C7D3×D411yescharacteristic
119C2×D7—22no—
120C2×D7—41no—
121C2×D7—41no—
122C4×C7—41no—
123C4×C7—121no—
124C4×C7#23711yescharacteristic
125C4×C7C2×#6711yescharacteristic
126Dic7—32no—
127Dic7—61no—
128Dic7—121no—
129C23×C3—141no—
130C3×Q8—21no—
131C3×Q8—21no—
132C3×D4—72no—
133C3×D4—141no—
134C2×C3×C4—21no—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—71no—
138C2×Dic3—71no—
139C2×Dic3—141no—
140C2×Dic3—282no—
141Dic6—141no—
142Dic6—141no—
143C3×C8—141no—
144C3⋊C8—21no—
145C3×C7#46111yescharacteristic, complemented, Hall
146C2×Q8—211no—
147C2×Q8D3×D711yescharacteristic
148C2×D4—71no—
149C22×C4—421no—
150SD16—142no—
151SD16—422no—
152C2×C8—31no—
153C2×C8—71no—
154C4⋊C4—31no—
155C4⋊C4—421no—
156C4⋊C4—421no—
157C22⋊C4—421no—
158C42—211no—
159C2×C7D3×SD1611yescharacteristic
160C2×C7#91511yescharacteristic
161C2×C7#93911yescharacteristic
162D7—42no—
163C22×C3—142no—
164C22×C3—281no—
165C22×C3—281no—
166C22×C3D4×D711yescharacteristic
167C3×C4—41no—
168C3×C4#115711yescharacteristic
169C3×C4D4×D711yescharacteristic
170Dic3—41no—
171Dic3—72no—
172Dic3—141no—
173Dic3—281no—
174C23—141no—
175Q8—21no—
176Q8—21no—
177Q8—421no—
178Q8—421no—
179D4—72no—
180D4—141no—
181C2×C4—21no—
182C2×C4—61no—
183C2×C4—211no—
184C2×C4—211no—
185C2×C4—421no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—61no—
189C8—141no—
190C7#516211yescharacteristic, complemented, Sylow
191C2×C3—282no—
192C2×C3#644711yescharacteristic
193C2×C3D7×SD1611yescharacteristic
194C2×C3#648111yescharacteristic
195C22—142no—
196C22—281no—
197C22—281no—
198C22#6669411yescharacteristic
199C4—41no—
200C4—121no—
201C4—212no—
202C4—421no—
203C4—841no—
204C4#6668311yescharacteristic
205C4D7×#6711yescharacteristic
206C3#8959011yescharacteristic, complemented, Sylow
207C2—282no—
208C2#1062106711yescharacteristic
209C2#1062108411yescharacteristic
210C2#1062111211yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D28C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#461
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
1. #10621427C2semidirect
5. C7×#835C2semidirect
6. #10620518C2semidirect
7. #10620510C2semidirect
63. #835D7semidirect
75. Dic21SD16semidirect
145. C3×C7#461semidirect
190. C7#5162semidirect
206. C3#89590semidirect

Maximal subgroups

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

subgroupindex
#106214272
C2×C3×#11432
#106209962
C2×#665652
C7×#8352
#106205182
#106205102
#895903
#51627