Finite Group Databasebeta

← #13054666

Subgroups — #13054666

Order 1344.

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

normal subgroups only

This group has 224 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
1Q8×Dic21C211yesmaximal, characteristic, complemented
2C2×C7×#216C211yesmaximal, characteristic, complemented
3C2×C3×#1143C211yesmaximal, characteristic
4#10620930C211yesmaximal, characteristic
5#10620518C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic
8#89957—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C4⋊Dic21—21no—
13C4⋊Dic21C2211yescharacteristic
14C4×Dic21—21no—
15C4×Dic21C2211yescharacteristic
16C7×#216—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1143—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21(C2×D7)⋊Dic3—21no—
22C2×C7×Q16—31no—
23#6510—31no—
24Q8×Dic7—31no—
25C2×#1143D311yescharacteristic
26#6409—31no—
27#6395—31no—
28Dic7⋊C8—31no—
29#5166—71nomaximal, Hall
30C22×C3×D7—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8D412yes—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38C2×C7×Dic3—21no—
39C7×Dic6—21no—
40C7×Dic6—21no—
41C3×D28—21no—
42C3×D28—21no—
43C3×C7⋊C8—21no—
44C7×C3⋊C8—21no—
45C2×C7×Q8—31no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28C2×D311yescharacteristic
48C7×Q16—62no—
49C2×C7×C8—31no—
50#1143—22no—
51(C2×D7)⋊C4—61no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—31no—
54C4⋊Dic7—61no—
55C4×Dic7—31no—
56C4×Dic7—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58C2×C3×SD16—71no—
59Q8×Dic3—71no—
60C2×#216D711yescharacteristic, complemented
61#936—71no—
62#832—71no—
63#816—71no—
64Dic3⋊C8—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×C3×D7—41no—
67C2×C3×D7—41no—
68C2×C3×D7—41no—
69C3×C4×C7—22no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—21no—
74Dic21—41no—
75C7×Dic3—41no—
76#464—211noSylow
77C22×D7—21no—
78C7×Q8—21no—
79C7×Q8—61no—
80C7×Q8—61no—
81C7×Q8#6712yes—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—31no—
86C2×Dic7—31no—
87C2×Dic7—61no—
88D28—21no—
89D28—21no—
90C7×C8—61no—
91C7⋊C8—21no—
92C2×C3×Q8C2×D711yescharacteristic
93C2×C3×D4—71no—
94C2×Dic6C2×D711yescharacteristic
95C3×SD16—142no—
96C2×C3×C8—71no—
97#217—141no—
98#217—141no—
99#216—22no—
100C4⋊Dic3—71no—
101C4⋊Dic3—141no—
102C4×Dic3—71no—
103C4×Dic3—141no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13511yescharacteristic
107C2×C3×C7#13711yescharacteristic
108C3×D7—81no—
109C2×Q16—31no—
110C2×SD16—71no—
111#124—211no—
112C4×Q8—211no—
113C4⋊C8—211no—
114#103—211no—
115#102—211no—
116C22×C7D3×D411yescharacteristic
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—22no—
121C4×C7—121no—
122C4×C7#23711yescharacteristic
123C4×C7C2×#6711yescharacteristic
124Dic7—61no—
125Dic7—61no—
126Dic7—121no—
127C23×C3—141no—
128C3×Q8—21no—
129C3×Q8C22⋊D712yes—
130C3×D4—141no—
131C3×D4—141no—
132C2×C3×C4—21no—
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—71no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138Dic6—21no—
139Dic6—21no—
140C3×C8—141no—
141C3⋊C8—21no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8D3×D711yescharacteristic
145C2×D4—71no—
146Q16—62no—
147SD16—142no—
148C2×C8—31no—
149C2×C8—71no—
150C4⋊C4—211no—
151C4⋊C4—421no—
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—211no—
155C42—421no—
156C2×C7#91511yescharacteristic
157C2×C7#91611yescharacteristic
158C2×C7#93911yescharacteristic
159D7—81no—
160C22×C3—281no—
161C22×C3—281no—
162C22×C3—281no—
163C22×C3D4×D711yescharacteristic
164C3×C4—22no—
165C3×C4#115911yescharacteristic
166C3×C4C2×C22⋊D711yescharacteristic
167Dic3—41no—
168Dic3—141no—
169Dic3—141no—
170Dic3—281no—
171C23—141no—
172Q8—21no—
173Q8—61no—
174Q8—61no—
175Q8#424512yes—
176D4—141no—
177D4—141no—
178C2×C4—21no—
179C2×C4—61no—
180C2×C4—211no—
181C2×C4—211no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—141no—
186C7#516611yescharacteristic, complemented, Sylow
187C2×C3—561no—
188C2×C3#648111yescharacteristic
189C2×C3#648211yescharacteristic
190C2×C3#650211yescharacteristic
191C22—281no—
192C22—281no—
193C22—281no—
194C22#6669411yescharacteristic
195C4—22no—
196C4—121no—
197C4—421no—
198C4—421no—
199C4—841no—
200C4#6668411yescharacteristic
201C4C2×#424511yescharacteristic
202C3#8995711yescharacteristic, complemented, Sylow
203C2—561no—
204C2#1062108311yescharacteristic
205C2#1062108411yescharacteristic
206C2#1062113311yescharacteristic
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#464
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. Q8×Dic21C2semidirect
2. C2×C7×#216C2semidirect
5. #10620518C2semidirect
6. #10620508C2semidirect
60. C2×#216D7semidirect
142. C3×C7#464semidirect
186. C7#5166semidirect
202. C3#89957semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic212
C2×C7×#2162
C2×C3×#11432
#106209302
#106205182
#106205082
#106205002
#899573
#51667